NEURAL NETWORKBASED CONTROL DESIGNS FOR COMPLEX
INDUSTRIAL PROCESS APPLICATIONS
By
WESLEY CURTIS LEFEBVRE
A DISSERTATION PRESENTED TO THE GRADUATE
SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL
FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE
OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA 2000
ACKNOWLEDGMENTS
When looking for a graduate school I was told to pick an advisor, not a school. My choice was Dr. Jose Principe, and his mentoring and friendship through a master's, Ph.D., and the startup of two companies has proven this advise to be sound. I would also like to thank Dr. Glen Johnson from IBM Yorktown for giving me this advice and first introducing me to the field of neural networks.
I would also like to thank Charles River Associates and Commonwealth Energy for believing in my work and this field, the members of my committee for the effort and time spent on my behalf, the staff at Canal Electric Generating Station, and the members of the Computational NeuroEngineering Laboratory (CNEL).
I would also like to thank my wife, Anna, for her patience as I struggled to juggle far too many commitments, and my family for their neverending support referring to us as Dr. and Mr. Lefebvre.
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TABLE OF CONTENTS
page
A CKN OW LED GM EN TS ............................................... ..........................................ii
LIST OF TABLES ...................... .. .... .. ....... ........................v
LIST OF FIGURES ......................................................................................................vi
CHAPTERS
1 INTRODU CTION ....................................................................................................1
Proposed W ork .....................................................................................................1..
D ocum ent Organization .........................................................................................7
2 LITERA TURE REV IEW ...........................................................................................9
Optim ization .......................... ..........................................................................9
N eural N etw orks ......................................................................................................16
Control Theory ................................................. ......................................................32
N Ox ......................................................................................................................... 40
FossilFired Pow er Generation .......................................... ....................................45
3 BOILER OPTIM IZATION .......................................................................................63
First Principles ........................................................................................................63
Fuel and Air D istribution ........................................................................................64
Boiler Tuning ..........................................................................................................65
The Role of CO .........................................................................................................66
4 CON TROL D ESIGN S ..............................................................................................67
V ariable D efinitions ................................................................................................67
Optim ization Objectives ........................................................................................68
Operating Constraints .............................................................................................70
Perform ance Criteria ..............................................................................................72
Controller D esigns ........................................................................... .......................73
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5 DATA PREPARATION .............................................................................................90
D ata M anagem ent ...................................................................................................91
V ariable Selection ...................................................................................................92
V alidation ................................................ ....... ....................................................... 93
Tim e Constants .......................................................................................................97
N orm alization .........................................................................................................97
6 M OD ELIN G ...........................................................................................................99
M ethodology ............................................................................ ...............................99
M odel D efinitions ...................................................................................................100
D atasets ..................................................................................................................... 104
Perform ance Criteria ...............................................................................................105
Learning A lgorithm .................................................................................................109
V ariable Pruning .....................................................................................................113
Architecture Selection .......................................... ...............................................118
Analysis .................................................................................................................... 136
7 CON TROLLER IM PLEM EN TATION S ...................................................................138
O ffline Q uantification ..............................................................................................138
Online Quantification ..............................................................................................143
8 PARAM ETERIZATION PROBLEM ........................ ............................................148
Search for a V alidation M etric .................................................................................149
Correlation Paradox .................................................................................................154
V alidation M etric .....................................................................................................160
Revised Representation Pruning A lgorithm ....................... ................................161
M odeling ...................................................................................................................168
Control Im plem entation ...........................................................................................169
9 CON CLU SION ..........................................................................................................176
Contributions ............................................................................................................176
A fterword ..................................................................................................................178
Future D irection ........................................................................................................178
APPEND IX .................................................................................................................... 180
REFEREN CES ................ ... .................................................................................181
BIO GRAPH ICA L SKETCH .........................................................................................187
iv
LIST OF TABLES
Table 1: Probability of ZScore exceeding value .......................................................96
Table 2: Types in order they were rem oved ...............................................................116
Table 3: Final variable selections after pruning ........................................................118
Table 4: Results of autoregressive tap search algorithm for all dynamic models .....121 Table 5: Results of moving average tap search algorithm for all dynamic models ....122 Table 6: Results of hidden layer #1 PE search algorithm for all steadystate models. 124 Table 7: Results of hidden layer #2 PE search algorithm for all steadystate models. 126 Table 8: Results of tap search algorithm for all dynamic models ..............................128
Table 9: Results of taps search algorithm for all dynamic models ............................130
Table 10: Results of hidden taps search algorithm for all dynamic models ................131
Table 11: Results of hidden states search algorithm for all dynamic models ..............133
Table 12: Results of state hidden PEs search algorithm for all dynamic models ........134 Table 13: Results of output hidden PEs search algorithm for all dynamic models ......136 Table 14: Final variable selections after revised pruning ............................................165
T able 15: E ssensial tag list ..................................................... .....................................180
v
LIST OF FIGURES
Figure 1: Multilayer perceptron model architecture ....................................................18
Figure 2: TDNN input PE connectivity .....................................................................19
Figure 3: Gamma memory processing element ...........................................................20
Figure 4: Nonlinear state space neural network configuration ....................................21
Figure 5: Combustion emissions characteristic versus air flow ..................................64
Figure 6: Effect of lower 02 on combustion emissions ..............................................65
Figure 7: Offline training and retuning configuration for steadystate optimizer ........75 Figure 8: Online control configuration for steadystate optimizer ..............................76
Figure 9: Offline training and retuning configuration for modelinverse controller ....79 Figure 10:Online control configuration for modelinverse controller ..........................80
Figure 11:Offline training and retuning configuration for steadystate optimizer ........82 Figure 12:Offline training and retuning configuration for model reference controller. .86 Figure 13:Online control configuration for model reference controller .......................89
Figure 14:Daily % missing across February dataset ....................................................94
Figure 15:Example of variables with large NMSE but high R .....................................108
Figure 16:Results of type pruning algorithm ...............................................................115
Figure 17:Representation pruning algorithm results ....................................................117
Figure 18:Results of autoregressive taps search algorithm for NOx ARMA Model ... 120 Figure 19:Results of moving average tap search algorithm for NOx ARMA Model ... 121 Figure 20:Results of hidden layer #1 PE search algorithm for the NOx MLP Model. ..124 vi
Figure 21 :Results of the hidden layer #2 PE search algorithm for NOx MLP Model. ..125 Figure 22:Results of tap search algorithm for NOx TDNN Model ..............................127
Figure 23:Results of taps search algorithm for NOx GNN model ...............................129
Figure 24:Results of hidden taps search algorithm for NOx GNN model ...................130
Figure 25:Results of hidden states search algorithm for NOx NLSS model ................133
Figure 26:Results of state hidden PEs search algorithm for NOx NLSS model ..........134 Figure 27:Results of output hidden PEs search algorithm for NOx NLSS model ........135 Figure 28:Best models for all model definitions by architecture .................................137
Figure 29:Average NOx reduction over testing dataset ..............................................141
Figure 30:Average CO above max over testing dataset ...............................................141
Figure 31:Average NOx reduction over testing dataset using train and test models ....142 Figure 32:Average CO above max over testing dataset using train and test models ....142 Figure 33:Change in NOx for steadystate controller experiments ..............................146
Figure 34:Final CO level for steadystate controller experiments ...............................147
Figure 35: Summary of validation metrics for MLP CV model ....................................150
Figure 36:NMSE and R for all 10 training results for MLP CV model .......................151
Figure 37: Summary of validation metrics for combined SV/CV model ......................153
Figure 38: Sensitivity results for all 10 training results for NOx CV model .................160
Figure 39:NOx CV model sensitivity with 95% confidence intervals .........................161
Figure 40:Results of revised representation pruning algorithm ...................................164
Figure 41:Sensitivity results for all 10 training results for revised NOx CV model .....167 vii
Figure 42:Revised NOx CV model sensitivity with 95% confidence intervals ............168
Figure 43: Best revised models for all model definitions by architecture ......................169
Figure 44:Average NOx reduction over testing dataset using old and revised models. .170 Figure 45:Average CO above max over testing dataset using old and revised models. .170 Figure 46: Change in NOx for revised steadystate controller experiments .................172
Figure 47:Final CO level for revised steadystate controller experiments ...................172
Figure 48:Average percent NOx reduction for 10 online experiments ........................173
Figure 49:Average percent CO reduction above 500ppm for 10 online experiments. ... 174
viii
Abstract of Dissertation Presented to the Graduate School of
the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy
NEURAL NETWORKBASED CONTROL DESIGNS FOR
COMPLEX INDUSTRIAL PROCESS APPLICATIONS By
Wesley Curtis Lefebvre
May 2000
Chairman: Jose C. Principe
Major Department: Electrical and Computer Engineering
Neural networks have successfully transitioned from an academic interest into a viable technology which is now being used in everyday products. To date, neural networks have been predominantly applied to forecasting or modeling applications. Based on their success in such applications, there has been significant interest in using neural networks in control applications, creating a new field called neurocontrol. Although there have been significant advances in the theory of neurocontrol, there are very few successful commercial applications using neurocontrollers. Commercial applications often provide the most challenging problems because the controllers are required to function robustly in complex and unknown environments. Realworld processes are complex and difficult to control because they contain a large number of highly interdependent variables, have highly nonlinear responses to these variables, and change their response over time.
This work identified two significant reasons why neurocontrol designs fail in realix
world applications: First, the controllable parameters over most industrial processes are highly correlated, often not for physical reasons but because of our process control strategies. Second, intermediate process states that affect the process output, which are also affected by the controllable parameters, have a significant impact on controller performance. When the controller changes the controllable parameters, the impact that this has on the process states, which will in turn affect the process output, is not accounted for in most neurocontrol designs in the literature.
This dissertation advances the field of neurocontrol by providing the following solutions: first, the use of statistical significance testing on the local linearized relationships extracted from nonlinear neural network models to avoid problems with correlated controllable parameters; second, augmenting neurocontrol designs to incorporate dependent state models. These enhancements have been applied to four distinct neurocontrol architectures. The new control architectures have been applied to the novel application of controlling NOx emission from an oil and gasfired electric power plant.
x
CHAPTER 1
INTRODUCTION
Neural networks have found an applications niche as robust predictors that have demonstrated the ability to outforecast more traditional methods in complex realworld applications. The vast majority of neural network applications to date rely solely on the model's ability to forecast without regard for what the model has learned about the relationships within the underlying process or the ability to affect it. Soulie and Gallinari [59] recently compiled 53 industrial applications of neural networks, of which only 4 make any attempt to make inferences about the underlying process or to control it.
This ratio does not apply to theoretical publications in the literature, however. The research into neural networkbased control has recently received widespread attention, coined neurocontrol [40]. Many authors have presented the abstract concepts behind neurocontrol [40][43][31][37], but the literature contains a disproportionately low number of papers presenting realworld neurocontrol applications. The vast majority of these papers have developed controllers for fabricated, simulated, or laboratory controlled processes.
1.1 Proposed Work
The goal of this project is to develop robust neurocontrol design methodologies for
complex industrial process control applications. Industrial control applications are characterized by nonlinear, noisy, nonGaussian, highly correlated, and nonstationary processes. This is an applications area where classical control designs have proven ineffective.
1
2
1.1.1 Case Study
As a case study, this work develops online advisory neurocontrollers designed to minimize the NOx emissions for an oil and gas cofired power plant. The combustion of fossil fuels inside a largescale boiler is a highly complex process; this complexity is a direct function of the boiler size. A typical electric power boiler maintains a "fireball" which is 3 to 5 stories tall, and there are hundreds of parameters which affect the injection of fuel and air at different locations within the furnace. The problem is our lack of understanding about how these combustion parameters affect NOx formation. This multivariate optimization problem requires a technology that can look at the process globally and determine the appropriate combination of combustion controls
The neurocontrollers developed in this study will forward control setpoints to operators through the plant's existing distributed control system (DCS). The neurocontrollers will be required to provide setpoints that minimize NOx emissions, while maintaining unit operating constraints. A demonstration system has been completed at Canal Electric's 580MW tangentially fired Unit 2. Charles River Associates (CRA) and Commonwealth Energy jointly funded this study.
1.1.2 Objectives
The author believes that neurocontrol strategies have not been more successful in realworld applications because they
1) are difficult to implement,
2) fail to account for dependent internal process states, and 3) have difficulty dealing with correlated process variables.
The scientific objectives for this work are to develop:
3
1) an applicationbased neurocontrol implementation methodology,
2) statespace neurocontrol architectures,
3) methods for dealing with correlated data,
4) accurate combustion models, and
5) a novel combustion controller.
1.1.2.1 ApplicationBased Neurocontrol Implementation Methodology
Most neurocontrol implementations in the literature have been ad hoc and application specific [40]. In many cases, the process has been simulated and thus known completely. Modem control theory has introduced many sophisticated control designs, but the fact remains that approximately 90% of the industrial control applications apply a simple proportional integral differential (PID) controller [37].
The PID controller only requires the process engineer to specify reasonable knowledge of the process. This applicationbased implementation methodology is largely responsible for the success of PID control in the industry. If neurocontrollers are to enter the mainstream process control market, there will have to be designs that do not require detailed knowledge of neural network or neurocontrol theory.
1.1.2.2 ApplicationBased Neurocontrol Implementation Methodology
We begin in Chapter 4 by defining a methodology for categorizing process variables into groups based on a set of objective criteria about their role in the process. Each controller will be implemented in Chapter 7 based on this labeling of the process variables, without additional process knowledge. This will allow process engineers with reasonable process knowledge and without any knowledge of neural network or neurocontrol theory to successfully deploy such technologies. Notice that achieving such a straightforward
4
implementation methodology does not imply that the details behind the neurocontroller implementation are easy, it simply requires that they can be automated.
1.1.2.3 StateSpace Neurocontrol Designs
Nonlinear statespace neural network architectures offer the greatest modeling potential, but the difficulties in their training have led investigators to reject their application [31]. In fact, most neurocontrol designs are based on input/output process models [31][43][40]. The use of input/output neural network architectures in the design of neurocontrollers assumes that all input variables are independent, a situation which in not likely to be found in realworld applications. Many of the process variables that affect the plant's output will depend on the same process setpoints that a controller is manipulating. These facts have limited the performance and complicated the implementation of neurocontrol designs. If neurocontrol is to be a viable methodology in industrial process situations, then the controllers will have to be extended with architectures capable of dealing with internal process states.
There has been limited success in applying statespace neural network models [31]. It is widely accepted, however, that statespace representations hold the most promise for modeling and controlling complex processes [34][35]. The literature seems to be treating the viability of statespace architectures as an allornothing affair. Most publications apply purely input/output architectures with overwhelming success, while a few investigates have tested purely statespace architectures where all states are treated as hidden and unknown with limited to no success.
This work will empirically investigate several shades of gray, ranging from purely
input/output to purely statespace controllers. These statespace controller designs are pre
5
sented in Chapter 4, and their performance is empirically investigated in chapters 7 and 8. The primary difference in the statespace representations proposed here is that the state variables are treated as not hidden and known. This will require an extension to the neurocontrol design strategies presented in the literature [31][43][40].
1.1.2.4 Methods for Dealing with Correlation
Industrial process applications are unique in that there is a massive amount of available data. The input variables for complex process models are typically highly correlated, a situation for which there are few solutions in the literature. This correlation can come from several sources: dependent states (as addressed above), physical linkages, and soft linkages through control strategies the lack of adequate system parameterization. Industrial process control applications will require the process engineer to select representative variables from a large set of available process variables. Because of issues like input correlation, the representative variables selected will have a significant impact on the performance of the resulting neurocontroller. A viable neurocontrol design methodology will have to be able to cope with correlated process variables. Linear control theory deals with this aspect through parameterization of the controller. Nonlinear control theory, however, has not solved this problem in general. In Chapter 8 we propose to compute sensitivities through a committee of trained neural models to select the best variables for system identification and control.
Statespace neurocontrol architectures will be able to explicitly deal with one source of correlation present in industrial processes, namely correlation produced by dependent state variables. As mentioned above there are several other sources of correlation, however. Our first goal is to empirically quantify the impact of this correlation on our model
6
ing and control objectives. We begin by ignoring other sources of correlation and investigate their impact in Chapter 7. Once specific problems have been identified, methods are developed for dealing with correlation during controller implementation in Chapter 8, and the performance of these methods are quantified with respect to the performance of the resulting controllers.
1.1.2.5 Accurate Combustion Models
Little is known about how NOx is formed from airbound nitrogen during combustion. To date, reliable models for NOx formation in electric power boilers have not been available [39]. There are not adequate models for many realworld industrial processes, a fact which has also limited the acceptance of modern control strategies. One reason that neurocontrol has received such widespread attention is because of its potential ability to deal with very complex processes that have escaped modeling.
Chapter 6 develops accurate combustion models according to accepted modeling performance metrics. Chapter 7 demonstrates the impact that correlation has on neurocontrol designs, and Chapter 8 investigates its impact on the accuracy of the underlying process models. Here it will be shown that accuracy is subjective, and that in fact no good metrics for model accuracy are available in the literature. A new metric is proposed and empirically compared against available metrics in the literature in Chapter 8. Applying this new metric, predictive combustion models are developed and used to shed light on which process variables have the greatest impact on NOx and CO formation.
7
1.1.2.6 Novel Combustion Controller
This project develops four neurocontrollers for the complex industrial process of NOx formation. We begin in Chapter 3 by looking at boiler optimization from a firstprinciples perspective, focusing on what a NOx controller is expected to achieve and why classical control methods are not able to achieve it. Predictive neural network combustion models are then developed in Chapter 6, and deployed within online neurocontrollers in chapters 7 and 8. The performance of each of these controllers is then quantified to compare and contrast the four control designs in Chapter 8.
To the authors knowledge, this work developed the first NOx controller for a gas and oil cofired electric power plant. New regulations and the restructuring of the electric power industry have combined to create a NOx trading market. The annual benefits to a gas and oil cofired electric power plant associated with a 25% NOx reduction will be in the range of $2,000,000 to $5,000,000. Clearly, a control strategy that uses existing plant capital investments and runs on a $2,000 pentium workstation has tremendous value.
1.2 Document Organization
Chapter 2 presents a summary of the required background and a literature review of the relevant work in neurocontrol, along with references to the literature for more detailed treatments. This chapter is the best place for readers to become familiar with the notation used throughout this work.
A strategy for reducing the NOx emissions from a fossilfired generating unit is provided in Chapter 3. The goal of this section is to provide a physical understanding for what we are asking the controllers to perform, thus providing justification that our objectives are feasible. Chapter 4 develops four detailed neurocontrol designs belonging to the model
8
predictive, model inverse, and model reference control families. The designs are presented as generalized methodologies that are applicable to any control application. Chapter 5 presents a management and preprocessing methodology for collecting data in support of these control designs and the required modeling.
Each control design considered requires accurate process models for its implementation. Chapter 6 presents a modeling methodology for developing these models. The controllers are then implemented in Chapter 7. The performance for each resulting controller is then quantified using offline simulations and online experiments. Significant problems are discovered with the controllers for which there are no solutions in the literature. These problems, along with proposed solutions, are investigated further in Chapter 8. This section additionally demonstrates the validity of these solutions by quantifying the performance of the revised controllers.
The "key learnings" and extensions to this work are summarized in Chapter 9.
CHAPTER 2
LITERATURE REVIEW
This chapter provides background for the rest of this document in the areas of:
1) optimization,
2) neural networks,
3) neurocontrol,
4) NOx, and
5) fossilfired power plants.
2.1 Optimization
Mathematical optimization methods are at the heart of modem modeling and control applications. Neural networks use optimization methods to facilitate learning, and control applications apply these methods to meet their control objectives. The notation and methods presented in this section will be used extensively throughout the rest of this document.
Optimization is defined as the process of finding the values of Nz decision variables
ffv NNz
z N that minimize a scalar performance objective J e N 91 [15]. Formally, this optimization task will be represented as ArgMin{J(1)} (1) where N is the decision variable space, which is most often taken to be euclidean N = 91. Optimization methods, also known as mathematical programming methods, can be classified according to the amount of a priori information available about the system
9
10
being optimized [54]. The following sections broadly categorize optimization methods into the following:
1) Classical Analytic Optimization: where the system being optimized is
known completely or nearly completely and a tractable analytic solution
exists
2) Descent Optimization: where first and/or secondorder partial derivatives are available everywhere for the parameters of the system being
optimized
3) Direct Optimization: where little to no a priori knowledge exists about
the physical structure of the system being optimized
If the optimization problem involves objective functions or constraints which cannot be stated as explicit functions of the design variables or are too complicated to manipulate, we cannot solve it by using classical analytic optimization methods. This work will be dealing with complex systems where little is known a priori and will therefore not consider analytic optimization methods.
2.1.1 Iterative Optimization Methods
All direct and descent optimization methods are iterative in nature, i.e., they start from an initial trial solution and proceed toward the minimum point in a sequential manner. An iterative optimization method is typically judged based on its rate of convergence [54]. In general, an optimization method is said to have convergence of order p if 1(n + 1) <*11 1 (2) II z(n) Z'l
where k(n) and k(n + 1) denote the points obtained at the end of iterations n and n + 1, respectively, '* represents the optimum point, and IIl denotes the length or norm of the vector &.
11
If p = 1 and 0
2.1.2 Direct Optimization
In problems where analytic solutions are not possible and the design variables are of mixed type there is little choice but to use some variation on a direct search methodology. Direct searches may be broken into the following broad categories.
2.1.2.1 Exhaustive methods
In most practical applications, the optimum solution is known to lie within restricted ranges of the design variables. Exhaustive search methods are applied to problems where the interval in which the optimum is known to lie is finite. Conceptually, these methods evaluate the objective function at a predetermined number of points in this interval and reduce the interval of uncertainty using the assumption of unimodality. Exhaustive methods include [54]:
1) Random Search
2) Grid Search
3) Pattern Directions
2.1.2.2 Elimination methods
The exhaustive search methods are similar to a larger class of algorithms known as
elimination methods, because they search by eliminating parts of the interval. Elimination methods differ in how they search and discard subintervals. The more common elimination methods include [54]:
1) Dichotomous Search
12
2) Interval Halving
3) Fibonacci Method
4) Golden Section Method
2.1.2.3 Interpolation methods
Interpolation methods iteratively fit the local performance surface with a simple polynomial form, and then approximate the minimum point of the system as the minimum point of the polynomial [65]. These methods are generally more efficient than elimination methods and can be accelerated if gradient information is available. Some of the more popular interpolation methods include [54]:
1) Quadratic Method
2) Cubic Method
3) Newton Method
4) QuasiNewton Method
5) Secant Method
2.1.2.4 Unrestricted methods
When the design variable range is not known the search must be performed without restrictions on the values of the variables. Most of these methods use a step size and move from an initial guess in favorable direction (positive or negative) [54]. The step size used must be small in relation to the final accuracy desired. This method is often accelerated by using a variable step size. These methods include [54]:
1) Simplex Method
2) Revised Simplex Method
3) Karmarkar's Method
4) Hook's and Jeeves' Method
13
5) Rosenbrock's Method
In addition, evolutionary computing techniques like genetic algorithms belong to this category.
2.1.2.5 Line search
All of the direct search methods presented above can be applied to both onedimensional or ndimensional searches. A onedimensional search is often referred to as a line search since we are searching along a line. The aim of all line searches is to find i* e 91 such that
n* = ArgMin {J( + 7 d)}, (3) where is the design vector, and d is a known search direction.
One of the most efficient, and hence most popular, line searches uses the Quadratic Method to find 4* [54]. This method has been applied in this work, using the following algorithm:
I: Normalize the search vector zd by dividing each component by the
absolute value of the element of zd with the maximum absolute value
II: Evaluate the function at the points A = 0 and D = o0, where 10 is
an initial step size
III: If JD> JA then set C = D andB = 10/2
IV: Else set B = D and evaluate at the point E = 2rI0. If JE > JD then
set C = E. Else set D = E and T0 = 20, and goto step III
V: Calculate
4JB 3JAJc (4) 4JB 2JC 2JA
14
VI: If J < AJn' then set T* = and quit 11
VII:If
VIII:Goto step V
where A'in is the minimum change in J to detect early stopping.
2.1.3 DescentBased Optimization
9z
When all values of e 9I are possible and the function J(1) has first and second partial derivatives everywhere, the necessary conditions for a local minimum are 8J
= 0, (5) by which we mean 8J/8zi = 0, Vi and a_2 J> (6)
2
by which we mean that the m x m matrix whose components are J/a8zizj must be positive semi definite, i.e., have eigenvalues that are zero or positive [15].
All points that satisfy (5) are called stationary points. Sufficient conditions for a local minimum are (5) and
a2
> 0, (7)
0 ,2
that is all eigenvalues must be positive. If(5) is satisfied but 892 jia2 = 0, that is, the determinant of the matrix is zero (meaning that one or more of its eigenvalues is zero),
15
additional information is needed to establish whether or not the point is a minimum. Such a point is called a singular point.
2.1.3.1 Methods
Classical analytic optimization methods use these conditions to solve for the optimal solution. If the optimization problem involves an objective function or constraints that can not be stated as explicit functions of the design variables, or which are too complicated to manipulate, then descent optimization methods provide efficient alternatives. In general, these methods will have significantly better convergence characteristics than direct methods [54].
Descent search methods are iterative algorithms for improving estimates of the decision variable, ,, so as to come closer to satisfying the conditions for a stationary point. The steps in using the descent method are as follows:
I: Set n = 0 and guess at the initial design vector ,(n), usually random
II: Determine the values of dJ/81(n)
III: Interpreting 8J/a(n) as the gradient vector, determine the search
direction d (n) = f (OJ/8k(n)) as a function of this gradient
IV: Determine the step size to be taken T(n) = f(zd(n)), as a function
of this direction
V: Update the estimates of Z(n + 1) = '(n) + (n)z'd(n) VI: Repeat II until (8H/8t(n))(8H/9(n))T is very small
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The variations in descentbased optimization can be expressed as variations in the
determination of the direction vector / and the step size f Some of the more common variations include [15][54][65]:
1) Steepest Descent:
dz (n) = 8H/a(n) (8) 1 (n) = constant (9) 2) Steepest Descent with Momentum:
zd(n) = (1 p)aH/a(n) + pz' d(n) (10) T (n) = constant (11) 3) Conjugate Gradients:
d (aH/8(n + 1))T[aH/a(n + 1) aH/a(n)] (12) (8H/8k(n + 1)) 8H/8~(n + 1)
1 (n) = LineSearch(Ad(n)) (13)
2.2 Neural Networks
Artificial neural networks (ANNs) are biologically motivated data processing structure that consist of a large number of relatively simple highly interconnected neurons or processing elements (PEs) [24]. In general, these structures provide an inductive mathematical model that can be represented by ) = f(, T), (14) where f:9N + 9 is the model's input/output map, and e 91 h~ E9 and 9Nw are its outputs, inputs, and coefficients, respectively. The coefficients in an
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ANN map are commonly referred to as weights. Artificial neural networks infer or learn the relationships between and b by observing actual process data. In this way, ANNs can be applied to generalized regression and classification inference problems.
ANN architectures possess two fundamental properties:
1) They are capable of approximating to arbitrary accuracy any continuous
function, i.e., they are universal mappers [24].
2) They have robust optimization convergence properties with respect to
the optimization of their coefficients, i.e., they are robust learners [26].
These properties make ANNs a useful tool for empirical modeling tasks where little to no a priori information is available about the underlying process.
2.2.1 Model Architecture
There are many types of ANNs in the literature, each with specific advantages when modeling various types of processes [14]. The two primary factors which differentiate between ANN models are their architecture and their learning rule. A model's architecture defines the way in which it processes input information to produce output information, i.e., the form of their mathematical input/output map f.
2.2.1.1 Multilayer perceptron
Most ANNs presented in the literature are static mappers, i.e., they are only capable of modeling static or steadystate process relationships. By far the most popular and widely applied ANN architecture is called the multilayer perceptron (MLP) [24]. This network consists of fullyinterconnected layers of PEs with logistic response characteristics. The MLP network is typically configured with one or two hidden layers ofPEs. A twolayer MLP is illustrated in Figure 1.
18
Figure 1: Multilayer perceptron model architecture.
Formally, using matrix algebra this architecture is given by
f/lp(h, o) = 0y(1h2 1(lTlh + hI)+ h2)+ Ay, (15) where le N,, x N" is the matrix of weights for the first hidden layer, h le N" is a vector of bias values for this layer, 2 C 2 X NhI is the matrix of weights for the second hidden layer, Ah2 e Nh2 is the bias vector for this layer, Y N x N12 is the matrix of weights for the output layer, A 9? NY is the corresponding bias vector, Nhl is the number of PEs in the first hidden layer, Nh2 is the number of PEs in the second hidden layer, acr is the tanh logistic function, and the set P = { T 1, h 1 2, h2, represents the model's weights.
When the process being modeled is dynamic, i.e., its current output is a function of its current state as well as previous process states, static models are not well suited. For such situations, models which are able to extract both static and temporal process relationships are required. The most common method for creating dynamic neural networks is to simply place dynamic PEs in the input layer of a static MLP [17]. These models have been referred to by many researchers as dynamic neural networks (DNNs). The dynamic PEs
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can have response characteristics based on a priori process knowledge or contain adaptive memory mechanisms or filters.
2.2.1.2 Timedelay neural network
The most common DNN is called the timedelay neural network (TDNN) [24]. This architecture consists of a MLP where each input PE has an adaptive linear FIR filter, as illustrated in Figure 2.
UP)3
(1W
Figure 2: TDNN input PE connectivity. Formally, the TDNN can be described by
ui (t) j = 1 (h(t), N = di j Vi,j (16) ij 1 (t 1) j E (2, N ] /dnn((t), ) = pdl( (t), NT), iP), (17) where :9 N N x N represents the tappeddelay line operator, and N is the number of taps in the delay line.
2.2.1.3 Gamma neural network
The main disadvantage to most DNNs is that they preprocess the input to extract fixed and known dynamics of the process data rather than learn these dynamics from this data. The TDNN can be considered as an exception to this rule, but here the process must have
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finite impulse response (FIR) dynamics of known order. The Gamma Neural Network (GNN) [17] represents an important class of dynamic ANN models that is able to learn infinite impulse response (IIR) process dynamics without a priori knowledge about the structure or order of these dynamics.
7 )Z '(1 y2
(1 71)z1 72
71
UI,/
ui(t)
Figure 3: Gamma memory processing element.
The GNN architecture is conceptually an MLP with adaptive Gamma Filters (GF)
placed at the output of its input and hidden layer PEs. A single GF is illustrated in Figure
3. Formally, the GNN is given by
T ui (t) j = 1
j( T(t), N) = Vi, j (18) l(t)+(1 yj) .j(t 1) j E (2,N T] till Wae phIT hl el h2)+A,
g (h(, t) = Ta(Jz2 J 1(hIlJ((t), N ) + ), I + ', (19) where Jg:9N" x 'NT
where : + N"x N represents the GF, N is the number of GF taps in the input layer, and N is the number of GF taps in the first hidden layer. This architecture has been presented without a GF in the second hidden layer, but such a configuration would be a straightforward extension.
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2.2.1.4 Nonlinear statespace model
The GNN uses the Gamma Filter to represent process dynamics. The GF approximates these dynamics from a Gamma memory kernel basis. The Gamma kernels are able to model an important class of dynamics but may not be the best representation for general process dynamics. An alternative approach to using Gamma kernels is to design the ANN architecture with an explicit state and data flow structure that is capable of learning universal process dynamics. This approach is the goal of the nonlinear statespace model (NLSS) which implements process dynamics directly as a nonlinear state evolution equation and an output observation equation [43], as given by
(t) = /x(t(t 1), h(t), tx) (20) Y(t) = ((t), (t), 4y), (21) where k(t) e 9< is the models state vector consisting of Nx hidden PEs, fj is an ANN map describing the time evolution of this state, tx are the weights of this state network, f is a second ANN map describing how outputs are produced from this state, and t are the weights of this output network. Figure 4 illustrates the configuration of a NLSS network.
Figure 4: Nonlinear state space neural network configuration.
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2.2.2 Learning Algorithms
The biological roots of neural networks are responsible for the widespread use of the term learning to describe the process during which the network parameters are changed to improve the performance of the neuralnetworkbased system. An ANN learning algorithm specifies how its weights are updated in response to training data. These algorithms are simply optimization methods applied to the task of finding the best model weights to minimize a specified modeling objective J, i.e., ArgMin,{J} In general, any of the optimization methods presented above can be used to solve this problem.
One of the most significant breakthroughs in the field of ANNs was the realization that the chain rule for ordered partial derivatives provides a mechanism for deriving the firstorder gradients for all weights in a model, even though the modeling objective is only an explicit function of the model's outputs [72]. Recall that the chain rule for ordered partial derivatives is given by
J J + _" Cxi (22) x ax a8x .ax .
J J ij 1 J
Applying the chain rule allows sensitivities to be calculated from the output of the
model back to its input, which is why the resulting algorithm has been coined "backpropagation" in the literature [40]. When the variables are temporally related, the chain rule has the following form
oJ ( J + a tJ xi(t + ) + (23)
8x(t) x(t) at )8xi(t + ) 8xj(t)
Here, in addition to backpropagating sensitivities from the model's output to its input, the sensitivities are backpropagated through time.
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The most common optimization method is simply steepestdescent with momentum, although many variations have been demonstrated to significantly improve convergence [10]. The issues leading to the selection of one optimization method over another are:
1) Convergence Rate
2) Implementation Complexity
3) Configuration Complexity
4) Avoidance of Local Minima
5) Sensitivity to Correlation
The most common modeling objective used in ANNs is the mean squared error (MSE) between the model's output ) and a specified desired response d e 91 as given by
J 1 (di(t) yi(t)) (24) Nt= l i= 1
where Nt is the number of samples in the training dataset. Learning rules which use the MSE criterion are commonly classified as supervised learning rules, because of the presence of a "teacher" implied by the explicit specification of a desired response. Learning rules without explicit reference to a desired response for the model in the objective function are commonly referred to as unsupervised learning rules.
2.2.3 Generalization
As universal mappers ANNs are almost always more complex than the relationships that we seek to uncover. The net result is that ANNs are notorious for overfitting a training dataset, i.e., performing well on training data but poorly on a blind test dataset [24]. It
24
is very important to optimize the complexity of the neural network in order to achieve the best generalization.
2.2.3.1 Bias and variance
Considerable insight into this phenomenon can be obtained by introducing the concept of the biasvariance tradeoff. Bishop [13] observes that the generalization error using the Euclidean norm, will depend on a particular dataset D on which the network was trained. The dependence on D can be eliminated by considering an average over the complete ensemble of datasets, which can be written as = ED[((W )f(, ))2], (25) where (Jh) denotes the conditional average, or regression, of the desired data given by,
(dI) = .p(Il)dt, (26) and p(d h) is the conditional density of the desired variable d conditioned on the input vector h Bishop [13] demonstrates that this generalization error can be decomposed into the sum of the bias squared plus the variance
= (ED f( t) h))2 + ED[f(, ) ED[f(, )]]. (27)
A model which is too simple, or too inflexible, will have a large bias, while one which has too much flexibility in relation to the particular dataset will have a large variance. Bias and variance are complementary quantities, and the best generalization is obtained when we have the best compromise between the conflicting requirements of small bias and small variance. The variance of the prediction will be further addressed below, Section
2.2.4 "Standard Errors."
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For any given dataset, there is some optimal balance between bias and variance which gives the smallest average generalization error. In order to improve the performance of the network further we need to be able to reduce the bias while simultaneously reducing the variance. The more straightforward way of achieving this is to use more data samples. As the number of data samples is increased we can afford to use more complex models, hence reducing the bias, while at the same time ensuring that each model is more heavily constrained by the data., thereby also reducing the variance. If the number of data samples is increased rapidly in relation to the model complexity we can find a sequence of models such that both bias and variance decrease. Models such as ANNs can in principle provide consistent estimators of arbitrary accuracy as the number of data points is increased to infinity. Note that, even if both the bias and variance can be reduced to zero, the generalization error will still be nonzero due to the intrinsic noise in the data.
One rarely has infinite data, and practical issues like training time make simply adding more data points impractical. There are several practical and practiced ways to improve model generalization, we start with regularization.
2.2.3.2 Regularization
Regularization was originally proposed by Tikhanov [62] as a method for solving illposed problems. The basic idea is to stabilize the solution by means of some auxiliary nonnegative functional that embeds prior information, e.g. smoothness constraints on the input/output mapping. Regularization is able to transform an illposed problem into a wellposed problem [48].
26
Tikhanov's regularization theory uses a regularization penalty term of the form 12
D = 2 Pf 1,, (28) where P is a linear (pseudo) differential operator. This penalty term is added to the objective function to give
J = J+ FO, (29) where F is the regularization parameter. Prior information about the form of the solution (i.e., the plant) is embedded in the operator P. The operator P is referred to as a stabilizer in the sense that it stabilizes the solution ', making it smooth.
The appropriate choice for P and the solution to (28) requires functional analysis and is beyond the scope of this work. The most commonly used form of regularizer, however, is quite simple to implement. Weight decay regularizer terms consist of the sum of squares of the adaptive parameters in the network = wi, (30)
I
where the sum runs over the weights and biases. In conventional curve fitting the use of this form of regularizer is called ridge regression. It has been found empirically that a regularizer of this form can lead to significant improvements in generalization [29].
2.2.3.3 Growing and pruning algorithms
The topology of a neural network, number of units and interconnections, can have a significant impact on its performance. Regularization helps to minimize this impact when the complexity of the network is larger than required for the particular application. Clearly, however, a better approach is to match the complexity of the model with the com
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plexity of the application. Various techniques have been developed for optimizing the topology, in some cases as part of the network training process itself [43]. It is important to distinguish between two distinct aspects of the topology selection problems. First, we need a systematic procedure for exploring some space of possible architectures. Second, we need some way of deciding which of the architectures considered should be selected.
A straightforward approach to network structure optimization involves an exhaustive search through a restricted class of network topologies. This approach requires significant computational effort and only searches a very restricted class of network topologies. Much of the computational burden can be lessened by considering a network which is relatively small and by allowing new units and connections to be added during training. This approach was shown to be successful by Bello [10] who used the weights from one network as the initial guess for training the next network (with the extra weights initialized randomly). Techniques of this form are called growing algorithms. An alternative approach is to start with a relatively large network and gradually remove units; these are known as pruning algorithms. Most of these procedures are ad hoc and tailor to specific applications, that is not to say, however, that they are ineffective.
More recent work has taken advantage of developments in discrete optimization using genetic algorithms [36]. Genetic algorithms provide a methodical way of searching large discrete spaces more efficiently.
2.2.3.4 Crossvalidation
An alternative to regularization as a way of controlling the effective complexity of a network is the procedure of crossvalidation [13]. The training of a nonlinear model corresponds to the iterative reduction of the error function defined with respect to the training
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dataset. During training, the error will generally monotonically decrease as a function of the number of presentations of the training dataset, i.e., epochs. However, the generalization error, with respect to an independent dataset called the validation dataset, often shows a decrease at first, followed by an increase as the network starts to overfit. Training can therefore be stopped at the point of smallest error with respect to the validation dataset as this produces a network with the smallest generalization error (or at least an approximation thereof).
2.2.3.5 Committees of networks
In practice, building neural network models requires the training of many different
candidate networks and then the selection of the best performer. Typically performance is based on the networks performance on a third dataset not used for training or crossvalidation. There are two disadvantages to this approach. First, all of the effort involved in training the remaining networks is wasted, and secondly, the generalization performance on the validation dataset has a random component due to the noise on the data [13]. The network which performed the best on this dataset might not be the one with the lowest generalization error. Recall that the generalization error is averaged over all datasets (25).
These limitations can be overcome by combining the networks together to form a
committee [47][46]. This approach was shown to provide significant improvements in the generalization error. Denote the committee prediction as Nc
(, = 1 y(,wit), (31) Nci= ,
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where Fi are the weights of committee member i, and { i } is the set of all weights for the committee.
Bishop [13] shows that if the errors of the individual committee member are decorrelated, then the committee will always have a lower generalization error than any of its individual members.
2.2.4 Standard Errors
Tibshirani [61] reviews a number of methods for estimating the standard error of predicted values from a multilayer perceptron. These include direct evaluation of maximum likelihoods based on the Hessian matrix, the "sandwich" estimator and the bootstrap method. Tibshirani offers the following observations:
1) The bootstrap methods provided the most accurate estimates of the standard errors of predicted values.
2) The nonsimulation methods (delta and sandwich) missed the substantial variability due to the random initial weights from the multiple training runs.
The nonsimulation methods are solved analytically, and therefore require unique
solutions for each network topology. Whereas the bootstrap methods apply to all network topologies, as well as nonneural network paradigms. The additional fact, as noted above, that the bootstrap methods account for local minima, provides strong argument for their use.
2.2.4.0.1 Bootstrap methods
Bootstrap methods work by creating many pseudoreplicates ("bootstrap datasets") from the training dataset and then reestimating the models weights t on each bootstrap dataset. There are two different approaches to bootstrapping [9]. One can consider each
30
training case as a sampling unit, and sample with replacement from the training dataset cases to create a bootstrap sample. This is often called the "bootstrap pairs" method. The bootstrap pairs sampling algorithm is given by:
I: Generate Nb samples, each one of size Ns drawn with replacement
from the Ns training observations { h (i), (i)}i= 1, and the b th sam~b ,b N'
ple by { Ab(i),d (i) }I 1
II: For each bootstrap sample b e [1, Nb], find
ArgMinb {Jb d f( b b )) } (32) III: Estimate the standard error of the i th prediction as
2 Yi) (33) where
1
Yi fi(d, ) (34) Nj= 1
On the other hand, one can consider the predictors as fixed, treat the model residuals
) as the sampling units, and create a bootstrap sample by adding residuals to the model fit )5. This is called the "bootstrap residuals" approach:
I: Find ArgMin0 {J(d f(k, t)) } from the N' training observations
{(i), (i.)} I= 1 and let r(i) = l(i)f(h(i), T)
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II: Generate Nb bootstrap samples, each one of size Ns drawn with
b
replacement from { (i) = 1, and the b th sample by { r (i) } i = i letting
b b '
i = {f(=, ) + (i)}i 1 (35) III: For each bootstrap sample be [1, Nb], find
ArgMin { Jb f( b)) } (36) IV: Estimate the standard error of the i th prediction as Nb
2 1 2 C (yi) 1 f(, ) i) (37) Nb 11=
j= 1
Note that both of these methods require fitting a model (retraining the network) Nb times. Typically Nb is in the range 20 Nb ___ 200. In simple linear least squares regression, it can be shown that the bootstrap methods both agree with the standard least squares formula as Nb > o.
The bootstrap methods will arrive at confidence intervals
2 2
Yi CconJ2 (i) Yi < i CcoJ (Yi) (38) where cconf depends on the desired confidence level 1 Ox. The factor cconf can be taken from a table with the percentage points of the Student's tdistribution with the number of degrees of freedom equal to the number of bootstrap runs Nb.
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2.3 Control Theory
An ANN is capable of modeling any process making them ideal candidates for complex process optimization and control strategies. Neurocontrol is but a subfield of classical control theory [31]. To put neurocontrol in perspective, it is important to consider its place within this field.
2.3.1 Classical Control Theory
Classical control theory is strongly biased towards linear timeinvariant systems [31]. General nonlinear systems simply do not allow us, because of their analytical intractability, to formulate a theory that is as strong as that of linear system theory. On the other hand, nonlinear systems can be qualitatively similar to linear ones under some circumstances.
2.3.1.1 Linear control
Linear control is concerned with systems of the form
& = +I (39) with state t, input h, measurable output
S= (40) and controllers of the form
= + (41) where i* is the reference state, that is, the state to which the plant is to be brought with the help of the controller [33].
The goals of linear control are:
1) Altering the closedloop behavior of the system to some userdefined
response characteristics.
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2) Controlling the closedloop stability, i.e., convergence back to an equilibrium point after disturbance.
The disadvantages of linear control designs:
1) Assume that the world is linear Gaussian and stationary, when in reality
the world is none of the above.
2) Require complete a priori knowledge of the process dynamics.
3) Require that the process is controllable and observable.
4) Cannot follow a reference trajectory produced by a system of lower
order that the process.
2.3.1.2 Robust control
Robust control addresses the problem of controlling a plant whose behavior is slightly different from that of a plant model [37]. The reasons for the difference are predominately the effect of the nonlinear, nonGaussian and nonstationary world. A popular pragmatic classical approach to robust control is concerned with preserving stability [2]. The closedloop eigenvalues are chosen so that they remain in the stability region even if the plant model should change in a defined range.
Although robust control strategies are primarily designed to compensate for differences between our linear timeinvariant assumptions and the real world, they are still developed based predominately on linear system theory. They are therefore, not able to cope with significant deviations from these assumptions.
2.3.1.3 Adaptive control
Adaptive control is another way to reach a goal similar to that of robust control [37]. Instead of designing robust controllers that work under conditions different from those for which they have been designed, adaptive controllers recognize the difference between the assumption and reality and change to perform better in the new conditions.
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Adaptation schemes can be based on both a reference model and a cost functional [2]. The approach called model reference adaptive control (MRAC) is, by its name, committed to the former. This approach is based on formulating the rules for computing the direction of change of controller parameters as a function of the difference between the behavior of the closedloop system and a reference model. Controller parameters can be adapted either directly or via the estimation of plant model parameters.
A more general approach is that of selftuning regulators (STR) [3] which consists in adaptive estimation of a plant model and applying a formalized controller design method to the plant model. This design method can be based on cost function optimization.
Like robust control, however, adaptive control implementations have been based on linear, or simple nonlinear parametric assumptions, about the process. As a result, adaptive control designs have not demonstrated significant successful with complex realworld processes.
2.3.1.4 Nonlinear control
Nonlinear control theory is concerned with general systems of the form [31]
x = f(&, ) (42) with measurable output
= g(t) (43) and controllers of the form
S= c(t' *). (44) The general formulation of nonlinear control holds promise to overcome all of the limitations of the classical control schemes presented above. The fields track record, however, does not deliver on this promise. The problem is that an analytical solution is known
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only for a restrictive subclass of nonlinear systems. The difficulties with genuine nonlinear controller designs have typically lead to a linearization approach, also known as gain scheduling.
2.3.1.5 Optimal control
The topic of optimal control theory is to design controllers that are optimized to a certain performance criterion [15]. Classical optimal control has primarily focused on applications where such optimality could be proven analytically. For example, for a linear plant and a quadratic performance criterion the Ricatti controller represents an explicit and global solution [31]. Dynamic optimization provides another example for state evaluation and selection of the optimal action, which can be proven optimal in certain applications. Alternatively, if each state at each sampling period is represented by a node in a directed graph and actions are represented by connecting edges of the subsequent states, then the task can also be transformed to the critical graph problem of graph theory [15].
In its most general form, the optimal control problem can be formulated as an optimization problem [15]. The plant can be generalized as in (42) and the goal is to find a controller described by (44) that minimizes
E[J(.* )], (45) where E[ ] is the mean value over time.
2.3.2 Neurocontrol
Most neurocontrol architectures are either explicit or disguised analogies of classical control design such as optimal control or numerical lyapunovfunctionbased design methods. It has been argued [14] that it is only the representation of functions by neural
36
networks that defines the field of neurocontrol in the broad sense and differentiates neurocontrol form classical control methods.
The author agrees that the use of nonparametric models does differentiate neurocontrol from classical controls, but would argue that the primary departure from, and extension to the potential of, control theory is neurocontrol's willingness to depart from a requirement for analytic solutions. Neurocontrol designs seek to realize the promise of general nonlinear control replacing analytic optimization methods with numeric ones. Researchers in the field of neural networks are accustomed to working in an intractable world, and have been willing to resolve important questions like stability, robustness and consistency empirically. The result has been, and will continue to be an important extension to control theory.
There are many types of neurocontrol architectures in the literature, each with specific advantages and disadvantages. The following sections review some of the conceptual neurocontrol strategies which have been proposed in the literature.
2.3.2.1 Modelpredictive control
When a model is used indirectly and offline the control scheme is usually referred to as modelpredictive control (MPC) [37]. In most industrial process control applications a priori knowledge about the process is hard to obtain and blackbox models must be used. The offline training phase performs supervised learning to develop an ANN model for the process to be controlled, i.e., the ANN attempts to mimic the process after being exposed to actual process data. This phase can be stated as ArgMin 0 { J(fp ffff.013 (46)
37
where fY and e are the process outputs to be controlled and inputs to be manipulated, respectively.
At the online control phase, the ANN model cannot be used alone; it must be incorporated with a modelbased control scheme [31]. This control scheme is once again an optimization problem, which can be stated as A rgMin { Ju* f(,P, P)) }, (47) where 9* 0 denotes the desired closedloop process output. This optimization is performed repeatedly at each time interval during the course of feedback control.
2.3.2.2 Modelinverse control
An ANN can be trained to develop an inverse model of a process [40]. Here, the model's input is the process output, and the model's output is the process input. The offline training phase can be stated as
A rgMin f{ J(P f(f", 0)) }. (48) Clearly, the inverse model is a steadystate model or the resulting controller would be noncausal. Given a desired process setpoint f*, the appropriate online control signal can be immediately calculated as
e = ff*, 4). (49) Successful applications of inverse modeling are discussed in [40] and [58]. Obviously, an inverse model exists only when the process behaves monotonically as a "feedforward" function at steadystate. If not, this approach is not applicable.
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2.3.2.3 Controller modeling
Another simple direct neurocontrol scheme is to use a neural network to model an
existing controller. The input to the existing controller is used as training input to the ANN model, and the controller output serves as the desired response. This approach is similar to the modelinverse control except that the desired response here is not a process but a controller. This approach can be formulated as ArgMinof{J(hc f(', W)) }, (50) where h c are the decision variables generated by an existing controller in response to the plant states p.
Like a process, a controller is generally dynamic and often comprises integrators or
differentiators. If an algebraic feedforward network is used to model the existing controller, dynamic information must be explicitly provided as input to the ANN model.
In general, this approach can result in controllers that are faster and/or cheaper than traditional controllers. Using this approach, for example, Pomerleau [50] presented an intriguing application where a neural network was used to replace a human operator, i.e., an existing controller.
2.3.2.4 Modelfree direct control
Without an existing controller or process knowledge, controllers have to be adapted or learn the way a human operator learns to control/operate a process for the first time. A modelfree neurocontrol design objective can be stated as ArgMinc{Jo* 1 ), P (P, c) (51)
39
where jc is an ANN that is directly controlling the process inputs, and tc are the weights of this network. Notice that the optimization criterion J is only a function of the actual and desired process outputs. This means that the optimization methodology employed must be able to learn Oc without an explicit desired response or even a mathematical linkage to the criterion.
The key feature of this direct adaptive approach is that a process model is neither known in advance nor explicitly developed during control design. This most common learning algorithm for this type of control design is referred to as reinforcement learning. The first work in this area was the "adaptive critic" algorithm proposed by Barto et al. [7]. Such an algorithm can be considered as an approximate version of dynamic programming [73][8], later coined as NeuroDynamic Programming [12].
Despite its historical importance and intuitive appeal, modelfree adaptive neurocontrol is not appropriate for most real world applications. The plant is most likely out of control during the learning process, and few industrial processes can tolerate the large number of failures required to adapt the controller.
2.3.2.5 Modelreference direct control
From a practical perspective, one would prefer to let failures take place in a simulated environment with a process model rather than in a real plant. Even if failures are not disastrous they can cause substantial losses. The performance of a controller could be evaluated based on a model for the process, rather than the process itself. The training stage of the control design can be given by
ArgMin m{J(pfn(, I C m)}, (52)
40
and the control design becomes
ArgMin p{Jo* _(e ,m A )), Ic = pc) }. (53)
In the course of modeling the plant, the plant must be operated "normally" instead of being driven out of control. After the modeling stage, the model can be used for controller design. If a process model is already available, an ANN controller can be developed in a simulation in which failures cannot cause any loss but that of computer time. A neural network controller after extensive training in the simulation can then be installed in the actual control system.
ModelReference direct control schemes have not only proven effective in several studies [41][63], but have also already produced notable economic benefits [60]. These approaches can be used for both offline control and for online adaptation.
2.4 NOx
The Clean Air Act Amendments of 1990 require that electric utilities make significant reductions in nitrogen oxide (NOx) emissions from their fossilfired power plants. To date, most efforts to reduce NOx emissions have come from expensive hardware retrofits with less than satisfactory performance. Further complicating matters, conditions that decrease NOx formation (lower temperature, excess fuel) result in the formation other polluting compounds, mainly carbon monoxide (CO). Similar emissions reductions are being required in Europe through local and European Economic Community (ECC) initiatives.
Nitrogen monoxide (NO) and nitrogen dioxide (NO2) are byproducts of the combustion process of virtually all fossil fuels. Historically, the quantity of these inorganic compounds in the products of combustion was not sufficient to affect boiler performance; their
41
presence was largely ignored. In recent years, oxides of nitrogen have been shown to be key constituents in the complex photochemical oxidant reaction with sunlight to form smog. Today, the emission of NO2 and NO (collectively referred to as NOx) is regulated by the 1990 Clean Air Act Amendments and has become an important consideration in the design of fuel firing equipment.
NOx is formed by two primary mechanisms: thermal NOx and fuelbound NOx. Thermal NOx formation occurs only at high flame temperatures when dissociated nitrogen from combustion air combines with oxygen atoms to produce oxides of nitrogen such as NO and NO2. The formation of thermal NOx increases exponentially with combustion temperature and increases by a squareroot relationship with the presence of oxygen in the combustion zone. Fuelbound NOx formation is not limited to high temperatures, but is dependent upon the nitrogen content of the fuel. The best way to minimize NOx formation is to reduce flame temperature, reduce excess oxygen, and/or to burn low nitrogencontaining fuels. Conditions that decrease NOx formation (lower temperature, excess fuel) can result in incomplete combustion. These conditions result in the formation other polluting compounds, mainly carbon monoxide (CO).
2.4.1 Reduction
The available NOx reduction technologies can be categorized into one of the following:
Before Combustion: Nitrogen is extracted from the fuel. This is relatively ineffective, since most of the nitrogen in the formation of NOx comes from the air
(containing N2).
42
After Combustion: NOx is chemically reduced before leaving the stack. This process is also expensive, requiring hardware retrofits.
During Combustion: Altering fuel and air flows and introducing them at different
points of the furnace can create several zones with different temperatures and
stoichiometry. These parameters significantly effect the rate of NOx formation.
The following section reviews available NOx reduction strategies and technologies for combustion sources.
Fuel Switching: Fuelbound NOx formation is most effectively reduced by
switching to a fuel with lower nitrogen content. No. 6 fuel oil or another residual fuel having a relatively high nitrogen content can be replaced with No. 2 fuel oil, another distillate oil or natural gas (which is essentially nitrogenfree) to reduce
NOx emissions.
Flue Gas Recirculation (FGR): Flue gas recirculation involves extraction of some
of the flue gas from the stack, and recirculation with the combustion air supplied to the burners. The process reduces both the oxygen concentrations at the burners and the temperature by diluting the combustion air with flue gas. CO can
become a significant problem here.
Low NOx Burners: Installation of burners especially designed to limit NOx formation can reduce NOx emissions. Higher reduction efficiencies can be
achieved by combining a low NOx burner with FGR. Low NOx burners are
designed to reduce the peak flame temperature by inducing recirculation zones,
staging combustion zones, and reducing local oxygen concentrations.
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* Derating: Some industrial boilers may be derated to produce a reduced quantity
of steam or hot water. Derating will decrease the flame temperature within the
unit, reducing formation of thermal NOx. Derating can be accomplished by
reducing the firing rate or by installing a permanent restriction, such as an orifice
plate, in the fuel line. Clearly this solution would have significant economic
impact on the unit.
* Steam or Water Injection: By injecting a small amount of water or steam into the
immediate vicinity of the flame, the flame temperature will be lowered and the local oxygen concentration reduced. The result would be to decrease the formation of thermal and fuelbound NOx. This process generally lowers the combustion efficiency of the unit by one or two percent.
* Staged Combustion: Either air or fuel injection can be staged, creating either a
fuelrich zone followed by an airrich zone, or an airrich zone followed by a
fuelrich zone. A low NOx burner utilizing staged combustion can be installed,
or the furnace itself can be retrofitted for staged combustion.
* Fuel Reburning: Staged combustion can be achieved through the fuel reburning
process. A Gas Reburning Zone (GRZ) is created above the primary combustion
zone. In the GRZ, additional natural gas is injected, creating a fuelrich region
where hydrocarbon radicals react with NOx to form molecular nitrogen.
* Reduced Oxygen Concentration: Decreasing excess air reduces the oxygen available in the combustion zone and lengthens the flame, resulting in a lower heat release rate per unit flame volume. NOx emissions are reduced in an approxi
44
mately linear fashion with decreasing excess air. However, as excess air is reduced beyond a threshold value, combustion efficiency will decrease due to incomplete mixing, and CO emissions will increase. Selective Catalytic Reduction (SCR): Selective catalytic reduction (SCR) is a postformation NOx control technology that uses a catalyst to facilitate a chemical reaction between NOx and ammonia to produce nitrogen and water. An ammonia/air or ammonia/steam mixture is injected into the exhaust gas, which then passes through a catalyst where NOx is reduced. To optimize the reaction, the temperature of the exhaust gas must be in a certain range when it passes through the catalyst bed. Among its disadvantages, SCR requires additional space for the catalyst and reactor vessel, as well as ammonia storage, distribution, and injection system. Precise control of ammonia injection is critical. An inadequate amount of ammonia can result in unacceptable high NOx emission rates, while excess ammonia can lead to ammonia "slip", or the venting of undesirable ammonia to the atmosphere.
SSelective NonCatalytic Reduction (SNCR): Selective noncatalytic NOx reduction involves injection of a nitrogenous agent, such as ammonia or urea, into the flue gas. The optimum injection temperature when using ammonia is 1850 degrees F, at which 60 percent NOx removal can be approached. The optimum temperature range is wider when using urea. Below the optimum temperature range, ammonia is formed, and above, NOx emissions actually increase. The success of NOx removal depends not only on the injection temperature, but also
45
on the ability of the agent to mix sufficiently with flue gas.
2.5 FossilFired Power Generation In general Canal Unit 2 is a large fossil fuel combustion engine. From an abstract perspective, the combustion process takes in air and fuel, and produces energy and exhaust; as described by:
1) Air: Fossil fuel combustion requires air, or more specifically the oxygen
contained in air. Subsystems within the plant measure, prepare and
introduce this air.
2) Fuel: Combustion also requires fuel. In the case of Canal Unit 2, the fuel
can be either #6 residual oil (leftover from the refining process) or natural gas. Canal Unit 2 can fire oil only, gas only or a mixture of both.
Both fuels must be measured, prepared and introduced to the furnace.
3) Energy: The energy released by the oxidation of fossil fuels during combustion is used to make steam. The properties of water still make it the
best choice when converting thermal energy to work. Canal uses the
radiative and convective heat from the combustion process to transform ultraclean water into superheated steam. The expansion of this steam is
used to turn a turbine that turns a coil in a magnetic field, producing
electric potential. The steam, having done this work flows through
ocean water filled condensers that convert it back to superclean water.
4) Exhaust: The gaseous products of combustion having contributed much
of their heat content to the production of steam are cleaned electrostatically and ejected into the atmosphere.
2.5.1 Process Variables
The specific process variables as they apply to the Canal generating unit are described in more detail in the following sections. These variables are also listed in Appendix, and will be referred to throughout this work.
2.5.1.1 Air
The air required for fossil fuel combustion is prepared and introduced in two ways.
Two large symmetrical Fans called Forced Draft fans push ambient air through a series of
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preheaters that warm this air to between 80 and 180 degrees F. This hot oxygen rich air is then pressed into a windbox that surrounds the furnace enclosing the burner ports. Through an array of vents called Primary and Secondary Air Shrouds around each burner and through secondary ports called Overfire Air Ports this pressurized air is vented into the combustion zone. In addition to this oxygen rich air, Canal Unit 2 has the ability to recycle exhaust gas into the combustion zone through a Gas Recirculation System.
The measurements of all this air are a function of boiler design, and fan capacities. To increase the output of the engine, additional air must be throttled through these devices.
2.5.1.1.1 Forced draft system
The forced draft fans are 2500 horsepower, 624,000 cfmn centrifugal fans with inlet vane throttles. They are constant speed fans meaning that the fan shaft turns at a constant speed while more or less air with more or less initial spin can be dumped into the blades by opening or closing the vanes. If the vanes are only slightly opened the flow volume of air available to the fans is small, and it takes less work to move it. Fan amps will be correspondingly low. If the inlet vanes are opened wider the flow is greater. Still the fan moves at a constant speed. More work is being done, and the amperage must increase. The output of the FD fans is derived from the boiler master signal. Forced draft output is specified along with fuel flow by the fuelair curve of the boiler. The fuelair curve gives a total air flow requirement, as well as a total fuel flow requirement for a given load.
These fans are symmetrical to the furnace like many other systems and they operate symmetrically, through their respective ducts unless biased. Bias represents an addition or subtraction of signal to the B side FD fan. These fans can also be trimmed to meet slightly less or slight more than the Total Air Flow demanded by the fuelair curve of the Boiler.
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The FD fans are the principal air throttles of the Boiler and so have a fundamental effect on nearly every other system.
2.5.1.1.2 Forced draft fan inlet vanes
Since the inlet vanes' positions represent the work being done by the fan and are the control most familiar to the operators, these tags were used to represent the FD fans.
2.5.1.1.3 02 trim
This tag represents the bias that operators set into the airflow demand predetermined by load. Functionally this control trims the response of the FD Fan to Air Demand. This tag gives the operators the ability to run the furnace slightly lean or rich overall.
2.5.1.1.4 Induced draft system
As mentioned in the section on the Forced Draft Fans, the function of the ID Fans is to take whatever gasses are present in the furnace, including air that has been introduced by the load following Forced Draft Fans, plus all products of combustion, and pull them out, maintaining a constant under pressure in the furnace of .5 inches of water column. The FD Fans' speed is kept constant while the volume of air they move is throttled with inlet vane controls. Canal is limited by the power of these fans. Current unit maximum output is frequently limited by the power of these fans to keep up with the increased air flows of the recently installed low NOx shroud and overfire air system.
2.5.1.1.5 Induce draft fan inlet vanes
The induced draft fan inlet vanes are the inlet throttles to the fans, they open in
response to request for increased output and as in the case of the FD fans, represent the work being done.
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2.5.1.1.6 Combustion air temps
The combustion air temperature tags represent the temperature of the incoming air
after the FD Fans. The temperature of this air is a direct result of energy added to ambient air by the Glycol Air Preheater (GAH), and the Combustion Air Preheater (CAH). Since density is a function of temperature, the temperature of this air can impact the combustion process that is sensitive to the Oxygen content of air as well as the operation of other volumetric systems like the Induced Draft (ID) Fans. It also has a primary impact on exhaust gas temperature and resultant stack gas velocity.
2.5.1.1.7 Primary air shrouds
The Primary Air (PA) Shrouds represent the circular articulating vents that surround the individual burner orifices. These are closest to the fuel gun concentrically inside of the Secondary Air (SA) Shrouds. They are responsible for supplying primary combustion air to the flame front. These tags represent actuator positions.
The PA shrouds are controlled by the Burner Management System (BMS) and they
move as a group from minimum position (5% open to provide cooling air) toward open as load increases. The signal that controls them is called the Primary Air Master Demand (PAMD). Separate PAMD signals exist for fuel gas primary air demand and for fuel oil primary air demand. Each specific burner effectively listens to the current fuel state. Having received this signal each burner's own PA shrouds responds to the PAMD in accordance with one of two functions that are unique to it a Burner Primary Air Shroud Function for oil operation and a Burner Primary Air Shroud Function for gas operation. The correct unique local shroud function is changed according to the correct master signal depending on the fuel state of the burner. These burner and fuel specific response func
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tions were set up to give roughly appropriate air flow to combustion at all load points and fuel states based on the air flow inherent to the furnace.
Aside from normal operation the PA shrouds can be biased from the fuel specific master signal or on an individual basis from their respective unique functions.
2.5.1.1.8 Secondary air shrouds
The secondary air shroud tags represent the broadcast actuator positions of the second, outer concentric set of circular articulating vents that surround the individual burner orifices.
The first function of the Secondary Air (SA) Shrouds is to introduce combustion air to the flame front following load. Their second function is to balance windbox pressure, and therefore total airflow, against the actuation of the Overfire Air Ports and the PA shrouds.
The SA Shrouds have a master signal against which a master bias can be set. In addition they have individual actuating functions and individual biases that can be set against these individual functions.
2.5.1.1.9 Over fire air ports
The Overfire Air (OFA) ports are rectangular louvered ports that pass combustion air from the Windbox to the Furnace above the top burner level. In doing this they reoxygenate the oxygen depleted flame front. The tags themselves represent the positions broadcast from the actuators that control the articulating louvers.
The OFA ports were installed as a part of the low NOx retrofit of 1996. The Forney low NOx burner system is designed to burn more coolly and incompletely than normal. NOx formation has been positively linked with time exposure to higher temperatures. After partial combustion has taken place, low in the flame front, extra oxygen rich com
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bustion air is introduced through the OA ports to complete the process. In this way the low NOx burner system stages offstoichiometric combustion to manage combustion products.
The OA port actuators receive their master signal from load. This signal can be biased. Each actuator's response is based on a unique function that was parametrically determined, in concert with the Primary and Secondary Air Shrouds during installation to give best airflow to combustion at all load points.
2.5.1.1.10 Air preheater temps
These represent the temperature of the exhaust gasses entering and leaving the ljungstrom combustion air heat exchanger. The ljungstrom is a large (30 ft. dia.) rotating wheel, arranged perpendicular to the gas flow. It is half enclosed by the exhaust ducts and half enclosed by fresh air ducts. As this wheel slowly rotates, heat is absorbed by a given area of the wheel exposed to exhaust gas. The absorbed heat is then imparted to the incoming air while that same section traverses the fresh air duct. Elaborate seals and pressurized sealing air keep the two gasses from mingling across this device.
The air preheater tags are somewhat redundant. The "In Temps" represent the temperature of the gas on its way in, while the "Out Temps" represent the temperature of the gas on the way out. The heat exchange of the air preheater is a function of the device and of the temperatures of the two gasses and is not controllable in the least. The gas temp after the air pre heater heat exchange was a more familiar control to the operators, however our ability to collect these signals was compromised by a failing thermocouple during a large part of the data collection for phasel. The gas temp before the air preheater was used to represent exit gas temp for the modeling instead.
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2.5.1.1.11 Windbox and furnace
The windbox is an enclosed volume that surrounds the waist of the furnace and the burner openings. Preheated, oxygen rich air is pressurized in this volume by the FD fans. From here this air can pass only into the furnace and only through vanes that surround the burner openings called primary and secondary sir shrouds, or through the overfire air ports above the burners. Canal Unit 2 is a balanced draft furnace which means that air flow through the furnace is controlled around a desired furnace pressure by both pushing and pulling fan systems. The pushing fans are the FD fans, while the pulling fans are the induced draft fans. The FD fans have the primary responsibility of getting the combustion zone all the oxygen it requires. The introduction of this pressurized air is accomplished not only by positively pressurizing the windbox but also by negatively pressurizing the furnace. With the windbox driven to a positive pressure and the furnace kept at a fixed relative negative pressure, the velocity of combustion airflow is assured. The induced draft fans have primary responsibility for maintaining the furnace at a negative pressure relative to the windbox. In the course of increasing unit output the FD fans increase air flow. Their aim is to maintain windbox pressure at +2 inches of water column while air transfer to the furnace increases through the widening overfire ports and primary and secondary air shrouds. The induced draft fans, trying to maintain a constant pressure of .5 inches of water column in the furnace despite this increasing flow of air from the windbox, also ramp up. The opposite happens for decreasing load. When the forced draft fans decrease their output in step with the fuelair demand, air flow from the windbox to the furnace decreases. In order to maintain a constant .5 inwc in the furnace the induced draft fans
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throttled back. Transient changes in the windbox to furnace pressure differential can also produce automated changes in the FD and ID fan flows.
2.5.1.1.12 WindBox pressure
This tags represents the positional average windbox air pressure. It is controlled around + 2 inwc
2.5.1.1.13 Furnace pressure
This tag represents the actual furnace air pressure.
2.5.1.2 Fuel
The Fuel required for Combustion may be either #6 Fuel Oil or Natural Gas. In both cases the fuel is taken from storage, filtered, heated to greater or lesser degree, pressurized, and injected. In the case of #6 Fuel Oil, the temperature required to achieve a pumpable consistency is usually around 200 degrees. Natural Gas comes from high pressure transmission lines and once stepped down to usable pressure is warmed up to around 80 degrees F. Both fuels are then pressurized in their respective headers. It is from these headers that burners, when they are lit, tap their fuel.
2.5.1.2.1 Burnerson/fuel
These tags represent the readings of an array of air cooled optical flame scanners
located in the furnace itself that observe the respective burner flames. Since each burner can fire either natural gas or fuel oil. A scanner, calibrated for each fuel specific flame is permanently assigned to each burner. Although these scanners are analog devices, their primary function is to confirm that the flame emanating from each lit burner is of a threshold quality. If the flame they are monitoring is not of a threshold quality the scanner has
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the will to declare a Master Fuel Trip and cut off all fuel to the furnace. This is to prevent the introduction of unburned fuel to the furnace. These are analog devices but because they are calibrated with the single purpose of either positively or negatively confirming this threshold they essentially read either 1 or 0. This specific set represents the flame quality of its burner if that burner is on natural gas. 2.5.1.2.2 Burner cells 18A & 18B MN gas flame
These are the signals for gas flame status for each burner.
2.5.1.2.3 Burner cells 18A & 18B MN oil flame
These are the signals for oil flame status for each burner.
2.5.1.2.4 Fuel type
As the Boiler Master request increased output BTUs are requested from the Fuel Supply Systems. As a default this request is divided evenly in proportion to burners in service, each of which have BTU content per unit of fuel settings. The total BTUs entering the furnace via the burners in service must equal this demand.
2.5.1.2.5 Fuel oil
The fuel oil introduction system consists of a main pressure generating pump that
ramps up in output as the unit master demand requests more output in the form of BTUs. This pump supplies an operating pressure to the fuel oil header. All oil burners once they are lit and placed into service tap a fixed orifice from this header. Since fuel oil pressure is fixed by the number of BTUs requested by load, and the orifice of each burner tip is a fixed diameter if open, the number of burners in service will dramatically affect Fuel Oil
54
Pressure. Changes in the number of burners lit can vary the fuel oil pressure in the header between 65 and 150 PSIG.
Fuel Temp Fired must be at least the temp required for pumpability, which is specific to the viscosity of the fuel oil being used.
2.5.1.2.6 Natural gas
In a fashion similar to the fuel oil introduction system, the unit master demand
requests BTUs from the gas system. Fuel gas from the pipeline is stepped down to operating pressure, filtered, warmed and supplied to a main gas header. All gas burners when lit tap a fixed orifice from this header. The number of burners lit on gas can affect the actual gas pressure indicated at the header.
2.5.1.2.7 Burner atomization
These tags represent the essential fuel oil atomizing steam parameters. Atomizing
steam is dry superheated steam extracted from the turbine or the reboiler and injected into the oil burner tips to atomize the fuel oil as it is introduced to the combustion zone.
Burner Atomizing Steam pressure runs at a specified 20psig over fuel oil pressure. Burner atomizing steam flow is modulated to maintain this constant difference from fuel oil pressure while the actual temperature fluctuates somewhat at the point of extraction. pv = nrt connects these three variables with temperature being somewhat variable, flow being the control, and pressure being the set point.
2.5.1.2.8 Fuel oil / fuel gas flow differential
This tag represents the ratio of BTUs contributed by the fuel oil system vs. the BTUs contributed by the fuel gas system to the total BTUs required for a given load.
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2.5.1.2.9 Energy
During operation at Canal Unit 2 feedwater, pressurized by a large parasitic turbine
driven pump, is circulated through series of preheaters and then through the very walls of the furnace. During this passage it is converted to steam. This steam is then collected in a pressure vessel called a Steam Drum located at the top of the boiler where it is "dried". From the Steam Drum this dry saturated steam is passed through radiator like Primary Superheater and Secondary Superheaters that hang at the top of the furnace where convective, and radiative heat transfer occurs. From the outlet of the Secondary Superheater the steam goes directly to the High Pressure inlet of the Turbine. Unit 2 is a single reheat boiler which means that the exhaust from the high pressure turbine, instead of being condensed, is passed back to the boiler and re superheated. This re superheated steam then turns the Intermediate and Low Turbine Stages. Attemperating sprays inject cool feedwater into the steam cycle between the Primary and Secondary Superheaters and also before the Reheat Superheater. These cooling sprays dampen thermal dynamics and keep steam temperature at the turbine roughly constant around 1000 degrees.
2.5.1.2.10 Generation
The Westinghouse turbine generator converts the expansion energy of superheated
steam to create rotational momentum in the turbine. This rotational energy is imparted to a coil enclosed in an induced electromagnetic field. The rotation of this coil in this excited field creates electric potential at the ends of the coil. This electric differential has roughly 560 megawatts of power with which to do work. Under normal operating conditions, and aside from throttling effects, the output of the turbine generator is in direct relationship to boiler output.
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This tag represents the actual instantaneous unit output in units of power.
2.5.1.2.11 Heat rate
This is a simple calculated tag representing the sum of BTUs flowing into combustion from oil and gas combined divided by the amount of power created. It can show the relative efficiency of combustionsteampower system in an energy in vs. energy out relationship. As load increases heat rate decreases due to the thermal properties of the steam loop.
2.5.1.2.12 Main steam
The main steam temperature, in concert with the throttle pressure is related via steam tables to volume, enthalpy and entropy and describes the output state of the steam generating system. Unit 2 is a sliding throttle unit capable of modulated steam temp output across different throttle valve configurations. Steam output is essentially controlled by flow. As the unit ramps up in load, more steam is generated from increased combustion. Steam temperature is held (roughly) steady via modulation of flow through the turbine throttle valves, which are sequentially opened. Once the unit reaches a certain level of output (@480MW) all throttle valves are set in the fully open position and steam flow is modulated by continuing to increasing steam output through combustion throttling. At all levels of output Steam temperature is controlled around 1000 degrees F for optimum turbine operation
2.5.1.2.13 Temperature
These tags represent the temperature of superheated steam as it exits the secondary superheater header and heads to the high pressure turbine throttle valves.
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2.5.1.2.14 Attemperation spray
These represent the amount of cool feedwater that is sprayed into main steam between the primary and secondary superheaters to control the temperature of the steam at the secondary superheater outlet to the turbine.
U28300 represents fine control. This valve responds automatically and in analog fashion to all changes in steam temperature at the secondary superheater outlet. U28301 represents bulk control. It responds only to changes in SSH outlet temp that are exceed preset deadband. These coarse and fine cooling controls are combined to dampen and control steam outlet temp against oscillations or imbalances inherent in the steam system.
All desuperheating sprays receive their volume of feedwater from total feedwater flow.
2.5.1.2.15 Reheat steam
Exhaust from the high pressure turbine stage is cycled back to the furnace via the reheat steam loop where it is sprayed then reintroduced to heat exchange in the reheat superheater. Through the reheat superheater this steam is brought back up to 1 000degF and 580pisa upon which it is sent to the intermediate stage of the turbine. Exhaust from the intermediate stage turbine flows to the low pressure turbine stage.
2.5.1.2.16 Temperature
This temperature represent the temperature of re superheated steam as it heads to the intermediate turbine stage inlet.
2.5.1.2.17 Attemperation sprays
These sprays function like the superheater sprays. They inject relatively cool feedwater into the reheat steam after it has been extracted from the turbine and before it is
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reheated. They function to control the temperature of the steam at the outlet of the reheat superheater. Unlike the superheat desuperheaters, these sprays do not have separate fine and coarse control functions.
2.5.1.2.18 Furnace metal temps
These tags represent an array of thermocouples installed on the vertical legs of the
pendant superheaters. Especially in gas burning the fire side material temperature of these heat exchangers can become problematic. Unit 2 has an especially large area of superheater, which is the heat exchange closest to the fire itself. Because gas burns at a cooler temperature than oil less radiant heat is absorbed by the waterwalls of the furnace and for the same output of steam more heat must be passed to the steam loop through the gas stream and the superheaters. This superheater weighted heat transfer zone in gas burning, combined with air flow stratification that seems to be inherent to this unit, make careful monitoring of these thermocouples necessary. Extended temps above 1100 degrees can increase material fatigue signifigantly.
2.5.1.2.19 Secondary superheater metal temps topbottom LR
These represent the temperature of the firesides of selected evenly spaced legs of the secondary superheater, which encounters hot gas second, after the primary superheater. They are alphabetized horizontally across the superheater surface with upper representing the trailing side and lower representing the leading side.
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2.5.1.2.20 Primary superheater metal temps topbottom LR
These represent the temperature of the firesides of selected evenly spaced legs of the primary superheater, which encounters hot gas first and is closest to the flame front. They are alphabetized horizontally across the superheater surface.
2.5.1.2.21 Reheat superheater metal temps topbottom LR
These represent the temperature of the firesides of selected evenly spaced legs of the reheat superheater, which encounters hot gas third, after the secondary superheater and before the feedwater economizer. They are alphabetized horizontally across the superheater surface with upper representing the trailing side and lower representing the leading side.
2.5.1.2.22 Exhaust
The gasses created by combustion flow upward through the furnace gas path across the primary and secondary superheaters, the reheat superheater, and a superheater like feedwater preheater called an economizer. In this pass all steam loop heat transfer occurs. After leaving the furnace these exhaust gasses flow into a ljungstrom air heat exchanger where heat is traded to the incoming combustion air. Under the pull of the induced draft fans this now 350 degree gas passes through the units robust electrostatic precipitator array, through the induced draft fans themselves and then up the stack.
The makeup of the fluegas at the point it leaves the furnace represents the overall quality of combustion. Key parameters include how much oxygen has been left by the combustion process, and how much CO has been created. The richness or leanness of combustion is directly evident.
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2.5.1.2.22.1 Flue gas
2.5.1.2.22.1.1 CO
These tags represent the CO contained in exhaust gasses as measured in the side A (U27814) and side B (U27813) furnace outlets to the exhaust ducts just after the economizer.
These are point measures of CO in a very large duct and may not capture exact CO content. They also display extreme side to side bias with side B showing higher CO content. Although peculiar, this side to side bias is believed to be a real feature of the Canal Unit 2 furnace draft. These tags are directly related to the quality of combustion and can serve as a non delayed approximation of CO as it will be seen at the stack.
2.5.1.2.22.1.2 02
These tags represent the 02 contained in exhaust gasses as measured in the side A and side B furnace outlets to the exhaust ducts just after the economizer. These tags are used in modeling to represent the richness or leanness of combustion. They are impacted by and can be used as a control reference for forced draft fan output trim on air demand. In addition these tags are used by Canal as a part of the CEM NOx calculation.
2.5.1.2.22.1.3 Temps
The temperature of the air being forced through the boiler at Canal Unit 2 impacts and represents many process parameters, from combustion quality, to heat transfer distribution, to induced draft fan output. It also is control reference for the temperature and velocity of exhaust leaving the stack.
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2.5.1.2.22.2 Stack
The CEM (Continuous Emissions Monitoring Unit) consists of an array of extraction gas analyzers in a computer room at the base of Canal's 500 ft. stack. The pitots of these analyzers sniff mixed exhaust from the top of the 18 foot wide Unit 2 flue. The specific amounts of certain compounds measured in this gas are entered into a database. This database serves as a binding legal history of Canals environmental compliance. Each violation of emissions limits placed on certain compounds like NOx and CO is recorded. If the unit is in danger of breaking its allowed daily average output of these regulated pollutants, measured from midnight to midnight, all steps must be taken to regain compliance, including dropping load. The cost of such a sacrifice is immense and in effect these hourly and daily emissions limits have become control variables of primary importance.
2.5.1.2.22.2.1 CO
This tag represents the CO content of stack gas in parts per million.
It is worth noting that CO and NOx represent conflicting states of combustion as they are currently understood and managed. To reduce NOx production, combustion is kept cool and rich. NOx formation has been shown to positively relate to increased exposure to combustion and increased temperature. Over fire air is used to complete this off stoichiometric combustion. Unfortunately such rich and cool (incomplete) combustion inherently produces increased CO.
2.5.1.2.22.2.2 NOx
This tag is calculated using a regulatory approved method and is used to represents the pounds of NOx produced by Canal Unit 2 per million BTUs.
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2.5.1.2.22.2.3 Temp
Stack temp is important to Canal for several reasons. Keeping stack temperature at a certain point guarantees that no condensation of sulphur products can occur in the exhaust ducts, precipitators, or in the stack itself. The products of sulphur condensation are acidic and over extended periods of time can be damaging to expensive capital equipment. As long as sulphur emissions are within limits, and they are not a problem at Canal, since Canal uses low sulphur fuel oil, it is beneficial to push them all the way out of the stack before they can condense. This requires sufficient stack gas temperatures and velocities.
Stack Temperature is controlled primarily by the amount of preheating that is done to the air before it even enters the windbox. Because of the relationship of final stack gas temperature to combustion air temperature, and the relationship of combustion air temperature to other properties of combustion, stack temperature can be an important and tricky control point. Since the heat losses to the exhaust through the exhaust ducts, precipitators and induced draft fans are fixed, fluegas temperature also represents stack temperature but without gas path travel delay.
CHAPTER 3
BOILER OPTIMIZATION
The most efficient method for reducing NOx emissions is clearly during the combustion process [39]. As presented in Section 2.4 "NOx," simply changing the combustion temperature and fuel/air distribution can dramatically affect NOx emissions. The combustion of fossil fuels inside a largescale boiler, however, is a highly complex process; this complexity is a direct function of the boiler size. A typical electric power boiler maintains a "fireball" which is 3 to 5 stories tall, and there are hundreds of parameters which affect the injection of fuel and air at different locations within the furnace.
The problem is our lack of understanding about how these combustion parameters
affect NOx formation. This multivariate optimization problem requires a technology that can look at the process globally and determine the appropriate combination of combustion controls.
3.1 First Principles
The concepts behind boiler optimization are relatively simple:
If the boiler operates in an oxygenrich environment, i.e., with unnecessary
excess air, boiler efficiency will decrease due to the loss of sensible heat up the
stack; NOx emissions will increase concurrently.
If the boiler operates in a fuelrich environment, i.e., with insufficient air, boiler
efficiency will decrease due to the loss of unburned fuel. In addition, insufficient
63
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air leads to CO formation which causes slagging and water wall corrosion, ultimately shortening boiler life.
Between these two airflow conditions there is a zone of optimum combustion. This is shown as a dark gray band in Figure 5.
3.2 Fuel and Air Distribution
Boilers for electric power and industrial steam typically have poor distribution of fuel and air within the furnace. This causes some regions of the firebox to be fuelrich and other regions to be oxygenrich. This situation is clearly undesirable as it leads not only to unnecessary NOx production and reduced efficiency, but reduced boiler life expectancy due to water wall corrosion and slagging.
The variability of the fuelair ratios at different locations throughout the boiler is represented as a light gray band in Figure 5. This variability determines the amount of aggregate air required to ensure that all regions inside the boiler avoid fuelrich combustion.
. .. . .. . . . ................. . !". .. .. .. . ... z  O f . . . . .. . . . ..
Zone of
Fuelrich environment Combustion Oxygenrich environment ....... ... .. . . . .. ....... ... ... .. ........ . ... .. ... .... .. . . . . ...........
i ... .. ... ... .. ... ... .. ... ... .. ..... .. .. .
CO Efficiency
. ...... ......... . ..... Unburnedi
........... uel N
... ............. .. . .. .....
Excess Fuel Increasing Air Excess Air ......... ...~ ~~ 1] 711 _...... . ... .... ......i.... . ....
Aggregate Air
Flow
Figure 5: Combustion emissions characteristic versus air flow.
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By improving the distribution of fuel and air in all parts of the firebox, it is possible to reduce the aggregate airflow while maintaining the same safety margin. This improvement is illustrated in Figure 6. The narrower darklyshaded band which represents the improved distribution of air and fuel moves closer to the zone of optimum combustion. Reducing the aggregate airflow simultaneously increases boiler efficiency and reduces NOx emissions.
Zone of ..... .........
Optimum
Fuelrich enironmen Combustion Oxygenrich environment . ... ......... i ciency
Efficiency
Unburned CO
Fuel NOx .... ........ .Nox reduced
............... ....... .. ............... .. .. . ... ..... ... ....... .... . ..............
!' ~ ~ i.............. . ..... ... ... ,i ..... ... .... ....... .............. . .............. .........E ; i r
Excess Fuel Air Excess Air Aggregate Air
Flow Reduced
Figure 6: Effect of lower 02 on combustion emissions.
The key challenge in boiler optimization is identifying which of the many controls affect performance and how they need to be manipulated to ensure optimal performance as process and economic conditions change.
3.3 Boiler Tuning
Boiler manufacturers and service companies offer boilertuning methodologies that use the above principles of combustion to identify a limited set of control settings which help lower NOx and increase efficiency without the need for substantial capital expenditure. Such boiler tuning improves unit performance but does not begin to generate the savings achievable through improved control.
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Unfortunately, the number of control variables available to optimize performance is too large for offline boiler tuning to predict the optimum settings. Optimum settings vary with load, fuel quality, boiler conditions, weather, and other factors making offline tuning difficult if not impossible.
3.4 The Role of CO
Figure 5 and 6 both show an exponential rise in CO as excess air is reduced and the boiler approaches peak efficiency. The steepness of the CO curve depends upon the degree of mixing of fuel and air within the furnace. Poor mixing broadens the CO curve by creating pockets of fuelrich and oxygenrich combustion. Together with 02, CO levels provide the best indication about combustion quality.
A model for CO will provide valuable information about:
how well mixed the fuel and air are in the furnace,
how individual setpoints can be used to improve this mixing, and
conditions which lead to slagging and waterwall corrosion.
The CO measurement serves as a key safety constraint when optimizing the boiler. By controlling to CO levels, the boiler can be optimized without compromising safety margins. Improved air and fuel distribution will merely tighten the CO curve, resulting in improved efficiency and lower NOx.
CHAPTER 4
CONTROL DESIGNS
This research investigates the applicability of neurocontrol techniques to complex process control problems, and develops a methodology for implementing them. Towards this end, this work will develop several detailed neural networkbased control designs and apply them to the reduction of NOx and the maintenance of acceptable CO levels in electric power plants. Subsequent sections implementation these control designs and use our NOx case study to compare and contrast them. The control methodology will be presented as follows:
1) A methodology for categorizing key process variables into groups that
are required for all control designs.
2) A methodology for formally stating the control optimization objectives
and operating constraints using the aforementioned variable definitions.
3) Performance criteria by which the various control designs will be
judged, based on these formal objectives and constraints.
4) Four formal control designs with explicitly account for state variable
dependencies.
4.1 Variable Definitions
When designing a controller for largescale industrial processes, there are a large number of variables to be considered. The physical processes are typically considered to have inputs, disturbances, states and outputs. The following variable definitions are proposed as a methodology for categorizing all process variables into subsets; these subsets will prove useful when designing controllers in general:
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1) Manipulated Variables (MVs): process inputs which have been selected
for our controller to manipulate. The MVs should be independent of one another, i.e., manipulating one will not cause a change in any of the others.
2) Disturbance Variables (DVs): process inputs or disturbances that affect
the state or output of the process, but we either cannot or have chosen not to manipulate. The DV should be independent of both each other
and the MVs.
3) Control Variables (CVs): the process state or output variables that the
controller will be designed to control. The CVs should be a function of
the MVs and DVs or there is little hope of the controller being able to
control them.
4) State Variables (SVs): process state variables, which are a function of
the MVs and/or DVs, that affect the CVs. Alternatively, the SVs may be process output variables that have not been selected for control but need
to be considered as constraints.
Notice that the MV, DV, SV and CV definitions categorize the process logically and not physically. These definitions divide variables based on how the controller will be configured, rather than how the physical process is configured. The MVs will always be process inputs, i.e., can be manipulated by operators, but the DVs can contain both process inputs and disturbances depending on which inputs are being manipulated. Likewise, SVs and CVs can each consist of any combination of process states and/or outputs, based on which will ultimately be controlled.
Notation: The categorization of variables into CVs, SVs, DVs and MVs
will be used extensively throughout this work, and is conceptually consistent with the literature on optimization and control [37].
4.2 Optimization Objectives
The control objective is to lower NOx. Formally this objective needs to be stated as an objective function for optimization. Since several of the controllers developed here are tra
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jectory (multistage) controllers, this objective function will be a function of time. Consider the single control variable NOx(to) e 9 as the measured value of NOx at time to. An optimal control objective with fixed terminal time T for minimizing NOx(t) over the interval t e (to, to + T] can be given by to+ T
J = NOx(t). (54) t = to
In general, there will be more than one CV. If all CVs are to have equal impact on this objective function, then two effects will have to be removed from the optimization objective: 1) the effect of power differences between these CVs, and 2) the current value of each CV. The following objective function extends (54) to multiple control variables
1 to + T N
J TC Pi(Z(cvi(t)) Z(cvi(to))), (55) I= toi= 1
where Ncv is the number of CVs, and pi e 91 is a priority weighting factor and Z(x) = (x tx)/cx is the zscore statistic [66]. Assuming that our controller is designed to minimize J; for pi > 0 the CV cvi will be minimized over the trajectory, while setting pi < 0 will maximize the output.
Equation (55) considers the case where CVs are to be maximized or minimized. In
general, the goal is to design a controller capable of maintaining a control setpoint. A generalized optimization objective is therefore presented as to+ T NcV
TC PiD (56)
t=toi= 1
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where Di is a desirability function that can be tailored for each CV to Diax = Z(cvi(t)) Z(cvi(to)), (57) Di = Z(cvi(to)) Z(cvi(t)), or (58) Dp = (Z(cvi(t)) Z(sp(t))) (59)
4.3 Operating Constraints
Constraints will be used to ensure that the optimizer produces a feasible solution. By feasible we mean: 1) the MV moves can be made, and 2) that when these MVs are applied the plant will end up in a desirable state. Feasible solutions will be guaranteed by designing controllers which are able to maintain MV and SV constraints.
4.3.1 Manipulated Variable Constraints
To ensure that the MV moves can be made, the controllers will maintain simple range constraints. A range constraint consists of the upper and lower limits that an MV will be allowed to move. Formally the range constraint for MV mvi will be given by C V = [C C' ], (60) where Cn is the MV's absolute minimum and Cipax is its maximum. Controllers will be to +T1
required to provide an optimal MV trajectory { mv (t) } t = to such that mvi*(t) e C'v Vi, t. (61)
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4.3.2 State Variable Constraints
Similarly, to ensure that controllers drive the plant to a desirable state, SV constraints will also have to be addressed. Formally, controllers will be required to provide optimal to+ T
MV trajectories that result in SV trajectories {fsv*(t)}t = to0+ 1 such that svi*(t) E Vi, t. (62)
4.3.3 Penalty Functions
Each control design considered will employ an optimization algorithm during some phase of its development. Some optimization algorithms are able to deal with constraints directly, i.e., given knowledge of the constraints they can ensure a feasible solutions. Others, however, will have to treat constraints indirectly by addressing them with the objective function. The most common method for addressing operating constraints in an objective function are through the use of penalty functions [54]. For example, SV constraints can be stated as penalty functions of the form {" 2ax 2 max
(svi Ca" ) svi> Cax
X(svi, CV) = 2 in (63) (svi C ) svi
K 0 else
Generalizing the penalty function to multiple SV constraints, differences in the energy of the respective signals will once again have to be normalized out. These effects can be compensated for using a generalized penalty function of the form (Z(svi) Z(Cmax))2 sVi > nax ,(svi, iv) = (Z(sI Z(cmin))2 svi < cin (64) 0(Z(sv) Z(Celse)) i
L0 else
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Given a set of N"v SV constraints, an optimizer may satisfy these constraints by appending their respective penalty functions to its criterion J#v
S J+ p "(svi, C'V) (65) i=1
where psv allows constraints to be individually prioritized. Similarly, both MV constraints can be appended to the optimizer's criterion by defining the penalty functions {k2(mvi nv) IV
Note that implementing constraints with penalty functions will not guarantee that the constraints are met precisely. If the constraints are properly prioritized relative to the optimization objectives, however, these constraints are easily maintained within the desired level of accuracy.
4.4 Performance Criteria
For the case study, controllers will be judged based on their ability to lower NOx while maintaining desired CO emissions. To this end, subsequent sections will measure the performance of controllers as a plant operator moves MVs according to their control laws. Comparing controller performance, however, will prove a difficult task, since the operator can only take the advice from one controller at a time and the plant is constantly changing state. Although the controllers may be able to deal with nonsteadystate conditions, it will be nearly impossible to separate the process responses to the state changes versus the control action.
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Further complicating matters, while onetime tests will provide useful results with which to judge the controllers, they are not the only criteria. The controllers studied will be judged by the following criteria:
1) Ability to control NOx and CO.
2) Ability of the operators to perform the recommended MV moves
3) Flexibility with respect to changing performance objectives and operating constraints
4) Ability to deal with changing operating states, e.g. load changes
4.5 Controller Designs
Four controller designs will be developed. The controller designs considered, fall into the broad categories of:
1) ModelPredictive Control
2) ModelInverse Control
3) ModelBased Direct Control
There are, however, no standard recipes for building these controllers. The field is still immature, and neurocontrol designs presented in the literature tend to be ad hoc. This work seeks to not only develop and test four neurocontrol designs, but also to develop a generalized methodology for implementing control designs belonging to the above abstract categories. Each controller must be able to deal with the MV and SV constraints, and will be judged by the performance criteria described above.
For each of the control designs considered, there are two distinct phases in the implementation:
1) Offline training.
2) Online control.
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4.5.1 SteadyState Optimizer
The simplest, and most prevalent, neurocontroller in the literature is the steadystate optimizer [43][40][3 1]. This controller belongs to the modelpredictive control family. Modelpredictive control (MPC) is not new to commercial applications in the process control industry. The advance proposed here is the application of neural network reference models within this controls methodology.
The concept of MPC is straight forward: combine a model for the process with an optimizer to obtain realtime optimal setpoints. Model predictive controllers can be steadystate or dynamic, depending on characteristics of their underlying process models. This section details the design of a neural networkbased steadystate MPC controller to meet the problem specifications presented in Sections 4.2, 4.3 and 4.4.
4.5.1.1 Offline training
Training a MPC controller follows the schematic outlined in Figure 7. Notice that
there are actually two reference models being trained: one SV model and one CV model. The details for how to train these models will be covered in Chapter 6. Notice, however, that the criterion J" is a model training criterion to be presented in Chapter 6, and not the control performance objective I presented in Section 4.4 "Performance Criteria."
The model definitions required by the steadystate optimizer are:
1) SteadyState SV Model: sv = ssSVModel(mv, dv)
2 SteadState CV Model: c = ssCVModel(m. dv. sv
2) SteadyState CV Model: cv =ssCVModel(mv, dv, sv)
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ssS/odel
dv sv
Cmy
01 Plant
sv dv
ssC ode v
Figure 7: Offline training and retuning configuration for steadystate optimizer.
The reason to have a CV model is obvious, it will provide the reference model that the optimizer uses to figure out its optimal MV setpoints. The motivation for having a SV model, however, is somewhat less apparent. The problem is that changes made to the MVs by the optimizer will not only change the CVs, but also the SVs. The optimizer will have to consider the effect that MVs will have on the SVs, if it is to accurately predict their effect on CVs. Note that the CV model has an input space that consist of MVs, DVs and SVs.
4.5.1.2 Online control
The online control configuration is illustrated in Figure 8. Here an optimizer calculates ArgMin {f } using the SV and CV reference models developed during model trainm v
ing. The optimizer starts with the current value of the MVs myv = my, uses the SV model to estimate the current SVs sv which are then used, along with the current value
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of the DVs d to estimate the current CVs cv The optimizer then iteratively updates its estimate for the optimal MVs myv* to minimize its objective function J.
{dy, sv }
MVs is small, then direct optimizer provides an efficient alternative. As the number ofc
MVs grows, however, direct optimization quickly becomes impractical. Descentbased optimization is possible because the SV and CV models are capable, via backpropagation, of calculating the gradient of f with respect to their inputs, i.e., their inputs sensitivities given the sensitivities at their outputs. In this manner, the optimizer calculates the CV senA ArgMin ,/
sitivities /sSVM from which the ssCVModel is able to calculate SV sensitivities
af/.~v* and partial MV sensitivities afv/av*, from which the SV model calculates
the remainFigure 8: Online control configuratio* and finor steadystate optimizer is able to updaBoth direct and descentbased optimalization can be used for MPC. If the number ofgradient of MVs is small, then direct optimizer provides an efficient alternative. As the number of MVs grows, however, direct optimization quickly becomes impractical. Descentbased optimization is possible because the SV and CV models are capable, via backpropagation, of calculating the gradient of Ic with respect to their inputs, i.e., their inputs sensitivities given the sensitivities at their outputs. In this manner, the optimizer calculates the CV sensitivities 8//cla' from which the CV Model is able to calculate SV sensitivities 8//8asv* and partial MV sensitivities 87 /8am ,_ from which the SV model calculates the remaining partial MV sensitivities 8/sv/8my ,V and finally the optimizer is able to update its optimal MV estimate using the MV gradient of af + aft (66) amy amy amy
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This is really just the backpropagation of backpropagations, a.k.a. more fun with the chain rule.
Several optimization methods were tested for the optimizer, along with various techniques for dealing with the constraints. The most effective combination identified was to use the unconstrained conjugate gradients method in combination with an objective function which included the SV and MV constraint penalty functions
NfV NVv Nv
i = 1 1 SV V p mvi, i (67)
cvi= 1 1viv i= I
where the MV, SV, DV and CV variable sets, along with their corresponding constraints, are defined in Section 6.6.3 "Final Variable Sets," and all priorities have been set to 1. The details of the conjugate gradients method will be presented in Section 6.5 "Learning Algorithm."
Penalty function can negatively impact the performance of a descentbased optimizer by adding complexity to the performance surface having little to do with the underlying problem. This is particularly true when the constrained variables lie outside of their constrained values. For the SV constraints, there is no choice but to use penalty functions for constraints. For MVs, however, there are alternatives, because the MVs always start at their current values which are always within the constraints. Hence, there is little to no overhead to using MV constraints for our online optimizer. It terms of the performance surface, the constraints can be thought of as placing a guardrail on both sides of our current position in weightspace along our path, while having little impact on the local topography of the road.
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4.5.2 SteadyState ModelInverse Controller
The next controller design belongs to the modelinverse control (MIC) family. Conceptually, modelinverse control is straightforward: train a model to predict the MVs from the current and known DVs, SVs and CVs, then, given a desired CV setpoint, this model can be used directly to obtain the required MVs. Implementing a MIC controller is also straightforward and can work reasonably well, given that the relationship between MVs and CVs is in fact invertible. This sections details the design of a neural networkbased MIC controller, designed to meet the problem specifications presented in Sections 4.2, 4.3 and 4.4.
4.5.2.1 Offline training
Training a MIC controller follows the schematic outlined in Figure 9. Once again, notice that the MV model is being implemented by separate inverseSV (ISV) and inverseMV (IMV) models. Once again, the details for how to train these models will be covered in Chapter 6. The model definitions required by the steadystate modelinverse controller are:
1) SteadyState ISV Model: sv = sslSVModel(cv, dv)
2) SteadyState IMV Model: my = sslMVModel(cv, sv, dv)
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S4 ssSV del
my Plant cy N
sv dv
7M ssI odel +mv H Plant C
Figure 9: Offline training and retuning configuration for modelinverse controller.
Analogous to our MPC controller, two models have been developed which when combined can invert the process. The reason to have a IMV model is obvious, it provides the inversemodel that the controller uses to figure out optimal MV setpoints. The problem is that not all CVSV combinations are feasible. Given a specified CV target, the ISV model estimates the corresponding SVs which are presented to the IMV model.
4.5.2.2 Online control
The online control configuration is illustrated in Figure 10. If a known target existed for the CVs, the online control implementation would actually be quite trivial. One complexity is that the exact value for the lowest achievable NOx from the controller for a given set of conditions is not known. Another complication with MIC is how to deal with constraints. If one applies a target CV to the input of the inversemodel, it will predict a set of inputs which it believes would have achieved this target. The problem is that the model does not understand the MV or SV constraints, and if one of the inputs it predicts falls out
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side these constraints the controller can not provide the required setpoints. This is analogous to the problem faced with SV or CV constraints for MPC. The implementation outlined in Figure 10, uses an optimizer in order to overcome both of these issues. Clearly, CV constraints are straightforward.
{dv, sv}
Op timizer mv* cv Plant
ArgMin_* P{ } "I
cyV
L sslSVModel ssIMVModelFigure 10: Online control configuration for modelinverse controller.
The MIC controller uses an optimizer to calculate ArgMin_* {J} using the ISV and
CV
IMV reference models developed during model training. The optimizer starts with the current value of the CVs cv = cv, uses the ISV model to estimate the current SVs sv which are then used, along with the current value of the DVs dv by the IMV model to estimate the current MVs my The optimizer then iteratively updates its estimate for the optimal MVs m v*, to minimize its objective function I.
Once again both direct and descentbased optimization can be used for MIC, and once again a conjugate gradientsbased optimizer was selected. Descentbased optimization is possible because the ISV and IMV models are capable, via backpropagation, of calculating the gradient of J with respect to their inputs, i.e., their inputs sensitivities given the
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sensitivities at their outputs. In this manner, the optimizer calculates the MV sensitivities a/mv*, from which the IMV Model is able to calculate SV sensitivities 8//sv* and partial CV sensitivities 8/Jf/acv*, from which the ISV model calculates the remaining partial CV sensitivities 8 sv/&* and finally the optimizer is able to update it optimal CV estimate using the CV gradient of / aJs', 8/'"
a +SV a (68) 8cy 8cy 8cy
The optimizer's objective function, which includes the SV and MV constraint penalty functions, is the same objective function used by our steadystate optimizer. The only difference is how the sensitivities flow through the system, as outlined above.
4.5.3 Dynamic ModelPredictive Controller
The steadystate optimizer considered above is a member of the MPC family. The vast majority of MPC applications use models which are firstprinciples based [37]. Since it is not possible to build an accurate firstprinciples model ofNOx, a new steadystate optimizer for MPC using neural network models was developed. The vast majority of MPC applications are dynamic, however. The steadystate optimizer only considers the effect that MV changes will have on the unit in steadystate conditions.
This section develops a dynamic neural networkbased MPC controller. The main differences between this controller and our steadystate optimizer is that it:
1) Understands the dynamics of the process.
2) Provides a trajectory of MV setpoints designed to optimize the path of
the unit into the future, rather than a optimal steadystate position. In
other words, the controller not only considers where your going but how
you'll get there.
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The concept behind this controller's operation is identical to that of the steadystate optimizer: combine a model for the process with an optimizer to obtain realtime optimal setpoints. The only difference is that the models are now dynamic, and the optimal setpoints become optimal setpoint trajectories.
This sections details the design of a neural networkbased dynamic MPC controller to meet the problem specifications presented in Sections 4.2, 4.3 and 4.4.
4.5.3.1 Offline training
Training a dynamic MPC controller follows a similar schematic as outlined in Figure 7, with the inclusion of each variables explicit dependence on time t, as illustrated in Figure 11. Here the SV and CV models are performing singlestage prediction with respect to the MVs and DVs; notice that the CV model uses the current value of the SVs sv(t + 1). The reasons for this configuration will become clear when we consider the online control implementation in the next section.
dS/Vodel f
dv(t) sv(t + 1)
m i (t) rPlant cy(t + 1) sV(t+ 1) d(t)
d1 Nodel E
Figure 11: Offline training and retuning configuration for steadystate optimizer.
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Refer to Chapter 6 for details on training the dynamic SV and CV reference models used by the dynamic MPC controller. For now we simply state the model definitions required by the dynamic MPC controller:
1) Dynamic SV Model: sv(t+ 1) = dSVModel(mv(t), dv(t))
2) Dynamic CV Model:
cv(t + 1) = dCVModel(mv(t), dv(t), sv(t + 1))
4.5.3.2 Online control
The online control configuration follows a similar configuration to the steadystate
optimizer presented in Figure 8. Here a dynamic optimizer is required, however. The optimizer calculates ArgMin {f(t) } using the dynamic SV and CV models developed mv*(t)
during model training. The steadystate optimizer used an application of the chain rule for ordered partial derivatives, which has been coined "backpropagation" [40]. From the perspective of the chain rule, our new optimizer is identical and only the criterion changes. From the perspective of the literature, this algorithm has been coined "backpropagation through time" [40][70].
The optimizer starts with the current value of the MVs m *(t) = mv(t); uses the SV model to estimate the resulting SVs sv (t + 1); which are then used, along with the current value of the DVs dv(t), to estimate the resulting CVs cv (t + 1). Notice that each estimate can rely on both present and past values of the inputs. The optimizer will then repeat this process over the time interval t e (t0, to + T] to produce the MV, SV and CV to+T ___, to +T to+ T trajectories {mv*(t)} = to {sv (t)}t = to+ 1 and I{cv*(t)}t = to + respectively.
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The objective function which included the SV and MV constraint penalty functions can now be calculated
1 +Tr IV +,
(t) = pi Z(cvi*(t)) p k(svi*(t), Civ) + p i v(mvi*(t), Civ) (69)
t=o i= 1 i=1 N i=
where the MV, SV, DV and CV variable sets, along with their corresponding constraints, are defined in Section 6.6.3 "Final Variable Sets," and all priorities have been set to 1.
The optimizer then iteratively updates its estimate for the optimal MV trajectories,
to+ T1 I
{m (t) = to to minimize its objective function f(t). Each step in the iteration performs the following, starting with t = to + T 1 and iterating down to t = to: first, the optimizer calculates the CV sensitivities af(t)/acv* (t), from which the CV Model is able to calculate SV sensitivities 8f(t)/8sv (t) and partial MV sensitivities afv(t)/lav*(t 1), from which the SV model calculates the remaining partial MV sensitivities oJsv(t)/8mv*(t 1), and finally the MV sensitivity at time t 1 can be calculated as
8/(t) Jv(t) + ajcv(t) (70) am* (t 1) mv* (t 1) m*(t l) Once the backward pass is complete, the optimizer is now able to update it's optimal to+ T1
MV trajectory estimate { m v*(t) } = to using the MV gradient trajectory
(t) t+ T 1
m V (t) t to(71)
8my (t)t=t
85
Notice that the sensitivities at time t depend on the sensitivities in the future. This is because the models variables at time t depend on the variables in thepast, i.e., the models are dynamic. Hence the term "backpropagation through time."
This entire optimization cycle is run at each time step to. The optimizer derives the next T 1 MV moves, and the first MV setpoint is applied to the unit mv*(to). At this point the entire process is repeated.
4.5.4 ModelReference Adaptive Controller
The final controller design considered belongs the modelreference adaptive control family (MRAC). Like the dynamic MPC controller, the MRAC controller understands process dynamics and provides a trajectory of MV setpoints which optimize both where you are going and how you get there. The fundamental difference between these two controllers is how this optimal trajectory is derived. The MPC design utilized an online optimizer to calculate this trajectory, while the MRAC design develops a neuralnetwork based controller which is able to calculate the optimal trajectory directly. Hence this is our first direct controller, i.e., calculates MV setpoints directly.
Notice that the MIC design would have provided a direct online controller, if it wasn't for:
1) the lack of a known target NOx level, and
2) the requirements for MV and SV constraints.
The MRAC design is able to overcome both of these hurdles by building knowledge of the best achievable NOx level and by building all of the constraints directly into the controller. The main advantage to the MRAC design is online response time. There is no opti
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mization to run, one simply presents the controller with the current, and past, state of the process, and it generates a MV setpoint as quickly as a neural network can think. These benefits do not come for free, however. The main drawbacks to the MRAC design are:
1) Extensive offline training and retuning requirements.
2) Inflexible online configuration, with respect to changing optimization
objectives and operating constraints.
This sections details the design of a neural networkbased dynamic MIRAC controller to meet the problem specifications presented in Sections 4.2, 4.3 and 4.4.
4.5.4.1 Offline training
Training a MRAC controller requires two stages. The first stage is identical to training and returning the dynamic MPC controller. Here, dynamic SV and CV models are developed using the same steps outlined in Figure 11. The second stage uses these models to train the controller with offline data, as illustrated in Figure 12. The offline training is similar to the online optimization which is performed for the MPC design, except this optimization is performed across the training dataset rather than online.
>vsv (t + 1)
dSVModel
di~t) cy(t + 1) Con ol dCVModel
WV
rm My (t)
11
zz
ArgMin CL{ (t)}
Figure 12: Offline training and retuning configuration for model reference controller.
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Once again a dynamic optimizer is required, and the objective function is given by
(69). To train the controller, the optimizer calculates A rgMin /L { f(t) } where ~CL are the weights of the control law neural network.
Training the control law (CL) model starts with the actual values for the DVs, SVs and CVs, and random initial weights for its CL model tCL Starting at time to = Ti + 1, where Ti is the first sample in the training dataset; training uses the CL model to estimate the resulting MVs mv*(t); which are then used to estimate the resulting SVs sv* (t + 1) and CVs cv (t + 1). This process is repeated over the time interval t e (to, to + T] to to+ T 1 to+ T
produce the MV, SV and CV trajectories {mv (t)} t = to {sv*(t)}t= t + 1 and
to+ T
{ cv (t) } t = to + I, respectively.
The training algorithm then iteratively updates its estimate for the optimal CL model weights, TPCL, to minimize its objective function 1(t). Each step in the iteration performs the following, starting with t = to + T and iterating down to t = to + 1: First, the training algorithm calculates the CV sensitivities 8f(t)/Oacv* (t), from which the CV Model is able to calculate SV sensitivities af (t)/8sv* (t) and partial MV sensitivities 8/v(t)/Dmv*(t 1), from which the SV model calculates the remaining partial MV sen
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sitivities afv(t)/Dmv*(t 1), and finally the MV sensitivity at time t 1 can be calculated as
Of(t) 8 ov(t) + afV(t) (72) 8vm (t 1) av (t 1) av (t1)
The MV sensitivities are finally passed to the CL model which backpropagates them to derive its weight gradients af /80cL, which the training algorithm is able to use to update its control law's weight estimate.
The training algorithm then increments to and repeats the entire process, until the
training algorithm has converged. When to = Tf T, to is reset to to =Ti + 1 The reason for training the CL model in increments of T is because the SV and CV have a limited prediction horizon, the time before their estimates are no longer valid. By resetting the state of these models to the actual state of the unit after T samples, we are able train the CL model within the prediction horizon of the SV and CV models.
4.5.4.2 Online control
The online control configuration for the MRAC design is straight forward, as illustrated in Figure 13. Simply supply the controller with the current SVs, DVs and CVs, and it outputs the next MV setpoint. This setpoints contains knowledge about the optimal achievable NOx, SV constraints, MV constraints and the trajectory through which it will drive the process into the future.
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S{sv(t), dv(t)}
Control my (t) cv(t) Law W1 Plant
r Law
Figure 13: Online control configuration for model reference controller.
Clearly, the controller is only as good as its underlying reference models. In addition, considerable care must be taken to ensure that the training data contains regions of the input space where the SV and MV constraints have been exercised. The design is easily augmented with limiters to guarantee that MV constraints are maintained. However, there is little that can be done to guarantee that the SV constraints are maintained.
CHAPTER 5
DATA PREPARATION
Given the detailed control designs just presented, the next step is to implement the
actual controllers by developing the required reference models. Both reference model and controller implementations require a significant amount of process data. Data collection is the most important aspect of any modeling or optimization project. There is a common saying "junk in, junk out," this study was relentless in reenforcing this lesson. Applying the most sophisticated modeling and/or optimal control algorithms in the world will not make up for problems with data preparation.
With the advanced distributed control systems (DDS) and supervisory control and data acquisition (SCADA) systems readily available in today's process plants, the relative quantity and quality of available data is overwhelming. Much of the statistics and modeling literature has been dedicated to the problems faced when drawing inferences from small sample spaces. Modem processing plants are anything but data limited. The relevant problems are just the opposite, how to draw meaningful inferences from a massive sample space.
The following section presents solutions for the most significant challenges faced in preparing data for modeling and optimization. Much of what is presented in this section was learned the hard way, during modeling and optimization.
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NEURAL NETWORKBASED CONTROL DESIGNS FOR COMPLEX
INDUSTRIAL PROCESS APPLICATIONS
By
WESLEY CURTIS LEFEBVRE
A DISSERTATION PRESENTED TO THE GRADUATE
SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL
FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE
OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2000
ACKNOWLEDGMENTS
When looking for a graduate school I was told to pick an advisor, not a school. My
choice was Dr. Jose Principe, and his mentoring and friendship through a masterâ€™s, Ph.D.,
and the startup of two companies has proven this advise to be sound. I would also like to
thank Dr. Glen Johnson from IBM Yorktown for giving me this advice and first introducÂ¬
ing me to the field of neural networks.
I would also like to thank Charles River Associates and Commonwealth Energy
for believing in my work and this field, the members of my committee for the effort and
time spent on my behalf, the staff at Canal Electric Generating Station, and the members
of the Computational NeuroEngineering Laboratory (CNEL).
I would also like to thank my wife, Anna, for her patience as I struggled to juggle
far too many commitments, and my family for their neverending support referring to us
as Dr. and Mr. Lefebvre.
n
TABLE OF CONTENTS
page
ACKNOWLEDGMENTS ii
LIST OF TABLES v
LIST OF FIGURES vi
CHAPTERS
1 INTRODUCTION 1
Proposed Work 1
Document Organization 7
2 LITERATURE REVIEW 9
Optimization 9
Neural Networks 16
Control Theory 32
NOx 40
FossilFired Power Generation 45
3 BOILER OPTIMIZATION 63
First Principles 63
Fuel and Air Distribution 64
Boiler Tuning 65
The Role of CO 66
4 CONTROL DESIGNS 67
Variable Definitions 67
Optimization Objectives 68
Operating Constraints 70
Performance Criteria 72
Controller Designs 73
iii
5DATA PREPARATION
90
Data Management 91
Variable Selection 92
Validation 93
Time Constants 97
Normalization 97
6 MODELING 99
Methodology 99
Model Definitions 100
Datasets 104
Performance Criteria 105
Learning Algorithm 109
Variable Pruning 113
Architecture Selection 118
Analysis 136
7 CONTROLLER IMPLEMENTATIONS 138
Offline Quantification 138
Online Quantification 143
8 PARAMETERIZATION PROBLEM 148
Search for a Validation Metric 149
Correlation Paradox 154
Validation Metric 160
Revised Representation Pruning Algorithm 161
Modeling 168
Control Implementation 169
9 CONCLUSION 176
Contributions 176
Afterword 178
Future Direction 178
APPENDIX 180
REFERENCES 181
BIOGRAPHICAL SKETCH 187
IV
LIST OF TABLES
Table 1: Probability of ZScore exceeding value 96
Table 2: Types in order they were removed 116
Table 3: Final variable selections after pruning 118
Table 4: Results of autoregressive tap search algorithm for all dynamic models 121
Table 5: Results of moving average tap search algorithm for all dynamic models 122
Table 6: Results of hidden layer #1 PE search algorithm for all steadystate models. 124
Table 7: Results of hidden layer #2 PE search algorithm for all steadystate models. 126
Table 8: Results of tap search algorithm for all dynamic models 128
Table 9: Results of taps search algorithm for all dynamic models 130
Table 10: Results of hidden taps search algorithm for all dynamic models 131
Table 11: Results of hidden states search algorithm for all dynamic models 133
Table 12: Results of state hidden PEs search algorithm for all dynamic models 134
Table 13: Results of output hidden PEs search algorithm for all dynamic models 136
Table 14: Final variable selections after revised pruning 165
Table 15: Essensial tag list 180
v
LIST OF FIGURES
Figure 1: Multilayer perceptron model architecture 18
Figure 2: TDNN input PE connectivity 19
Figure 3: Gamma memory processing element 20
Figure 4: Nonlinear state space neural network configuration 21
Figure 5: Combustion emissions characteristic versus air flow 64
Figure 6: Effect of lower 02 on combustion emissions 65
Figure 7: Offline training and retuning configuration for steadystate optimizer 75
Figure 8: Online control configuration for steadystate optimizer 76
Figure 9: Offline training and retuning configuration for modelinverse controller 79
Figure 10: Online control configuration for modelinverse controller 80
Figure 11: Offline training and retuning configuration for steadystate optimizer 82
Figure 12: Offline training and retuning configuration for model reference controller. .86
Figure 13:Online control configuration for model reference controller 89
Figure 14: Daily % missing across February dataset 94
Figure 15:Example of variables with large NMSE but high R 108
Figure 16: Results of type pruning algorithm 115
Figure 17: Representation pruning algorithm results 117
Figure 18: Results of autoregressive taps search algorithm for NOx ARMA Model. ...120
Figure 19:Results of moving average tap search algorithm for NOx ARMA Model. ...121
Figure 20:Results of hidden layer #1 PE search algorithm for the NOx MLP Model. ..124
vi
Figure 21 :Results of the hidden layer #2 PE search algorithm for NOx MLP Model. ..125
Figure 22:Results of tap search algorithm for NOx TDNN Model 127
Figure 23: Results of taps search algorithm for NOx GNN model 129
Figure 24:Results of hidden taps search algorithm for NOx GNN model 130
Figure 25: Results of hidden states search algorithm for NOx NLSS model 133
Figure 26: Results of state hidden PEs search algorithm for NOx NLSS model 134
Figure 27:Results of output hidden PEs search algorithm for NOx NLSS model 135
Figure 28:Best models for all model definitions by architecture 137
Figure 29: Average NOx reduction over testing dataset 141
Figure 30: Average CO above max over testing dataset 141
Figure 31: Average NOx reduction over testing dataset using train and test models 142
Figure 32: Average CO above max over testing dataset using train and test models 142
Figure 33: Change in NOx for steadystate controller experiments 146
Figure 34:Final CO level for steadystate controller experiments 147
Figure 35: Summary of validation metrics for MLP CV model 150
Figure 36:NMSE and R for all 10 training results for MLP CV model 151
Figure 37: Summary of validation metrics for combined SV/CV model 153
Figure 38: Sensitivity results for all 10 training results for NOx CV model 160
Figure 39:NOx CV model sensitivity with 95% confidence intervals 161
Figure 40: Results of revised representation pruning algorithm 164
Figure 41: Sensitivity results for all 10 training results for revised NOx CV model 167
vii
Figure 42:Revised NOx CV model sensitivity with 95% confidence intervals 168
Figure 43: Best revised models for all model definitions by architecture 169
Figure 44: Average NOx reduction over testing dataset using old and revised models. .170
Figure 45: Average CO above max over testing dataset using old and revised models. .170
Figure 46: Change in NOx for revised steadystate controller experiments 172
Figure 47:Final CO level for revised steadystate controller experiments 172
Figure 48: Average percent NOx reduction for 10 online experiments 173
Figure 49:Average percent CO reduction above 500ppm for 10 online experiments. ...174
viii
Abstract of Dissertation Presented to the Graduate School of
the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
NEURAL NETWORKBASED CONTROL DESIGNS FOR
COMPLEX INDUSTRIAL PROCESS APPLICATIONS
By
Wesley Curtis Lefebvre
May 2000
Chairman: Jose C. Principe
Major Department: Electrical and Computer Engineering
Neural networks have successfully transitioned from an academic interest into a
viable technology which is now being used in everyday products. To date, neural netÂ¬
works have been predominantly applied to forecasting or modeling applications. Based on
their success in such applications, there has been significant interest in using neural netÂ¬
works in control applications, creating a new field called neurocontrol. Although there
have been significant advances in the theory of neurocontrol, there are very few successful
commercial applications using neurocontrollers. Commercial applications often provide
the most challenging problems because the controllers are required to function robustly in
complex and unknown environments. Realworld processes are complex and difficult to
control because they contain a large number of highly interdependent variables, have
highly nonlinear responses to these variables, and change their response over time.
This work identified two significant reasons why neurocontrol designs fail in real
IX
world applications: First, the controllable parameters over most industrial processes are
highly correlated, often not for physical reasons but because of our process control strateÂ¬
gies. Second, intermediate process states that affect the process output, which are also
affected by the controllable parameters, have a significant impact on controller perforÂ¬
mance. When the controller changes the controllable parameters, the impact that this has
on the process states, which will in turn affect the process output, is not accounted for in
most neurocontrol designs in the literature.
This dissertation advances the field of neurocontrol by providing the following
solutions: first, the use of statistical significance testing on the local linearized relationÂ¬
ships extracted from nonlinear neural network models to avoid problems with correlated
controllable parameters; second, augmenting neurocontrol designs to incorporate depenÂ¬
dent state models. These enhancements have been applied to four distinct neurocontrol
architectures. The new control architectures have been applied to the novel application of
controlling NOx emission from an oil and gasfired electric power plant.
x
CHAPTER 1
INTRODUCTION
Neural networks have found an applications niche as robust predictors that have demÂ¬
onstrated the ability to outforecast more traditional methods in complex realworld appliÂ¬
cations. The vast majority of neural network applications to date rely solely on the
modelâ€™s ability to forecast without regard for what the model has learned about the relaÂ¬
tionships within the underlying process or the ability to affect it. Soulie and Gallinari [59]
recently compiled 53 industrial applications of neural networks, of which only 4 make any
attempt to make inferences about the underlying process or to control it.
This ratio does not apply to theoretical publications in the literature, however. The
research into neural networkbased control has recently received widespread attention,
coined neurocontrol [40]. Many authors have presented the abstract concepts behind neuÂ¬
rocontrol [40][43][31 ] [37], but the literature contains a disproportionately low number of
papers presenting realworld neurocontrol applications. The vast majority of these papers
have developed controllers for fabricated, simulated, or laboratory controlled processes.
1.1 Proposed Work
The goal of this project is to develop robust neurocontrol design methodologies for
complex industrial process control applications. Industrial control applications are characÂ¬
terized by nonlinear, noisy, nonGaussian, highly correlated, and nonstationary processes.
This is an applications area where classical control designs have proven ineffective.
1
2
1.1.1 Case Study
As a case study, this work develops online advisory neurocontrollers designed to minÂ¬
imize the NOx emissions for an oil and gas cofired power plant. The combustion of fossil
fuels inside a largescale boiler is a highly complex process; this complexity is a direct
function of the boiler size. A typical electric power boiler maintains a "fireball" which is 3
to 5 stories tall, and there are hundreds of parameters which affect the injection of fuel and
air at different locations within the furnace. The problem is our lack of understanding
about liow these combustion parameters affect NOx formation. This multivariate optimiÂ¬
zation problem requires a technology that can look at the process globally and determine
the appropriate combination of combustion controls
The neurocontrollers developed in this study will forward control setpoints to operaÂ¬
tors through the plant's existing distributed control system (DCS). The neurocontrollers
will be required to provide setpoints that minimize NOx emissions, while maintaining unit
operating constraints. A demonstration system has been completed at Canal Electricâ€™s
580MW tangentially fired Unit 2. Charles River Associates (CRA) and Commonwealth
Energy jointly funded this study.
1.1.2 Objectives
The author believes that neurocontrol strategies have not been more successful in real
world applications because they
1) are difficult to implement,
2) fail to account for dependent internal process states, and
3) have difficulty dealing with correlated process variables.
The scientific objectives for this work are to develop:
3
1) an applicationbased neurocontrol implementation methodology,
2) statespace neurocontrol architectures,
3) methods for dealing with correlated data,
4) accurate combustion models, and
5) a novel combustion controller.
1.1.2.1 ApplicationBased Neurocontrol Implementation Methodology
Most neurocontrol implementations in the literature have been ad hoc and application
specific [40], In many cases, the process has been simulated and thus known completely.
Modem control theory has introduced many sophisticated control designs, but the fact
remains that approximately 90% of the industrial control applications apply a simple proÂ¬
portional integral differential (PID) controller [37],
The PID controller only requires the process engineer to specify reasonable knowlÂ¬
edge of the process. This applicationbased implementation methodology is largely
responsible for the success of PID control in the industry. If neurocontrollers are to enter
the mainstream process control market, there will have to be designs that do not require
detailed knowledge of neural network or neurocontrol theory.
1.1.2.2 ApplicationBased Neurocontrol Implementation Methodology
We begin in Chapter 4 by defining a methodology for categorizing process variables
into groups based on a set of objective criteria about their role in the process. Each conÂ¬
troller will be implemented in Chapter 7 based on this labeling of the process variables,
without additional process knowledge. This will allow process engineers with reasonable
process knowledge and without any knowledge of neural network or neurocontrol theory
to successfully deploy such technologies. Notice that achieving such a straightforward
4
implementation methodology does not imply that the details behind the neurocontroller
implementation are easy, it simply requires that they can be automated.
1.1.2.3 StateSpace Neurocontrol Designs
Nonlinear statespace neural network architectures offer the greatest modeling potenÂ¬
tial, but the difficulties in their training have led investigators to reject their application
[31]. In fact, most neurocontrol designs are based on input/output process models
[31][43][40], The use of input/output neural network architectures in the design of neuroÂ¬
controllers assumes that all input variables are independent, a situation which in not likely
to be found in realworld applications. Many of the process variables that affect the plantâ€™s
output will depend on the same process setpoints that a controller is manipulating. These
facts have limited the performance and complicated the implementation of neurocontrol
designs. If neurocontrol is to be a viable methodology in industrial process situations, then
the controllers will have to be extended with architectures capable of dealing with internal
process states.
There has been limited success in applying statespace neural network models [31]. It
is widely accepted, however, that statespace representations hold the most promise for
modeling and controlling complex processes [34][35]. The literature seems to be treating
the viability of statespace architectures as an allornothing affair. Most publications
apply purely input/output architectures with overwhelming success, while a few investiÂ¬
gates have tested purely statespace architectures where all states are treated as hidden and
unknown with limited to no success.
This work will empirically investigate several shades of gray, ranging from purely
mput/output to purely statespace controllers. These statespace controller designs are pre
5
sented in Chapter 4, and their performance is empirically investigated in chapters 7 and 8.
The primary difference in the statespace representations proposed here is that the state
variables are treated as not hidden and known. This will require an extension to the neuroÂ¬
control design strategies presented in the literature [31][43][40].
1.1.2.4 Methods for Dealing with Correlation
Industrial process applications are unique in that there is a massive amount of availÂ¬
able data. The input variables for complex process models are typically highly correlated,
a situation for which there are few solutions in the literature. This correlation can come
from several sources: dependent states (as addressed above), physical linkages, and soft
linkages through control strategies the lack of adequate system parameterization. IndusÂ¬
trial process control applications will require the process engineer to select representative
variables from a large set of available process variables. Because of issues like input corÂ¬
relation, the representative variables selected will have a significant impact on the perforÂ¬
mance of the resulting neurocontroller. A viable neurocontrol design methodology will
have to be able to cope with correlated process variables. Linear control theory deals with
this aspect through parameterization of the controller. Nonlinear control theory, however,
has not solved this problem in general. In Chapter 8 we propose to compute sensitivities
through a committee of trained neural models to select the best variables for system idenÂ¬
tification and control.
Statespace neurocontrol architectures will be able to explicitly deal with one source
of correlation present in industrial processes, namely correlation produced by dependent
state variables. As mentioned above there are several other sources of correlation, howÂ¬
ever. Our first goal is to empirically quantify the impact of this correlation on our model
6
ing and control objectives. We begin by ignoring other sources of correlation and
investigate their impact in Chapter 7. Once specific problems have been identified, methÂ¬
ods are developed for dealing with correlation during controller implementation in ChapÂ¬
ter 8, and the performance of these methods are quantified with respect to the performance
of the resulting controllers.
1.1.2.5 Accurate Combustion Models
Little is known about how NOx is formed from airbound nitrogen during combustion.
To date, reliable models for NOx formation in electric power boilers have not been availÂ¬
able [39], There are not adequate models for many realworld industrial processes, a fact
which has also limited the acceptance of modem control strategies. One reason that neuroÂ¬
control has received such widespread attention is because of its potential ability to deal
with very complex processes that have escaped modeling.
Chapter 6 develops accurate combustion models according to accepted modeling perÂ¬
formance metrics. Chapter 7 demonstrates the impact that correlation has on neurocontrol
designs, and Chapter 8 investigates its impact on the accuracy of the underlying process
models. Here it will be shown that accuracy is subjective, and that in fact no good metrics
for model accuracy are available in the literature. A new metric is proposed and empiriÂ¬
cally compared against available metrics in the literature in Chapter 8. Applying this new
metric, predictive combustion models are developed and used to shed light on which proÂ¬
cess variables have the greatest impact on NOx and CO formation.
7
1.1.2.6 Novel Combustion Controller
This project develops four neurocontrollers for the complex industrial process of NOx
formation. We begin in Chapter 3 by looking at boiler optimization from a firstprinciples
perspective, focusing on what a NOx controller is expected to achieve and why classical
control methods are not able to achieve it. Predictive neural network combustion models
are then developed in Chapter 6, and deployed within online neurocontrollers in chapters 7
and 8. The performance of each of these controllers is then quantified to compare and conÂ¬
trast the four control designs in Chapter 8.
To the authors knowledge, this work developed the first NOx controller for a gas and
oil cofired electric power plant. New regulations and the restructuring of the electric
power industry have combined to create a NOx trading market. The annual benefits to a
gas and oil cofired electric power plant associated with a 25% NOx reduction will be in
the range of $2,000,000 to $5,000,000. Clearly, a control strategy that uses existing plant
capital investments and runs on a $2,000 pentium workstation has tremendous value.
1.2 Document Organization
Chapter 2 presents a summary of the required background and a literature review of
the relevant work in neurocontrol, along with references to the literature for more detailed
treatments. This chapter is the best place for readers to become familiar with the notation
used throughout this work.
A strategy for reducing the NOx emissions from a fossilfired generating unit is proÂ¬
vided in Chapter 3. The goal of this section is to provide a physical understanding for what
we are asking the controllers to perform, thus providing justification that our objectives
are feasible. Chapter 4 develops four detailed neurocontrol designs belonging to the model
8
predictive, model inverse, and model reference control families. The designs are presented
as generalized methodologies that are applicable to any control application. Chapter 5 preÂ¬
sents a management and preprocessing methodology for collecting data in support of these
control designs and the required modeling.
Each control design considered requires accurate process models for its implementaÂ¬
tion. Chapter 6 presents a modeling methodology for developing these models. The conÂ¬
trollers are then implemented in Chapter 7. The performance for each resulting controller
is then quantified using offline simulations and online experiments. Significant problems
are discovered with the controllers for which there are no solutions in the literature. These
problems, along with proposed solutions, are investigated further in Chapter 8. This secÂ¬
tion additionally demonstrates the validity of these solutions by quantifying the perforÂ¬
mance of the revised controllers.
The "key learnings" and extensions to this work are summarized in Chapter 9.
CHAPTER 2
LITERATURE REVIEW
This chapter provides background for the rest of this document in the areas of:
1) optimization,
2) neural networks,
3) neurocontrol,
4) NOx, and
5) fossilfired power plants.
2.1 Optimization
Mathematical optimization methods are at the heart of modem modeling and control
applications. Neural networks use optimization methods to facilitate learning, and control
applications apply these methods to meet their control objectives. The notation and methÂ¬
ods presented in this section will be used extensively throughout the rest of this document.
Optimization is defined as the process of finding the values of N2 decision variables
z e N that minimize a scalar performance objective J e 'is â€”> 9? [15]. Formally, this
optimization task will be represented as
ArgMini{J(z)}, (1)
where hi is the decision variable space, which is most often taken to be euclidean
N = 91. Optimization methods, also known as mathematical programming methods, can
be classified according to the amount of a priori information available about the system
9
being optimized [54], The following sections broadly categorize optimization methods
into the following:
10
1) Classical Analytic Optimization: where the system being optimized is
known completely or nearly completely and a tractable analytic solution
exists
2) Descent Optimization: where first and/or secondorder partial derivaÂ¬
tives are available everywhere for the parameters of the system being
optimized
3) Direct Optimization: where little to no a priori knowledge exists about
the physical structure of the system being optimized
If the optimization problem involves objective functions or constraints which cannot
be stated as explicit functions of the design variables or are too complicated to manipulate,
we cannot solve it by using classical analytic optimization methods. This work will be
dealing with complex systems where little is known a priori and will therefore not conÂ¬
sider analytic optimization methods.
2.1.1 Iterative Optimization Methods
All direct and descent optimization methods are iterative in nature, i.e., they start from
an initial trial solution and proceed toward the minimum point in a sequential manner. An
iterative optimization method is typically judged based on its rate of convergence [54], In
general, an optimization method is said to have convergence of order p if
Â£(n+ 1)HI Â¿>0 p> 1, (2)
iÂ¿(rc)  nr
where z(n) and + 1) denote the points obtained at the end of iterations n and n + 1,
respectively, z* represents the optimum point, and x denotes the length or norm of the
vector x.
11
If p = 1 and 0 < k < 1, the method is said to be linearly convergent (corresponds to
slow convergence). If p = 2, the method is said to be quadratically convergent (correÂ¬
sponds to faster convergence) [65].
2.1.2 Direct Optimization
In problems where analytic solutions are not possible and the design variables are of
mixed type there is little choice but to use some variation on a direct search methodology.
Direct searches may be broken into the following broad categories.
2.1.2.1 Exhaustive methods
In most practical applications, the optimum solution is known to lie within restricted
ranges of the design variables. Exhaustive search methods are applied to problems where
the interval in which the optimum is known to lie is finite. Conceptually, these methods
evaluate the objective function at a predetermined number of points in this interval and
reduce the interval of uncertainty using the assumption of unimodality. Exhaustive methÂ¬
ods include [54]:
1) Random Search
2) Grid Search
3) Pattern Directions
2.1.2.2 Elimination methods
The exhaustive search methods are similar to a larger class of algorithms known as
elimination methods, because they search by eliminating parts of the interval. Elimination
methods differ in how they search and discard subintervals. The more common eliminaÂ¬
tion methods include [54]:
1) Dichotomous Search
12
2) Interval Halving
3) Fibonacci Method
4) Golden Section Method
2.1.2.3 Interpolation methods
Interpolation methods iteratively fit the local performance surface with a simple polyÂ¬
nomial form, and then approximate the minimum point of the system as the minimum
point of the polynomial [65]. These methods are generally more efficient than elimination
methods and can be accelerated if gradient information is available. Some of the more
popular interpolation methods include [54]:
1) Quadratic Method
2) Cubic Method
3) Newton Method
4) QuasiNewton Method
5) Secant Method
2.1.2.4 Unrestricted methods
When the design variable range is not known the search must be performed without
restrictions on the values of the variables. Most of these methods use a step size and move
from an initial guess in favorable direction (positive or negative) [54], The step size used
must be small in relation to the final accuracy desired. This method is often accelerated by
using a variable step size. These methods include [54]:
1) Simplex Method
2) Revised Simplex Method
3) Karmarkarâ€™s Method
4) Hookâ€™s and Jeevesâ€™ Method
13
5) Rosenbrockâ€™s Method
In addition, evolutionary computing techniques like genetic algorithms belong to this
category.
2.1.2.5 Line search
All of the direct search methods presented above can be applied to both onedimenÂ¬
sional or ndimensional searches. A onedimensional search is often referred to as a line
search since we are searching along a line. The aim of all line searches is to find q* e 91
such that
r* = ArgMinr[{J(z + r\zd)},
(3)
where z is the design vector, and zd is a known search direction.
One of the most efficient, and hence most popular, line searches uses the Quadratic
Method to find r* [54]. This method has been applied in this work, using the following
algorithm:
I: Normalize the search vector z,d by dividing each component by the
absolute value of the element of zd with the maximum absolute value
II: Evaluate the function at the points A = 0 and D = q0, where q0 is
an initial step size
III: If JD>JA then set C = D and B = q0/2
IV: Else set B = D and evaluate at the point E = 2q0. If JE > JD then
set C = E. Else set D = E and q0 = 2q0, and goto step III
V: Calculate
rf
4Jb 3 JA Jc
4JB ~ 2JC  2JA
(4)
14
VI: If JB  < A/"" then set r* = rj* and quit
VII: If f* < B then set C = B and B = rj* . Else set A = B and B = fj*
VIII: Goto step V
where A fu" is the minimum change in J to detect early stopping.
2.1.3 DescentBased Optimization
z
When all values of z e 91 are possible and the function J(z) has first and second parÂ¬
tial derivatives everywhere, the necessary conditions for a local minimum are
^0.
Si
by which we mean dJ!dzi = 0, Vi and
(5)
(6)
by which we mean that the m x m matrix whose components are dJ2/dzfizj must be posÂ¬
itive semi definite, i.e., have eigenvalues that are zero or positive [15],
All points that satisfy (5) are called stationary points. Sufficient conditions for a local
minimum are (5) and
q>o, (7)
sr
2 2
that is all eigenvalues must be positive. If (5) is satisfied but d J/dz~ = 0, that is, the
determinant of the matrix is zero (meaning that one or more of its eigenvalues is zero),
15
additional information is needed to establish whether or not the point is a minimum. Such
a point is called a singular point.
2.1.3.1 Methods
Classical analytic optimization methods use these conditions to solve for the optimal
solution. If the optimization problem involves an objective function or constraints that can
not be stated as explicit functions of the design variables, or which are too complicated to
manipulate, then descent optimization methods provide efficient alternatives. In general,
these methods will have significantly better convergence characteristics than direct methÂ¬
ods [54],
Descent search methods are iterative algorithms for improving estimates of the deciÂ¬
sion variable, z, so as to come closer to satisfying the conditions for a stationary point.
The steps in using the descent method are as follows:
I: Set n = 0 and guess at the initial design vector z(n), usually random
II: Determine the values of dJ/dz(n)
III: Interpreting dJ/dz(n) as the gradient vector, determine the search
d
direction z (n) = f (dJ/dz(n)) as a function of this gradient
IV: Determine the step size to be taken r(Â«) = , as a function
of this direction
V: Update the estimates of z(n + 1) = z(n) + r\(n)zd(n)
VI: Repeat II until (dH/dz(n))(dH/dz(n))T is very small
16
The variations in descentbased optimization can be expressed as variations in the
determination of the direction vector f and the step size/1. Some of the more common
variations include [15] [54][65]:
1) Steepest Descent:
zd{n) = dH/dz(n) (8)
r(Â«) = constant (9)
2) Steepest Descent with Momentum:
zd(n) =  (1  p)dH/di(n) + pzd(n) (10)
r\ (n) = constant (11)
3) Conjugate Gradients:
= (dH/dzjn + 1 ))T\dH/dz(n + 1 )dH/di(n)]
(dH/dz(n + 1 ))TdH/dz{n + 1)
r(Â«) = LineSearch(z/Â«))
(12)
(13)
2.2 Neural Networks
Artificial neural networks (ANNs) are biologically motivated data processing structure
that consist of a large number of relatively simple highly interconnected neurons or proÂ¬
cessing elements (PEs) [24], In general, these structures provide an inductive mathematiÂ¬
cal model that can be represented by
(14)
y^
where/:91 Â» 91 is the modelâ€™s input/output map, and j> e 91 , Ãº e 91 , and
tv e 91^ are its outputs, inputs, and coefficients, respectively. The coefficients in an
17
ANN map are commonly referred to as weights. Artificial neural networks infer or learn
the relationships between p and w by observing actual process data. In this way, ANNs
can be applied to generalized regression and classification inference problems.
ANN architectures possess two fundamental properties:
1) They are capable of approximating to arbitrary accuracy any continuous
function, i.e., they are universal mappers [24],
2) They have robust optimization convergence properties with respect to
the optimization of their coefficients, i.e., they are robust learners [26],
These properties make ANNs a useful tool for empirical modeling tasks where little to
no a priori information is available about the underlying process.
2.2.1 Model Architecture
There are many types of ANNs in the literature, each with specific advantages when
modeling various types of processes [14]. The two primary factors which differentiate
between ANN models are their architecture and their learning rule. A modelâ€™s architecture
defines the way in which it processes input information to produce output information,
i.e., the form of their mathematical input/output map /.
2.2.1.1 Multilayer perceptron
Most ANNs presented in the literature are static mappers, i.e., they are only capable of
modeling static or steadystate process relationships. By far the most popular and widely
applied ANN architecture is called the multilayer perceptron (MLP) [24], This network
consists of fullyinterconnected layers of PEs with logistic response characteristics. The
MLP network is typically configured with one or two hidden layers of PEs. A twolayer
MLP is illustrated in Figure 1.
18
Figure 1: Multilayer perceptron model architecture.
Formally, using matrix algebra this architecture is given by
(15)
where Jp^1 e 91^'1
is the matrix of weights for the first hidden layer, t)h* e 91 ^ is a
vector of bias values for this layer, e 91 ^ Nl'' is the matrix of weights for the secÂ¬
ond hidden layer, t)'2 e 91^'2 is the bias vector for this layer, JpV e 91 ^ ^hl is the matrix
NyxNh2
At
of weights for the output layer, b e 91 ; is the corresponding bias vector, N,, is the
number of PEs in the first hidden layer, Nhl is the number of PEs in the second hidden
layer, a is the tanh logistic function, and the set Tv = %hl, JPy, b' } repreÂ¬
sents the modelâ€™s weights.
When the process being modeled is dynamic, i.e., its current output is a function of its
current state as well as previous process states, static models are not well suited. For such
situations, models which are able to extract both static and temporal process relationships
are required. The most common method for creating dynamic neural networks is to simply
place dynamic PEs in the input layer of a static MLP [17]. These models have been
referred to by many researchers as dynamic neural networks (DNNs). The dynamic PEs
19
can have response characteristics based on a priori process knowledge or contain adaptive
memory mechanisms or fdters.
2.2.1.2 Timedelay neural network
The most common DNN is called the timedelay neural network (TDNN) [24]. This
architecture consists of a MLP where each input PE has an adaptive linear FIR filter, as
illustrated in Figure 2.
Figure 2: TDNN input PE connectivity.
Formally, the TDNN can be described by
/'(Â«(/), Nr) =
'IJ
Uj (0
J = 1
j e (2,Nr]
V/,y
J
fdn'\Ã¼{t), NT), u>)
nip, Jdl.
(16)
(17)
Adi Nu x j1
where / : 91 â€”Â» 91 x represents the tappeddelay line operator, and N
ber of taps in the delay line.
is the num
2.2.1.3 Gamma neural network
The main disadvantage to most DNNs is that they preprocess the input to extract fixed
and known dynamics of the process data rather than learn these dynamics from this data.
The TDNN can be considered as an exception to this rule, but here the process must have
20
finite impulse response (FIR) dynamics of known order. The Gamma Neural Network
(GNN) [17] represents an important class of dynamic ANN models that is able to leam
infinite impulse response (HR) process dynamics without a priori knowledge about the
structure or order of these dynamics.
Figure 3: Gamma memory processing element.
The GNN architecture is conceptually an MLP with adaptive Gamma Filters (GF)
placed at the output of its input and hidden layer PEs. A single GF is illustrated in Figure
3. Formally, the GNN is given by
jfjWO, nt)
Â«/ (0
j = 1
V/J (18)
*1_,(<) + (! rÂ¡)ffp 1) j e (2, NT]
tÃ) = /Vr) T l/"' ),Nr ) i t!â€™2) . U , (19)
of Nâ€œ Nâ€œ X Nâ€˜ . V .
where /:iK â€”> 5? represents the GF, N is the number of GF taps in the input
f .
layer, and N is the number of GF taps in the first hidden layer. This architecture has
been presented without a GF in the second hidden layer, but such a configuration would be
a straightforward extension.
21
2.2.1.4 Nonlinear statespace model
The GNN uses the Gamma Filter to represent process dynamics. The GF approximates
these dynamics from a Gamma memory kernel basis. The Gamma kernels are able to
model an important class of dynamics but may not be the best representation for general
process dynamics. An alternative approach to using Gamma kernels is to design the ANN
architecture with an explicit state and data flow structure that is capable of learning uniÂ¬
versal process dynamics. This approach is the goal of the nonlinear statespace model
(NLSS) which implements process dynamics directly as a nonlinear state evolution equaÂ¬
tion and an output observation equation [43], as given by
i(o =/(ioi),Â¿(o,tfg
(20)
y(t) = /(l(0. *K0> fy).
(21)
where 1(7) e 91^ is the models state vector consisting of N* hidden PEs,/* is an ANN
map describing the time evolution of this state, tvv are the weights of this state network,/
is a second ANN map describing how outputs are produced from this state, and $>y are the
weights of this output network. Figure 4 illustrates the configuration of a NLSS network.
Figure 4: Nonlinear state space neural network configuration.
22
2.2.2 Learning Algorithms
The biological roots of neural networks are responsible for the widespread use of the
term learning to describe the process during which the network parameters are changed to
improve the performance of the neuralnetworkbased system. An ANN learning algoÂ¬
rithm specifies how its weights are updated in response to training data. These algorithms
are simply optimization methods applied to the task of finding the best model weights to
minimize a specified modeling objective J, i.e., ArgMin^{J). In general, any of the
optimization methods presented above can be used to solve this problem.
One of the most significant breakthroughs in the field of ANNs was the realization that
the chain rule for ordered partial derivatives provides a mechanism for deriving the first
order gradients for all weights in a model, even though the modeling objective is only an
explicit function of the modelâ€™s outputs [72], Recall that the chain rule for ordered partial
derivatives is given by
iio r XT xÂ° t fix.
(22)
Applying the chain rule allows sensitivities to be calculated from the output of the
model back to its input, which is why the resulting algorithm has been coined â€œbackpropa
gationâ€ in the literature [40], When the variables are temporally related, the chain rule has
the following form
(23)
Here, in addition to backpropagating sensitivities from the modelâ€™s output to its input,
the sensitivities are backpropagated through time.
23
The most common optimization method is simply steepestdescent with momentum,
although many variations have been demonstrated to significantly improve convergence
[10]. The issues leading to the selection of one optimization method over another are:
1) Convergence Rate
2) Implementation Complexity
3) Configuration Complexity
4) Avoidance of Local Minima
5) Sensitivity to Correlation
The most common modeling objective used in ANNs is the mean squared error (MSE)
a.
between the modelâ€™s output y and a specified desired response d e 91 , as given by
J =
1
tvW
N' Ã‘*
X iwoMO)2,
t = 1 i = 1
(24)
where TV* is the number of samples in the training dataset. Learning rules which use the
MSE criterion are commonly classified as supervised learning rules, because of the presÂ¬
ence of a â€œteacherâ€ implied by the explicit specification of a desired response. Learning
rules without explicit reference to a desired response for the model in the objective funcÂ¬
tion are commonly referred to as unsupervised learning rules.
2.2.3 Generalization
As universal mappers ANNs are almost always more complex than the relationships
that we seek to uncover. The net result is that ANNs are notorious for overfitting a trainÂ¬
ing dataset, i.e., performing well on training data but poorly on a blind test dataset [24], It
24
is very important to optimize the complexity of the neural network in order to achieve the
best generalization.
2.2.3.1 Bias and variance
Considerable insight into this phenomenon can be obtained by introducing the concept
of the biasvariance tradeoff. Bishop [13] observes that the generalization error Â£,, using
the Euclidean norm, will depend on a particular dataset D on which the network was
trained. The dependence on D can be eliminated by considering an average over the comÂ¬
plete ensemble of datasets, which can be written as
<; = Â£0[Â«Ã¡a>/(Ã¡,d>))2], (25)
where (d\u) denotes the conditional average, or regression, of the desired data given by,
(d\Ãš) = fÃp(Ãœ\Ã±)dt, (26)
and p(d\u) is the conditional density of the desired variable d conditioned on the input
vector u . Bishop [13] demonstrates that this generalization error can be decomposed into
the sum of the bias squared plus the variance
; = (ED[f(Ãº,Ã±)]$\Ãº))2 + ED[f(Ã±,Ã±)ED[f(Ã¼,Ã±)]]. (27)
A model which is too simple, or too inflexible, will have a large bias, while one which
has too much flexibility in relation to the particular dataset will have a large variance. Bias
and variance are complementary quantities, and the best generalization is obtained when
we have the best compromise between the conflicting requirements of small bias and
small variance. The variance of the prediction will be further addressed below, Section
2.2.4 "Standard Errors.
25
For any given dataset, there is some optimal balance between bias and variance which
gives the smallest average generalization error. In order to improve the performance of the
network further we need to be able to reduce the bias while simultaneously reducing the
variance. The more straightforward way of achieving this is to use more data samples. As
the number of data samples is increased we can afford to use more complex models, hence
reducing the bias, while at the same time ensuring that each model is more heavily conÂ¬
strained by the data., thereby also reducing the variance. If the number of data samples is
increased rapidly in relation to the model complexity we can find a sequence of models
such that both bias and variance decrease. Models such as ANNs can in principle provide
consistent estimators of arbitrary accuracy as the number of data points is increased to
infinity. Note that, even if both the bias and variance can be reduced to zero, the generaliÂ¬
zation error will still be nonzero due to the intrinsic noise in the data.
One rarely has infinite data, and practical issues like training time make simply adding
more data points impractical. There are several practical and practiced ways to improve
model generalization, we start with regularization.
2.2.3.2 Regularization
Regularization was originally proposed by Tikhanov [62] as a method for solving ill
posed problems. The basic idea is to stabilize the solution by means of some auxiliary
nonnegative functional that embeds prior information, e.g. smoothness constraints on the
input/output mapping. Regularization is able to transform an illposed problem into a
wellposed problem [48],
26
Tikhanovâ€™s regularization theory uses a regularization penalty term of the form
Â®  jFfJ2
(28)
where P is a linear (pseudo) differential operator. This penalty term is added to the objecÂ¬
tive function to give
j= y+ro,
(29)
where Y is the regularization parameter. Prior information about the form of the solution
(i.e., the plant) is embedded in the operator P. The operator P is referred to as a stabilizer
in the sense that it stabilizes the solution y, making it smooth.
The appropriate choice for P and the solution to (28) requires functional analysis and
is beyond the scope of this work. The most commonly used form of regularizer, however,
is quite simple to implement. Weight decay regularizer terms consist of the sum of squares
of the adaptive parameters in the network
(30)
where the sum runs over the weights and biases. In conventional curve fitting the use of
this form of regularizer is called ridge regression. It has been found empirically that a regÂ¬
ularizer of this form can lead to significant improvements in generalization [29].
2.2.3.3 Growing and pruning algorithms
The topology of a neural network, number of units and interconnections, can have a
significant impact on its performance. Regularization helps to minimize this impact when
the complexity of the network is larger than required for the particular application.
Clearly, however, a better approach is to match the complexity of the model with the com
27
plexity of the application. Various techniques have been developed for optimizing the
topology, in some cases as part of the network training process itself [43]. It is important
to distinguish between two distinct aspects of the topology selection problems. First, we
need a systematic procedure for exploring some space of possible architectures. Second,
we need some way of deciding which of the architectures considered should be selected.
A straightforward approach to network structure optimization involves an exhaustive
search through a restricted class of network topologies. This approach requires significant
computational effort and only searches a very restricted class of network topologies. Much
of the computational burden can be lessened by considering a network which is relatively
small and by allowing new units and connections to be added during training. This
approach was shown to be successful by Bello [10] who used the weights from one netÂ¬
work as the initial guess for training the next network (with the extra weights initialized
randomly). Techniques of this form are called growing algorithms. An alternative
approach is to start with a relatively large network and gradually remove units; these are
known as pruning algorithms. Most of these procedures are ad hoc and tailor to specific
applications, that is not to say, however, that they are ineffective.
More recent work has taken advantage of developments in discrete optimization using
genetic algorithms [36]. Genetic algorithms provide a methodical way of searching large
discrete spaces more efficiently.
2.2.3.4 Crossvalidation
An alternative to regularization as a way of controlling the effective complexity of a
network is the procedure of crossvalidation [13]. The training of a nonlinear model correÂ¬
sponds to the iterative reduction of the error function defined with respect to the training
28
dataset. During training, the error will generally monotonically decrease as a function of
the number of presentations of the training dataset, i.e., epochs. However, the generalizaÂ¬
tion error, with respect to an independent dataset called the validation dataset, often shows
a decrease at first, followed by an increase as the network starts to overfit. Training can
therefore be stopped at the point of smallest error with respect to the validation dataset as
this produces a network with the smallest generalization error (or at least an approximaÂ¬
tion thereof).
2.2.3.5 Committees of networks
In practice, building neural network models requires the training of many different
candidate networks and then the selection of the best performer. Typically performance is
based on the networks performance on a third dataset not used for training or crossvalidaÂ¬
tion. There are two disadvantages to this approach. First, all of the effort involved in trainÂ¬
ing the remaining networks is wasted, and secondly, the generalization performance on
the validation dataset has a random component due to the noise on the data [13]. The netÂ¬
work which performed the best on this dataset might not be the one with the lowest generÂ¬
alization error. Recall that the generalization error is averaged over all datasets (25).
These limitations can be overcome by combining the networks together to form a
committee [47][46]. This approach was shown to provide significant improvements in the
generalization error. Denote the committee prediction as
Y(u, Jfr) = ~ Â£ y,(u, tv(.) ,
A
(31)
29
where tv, are the weights of committee member i, and jfr = ] is the set of all
weights for the committee.
Bishop [13] shows that if the errors of the individual committee member are decorre
lated, then the committee will always have a lower generalization error than any of its
individual members.
2.2.4 Standard Errors
Tibshirani [61] reviews a number of methods for estimating the standard error of preÂ¬
dicted values from a multilayer perceptron. These include direct evaluation of maximum
likelihoods based on the Hessian matrix, the â€œsandwichâ€ estimator and the bootstrap
method. Tibshirani offers the following observations:
1) The bootstrap methods provided the most accurate estimates of the stanÂ¬
dard errors of predicted values.
2) The nonsimulation methods (delta and sandwich) missed the substanÂ¬
tial variability due to the random initial weights from the multiple trainÂ¬
ing runs.
The nonsimulation methods are solved analytically, and therefore require unique
solutions for each network topology. Whereas the bootstrap methods apply to all network
topologies, as well as nonneural network paradigms. The additional fact, as noted above,
that the bootstrap methods account for local minima, provides strong argument for their
use.
2.2.4.0.1 Bootstrap methods
Bootstrap methods work by creating many pseudoreplicates (â€œbootstrap datasetsâ€)
from the training dataset and then reestimating the models weights tv on each bootstrap
dataset. There are two different approaches to bootstrapping [9], One can consider each
30
training case as a sampling unit, and sample with replacement from the training dataset
cases to create a bootstrap sample. This is often called the â€œbootstrap pairsâ€ method. The
bootstrap pairs sampling algorithm is given by:
I: Generate Nb samples, each one of size TV5 drawn with replacement
* ^ AT5
from the A' training observations {u(i), d(i)}Â¡ = i, and the b th samÂ¬
ple by {ub(i),d\i)}i= ,
II: For each bootstrap sample b e [ 1, //], find
ArgMin^h{jCdbf{Ãœb,A)} (32)
III: Estimate the standard error of the i th prediction as
 2
c2(yÂ¡) = J1â€” 2 )y,) , (33)
"  'y1
where
Nb
f/ = 7 ^ <34)
On the other hand, one can consider the predictors as fixed, treat the model residuals
A
d
y as the sampling units, and create a bootstrap sample by adding residuals to the
model fit p. This is called the â€œbootstrap residualsâ€ approach:
I:
Find ArgMin^{J(df(x, tv))} from the /Vs training observations
{Ã¼(i), d(i)}f= i and let r(i) = d(i) f(Ãº(i), tv)
31
II: Generate A^ bootstrap samples, each one of size ]\f drawn with
replacement from {r(i)= x, and the b th sample by {r (/)},â– = ] letÂ¬
ting
yb = {AÃ±,Ã±) + rb(i)}f=l (35)
III: For each bootstrap sample b e [ 1, A^], find
ArgMin^h{J(yb f(u, TvÂ¿))} (36)
IV: Estimate the standard error of the i th prediction as
V
^(F/)  Jâ€” X (W yt) (37)
A' _ = i
Note that both of these methods require fitting a model (retraining the network) V*
times. Typically A^ is in the range 20 < 7Va < 200. In simple linear least squares regresÂ¬
sion, it can be shown that the bootstrap methods both agree with the standard least squares
formula as A^ â€”Â» co.
The bootstrap methods will arrive at confidence intervals
yÂ¡ ~ cconf2(yÂ¡)  y i  cCOnfÂ¡2 (y i). (38)
where cconj depends on the desired confidence level 1  a . The factor cC0nj can be taken
from a table with the percentage points of the Studentâ€™s t distribution with the number of
degrees of freedom equal to the number of bootstrap runs A^.
32
2.3 Control Theory
An ANN is capable of modeling any process making them ideal candidates for comÂ¬
plex process optimization and control strategies. Neurocontrol is but a subfield of classiÂ¬
cal control theory [31]. To put neurocontrol in perspective, it is important to consider its
place within this field.
2.3.1 Classical Control Theory
Classical control theory is strongly biased towards linear timeinvariant systems [31].
General nonlinear systems simply do not allow us, because of their analytical intractabilÂ¬
ity, to formulate a theory that is as strong as that of linear system theory. On the other
hand, nonlinear systems can be qualitatively similar to linear ones under some circumÂ¬
stances.
2.3.1.1 Linear control
Linear control is concerned with systems of the form
x = + Ã‰Ãº (39)
with state x, input u, measurable output
P = Â£*, (40)
and controllers of the form
u=Fx + Ã“x*, (41)
where is the reference state, that is, the state to which the plant is to be brought with the
help of the controller [33].
The goals of linear control are:
1) Altering the closedloop behavior of the system to some userdefined
response characteristics.
33
2)Controlling the closedloop stability, i.e., convergence back to an equiÂ¬
librium point after disturbance.
The disadvantages of linear control designs:
1) Assume that the world is linear Gaussian and stationary, when in reality
the world is none of the above.
2) Require complete a priori knowledge of the process dynamics.
3) Require that the process is controllable and observable.
4) Cannot follow a reference trajectory produced by a system of lower
order that the process.
2.3.1.2 Robust control
Robust control addresses the problem of controlling a plant whose behavior is slightly
different from that of a plant model [37], The reasons for the difference are predominately
the effect of the nonlinear, nonGaussian and nonstationary world. A popular pragmatic
classical approach to robust control is concerned with preserving stability [2]. The closed
loop eigenvalues are chosen so that they remain in the stability region even if the plant
model should change in a defined range.
Although robust control strategies are primarily designed to compensate for differÂ¬
ences between our linear timeinvariant assumptions and the real world, they are still
developed based predominately on linear system theory. They are therefore, not able to
cope with significant deviations from these assumptions.
2.3.1.3 Adaptive control
Adaptive control is another way to reach a goal similar to that of robust control [37].
Instead of designing robust controllers that work under conditions different from those for
which they have been designed, adaptive controllers recognize the difference between the
assumption and reality and change to perform better in the new conditions.
34
Adaptation schemes can be based on both a reference model and a cost functional [2].
The approach called model reference adaptive control (MRAC) is, by its name, committed
to the former. This approach is based on formulating the rules for computing the direction
of change of controller parameters as a function of the difference between the behavior of
the closedloop system and a reference model. Controller parameters can be adapted either
directly or via the estimation of plant model parameters.
A more general approach is that of selftuning regulators (STR) [3] which consists in
adaptive estimation of a plant model and applying a formalized controller design method
to the plant model. This design method can be based on cost function optimization.
Like robust control, however, adaptive control implementations have been based on
linear, or simple nonlinear parametric assumptions, about the process. As a result, adapÂ¬
tive control designs have not demonstrated significant successful with complex realworld
processes.
2.3.1.4 Nonlinear control
Nonlinear control theory is concerned with general systems of the form [31]
i = /(*, Ã“) (42)
with measurable output
P = g(*) (43)
and controllers of the form
Ãºc = c(^, Â£*). (44)
The general formulation of nonlinear control holds promise to overcome all of the limÂ¬
itations of the classical control schemes presented above. The fields track record, howÂ¬
ever, does not deliver on this promise. The problem is that an analytical solution is known
35
only for a restrictive subclass of nonlinear systems. The difficulties with genuine nonlinÂ¬
ear controller designs have typically lead to a linearization approach, also known as gain
scheduling.
2.3.1.5 Optimal control
The topic of optimal control theory is to design controllers that are optimized to a cerÂ¬
tain performance criterion [15]. Classical optimal control has primarily focused on appliÂ¬
cations where such optimality could be proven analytically. For example, for a linear plant
and a quadratic performance criterion the Ricatti controller represents an explicit and gloÂ¬
bal solution [31]. Dynamic optimization provides another example for state evaluation and
selection of the optimal action, which can be proven optimal in certain applications. AlterÂ¬
natively, if each state at each sampling period is represented by a node in a directed graph
and actions are represented by connecting edges of the subsequent states, then the task can
also be transformed to the critical graph problem of graph theory [15].
In its most general form, the optimal control problem can be formulated as an optimiÂ¬
zation problem [15]. The plant can be generalized as in (42) and the goal is to find a conÂ¬
troller described by (44) that minimizes
Â£[/(***)], (45)
where E[ ] is the mean value over time.
2.3.2 Neurocontrol
Most neurocontrol architectures are either explicit or disguised analogies of classical
control design such as optimal control or numerical lyapunovfunctionbased design
methods. It has been argued [14] that it is only the representation of functions by neural
36
networks that defines the field of neurocontrol in the broad sense and differentiates neuroÂ¬
control form classical control methods.
The author agrees that the use of nonparametric models does differentiate neuroconÂ¬
trol from classical controls, but would argue that the primary departure from, and extenÂ¬
sion to the potential of, control theory is neurocontrolâ€™s willingness to depart from a
requirement for analytic solutions. Neurocontrol designs seek to realize the promise of
general nonlinear control replacing analytic optimization methods with numeric ones.
Researchers in the field of neural networks are accustomed to working in an intractable
world, and have been willing to resolve important questions like stability, robustness and
consistency empirically. The result has been, and will continue to be an important extenÂ¬
sion to control theory.
There are many types of neurocontrol architectures in the literature, each with specific
advantages and disadvantages. The following sections review some of the conceptual neuÂ¬
rocontrol strategies which have been proposed in the literature.
2.3.2.1 Modelpredictive control
When a model is used indirectly and offline the control scheme is usually referred to
as modelpredictive control (MPC) [37], In most industrial process control applications a
priori knowledge about the process is hard to obtain and blackbox models must be used.
The offline training phase performs supervised learning to develop an ANN model for the
process to be controlled, i.e., the ANN attempts to mimic the process after being exposed
to actual process data. This phase can be stated as
ArgMin${J(f /(if, tv))},
(46)
37
where and up are the process outputs to be controlled and inputs to be manipulated,
respectively.
At the online control phase, the ANN model cannot be used alone; it must be incorpoÂ¬
rated with a modelbased control scheme [31]. This control scheme is once again an optiÂ¬
mization problem, which can be stated as
ArgMin^Jip* /(if, *>))}, (47)
where y* g denotes the desired closedloop process output. This optimization is perÂ¬
formed repeatedly at each time interval during the course of feedback control.
2.3.2.2 Modelinverse control
An ANN can be trained to develop an inverse model of a process [40]. Here, the
modelâ€™s input is the process output, and the modelâ€™s output is the process input. The
offline training phase can be stated as
ArgMin^Jitf f(f, . (48)
Clearly, the inverse model is a steadystate model or the resulting controller would be
noncausal. Given a desired process setpoint y* , the appropriate online control signal tip
can be immediately calculated as
(49)
Successful applications of inverse modeling are discussed in [40] and [58]. Obviously,
an inverse model exists only when the process behaves monotonically as a â€œfeedforwardâ€
function at steadystate. If not, this approach is not applicable.
38
2.3.2.3 Controller modeling
Another simple direct neurocontrol scheme is to use a neural network to model an
existing controller. The input to the existing controller is used as training input to the ANN
model, and the controller output serves as the desired response. This approach is similar to
the modelinverse control except that the desired response here is not a process but a conÂ¬
troller. This approach can be formulated as
ArgMin^jtf (50)
where uc are the decision variables generated by an existing controller in response to the
plant states y?.
Like a process, a controller is generally dynamic and often comprises integrators or
differentiators. If an algebraic feedforward network is used to model the existing controlÂ¬
ler, dynamic information must be explicitly provided as input to the ANN model.
In general, this approach can result in controllers that are faster and/or cheaper than
traditional controllers. Using this approach, for example, Pomerleau [50] presented an
intriguing application where a neural network was used to replace a human operator, i.e.,
an existing controller.
2.3.2.4 Modelfree direct control
Without an existing controller or process knowledge, controllers have to be adapted or
learn the way a human operator leams to control/operate a process for the first time. A
modelfree neurocontrol design objective can be stated as
ArgMin^c{J(y* f), tf =/(xp, tvc)},
(51)
39
where f is an ANN that is directly controlling the process inputs, and tvc are the weights
of this network. Notice that the optimization criterion J is only a function of the actual
and desired process outputs. This means that the optimization methodology employed
must be able to learn tvc without an explicit desired response or even a mathematical linkÂ¬
age to the criterion.
The key feature of this direct adaptive approach is that a process model is neither
known in advance nor explicitly developed during control design. This most common
learning algorithm for this type of control design is referred to as reinforcement learning.
The first work in this area was the â€œadaptive criticâ€ algorithm proposed by Barto et al. [7].
Such an algorithm can be considered as an approximate version of dynamic programming
[73][8], later coined as NeuroDynamic Programming [12].
Despite its historical importance and intuitive appeal, modelfree adaptive neuroconÂ¬
trol is not appropriate for most real world applications. The plant is most likely out of conÂ¬
trol during the learning process, and few industrial processes can tolerate the large number
of failures required to adapt the controller.
2.3.2.5 Modelreference direct control
From a practical perspective, one would prefer to let failures take place in a simulated
environment with a process model rather than in a real plant. Even if failures are not disasÂ¬
trous they can cause substantial losses. The performance of a controller could be evaluated
based on a model for the process, rather than the process itself. The training stage of the
control design can be given by
ArgMin ^m{J(ypf\u, tv'"))},
(52)
40
and the control design becomes
ArgMin^c{J(y* f\u , tv'")), Ãºc =/(*?, tvc)}. (53)
In the course of modeling the plant, the plant must be operated â€œnormallyâ€ instead of
being driven out of control. After the modeling stage, the model can be used for controller
design. If a process model is already available, an ANN controller can be developed in a
simulation in which failures cannot cause any loss but that of computer time. A neural netÂ¬
work controller after extensive training in the simulation can then be installed in the actual
control system.
ModelReference direct control schemes have not only proven effective in several
studies [41 ][63], but have also already produced notable economic benefits [60]. These
approaches can be used for both offline control and for online adaptation.
2.4 NOx
The Clean Air Act Amendments of 1990 require that electric utilities make significant
reductions in nitrogen oxide (NOx) emissions from their fossilfired power plants. To
date, most efforts to reduce NOx emissions have come from expensive hardware retrofits
with less than satisfactory performance. Further complicating matters, conditions that
decrease NOx formation (lower temperature, excess fuel) result in the formation other polÂ¬
luting compounds, mainly carbon monoxide (CO). Similar emissions reductions are being
required in Europe through local and European Economic Community (ECC) initiatives.
Nitrogen monoxide (NO) and nitrogen dioxide (N02) are byproducts of the combusÂ¬
tion process of virtually all fossil fuels. Historically, the quantity of these inorganic comÂ¬
pounds in the products of combustion was not sufficient to affect boiler performance; their
41
presence was largely ignored. In recent years, oxides of nitrogen have been shown to be
key constituents in the complex photochemical oxidant reaction with sunlight to form
smog. Today, the emission of N02 and NO (collectively referred to as NOx) is regulated
by the 1990 Clean Air Act Amendments and has become an important consideration in the
design of fuel firing equipment.
NOx is formed by two primary mechanisms: thermal NOx and fuelbound NOx. TherÂ¬
mal NOx formation occurs only at high flame temperatures when dissociated nitrogen
from combustion air combines with oxygen atoms to produce oxides of nitrogen such as
NO and N02. The formation of thermal NOx increases exponentially with combustion
temperature and increases by a squareroot relationship with the presence of oxygen in the
combustion zone. Fuelbound NOx formation is not limited to high temperatures, but is
dependent upon the nitrogen content of the fuel. The best way to minimize NOx formation
is to reduce flame temperature, reduce excess oxygen, and/or to bum low nitrogenconÂ¬
taining fuels. Conditions that decrease NOx formation (lower temperature, excess fuel)
can result in incomplete combustion. These conditions result in the formation other pollutÂ¬
ing compounds, mainly carbon monoxide (CO).
2.4.1 Reduction
The available NOx reduction technologies can be categorized into one of the followÂ¬
ing:
â€¢ Before Combustion: Nitrogen is extracted from the fuel. This is relatively inefÂ¬
fective, since most of the nitrogen in the formation of NOx comes from the air
(containing N2).
42
â€¢ After Combustion: NOx is chemically reduced before leaving the stack. This proÂ¬
cess is also expensive, requiring hardware retrofits.
â€¢ During Combustion: Altering fuel and air flows and introducing them at different
points of the furnace can create several zones with different temperatures and
stoichiometry. These parameters significantly effect the rate of NOx formation.
The following section reviews available NOx reduction strategies and technologies for
combustion sources.
â€¢ Fuel Switching: Fuelbound NOx formation is most effectively reduced by
switching to a fuel with lower nitrogen content. No. 6 fuel oil or another residual
fuel having a relatively high nitrogen content can be replaced with No. 2 fuel oil,
another distillate oil or natural gas (which is essentially nitrogenfree) to reduce
NOx emissions.
â€¢ Flue Gas Recirculation (FGR): Flue gas recirculation involves extraction of some
of the flue gas from the stack, and recirculation with the combustion air supplied
to the burners. The process reduces both the oxygen concentrations at the burnÂ¬
ers and the temperature by diluting the combustion air with flue gas. CO can
become a significant problem here.
â€¢ Low NOx Burners: Installation of burners especially designed to limit NOx forÂ¬
mation can reduce NOx emissions. Higher reduction efficiencies can be
achieved by combining a low NOx burner with FGR. Low NOx burners are
designed to reduce the peak flame temperature by inducing recirculation zones,
staging combustion zones, and reducing local oxygen concentrations.
43
â€¢ Derating: Some industrial boilers may be derated to produce a reduced quantity
of steam or hot water. Derating will decrease the flame temperature within the
unit, reducing formation of thermal NOx. Derating can be accomplished by
reducing the firing rate or by installing a permanent restriction, such as an orifice
plate, in the fuel line. Clearly this solution would have significant economic
impact on the unit.
â€¢ Steam or Water Injection: By injecting a small amount of water or steam into the
immediate vicinity of the flame, the flame temperature will be lowered and the
local oxygen concentration reduced. The result would be to decrease the formaÂ¬
tion of thermal and fuelbound NOx. This process generally lowers the combusÂ¬
tion efficiency of the unit by one or two percent.
â€¢ Staged Combustion: Either air or fuel injection can be staged, creating either a
fuelrich zone followed by an airrich zone, or an airrich zone followed by a
fuelrich zone. A low NOx burner utilizing staged combustion can be installed,
or the furnace itself can be retrofitted for staged combustion.
â€¢ Fuel Rebuming: Staged combustion can be achieved through the fuel rebuming
process. A Gas Rebuming Zone (GRZ) is created above the primary combustion
zone. In the GRZ, additional natural gas is injected, creating a fuelrich region
where hydrocarbon radicals react with NOx to form molecular nitrogen.
â€¢ Reduced Oxygen Concentration: Decreasing excess air reduces the oxygen availÂ¬
able in the combustion zone and lengthens the flame, resulting in a lower heat
release rate per unit flame volume. NOx emissions are reduced in an approxi
44
mately linear fashion with decreasing excess air. However, as excess air is
reduced beyond a threshold value, combustion efficiency will decrease due to
incomplete mixing, and CO emissions will increase.
â€¢ Selective Catalytic Reduction (SCR): Selective catalytic reduction (SCR) is a
postformation NOx control technology that uses a catalyst to facilitate a chemiÂ¬
cal reaction between NOx and ammonia to produce nitrogen and water. An
ammonia/air or ammonia/steam mixture is injected into the exhaust gas, which
then passes through a catalyst where NOx is reduced. To optimize the reaction,
the temperature of the exhaust gas must be in a certain range when it passes
through the catalyst bed. Among its disadvantages, SCR requires additional
space for the catalyst and reactor vessel, as well as ammonia storage, distribuÂ¬
tion, and injection system. Precise control of ammonia injection is critical. An
inadequate amount of ammonia can result in unacceptable high NOx emission
rates, while excess ammonia can lead to ammonia â€œslipâ€, or the venting of undeÂ¬
sirable ammonia to the atmosphere.
â€¢ Selective NonCatalytic Reduction (SNCR): Selective noncatalytic NOx reducÂ¬
tion involves injection of a nitrogenous agent, such as ammonia or urea, into the
flue gas. The optimum injection temperature when using ammonia is 1850
degrees F, at which 60 percent NOx removal can be approached. The optimum
temperature range is wider when using urea. Below the optimum temperature
range, ammonia is formed, and above, NOx emissions actually increase. The
success of NOx removal depends not only on the injection temperature, but also
45
on the ability of the agent to mix sufficiently with flue gas.
2.5 FossilFired Power Generation
In general Canal Unit 2 is a large fossil fuel combustion engine. From an abstract perÂ¬
spective, the combustion process takes in air and fuel, and produces energy and exhaust;
as described by:
1) Air: Fossil fuel combustion requires air, or more specifically the oxygen
contained in air. Subsystems within the plant measure, prepare and
introduce this air.
2) Fuel: Combustion also requires fuel. In the case of Canal Unit 2, the fuel
can be either #6 residual oil (leftover from the refining process) or natuÂ¬
ral gas. Canal Unit 2 can fire oil only, gas only or a mixture of both.
Both fuels must be measured, prepared and introduced to the furnace.
3) Energy: The energy released by the oxidation of fossil fuels during comÂ¬
bustion is used to make steam. The properties of water still make it the
best choice when converting thermal energy to work. Canal uses the
radiative and convective heat from the combustion process to transform
ultraclean water into superheated steam. The expansion of this steam is
used to turn a turbine that turns a coil in a magnetic field, producing
electric potential. The steam, having done this work flows through
ocean water filled condensers that convert it back to superclean water.
4) Exhaust: The gaseous products of combustion having contributed much
of their heat content to the production of steam are cleaned electrostatiÂ¬
cally and ejected into the atmosphere.
2.5.1 Process Variables
The specific process variables as they apply to the Canal generating unit are described
in more detail in the following sections. These variables are also listed in Appendix, and
will be referred to throughout this work.
2.5.1.1 Air
The air required for fossil fuel combustion is prepared and introduced in two ways.
Two large symmetrical Fans called Forced Draft fans push ambient air through a series of
46
preheaters that warm this air to between 80 and 180 degrees F. This hot oxygen rich air is
then pressed into a windbox that surrounds the furnace enclosing the burner ports.
Through an array of vents called Primary and Secondary Air Shrouds around each burner
and through secondary ports called Overfire Air Ports this pressurized air is vented into
the combustion zone. In addition to this oxygen rich air, Canal Unit 2 has the ability to
recycle exhaust gas into the combustion zone through a Gas Recirculation System.
The measurements of all this air are a function of boiler design, and fan capacities. To
increase the output of the engine, additional air must be throttled through these devices.
2.5.1.1.1 Forced draft system
The forced draft fans are 2500 horsepower, 624,000 cfm centrifugal fans with inlet
vane throttles. They are constant speed fans meaning that the fan shaft turns at a constant
speed while more or less air with more or less initial spin can be dumped into the blades
by opening or closing the vanes. If the vanes are only slightly opened the flow volume of
air available to the fans is small, and it takes less work to move it. Fan amps will be correÂ¬
spondingly low. If the inlet vanes are opened wider the flow is greater. Still the fan moves
at a constant speed. More work is being done, and the amperage must increase. The output
of the FD fans is derived from the boiler master signal. Forced draft output is specified
along with fuel flow by the fuelair curve of the boiler. The fuelair curve gives a total air
flow requirement, as well as a total fuel flow requirement for a given load.
These fans are symmetrical to the furnace like many other systems and they operate
symmetrically, through their respective ducts unless biased. Bias represents an addition or
subtraction of signal to the B side FD fan. These fans can also be trimmed to meet slightly
less or slight more than the Total Air Flow demanded by the fuelair curve of the Boiler.
47
The FD fans are the principal air throttles of the Boiler and so have a fundamental effect
on nearly every other system.
2.5.1.1.2 Forced draft fan inlet vanes
Since the inlet vanesâ€™ positions represent the work being done by the fan and are the
control most familiar to the operators, these tags were used to represent the FD fans.
2.5.1.1.3 02 trim
This tag represents the bias that operators set into the airflow demand predetermined
by load. Functionally this control trims the response of the FD Fan to Air Demand. This
tag gives the operators the ability to run the furnace slightly lean or rich overall.
2.5.1.1.4 Induced draft system
As mentioned in the section on the Forced Draft Fans, the function of the ID Fans is to
take whatever gasses are present in the furnace, including air that has been introduced by
the load following Forced Draft Fans, plus all products of combustion, and pull them out,
maintaining a constant under pressure in the furnace of .5 inches of water column. The
FD Fansâ€™ speed is kept constant while the volume of air they move is throttled with inlet
vane controls. Canal is limited by the power of these fans. Current unit maximum output is
frequently limited by the power of these fans to keep up with the increased air flows of the
recently installed low NOx shroud and overfire air system.
2.5.1.1.5 Induce draft fan inlet vanes
The induced draft fan inlet vanes are the inlet throttles to the fans, they open in
response to request for increased output and as in the case of the FD fans, represent the
work being done.
48
2.5.1.1.6 Combustion air temps
The combustion air temperature tags represent the temperature of the incoming air
after the FD Fans. The temperature of this air is a direct result of energy added to ambient
air by the Glycol Air Preheater (GAH), and the Combustion Air Preheater (CAFI). Since
density is a function of temperature, the temperature of this air can impact the combustion
process that is sensitive to the Oxygen content of air as well as the operation of other volÂ¬
umetric systems like the Induced Draft (ID) Fans. It also has a primary impact on exhaust
gas temperature and resultant stack gas velocity.
2.5.1.1.7 Primary air shrouds
The Primary Air (PA) Shrouds represent the circular articulating vents that surround
the individual burner orifices. These are closest to the fuel gun concentrically inside of the
Secondary Air (SA) Shrouds. They are responsible for supplying primary combustion air
to the flame front. These tags represent actuator positions.
The PA shrouds are controlled by the Burner Management System (BMS) and they
move as a group from minimum position (5% open to provide cooling air) toward open as
load increases. The signal that controls them is called the Primary Air Master Demand
(PAMD). Separate PAMD signals exist for fuel gas primary air demand and for fuel oil
primary air demand. Each specific burner effectively listens to the current fuel state. HavÂ¬
ing received this signal each burnerâ€™s own PA shrouds responds to the PAMD in accorÂ¬
dance with one of two functions that are unique to it  a Burner Primary Air Shroud
Function for oil operation and a Burner Primary Air Shroud Function for gas operation.
The correct unique local shroud function is changed according to the correct master signal
depending on the fuel state of the burner. These burner and fuel specific response func
49
tions were set up to give roughly appropriate air flow to combustion at all load points and
fuel states based on the air flow inherent to the furnace.
Aside from normal operation the PA shrouds can be biased from the fuel specific masÂ¬
ter signal or on an individual basis from their respective unique functions.
2.5.1.1.8 Secondary air shrouds
The secondary air shroud tags represent the broadcast actuator positions of the second,
outer concentric set of circular articulating vents that surround the individual burner oriÂ¬
fices.
The first function of the Secondary Air (SA) Shrouds is to introduce combustion air to
the flame front following load. Their second function is to balance windbox pressure, and
therefore total airflow, against the actuation of the Overfire Air Ports and the PA shrouds.
The SA Shrouds have a master signal against which a master bias can be set. In addiÂ¬
tion they have individual actuating functions and individual biases that can be set against
these individual functions.
2.5.1.1.9 Over fire air ports
The Overfire Air (OFA) ports are rectangular louvered ports that pass combustion air
from the Windbox to the Furnace above the top burner level. In doing this they reoxygen
ate the oxygen depleted flame front. The tags themselves represent the positions broadcast
from the actuators that control the articulating louvers.
The OFA ports were installed as a part of the low NOx retrofit of 1996. The Forney
low NOx burner system is designed to bum more coolly and incompletely than normal.
NOx formation has been positively linked with time exposure to higher temperatures.
After partial combustion has taken place, low in the flame front, extra oxygen rich com
50
bustion air is introduced through the OA ports to complete the process. In this way the low
NOx burner system stages offstoichiometric combustion to manage combustion products.
The OA port actuators receive their master signal from load. This signal can be biased.
Each actuatorâ€™s response is based on a unique function that was parametrically deterÂ¬
mined, in concert with the Primary and Secondary Air Shrouds during installation to give
best airflow to combustion at all load points.
2.5.1.1.10 Air preheater temps
These represent the temperature of the exhaust gasses entering and leaving the ljung
strom combustion air heat exchanger. The ljungstrom is a large (30 ft. dia.) rotating wheel,
arranged perpendicular to the gas flow. It is half enclosed by the exhaust ducts and half
enclosed by fresh air ducts. As this wheel slowly rotates, heat is absorbed by a given area
of the wheel exposed to exhaust gas. The absorbed heat is then imparted to the incoming
air while that same section traverses the fresh air duct. Elaborate seals and pressurized
sealing air keep the two gasses from mingling across this device.
The air preheater tags are somewhat redundant. The â€œIn Tempsâ€ represent the temperÂ¬
ature of the gas on its way in, while the â€œOut Tempsâ€ represent the temperature of the gas
on the way out. The heat exchange of the air preheater is a function of the device and of
the temperatures of the two gasses and is not controllable in the least. The gas temp after
the air pre heater heat exchange was a more familiar control to the operators, however our
ability to collect these signals was compromised by a failing thermocouple during a large
part of the data collection for phasel. The gas temp before the air preheater was used to
represent exit gas temp for the modeling instead.
51
2.5.1.1.11 Windbox and furnace
The windbox is an enclosed volume that surrounds the waist of the furnace and the
burner openings. Preheated, oxygen rich air is pressurized in this volume by the FD fans.
From here this air can pass only into the furnace and only through vanes that surround the
burner openings called primary and secondary sir shrouds, or through the overfire air ports
above the burners. Canal Unit 2 is a balanced draft furnace which means that air flow
through the furnace is controlled around a desired furnace pressure by both pushing and
pulling fan systems. The pushing fans are the FD fans, while the pulling fans are the
induced draft fans. The FD fans have the primary responsibility of getting the combustion
zone all the oxygen it requires. The introduction of this pressurized air is accomplished not
only by positively pressurizing the windbox but also by negatively pressurizing the furÂ¬
nace. With the windbox driven to a positive pressure and the furnace kept at a fixed relaÂ¬
tive negative pressure, the velocity of combustion airflow is assured. The induced draft
fans have primary responsibility for maintaining the furnace at a negative pressure relative
to the windbox. In the course of increasing unit output the FD fans increase air flow. Their
aim is to maintain windbox pressure at +2 inches of water column while air transfer to the
furnace increases through the widening overfire ports and primary and secondary air
shrouds. The induced draft fans, trying to maintain a constant pressure of.5 inches of
water column in the furnace despite this increasing flow of air from the windbox, also
ramp up. The opposite happens for decreasing load. When the forced draft fans decrease
their output in step with the fuelair demand, air flow from the windbox to the furnace
decreases. In order to maintain a constant .5 inwc in the furnace the induced draft fans
52
throttled back. Transient changes in the windbox to furnace pressure differential can also
produce automated changes in the FD and ID fan flows.
2.5.1.1.12 WindBox pressure
This tags represents the positional average windbox air pressure. It is controlled
around + 2 inwc
2.5.1.1.13 Furnace pressure
This tag represents the actual furnace air pressure.
2.5.1.2 Fuel
The Fuel required for Combustion may be either #6 Fuel Oil or Natural Gas. In both
cases the fuel is taken from storage, filtered, heated to greater or lesser degree, pressurÂ¬
ized, and injected. In the case of #6 Fuel Oil, the temperature required to achieve a pump
able consistency is usually around 200 degrees. Natural Gas comes from high pressure
transmission lines and once stepped down to usable pressure is warmed up to around 80
degrees F. Both fuels are then pressurized in their respective headers. It is from these
headers that burners, when they are lit, tap their fuel.
2.5.1.2.1 Burnerson/fuel
These tags represent the readings of an array of air cooled optical flame scanners
located in the furnace itself that observe the respective burner flames. Since each burner
can fire either natural gas or fuel oil. A scanner, calibrated for each fuel specific flame is
permanently assigned to each burner. Although these scanners are analog devices, their
primary function is to confirm that the flame emanating from each lit burner is of a threshÂ¬
old quality. If the flame they are monitoring is not of a threshold quality the scanner has
53
the will to declare a Master Fuel Trip and cut off all fuel to the furnace. This is to prevent
the introduction of unbumed fuel to the furnace. These are analog devices but because
they are calibrated with the single purpose of either positively or negatively confirming
this threshold they essentially read either 1 or 0. This specific set represents the flame
quality of its burner if that burner is on natural gas.
2.5.1.2.2 Burner cells 18A & 18B MN gas flame
These are the signals for gas flame status for each burner.
2.5.1.2.3 Burner cells 18A & 18B MN oil flame
These are the signals for oil flame status for each burner.
2.5.1.2.4 Fuel type
As the Boiler Master request increased output BTUs are requested from the Fuel SupÂ¬
ply Systems. As a default this request is divided evenly in proportion to burners in service,
each of which have BTU content per unit of fuel settings. The total BTUs entering the furÂ¬
nace via the burners in service must equal this demand.
2.5.1.2.5 Fuel oil
The fuel oil introduction system consists of a main pressure generating pump that
ramps up in output as the unit master demand requests more output in the form of BTUs.
This pump supplies an operating pressure to the fuel oil header. All oil burners once they
are lit and placed into service tap a fixed orifice from this header. Since fuel oil pressure is
fixed by the number of BTUs requested by load, and the orifice of each burner tip is a
fixed diameter if open, the number of burners in service will dramatically affect Fuel Oil
54
Pressure. Changes in the number of burners lit can vary the fuel oil pressure in the header
between 65 and 150 PSIG.
Fuel Temp Fired must be at least the temp required for pumpability, which is specific
to the viscosity of the fuel oil being used.
2.5.1.2.6 Natural gas
In a fashion similar to the fuel oil introduction system, the unit master demand
requests BTUs from the gas system. Fuel gas from the pipeline is stepped down to operatÂ¬
ing pressure, filtered, warmed and supplied to a main gas header. All gas burners when lit
tap a fixed orifice from this header. The number of burners lit on gas can affect the actual
gas pressure indicated at the header.
2.5.1.2.7 Burner atomization
These tags represent the essential fuel oil atomizing steam parameters. Atomizing
steam is dry superheated steam extracted from the turbine or the reboiler and injected into
the oil burner tips to atomize the fuel oil as it is introduced to the combustion zone.
Burner Atomizing Steam pressure runs at a specified 20psig over fuel oil pressure.
Burner atomizing steam flow is modulated to maintain this constant difference from fuel
oil pressure while the actual temperature fluctuates somewhat at the point of extraction.
pv = nrt connects these three variables with temperature being somewhat variable, flow
being the control, and pressure being the set point.
2.5.1.2.8 Fuel oil / fuel gas flow differential
This tag represents the ratio of BTUs contributed by the fuel oil system vs. the BTUs
contributed by the fuel gas system to the total BTUs required for a given load.
55
2.5.1.2.9 Energy
During operation at Canal Unit 2 feedwater, pressurized by a large parasitic turbine
driven pump, is circulated through series of preheaters and then through the very walls of
the furnace. During this passage it is converted to steam. This steam is then collected in a
pressure vessel called a Steam Drum located at the top of the boiler where it is â€œdriedâ€.
From the Steam Drum this dry saturated steam is passed through radiator like Primary
Superheater and Secondary Superheaters that hang at the top of the furnace where convecÂ¬
tive, and radiative heat transfer occurs. From the outlet of the Secondary Superheater the
steam goes directly to the High Pressure inlet of the Turbine. Unit 2 is a single reheat
boiler which means that the exhaust from the high pressure turbine, instead of being conÂ¬
densed, is passed back to the boiler and re superheated. This re superheated steam then
turns the Intermediate and Low Turbine Stages. Attemperating sprays inject cool feedwaÂ¬
ter into the steam cycle between the Primary and Secondary Superheaters and also before
the Reheat Superheater. These cooling sprays dampen thermal dynamics and keep steam
temperature at the turbine roughly constant around 1000 degrees.
2.5.1.2.10 Generation
The Westinghouse turbine generator converts the expansion energy of superheated
steam to create rotational momentum in the turbine. This rotational energy is imparted to a
coil enclosed in an induced electromagnetic field. The rotation of this coil in this excited
field creates electric potential at the ends of the coil. This electric differential has roughly
560 megawatts of power with which to do work. Under normal operating conditions, and
aside from throttling effects, the output of the turbine generator is in direct relationship to
boiler output.
56
This tag represents the actual instantaneous unit output in units of power.
2.5.1.2.11 Heat rate
This is a simple calculated tag representing the sum of BTUs flowing into combustion
from oil and gas combined divided by the amount of power created. It can show the relaÂ¬
tive efficiency of combustionsteampower system in an energy in vs. energy out relationÂ¬
ship. As load increases heat rate decreases due to the thermal properties of the steam loop.
2.5.1.2.12 Main steam
The main steam temperature, in concert with the throttle pressure is related via steam
tables to volume, enthalpy and entropy and describes the output state of the steam generatÂ¬
ing system. Unit 2 is a sliding throttle unit capable of modulated steam temp output across
different throttle valve configurations. Steam output is essentially controlled by flow. As
the unit ramps up in load, more steam is generated from increased combustion. Steam
temperature is held (roughly) steady via modulation of flow through the turbine throttle
valves, which are sequentially opened. Once the unit reaches a certain level of output
(@480MW) all throttle valves are set in the fully open position and steam flow is moduÂ¬
lated by continuing to increasing steam output through combustion throttling. At all levels
of output Steam temperature is controlled around 1000 degrees F for optimum turbine
operation
2.5.1.2.13 Temperature
These tags represent the temperature of superheated steam as it exits the secondary
superheater header and heads to the high pressure turbine throttle valves.
57
2.5.1.2.14 Attemperation spray
These represent the amount of cool feedwater that is sprayed into main steam between
the primary and secondary superheaters to control the temperature of the steam at the secÂ¬
ondary superheater outlet to the turbine.
U28300 represents fine control. This valve responds automatically and in analog fashÂ¬
ion to all changes in steam temperature at the secondary superheater outlet. U28301 repreÂ¬
sents bulk control. It responds only to changes in SSH outlet temp that are exceed preset
deadband. These coarse and fine cooling controls are combined to dampen and control
steam outlet temp against oscillations or imbalances inherent in the steam system.
All desuperheating sprays receive their volume of feedwater from total feedwater
flow.
2.5.1.2.15 Reheat steam
Exhaust from the high pressure turbine stage is cycled back to the furnace via the
reheat steam loop where it is sprayed then reintroduced to heat exchange in the reheat
superheater. Through the reheat superheater this steam is brought back up to lOOOdegF
and 580pisa upon which it is sent to the intermediate stage of the turbine. Exhaust from the
intermediate stage turbine flows to the low pressure turbine stage.
2.5.1.2.16 Temperature
This temperature represent the temperature of re superheated steam as it heads to the
intermediate turbine stage inlet.
2.5.1.2.17 Attemperation sprays
These sprays function like the superheater sprays. They inject relatively cool feedwaÂ¬
ter into the reheat steam after it has been extracted from the turbine and before it is
58
reheated. They function to control the temperature of the steam at the outlet of the reheat
superheater. Unlike the superheat desuperheaters, these sprays do not have separate fine
and coarse control functions.
2.5.1.2.18 Furnace metal temps
These tags represent an array of thermocouples installed on the vertical legs of the
pendant superheaters. Especially in gas burning the fire side material temperature of these
heat exchangers can become problematic. Unit 2 has an especially large area of superÂ¬
heater, which is the heat exchange closest to the fire itself. Because gas bums at a cooler
temperature than oil less radiant heat is absorbed by the waterwalls of the furnace and for
the same output of steam more heat must be passed to the steam loop through the gas
stream and the superheaters. This superheater weighted heat transfer zone in gas burning,
combined with air flow stratification that seems to be inherent to this unit, make careful
monitoring of these thermocouples necessary. Extended temps above 1100 degrees can
increase material fatigue signifigantly.
2.5.1.2.19 Secondary superheater metal temps topbottom LR
These represent the temperature of the firesides of selected evenly spaced legs of the
secondary superheater, which encounters hot gas second, after the primary superheater.
They are alphabetized horizontally across the superheater surface with upper representing
the trailing side and lower representing the leading side.
59
2.5.1.2.20 Primary superheater metal temps topbottom LR
These represent the temperature of the firesides of selected evenly spaced legs of the
primary superheater, which encounters hot gas first and is closest to the flame front. They
are alphabetized horizontally across the superheater surface.
2.5.1.2.21 Reheat superheater metal temps topbottom LR
These represent the temperature of the firesides of selected evenly spaced legs of the
reheat superheater, which encounters hot gas third, after the secondary superheater and
before the feedwater economizer. They are alphabetized horizontally across the superÂ¬
heater surface with upper representing the trailing side and lower representing the leading
side.
2.5.1.2.22 Exhaust
The gasses created by combustion flow upward through the furnace gas path across
the primary and secondary superheaters, the reheat superheater, and a superheater like
feedwater preheater called an economizer. In this pass all steam loop heat transfer occurs.
After leaving the furnace these exhaust gasses flow into a ljungstrom air heat exchanger
where heat is traded to the incoming combustion air. Under the pull of the induced draft
fans this now 350 degree gas passes through the units robust electrostatic precipitator
array, through the induced draft fans themselves and then up the stack.
The makeup of the fluegas at the point it leaves the furnace represents the overall qualÂ¬
ity of combustion. Key parameters include how much oxygen has been left by the comÂ¬
bustion process, and how much CO has been created. The richness or leanness of
combustion is directly evident.
60
2.5.1.2.22.1 Flue gas
2.5.1.2.22.1.1 CO
These tags represent the CO contained in exhaust gasses as measured in the side A
(U27814) and side B (U27813) furnace outlets to the exhaust ducts just after the econoÂ¬
mizer.
These are point measures of CO in a very large duct and may not capture exact CO
content. They also display extreme side to side bias with side B showing higher CO conÂ¬
tent. Although peculiar, this side to side bias is believed to be a real feature of the Canal
Unit 2 furnace draft. These tags are directly related to the quality of combustion and can
serve as a non delayed approximation of CO as it will be seen at the stack.
2.5.1.2.22.1.2 02
These tags represent the 02 contained in exhaust gasses as measured in the side A and
side B furnace outlets to the exhaust ducts just after the economizer. These tags are used in
modeling to represent the richness or leanness of combustion. They are impacted by and
can be used as a control reference for forced draft fan output trim on air demand. In addiÂ¬
tion these tags are used by Canal as a part of the CEM NOx calculation.
2.5.1.2.22.1.3 Temps
The temperature of the air being forced through the boiler at Canal Unit 2 impacts and
represents many process parameters, from combustion quality, to heat transfer distribuÂ¬
tion, to induced draft fan output. It also is control reference for the temperature and velocÂ¬
ity of exhaust leaving the stack.
61
2.5.1.2.22.2 Stack
The CEM (Continuous Emissions Monitoring Unit) consists of an array of extraction
gas analyzers in a computer room at the base of Canalâ€™s 500 ft. stack. The pitots of these
analyzers sniff mixed exhaust from the top of the 18 foot wide Unit 2 flue. The specific
amounts of certain compounds measured in this gas are entered into a database. This dataÂ¬
base serves as a binding legal history of Canals environmental compliance. Each violation
of emissions limits placed on certain compounds like NOx and CO is recorded. If the unit
is in danger of breaking its allowed daily average output of these regulated pollutants,
measured from midnight to midnight, all steps must be taken to regain compliance, includÂ¬
ing dropping load. The cost of such a sacrifice is immense and in effect these hourly and
daily emissions limits have become control variables of primary importance.
2.5.1.2.22.2.1 CO
This tag represents the CO content of stack gas in parts per million.
It is worth noting that CO and NOx represent conflicting states of combustion as they
are currently understood and managed. To reduce NOx production, combustion is kept
cool and rich. NOx formation has been shown to positively relate to increased exposure to
combustion and increased temperature. Over fire air is used to complete this off  stoichiÂ¬
ometric combustion. Unfortunately such rich and cool (incomplete) combustion inherently
produces increased CO.
2.5.1.2.22.2.2 NOx
This tag is calculated using a regulatory approved method and is used to represents the
pounds of NOx produced by Canal Unit 2 per million BTUs.
62
2.5.1.2.22.2.3 Temp
Stack temp is important to Canal for several reasons. Keeping stack temperature at a
certain point guarantees that no condensation of sulphur products can occur in the exhaust
ducts, precipitators, or in the stack itself. The products of sulphur condensation are acidic
and over extended periods of time can be damaging to expensive capital equipment. As
long as sulphur emissions are within limits, and they are not a problem at Canal, since
Canal uses low sulphur fuel oil, it is beneficial to push them all the way out of the stack
before they can condense. This requires sufficient stack gas temperatures and velocities.
Stack Temperature is controlled primarily by the amount of preheating that is done to
the air before it even enters the windbox. Because of the relationship of final stack gas
temperature to combustion air temperature, and the relationship of combustion air temperÂ¬
ature to other properties of combustion, stack temperature can be an important and tricky
control point. Since the heat losses to the exhaust through the exhaust ducts, precipitators
and induced draft fans are fixed, fluegas temperature also represents stack temperature but
without gas path travel delay.
CHAPTER 3
BOILER OPTIMIZATION
The most efficient method for reducing NOx emissions is clearly during the combusÂ¬
tion process [39]. As presented in Section 2.4 "NOx," simply changing the combustion
temperature and fuel/air distribution can dramatically affect NOx emissions. The combusÂ¬
tion of fossil fuels inside a largescale boiler, however, is a highly complex process; this
complexity is a direct function of the boiler size. A typical electric power boiler maintains
a "fireball" which is 3 to 5 stories tall, and there are hundreds of parameters which affect
the injection of fuel and air at different locations within the furnace.
The problem is our lack of understanding about how these combustion parameters
affect NOx formation. This multivariate optimization problem requires a technology that
can look at the process globally and determine the appropriate combination of combustion
controls.
3.1 First Principles
The concepts behind boiler optimization are relatively simple:
â€¢ If the boiler operates in an oxygenrich environment, i.e., with unnecessary
excess air, boiler efficiency will decrease due to the loss of sensible heat up the
stack; NOx emissions will increase concurrently.
â€¢ If the boiler operates in a fuelrich environment, i.e., with insufficient air, boiler
efficiency will decrease due to the loss of unbumed fuel. In addition, insufficient
63
64
air leads to CO formation which causes slagging and water wall corrosion, ultiÂ¬
mately shortening boiler life.
Between these two airflow conditions there is a zone of optimum combustion. This is
shown as a dark gray band in Figure 5.
3.2 Fuel and Air Distribution
Boilers for electric power and industrial steam typically have poor distribution of fuel
and air within the furnace. This causes some regions of the firebox to be fuelrich and
other regions to be oxygenrich. This situation is clearly undesirable as it leads not only to
unnecessary NOx production and reduced efficiency, but reduced boiler life expectancy
due to water wall corrosion and slagging.
The variability of the fuelair ratios at different locations throughout the boiler is repÂ¬
resented as a light gray band in Figure 5. This variability determines the amount of aggreÂ¬
gate air required to ensure that all regions inside the boiler avoid fuelrich combustion.
Figure 5: Combustion emissions characteristic versus air flow.
65
By improving the distribution of fuel and air in all parts of the firebox, it is possible to
reduce the aggregate airflow while maintaining the same safety margin. This improvement
is illustrated in Figure 6. The narrower darklyshaded band which represents the improved
distribution of air and fuel moves closer to the zone of optimum combustion. Reducing the
aggregate airflow simultaneously increases boiler efficiency and reduces NOx emissions.
Zone of
Optimum
Fuelrich environmen Combustion Oxygenrich environment;
Figure 6: Effect of lower 02 on combustion emissions.
The key challenge in boiler optimization is identifying which of the many controls
affect performance and how they need to be manipulated to ensure optimal performance
as process and economic conditions change.
3.3 Boiler Tuning
Boiler manufacturers and service companies offer boilertuning methodologies that
use the above principles of combustion to identify a limited set of control settings which
help lower NOx and increase efficiency without the need for substantial capital expendiÂ¬
ture. Such boiler tuning improves unit performance but does not begin to generate the savÂ¬
ings achievable through improved control.
66
Unfortunately, the number of control variables available to optimize performance is
too large for offline boiler tuning to predict the optimum settings. Optimum settings vary
with load, fuel quality, boiler conditions, weather, and other factors making offline tuning
difficult if not impossible.
3.4 The Role of CO
Figure 5 and 6 both show an exponential rise in CO as excess air is reduced and the
boiler approaches peak efficiency. The steepness of the CO curve depends upon the
degree of mixing of fuel and air within the furnace. Poor mixing broadens the CO curve by
creating pockets of fuelrich and oxygenrich combustion. Together with 02, CO levels
provide the best indication about combustion quality.
A model for CO will provide valuable information about:
â€¢ how well mixed the fuel and air are in the furnace,
â€¢ how individual setpoints can be used to improve this mixing, and
â€¢ conditions which lead to slagging and waterwall corrosion.
The CO measurement serves as a key safety constraint when optimizing the boiler. By
controlling to CO levels, the boiler can be optimized without compromising safety marÂ¬
gins. Improved air and fuel distribution will merely tighten the CO curve, resulting in
improved efficiency and lower NOx.
CHAPTER 4
CONTROL DESIGNS
This research investigates the applicability of neurocontrol techniques to complex proÂ¬
cess control problems, and develops a methodology for implementing them. Towards this
end, this work will develop several detailed neural networkbased control designs and
apply them to the reduction of NOx and the maintenance of acceptable CO levels in elecÂ¬
tric power plants. Subsequent sections implementation these control designs and use our
NOx case study to compare and contrast them. The control methodology will be presented
as follows:
1) A methodology for categorizing key process variables into groups that
are required for all control designs.
2) A methodology for formally stating the control optimization objectives
and operating constraints using the aforementioned variable definitions.
3) Performance criteria by which the various control designs will be
judged, based on these formal objectives and constraints.
4) Four formal control designs with explicitly account for state variable
dependencies.
4.1 Variable Definitions
When designing a controller for largescale industrial processes, there are a large numÂ¬
ber of variables to be considered. The physical processes are typically considered to have
inputs, disturbances, states and outputs. The following variable definitions are proposed
as a methodology for categorizing all process variables into subsets; these subsets will
prove useful when designing controllers in general:
67
68
1) Manipulated Variables (MVs): process inputs which have been selected
for our controller to manipulate. The MVs should be independent of one
another, i.e., manipulating one will not cause a change in any of the othÂ¬
ers.
2) Disturbance Variables (DVs): process inputs or disturbances that affect
the state or output of the process, but we either cannot or have chosen
not to manipulate. The DV should be independent of both each other
and the MVs.
3) Control Variables (CVs): the process state or output variables that the
controller will be designed to control. The CVs should be a function of
the MVs and DVs or there is little hope of the controller being able to
control them.
4) State Variables (SVs): process state variables, which are a function of
the MVs and/or DVs, that affect the CVs. Alternatively, the SVs may be
process output variables that have not been selected for control but need
to be considered as constraints.
Notice that the MV, DV, SV and CV definitions categorize the process logically and
not physically. These definitions divide variables based on how the controller will be conÂ¬
figured, rather than how the physical process is configured. The MVs will always be proÂ¬
cess inputs, i.e., can be manipulated by operators, but the DVs can contain both process
inputs and disturbances depending on which inputs are being manipulated. Likewise, SVs
and CVs can each consist of any combination of process states and/or outputs, based on
which will ultimately be controlled.
Notation: The categorization of variables into CVs, SVs, DVs and MVs
will be used extensively throughout this work, and is conceptually consisÂ¬
tent with the literature on optimization and control [37].
4.2 Optimization Objectives
The control objective is to lower NOx. Formally this objective needs to be stated as an
objective function for optimization. Since several of the controllers developed here are tra
69
jectory (multistage) controllers, this objective function will be a function of time. ConÂ¬
sider the single control variable NOx(t0) e 91 as the measured value of NOx at time t0.
An optimal control objective with fixed terminal time T for minimizing NOx(t) over the
interval t e (t0, t0 + T] can be given by
(54)
t = t.
In general, there will be more than one CV. If all CVs are to have equal impact on this
objective function, then two effects will have to be removed from the optimization objecÂ¬
tive: 1) the effect of power differences between these CVs, and 2) the current value of
each CV. The following objective function extends (54) to multiple control variables
(55)
where Ncv is the number of CVs, and p( 6 91 is a priority weighting factor and
Z(x)  (x pY)/av is the zscore statistic [66], Assuming that our controller is designed
to minimize J\ for p( > 0 the CV cvt will be minimized over the trajectory, while setting
p( < 0 will maximize the output.
Equation (55) considers the case where CVs are to be maximized or minimized. In
general, the goal is to design a controller capable of maintaining a control setpoint. A genÂ¬
eralized optimization objective is therefore presented as
(56)
70
where DÂ¡ is a desirability function that can be tailored for each CV to
D"â€˜ax = Z(cv,(0)Z(cv,(t 0)),
(57)
D'fn = Z(cv,.(i0))Z(cv.( 0),or
(58)
Dâ€™p = J(Z(cVi(t))  Z(sp(t)))2 .
(59)
4.3 Operating Constraints
Constraints will be used to ensure that the optimizer produces a feasible solution. By
feasible we mean: 1) the MV moves can be made, and 2) that when these MVs are applied
the plant will end up in a desirable state. Feasible solutions will be guaranteed by designÂ¬
ing controllers which are able to maintain MV and SV constraints.
4.3.1 Manipulated Variable Constraints
To ensure that the MV moves can be made, the controllers will maintain simple range
constraints. A range constraint consists of the upper and lower limits that an MV will be
allowed to move. Formally the range constraint for MV mvÂ¡ will be given by
Cmv _ r^min max,
i â€” 5 J 5
(60)
where Câ„¢"' is the MVâ€™s absolute minimum and is its maximum. Controllers will be
He â€˜o+T
required to provide an optimal MV trajectory {mv (t)} t = / such that
Vi, t.
e Cfv
(61)
71
4.3.2 State Variable Constraints
Similarly, to ensure that controllers drive the plant to a desirable state, SV constraints
will also have to be addressed. Formally, controllers will be required to provide optimal
t0+ T
MV trajectories that result in SV trajectories {sv (t)}t = ,0+ i such that
SV;
^,*(0 eCf Vi,f. (62)
4.3.3 Penalty Functions
Each control design considered will employ an optimization algorithm during some
phase of its development. Some optimization algorithms are able to deal with constraints
directly, i.e., given knowledge of the constraints they can ensure a feasible solutions. OthÂ¬
ers, however, will have to treat constraints indirectly by addressing them with the objecÂ¬
tive function. The most common method for addressing operating constraints in an
objective function are through the use of penalty functions [54], For example, SV conÂ¬
straints can be stated as penalty functions of the form
Msv,, c;v) =
Ã 2
, â€žmaxÂ¿ â€žn
(svÂ¿  C ) svÂ¿ > C
, â€ž/>
(sv,  C ) sv, < C
V 0 else
(63)
Generalizing the penalty function to multiple SV constraints, differences in the energy
of the respective signals will once again have to be normalized out. These effects can be
compensated for using a generalized penalty function of the form
Hsvi, c;v)
\z(svi)  Z(C"â€˜ax))2 sv > Cmi
(Z(5v,.)Z(Cmi"))2 sv, < C""
(64)
V
0
else
72
Given a set of N6' SV constraints, an optimizer may satisfy these constraints by
appending their respective penalty functions to its criterion
/=/+^pâ€™vMivi,C'v), (65)
(= 1
where p*' allows constraints to be individually prioritized. Similarly, both MV conÂ¬
straints can be appended to the optimizerâ€™s criterion by defining the penalty functions
{X(mvÂ¡,CÂ¡ )}i=.
Note that implementing constraints with penalty functions will not guarantee that the
constraints are met precisely. If the constraints are properly prioritized relative to the optiÂ¬
mization objectives, however, these constraints are easily maintained within the desired
level of accuracy.
4.4 Performance Criteria
For the case study, controllers will be judged based on their ability to lower NOx while
maintaining desired CO emissions. To this end, subsequent sections will measure the perÂ¬
formance of controllers as a plant operator moves MVs according to their control laws.
Comparing controller performance, however, will prove a difficult task, since the operator
can only take the advice from one controller at a time and the plant is constantly changing
state. Although the controllers may be able to deal with nonsteadystate conditions, it will
be nearly impossible to separate the process responses to the state changes versus the conÂ¬
trol action.
73
Further complicating matters, while onetime tests will provide useful results with
which to judge the controllers, they are not the only criteria. The controllers studied will
be judged by the following criteria:
1) Ability to control NOx and CO.
2) Ability of the operators to perform the recommended MV moves
3) Flexibility with respect to changing performance objectives and operatÂ¬
ing constraints
4) Ability to deal with changing operating states, e.g. load changes
4.5 Controller Designs
Four controller designs will be developed. The controller designs considered, fall into
the broad categories of:
1) ModelPredictive Control
2) ModelInverse Control
3) ModelBased Direct Control
There are, however, no standard recipes for building these controllers. The field is still
immature, and neurocontrol designs presented in the literature tend to be ad hoc. This
work seeks to not only develop and test four neurocontrol designs, but also to develop a
generalized methodology for implementing control designs belonging to the above
abstract categories. Each controller must be able to deal with the MV and SV constraints,
and will be judged by the performance criteria described above.
For each of the control designs considered, there are two distinct phases in the impleÂ¬
mentation:
1) Offline training.
2) Online control.
74
4.5.1 SteadyState Optimizer
The simplest, and most prevalent, neurocontroller in the literature is the steadystate
optimizer [43][40][31 ]. This controller belongs to the modelpredictive control family.
Modelpredictive control (MPC) is not new to commercial applications in the process conÂ¬
trol industry. The advance proposed here is the application of neural network reference
models within this controls methodology.
The concept of MPC is straight forward: combine a model for the process with an optiÂ¬
mizer to obtain realtime optimal setpoints. Model predictive controllers can be steady
state or dynamic, depending on characteristics of their underlying process models. This
section details the design of a neural networkbased steadystate MPC controller to meet
the problem specifications presented in Sections 4.2, 4.3 and 4.4.
4.5.1.1 Offline training
Training a MPC controller follows the schematic outlined in Figure 7. Notice that
there are actually two reference models being trained: one SV model and one CV model.
The details for how to train these models will be covered in Chapter 6. Notice, however,
that the criterion j" is a model training criterion to be presented in Chapter 6, and not the
control performance objective f presented in Section 4.4 "Performance Criteria."
The model definitions required by the steadystate optimizer are:
1) SteadyState SV Model: sv = ssSVModel(mv, dv)
2) SteadyState CV Model: cv = ssCVModel(mv, dv, sv)
75
Figure 7: Offline training and retuning configuration for steadystate optimizer.
The reason to have a CV model is obvious, it will provide the reference model that the
optimizer uses to figure out its optimal MV setpoints. The motivation for having a SV
model, however, is somewhat less apparent. The problem is that changes made to the MVs
by the optimizer will not only change the CVs, but also the SVs. The optimizer will have
to consider the effect that MVs will have on the SVs, if it is to accurately predict their
effect on CVs. Note that the CV model has an input space that consist of MVs, DVs and
SVs.
4.5.1.2 Online control
The online control configuration is illustrated in Figure 8. Here an optimizer calculates
ArgMin_Â±*{f} using the SV and CV reference models developed during model train
mv
ing. The optimizer starts with the current value of the MVs mv* = mv, uses the SV
. \ j{ <
model to estimate the current SVs sv , which are then used, along with the current value
76
of the DVs dv , to estimate the current CVs cv* . The optimizer then iteratively updates
â€¢ â€¢ â€¢ t t
its estimate for the optimal MVs mv to minimize its objective function j .
Figure 8: Online control configuration for steadystate optimizer.
Both direct and descentbased optimization can be used for MPC. If the number of
MVs is small, then direct optimizer provides an efficient alternative. As the number of
MVs grows, however, direct optimization quickly becomes impractical. Descentbased
optimization is possible because the SV and CV models are capable, via backpropagation,
of calculating the gradient of f with respect to their inputs, i.e., their inputs sensitivities
given the sensitivities at their outputs. In this manner, the optimizer calculates the CV senÂ¬
sitivities df /dev* , from which the CV Model is able to calculate SV sensitivities
df /dsv* and partial MV sensitivities df'/dmv* , from which the SV model calculates
the remaining partial MV sensitivities df' /dmv* , and finally the optimizer is able to
update its optimal MV estimate using the MV gradient of
df _ dfv +dfv
dmv* dmv* dmv*
(66)
77
This is really just the backpropagation of backpropagations, a.k.a. more fun with the
chain rule.
Several optimization methods were tested for the optimizer, along with various techÂ¬
niques for dealing with the constraints. The most effective combination identified was to
use the unconstrained conjugate gradients method in combination with an objective funcÂ¬
tion which included the SV and MV constraint penalty functions
Vv
V"v
f  Â± Z pÃâ€™Z(Â«v,*> + Â± E pTHsv,*, CD + L, E PrMv,*, Cf), (67)
aT;
i= 1
NSV ; =
N"â€˜v,
i = 1
where the MV, SV, DV and CV variable sets, along with their corresponding constraints,
are defined in Section 6.6.3 "Final Variable Sets," and all priorities have been set to 1. The
details of the conjugate gradients method will be presented in Section 6.5 "Learning AlgoÂ¬
rithm."
Penalty function can negatively impact the performance of a descentbased optimizer
by adding complexity to the performance surface having little to do with the underlying
problem. This is particularly true when the constrained variables lie outside of their conÂ¬
strained values. For the SV constraints, there is no choice but to use penalty functions for
constraints. For MVs, however, there are alternatives, because the MVs always start at
their current values which are always within the constraints. Hence, there is little to no
overhead to using MV constraints for our online optimizer. It terms of the performance
surface, the constraints can be thought of as placing a guardrail on both sides of our curÂ¬
rent position in weightspace along our path, while having little impact on the local topogÂ¬
raphy of the road.
78
4.5.2 SteadyState ModelInverse Controller
The next controller design belongs to the modelinverse control (MIC) family. ConÂ¬
ceptually, modelinverse control is straightforward: train a model to predict the MVs from
the current and known DVs, SVs and CVs, then, given a desired CV setpoint, this model
can be used directly to obtain the required MVs. Implementing a MIC controller is also
straightforward and can work reasonably well, given that the relationship between MVs
and CVs is in fact invertible. This sections details the design of a neural networkbased
MIC controller, designed to meet the problem specifications presented in Sections 4.2, 4.3
and 4.4.
4.5.2.1 Offline training
Training a MIC controller follows the schematic outlined in Figure 9. Once again,
notice that the MV model is being implemented by separate inverseSV (ISV) and
inverseMV (IMV) models. Once again, the details for how to train these models will be
covered in Chapter 6. The model definitions required by the steadystate modelinverse
controller are:
1) SteadyState ISV Model: sv = ssISVModel(cv, dv)
2) SteadyState IMV Model: mv = ssIMVModel(cv, sv, dv)
79
Figure 9: Offline training and retuning configuration for modelinverse controller.
Analogous to our MPC controller, two models have been developed which when comÂ¬
bined can invert the process. The reason to have a IMV model is obvious, it provides the
inversemodel that the controller uses to figure out optimal MV setpoints. The problem is
that not all CVSV combinations are feasible. Given a specified CV target, the ISV model
estimates the corresponding SVs which are presented to the IMV model.
4.5.2.2 Online control
The online control configuration is illustrated in Figure 10. If a known target existed
for the CVs, the online control implementation would actually be quite trivial. One comÂ¬
plexity is that the exact value for the lowest achievable NOx from the controller for a
given set of conditions is not known. Another complication with MIC is how to deal with
constraints. If one applies a target CV to the input of the inversemodel, it will predict a set
of inputs which it believes would have achieved this target. The problem is that the model
does not understand the MV or SV constraints, and if one of the inputs it predicts falls out
80
side these constraints the controller can not provide the required setpoints. This is analoÂ¬
gous to the problem faced with SV or CV constraints for MPC. The implementation
outlined in Figure 10, uses an optimizer in order to overcome both of these issues. Clearly,
CV constraints are straightforward.
Figure 10: Online control configuration for modelinverse controller.
The MIC controller uses an optimizer to calculate ArgMin^{f) using the ISV and
CV
IMV reference models developed during model training. The optimizer starts with the curÂ¬
rent value of the CVs cv* = cv, uses the ISV model to estimate the current SVs sv* ,
which are then used, along with the current value of the DVs dv , by the IMV model to
â€”^
estimate the current MVs mv . The optimizer then iteratively updates its estimate for the
optimal MVs mv* , to minimize its objective function f.
Once again both direct and descentbased optimization can be used for MIC, and once
again a conjugate gradientsbased optimizer was selected. Descentbased optimization is
possible because the ISV and IMV models are capable, via backpropagation, of calculatÂ¬
ing the gradient of f with respect to their inputs, i.e., their inputs sensitivities given the
81
sensitivities at their outputs. In this manner, the optimizer calculates the MV sensitivities
df Idmv* , from which the IMV Model is able to calculate SV sensitivities dfIdsv* and
partial CV sensitivities dfâ€˜v/dcv* , from which the ISV model calculates the remaining
partial CV sensitivities df*' /dev* , and finally the optimizer is able to update it optimal
CV estimate using the CV gradient of
df = 8fsv ar
3^* a~^*
ocv ocv ocv
(68)
The optimizerâ€™s objective function, which includes the SV and MV constraint penalty
functions, is the same objective function used by our steadystate optimizer. The only difÂ¬
ference is how the sensitivities flow through the system, as outlined above.
4.5.3 Dynamic ModelPredictive Controller
The steadystate optimizer considered above is a member of the MPC family. The vast
majority of MPC applications use models which are firstprinciples based [37], Since it is
not possible to build an accurate firstprinciples model of NOx, a new steadystate optiÂ¬
mizer for MPC using neural network models was developed. The vast majority of MPC
applications are dynamic, however. The steadystate optimizer only considers the effect
that MV changes will have on the unit in steadystate conditions.
This section develops a dynamic neural networkbased MPC controller. The main difÂ¬
ferences between this controller and our steadystate optimizer is that it:
1) Understands the dynamics of the process.
2) Provides a trajectory of MV setpoints designed to optimize the path of
the unit into the future, rather than a optimal steadystate position. In
other words, the controller not only considers where your going but how
youâ€™ll get there.
82
The concept behind this controllerâ€™s operation is identical to that of the steadystate
optimizer: combine a model for the process with an optimizer to obtain realtime optimal
setpoints. The only difference is that the models are now dynamic, and the optimal set
points become optimal setpoint trajectories.
This sections details the design of a neural networkbased dynamic MPC controller to
meet the problem specifications presented in Sections 4.2, 4.3 and 4.4.
4.5.3.1 Offline training
Training a dynamic MPC controller follows a similar schematic as outlined in Figure
7, with the inclusion of each variables explicit dependence on time t, as illustrated in FigÂ¬
ure 11. Here the SV and CV models are performing singlestage prediction with respect to
the MVs and DVs; notice that the CV model uses the current value of the SVs sv(t + 1).
The reasons for this configuration will become clear when we consider the online control
implementation in the next section.
Figure 11: Offline training and retuning configuration for steadystate optimizer.
83
Refer to Chapter 6 for details on training the dynamic SV and CV reference models
used by the dynamic MPC controller. For now we simply state the model definitions
required by the dynamic MPC controller:
1) Dynamic SV Model: sv(t+ 1) = dSVModel(mv(t), dv(t))
2) Dynamic CV Model:
cv(t+l) = dCVModel(mv(t), dv(t), sv(t + 1))
4.5.3.2 Online control
The online control configuration follows a similar configuration to the steadystate
optimizer presented in Figure 8. Here a dynamic optimizer is required, however. The optiÂ¬
mizer calculates ArgMin > {./(t)} using the dynamic SV and CV models developed
mv*(t)
during model training. The steadystate optimizer used an application of the chain rule for
ordered partial derivatives, which has been coined â€œbackpropagationâ€ [40], From the perÂ¬
spective of the chain rule, our new optimizer is identical and only the criterion changes.
From the perspective of the literature, this algorithm has been coined â€œbackpropagation
through timeâ€ [40][70].
The optimizer starts with the current value of the MVs mv*(t) = mv(t); uses the SV
^ )(
model to estimate the resulting SVs sv (t + 1); which are then used, along with the curÂ¬
rent value of the DVs dv(t), to estimate the resulting CVs cv*(t + 1) . Notice that each
estimate can rely on both present and past values of the inputs. The optimizer will then
repeat this process over the time interval t e (tQ, t0 + T] to produce the MV, SV and CV
^ t0+Tâ€” 1  _! t0+T x* t0+T
trajectories {mv (0}/ = /â€ž > {5V (0}/ = f0+i and icv (0}/ = /â€ž+ l > respectively.
84
The objective function which included the SV and MV constraint penalty functions
can now be calculated
Atâ€ž) =
in + Tr Ncv
1
N!v
N"
1 p>i*w)+Tv I p?v ww. o+x Prvx(Â«v,( o.
yv/f, ^ , ,
i = 1
(69)
where the MV, SV, DV and CV variable sets, along with their corresponding constraints,
are defined in Section 6.6.3 "Final Variable Sets," and all priorities have been set to 1.
The optimizer then iteratively updates its estimate for the optimal MV trajectories,
>.* t0+T\
{mv = tg , to minimize its objective function J (t) . Each step in the iteration perÂ¬
forms the following, starting with t = tQ + T 1 and iterating down to t = t0 : first, the
optimizer calculates the CV sensitivities dJÂ°(t)/dev* (t), from which the CV Model is
able to calculate SV sensitivities df \t)!dsv* (t) and partial MV sensitivities
dfv(t)/dmv*(t  1), from which the SV model calculates the remaining partial MV senÂ¬
sitivities dfv(t)ldmv (tâ€” 1), and finally the MV sensitivity at time tâ€” 1 can be calcuÂ¬
lated as
df(t) = df\t) df\t)
dmv*(t 1) dmv*(Jâ€” 1) dmv*(t\)
(70)
Once the backward pass is complete, the optimizer is now able to update itâ€™s optimal
ta+T~.
MV trajectory estimate {mv (t)}t = to using the MV gradient trajectory
sao rr
.dmv (0
t = ta
(71)
85
Notice that the sensitivities at time t depend on the sensitivities in the future. This is
because the models variables at time t depend on the variables in the past, i.e., the models
are dynamic. Hence the term â€œbackpropagation through time.â€
This entire optimization cycle is run at each time step tQ. The optimizer derives the
next T 1 MV moves, and the first MV setpoint is applied to the unit mv (/0). At this
point the entire process is repeated.
4.5.4 ModelReference Adaptive Controller
The final controller design considered belongs the modelreference adaptive control
family (MRAC). Like the dynamic MPC controller, the MRAC controller understands
process dynamics and provides a trajectory of MV setpoints which optimize both where
you are going and how you get there. The fundamental difference between these two conÂ¬
trollers is how this optimal trajectory is derived. The MPC design utilized an online optiÂ¬
mizer to calculate this trajectory, while the MRAC design develops a neuralnetwork
based controller which is able to calculate the optimal trajectory directly. Hence this is our
first direct controller, i.e., calculates MV setpoints directly.
Notice that the MIC design would have provided a direct online controller, if it wasnâ€™t
for:
1) the lack of a known target NOx level, and
2) the requirements for MV and SV constraints.
The MRAC design is able to overcome both of these hurdles by building knowledge of
the best achievable NOx level and by building all of the constraints directly into the conÂ¬
troller. The main advantage to the MRAC design is online response time. There is no opti
86
mization to run, one simply presents the controller with the current, and past, state of the
process, and it generates a MV setpoint as quickly as a neural network can think. These
benefits do not come for free, however. The main drawbacks to the MRAC design are:
1) Extensive offline training and retuning requirements.
2) Inflexible online configuration, with respect to changing optimization
objectives and operating constraints.
This sections details the design of a neural networkbased dynamic MRAC controller
to meet the problem specifications presented in Sections 4.2, 4.3 and 4.4.
4.5.4.1 Offline training
Training a MRAC controller requires two stages. The first stage is identical to training
and returning the dynamic MPC controller. Here, dynamic SV and CV models are develÂ¬
oped using the same steps outlined in Figure 11. The second stage uses these models to
train the controller with offline data, as illustrated in Figure 12. The offline training is simÂ¬
ilar to the online optimization which is performed for the MPC design, except this optimiÂ¬
zation is performed across the training dataset rather than online.
Figure 12: Offline training and retuning configuration for model reference controller.
87
Once again a dynamic optimizer is required, and the objective function is given by
CL
(69). To train the controller, the optimizer calculates ArgMin^CL{f (t)}, where Jp are
the weights of the control law neural network.
Training the control law (CL) model starts with the actual values for the DVs, SVs and
CVs, and random initial weights for its CL model Jp . Starting at time t0 = TÂ¡+ 1,
where TÂ¡ is the first sample in the training dataset; training uses the CL model to estimate
^ j( f \ jj(
the resulting MVs mv (t); which are then used to estimate the resulting SVs sv (t + 1)
â€”^
and CVs cv (t + 1) . This process is repeated over the time interval t e (t0, tQ + T] to
t0+T\ v :. t0+T
produce the MV, SV and CV trajectories {mv (t)}i = io , {sv (0}i = i0+i an(l
â€˜o+T
{cv (t)}t = to+\, respectively.
The training algorithm then iteratively updates its estimate for the optimal CL model
weights, JPCL, to minimize its objective function f \t). Each step in the iteration perÂ¬
forms the following, starting with t = tQ + T and iterating down to t  t0 + 1: First, the
training algorithm calculates the CV sensitivities df (t)/dcv* (t), from which the CV
Model is able to calculate SV sensitivities df (t)/dsv*(t) and partial MV sensitivities
dfv(t)/dmv*(t  1), from which the SV model calculates the remaining partial MV sen
88
sitivities dfv(t)/dmv*(t  1), and finally the MV sensitivity at time t  1 can be calcuÂ¬
lated as
df(t) = df\t) + df\t) (7T
dmv*(t 1) dmv*(t 1) dmv*(tâ€”\)
The MV sensitivities are finally passed to the CL model which backpropagates them
to derive its weight gradients df /dW , which the training algorithm is able to use to
update its control lawâ€™s weight estimate.
The training algorithm then increments tQ and repeats the entire process, until the
training algorithm has converged. When t0 = Tjâ€” T, tQ is reset to tQ = 7j. + 1. The reaÂ¬
son for training the CL model in increments of T is because the SV and CV have a limited
prediction horizon, the time before their estimates are no longer valid. By resetting the
state of these models to the actual state of the unit after T samples, we are able train the
CL model within the prediction horizon of the SV and CV models.
4.5.4.2 Online control
The online control configuration for the MRAC design is straight forward, as illusÂ¬
trated in Figure 13. Simply supply the controller with the current SVs, DVs and CVs, and
it outputs the next MV setpoint. This setpoints contains knowledge about the optimal
achievable NOx, SV constraints, MV constraints and the trajectory through which it will
drive the process into the future.
89
{sv(t), dv(t)}
Control
mv*(t)
Plant
cv(t)
f^
Law
Figure 13: Online control configuration for model reference controller.
Clearly, the controller is only as good as its underlying reference models. In addition,
considerable care must be taken to ensure that the training data contains regions of the
input space where the SV and MV constraints have been exercised. The design is easily
augmented with limiters to guarantee that MV constraints are maintained. However, there
is little that can be done to guarantee that the SV constraints are maintained.
CHAPTER 5
DATA PREPARATION
Given the detailed control designs just presented, the next step is to implement the
actual controllers by developing the required reference models. Both reference model and
controller implementations require a significant amount of process data. Data collection is
the most important aspect of any modeling or optimization project. There is a common
saying â€œjunk in, junk out,â€ this study was relentless in reenforcing this lesson. Applying
the most sophisticated modeling and/or optimal control algorithms in the world will not
make up for problems with data preparation.
With the advanced distributed control systems (DDS) and supervisory control and data
acquisition (SC AD A) systems readily available in todayâ€™s process plants, the relative
quantity and quality of available data is overwhelming. Much of the statistics and modelÂ¬
ing literature has been dedicated to the problems faced when drawing inferences from
small sample spaces. Modem processing plants are anything but data limited. The relevant
problems are just the opposite, how to draw meaningful inferences from a massive sample
space.
The following section presents solutions for the most significant challenges faced in
preparing data for modeling and optimization. Much of what is presented in this section
was learned the hard way, during modeling and optimization.
90
91
5.1 Data Management
The power plant treated in this work collects and stores tens of thousands of variables
from various sensors and actuators throughout the plant. These variables were collected by
the DCS and forwarded to a data historian called PI, marketed by OSI, Incorporated.
Notation: The author will reserve the term â€œvariableâ€ to represent process
states with a specific physical interpretation, which are considered relevant
for modeling. The DCS will collect information from many sensors or
actuators each of which represent a single process variable. The specific
points collected by the DCS will be referred to as â€œtags.â€
The DCS works with exceptionbased sampling, a nonuniform sampling scheme.
Each tag is given an absolute deviation above which an exception is raised. Raised excepÂ¬
tions are forwarded to the process control algorithms within the DCS, as well as to PI. The
PI data historian then applies a time stamp and stores the exception. The time resolution of
a sample can be trusted within about Â±2 seconds, and the quantization error is approxiÂ¬
mately equal to the deviation settings for each tag in the DCS.
This exceptionbased data acquisition scheme allows the data historian to store an
impressive amount of data. Uniformly sampled trends are provided through simple funcÂ¬
tion calls to Piâ€™s API. Any subset of the 10,000+ tags can be easily recalled for arbitrary
time ranges over the last year of continuous plant operation. Initial concerns over the data
quality given this nonuniform sampling scheme were quickly dismissed. However, sigÂ¬
nificant tuning of the tag deviations was required however.
92
5.2 Variable Selection
For the case study of boiler optimization, it is clear that we are looking for process
variables that are related to the measurement of NOx or CO, and impact the combustion
process with respect to these measurements. With the massive number of variables to
choose from in the database, the first line of defense is to use our firstprinciples knowlÂ¬
edge of the process.
Interviews were conducted with engineering and operations personnel from the plant.
They were asked to:
1) Identify all tags that represent the variables NOx and CO.
2) Identify all variables that have any affect on combustion parameters like
emissions, fuel and air flows, temperatures or pressures inside the
boiler.
3) Rank these variables with respect to their effect on NOx and CO as
essential, secondary or minimal.
4) Identify all of the tags that represent or directly impact the listed variÂ¬
ables.
5) Classify each of these tags as either:
â€¢Setpoint: can be manipulated by operators via the DCS.
â€¢Tunable Parameter (TP): can be manually manipulated by â€œengiÂ¬
neers with wrenchesâ€.
â€¢Disturbance: can not be manipulated but has an affect on comÂ¬
bustion.
â€¢State: represents a particular state of combustion which will have
an affect on emissions, can be a function of setpoints and/or dis
93
turbances but cannot be manipulated directly.
6) Identify operating constraints and concerns that require monitoring
when manipulating any of the setpoints.
This process, and subsequent iterations, produced 76 essential, 100 secondary and 300
minimal tags for Canal Electric Generating Station. The essential tags are listed in AppenÂ¬
dix. The operators and engineers identified CO as their overwhelming operating conÂ¬
straint. The process of variable selection shall be continued in Section 6.2 "Model
Definitions."
5.3 Validation
In order to get a handle on the quality of data being collected by the data historian, for
each of the essential tags above, one month of data were extracted between 1/1/1998 and
2/1/1998. The sampling rate was set to Ts = 5 seconds. One of the most useful data valiÂ¬
dation tools was to simply trend each tag over various intervals. The eye is able to spot
most data integrity issues that are commonly missed by statistical indicators.
The standard descriptive statistics of mean, variance, max and standard error were calÂ¬
culated for each tag over the datasets. These statistics were compared against firstprinciÂ¬
ples knowledge to look for the data integrity issues considered below.
5.3.1 Quantization and Clipping
The most common data integrity problem encountered with industrial data historians
is having invalid range settings. Range settings for event based data acquisition systems
are analogous to sampling rate for uniform sampling. There are three important settings,
the data minimum, data maximum and absolute deviation. When either the data minimum
94
settings are too high or the data maximum settings are too low, the archived data is
clipped. In addition, if the deviation setting is too large, quantization errors can signifiÂ¬
cantly corrupt the archived data. The best way to detect these errors is simple visual
inspection, and the only remedy is to correct the settings.
5.3.2 Missing Data
Along with each sample, the historian provided status information indicating any
errors that were detected by the DCS when collecting the sample. Considering any error as
missing, the dataset contained 10.46% missing data. Since one of the goals of this project
is to evaluate the performance of both static and dynamic models, all missing values will
need to be accounted for despite the fact that there numbers are quite low.
60.00%












â– Good
â–¡ Missing
40.00%
30.00%
20.00%
10.00%



















K4 vS4 fC4 nS4 .K4 K4
A ^ ^ ^ ^ /
Figure 14: Daily % missing across February dataset.
Figures 14 illustrates the daily sum of missing values across the dataset. One can easÂ¬
ily identify a region of four consecutive days where the plant was missing data. This
region corresponds to a unit shutdown. Note that removing large blocks of time from the
95
data will not be an issue for dynamic modeling, as long as the valid data falls into a relaÂ¬
tively small number of blocks with a large number of samples that are free of missing valÂ¬
ues. Removing this region from our analysis, the missing percentages become less than
1%.
Since the remaining errors are highly sporadic and down sampling of the data is
required, there exist a solution for dealing with the remaining errors. The collected data
was sampled at Ts = 5 seconds. Downsampling to Ts = 60 seconds will be applied by
averaging 12 consecutive samples and decimating. If 6 or fewer of the samples are in
error, then the remaining will be averages and the status of the decimated sampleâ€™s status
will be set to â€œgoodâ€. If more than 6 samples are in error, then the decimated sampleâ€™s staÂ¬
tus will be set to â€œmissingâ€. This procedure removed all remaining errors from this
dataset.
Although the above procedure was able to clean all of the data for this dataset, it is still
possible that future datasets may still contain missing values. For all other cases, missing
data will be replaced with interpolated data between the nearest surrounding valid samÂ¬
ples.
5.3.3 Outliers
Treating each tag as a random variable, the tags were standardized to a ZScore by
subtracting the mean and dividing by their standard deviation
(73)
96
The Zscore removes all effects of offset and measurement scale. They can approach
positive and negative infinity. Table 1 shows the probability of the absolute value of a Z
score exceeding some limit for normally distributed variables.
Table 1: Probability of ZScore exceeding value.
ZScore
P(Exceeding)
1.28
0.2
1.64
0.1
1.96
0.05
2.58
0.01
Calculating the probabilities of a Zscore value exceeding a for our datasets as
N
P(Z, > a) =  y \
tffi lo
\ZÂ¡(Xj)\ > a
else
(74)
where N is the total number of samples in the dataset. Tags with P(ZÂ¿ > 1.64) > 0.20 we
considered candidates for filtering or smoothing operations, and tags with
P(ZÂ¡ > 2.58) > 0.02 were considered candidates for outlier removal.
Filtering operations were most commonly applied to tags with sensor cleaning and calÂ¬
ibration spikes. Sensor calibration can occur as often as hourly for sensors exposed to the
fluegas. Cleaning and calibration will corrupt the data with shortduration spikes within
the range of normal data for the tag. Lowpass filtering was used to remove these effects
when encountered.
Tags with outliers had large spikes, well outside the range of normal data. Outliers
were most often caused by glitches during data acquisition. Errors in the data acquisition
system characteristically caused large instantaneous changes in the tagâ€™s value for a short
97
period; after this period the tagâ€™s value would instantaneously return to its true value. OutÂ¬
liers were removed by replacing the missing data with linearly interpolated data.
5.4 Time Constants
There are many delays or deadtimes inherent to the process. The natural response
time of most MV actuators, however, is less than 1 second. Their effect on combustion is
felt within seconds. Much of this will take place faster than it can resolved. If its effect can
be measured within our sampling resolution, then the dynamic models should have no
trouble extracting the temporal relationships.
There are two temporal lags that will give the modeling effort trouble. First, some set
point actuators are driven with PID loops which have been dampened to prevent operators
from over reacting. The steadystate settling time of such loops can be as long as 90 minÂ¬
utes. This situation has been compensated for by using a simple firstorder low pass filter
to dampen the actuator setpoint signal to match the actuators response characteristics.
Second, some sensors, particularly the continuous emissions monitors (CEMs), can
have significant extraction times. The Canal Electric Generating Station CEMs measuring
NOx and CO add a 8 to 10 minute deadtime between the gases formation in the boiler
until they are recorded. This situation has been taken care of by shifting these tags within
the dataset, such that setpoints and process outputs are aligned.
5.5 Normalization
In support of modeling, all tags were normalized. Variables applied to a neural netÂ¬
work should fall within the neuronâ€™s activation limits. All of the neural networks considÂ¬
ered in this study utilize a tank activation function, therefore all variables were normalized
to fall within the range [0.9,0.9]. Assuming normal distributions, the tag Zscoreâ€™s were
used for normalization such that only 1% of the data would fall outside the neuronâ€™s actiÂ¬
vation limits, according to
x
2.58,
(75)
where r =
2.58(2.58)
0.9(0.9)
2.867 is the ratio between acceptable Zscore range and the
neuronâ€™s activation range. Denormalization was then preformed according to
CHAPTER 6
MODELING
The modeling objectives for this work are inherently tied to the control designs and the
case study presented in the proceeding sections. To this end, the objective of this section is
to determine the best model architecture for each of the model definitions required by the
various control designs considered. Candidate model architectures will be judged based on
their ability to predict process dynamics. The following architectures will be considered:
1) AutoRegressive Moving Average Model (ARMA)
2) Multilayer Perceptron (MLP)
3) TimeDelay Neural Network (TDNN)
4) Gamma Neural Network (GNN)
5) Nonlinear StateSpace Model (NLSS)
Notice that the ARMA model has been included to provide a benchmark and to valiÂ¬
date the application of nonlinear control strategies for the case study application.
6.1 Methodology
The methodology for developing the â€œbestâ€ models will be as follows:
1) Model Definitions: The ultimate goal is to find the best possible models
for NOx and CO. The control designs presented require that these proÂ¬
cesses are represented using specific model definitions. Detailed speciÂ¬
fications for the models are developed, i.e., identifying the specific
process inputs and outputs to be used by the models.
2) Datasets: A 3 month dataset will be generated using the methodology
presented in Chapter 5, and divided into disjoint training, crossvalidaÂ¬
tion and testing regions.
99
100
3) Learning Algorithm: The learning algorithm used for model training
combines the PolakRibiere algorithm [49] with a line search as preÂ¬
sented by Brent [14]. This algorithm is presented along with the details
describing its application.
4) Performance Criteria: The criteria for selecting the â€œbestâ€ model are
presented.
5) Variable Pruning: A MLP is constructed, and starting from the input
sets determined from firstprinciples knowledge in Section 5.2 "VariÂ¬
able Selection" the input sets are pruned to the smallest possible set of
relevant variables.
6) Architecture Selection: Optimal parameters (e.g. number of hidden layÂ¬
ers, processing elements and memory taps) will be individually deterÂ¬
mined for each combination of architecture and process output using a
direct search methodology.
7) Analysis: The results will be analyzed to find the best steadystate and
dynamic models for each model definition.
6.2 Model Definitions
In support of the control designs presented in Chapter 4, models will be developed
according to the following model definitions:
1) SteadyState SV Model: sv = ssSVModel(mv, dv)
2) SteadyState CV Model: cv = ssCVModel(mv, dv, sv)
3) SteadyState ISV Model: sv = ssISVModel(cv, dv)
4) SteadyState IMV Model: mv = ssIMVModel(cv, sv, dv)
5) Dynamic SV Model: sv(Ã+ 1) = dSVModel(mv(t), dv(t))
6) Dynamic CV Model:
cv(t + 1) = dCVModel(mv(t), dv(t), sv(t + 1))
where the vectors mv, dv, sv and cv represent the steady state values of the MV,
DV, SV and CV variable sets, respectively; mv(t), dv(t) , sv(t) and cv(t) represent
101
their respective values at time t; and by cv = ssCVModel{mv, dv, sv) we mean that
ssCVModel is a model which takes on vectors mv, dv and sv as inputs, and produces
the vector cv as an output.
Notation: The term model definition will be used to describe the input/outÂ¬
put space of a model along with whether it is steadystate or dynamic,
where model refers to a particular realization of a predictor which impleÂ¬
ments the model definition. There will be many models developed which
implement each of the above model definitions.
6.2.1 Variable Definitions
The model definitions were based on definitions for the MV, DV, SV and CV variable
sets. The tag list in Appendix is categorized according to the variable definitions, which
shall then be used to implement the model definitions.
6.2.1.1 Control variables
The case study considers a single control variable, NOx. While CO is a process output
and maintaining appropriate levels of CO is also an objective, CO shall be considered a
constrained SV. Notice that NOx has been marked as a CV in the Essential Tag List under
the field labeled â€œType.â€
6.2.1.2 Manipulated variables
The MVs are defined as the variables that we want the controller to manipulate. There
are many inputs that could be manipulated, but there is a cost associated with the number
of manipulated inputs. These costs include:
1) generalization costs associated with the â€œcurse of dimensionality,â€
102
2) computational costs associated with the optimization during both modÂ¬
eling and control, and
3) operational costs associated with getting the operators to implement the
MV setpoints.
The goal is for the controller to manipulate those variables which have the greatest
impact on combustion, but the number of inputs should be restricted. Beginning with a
wide list of all potential MV candidates developed from firstprinciples knowledge of the
process, variables with the least impact on the SVs and CVs are pruned. The pruning
methodology will be presented in Section 6.6 "Variable Pruning." The complete list of
potential MVs is presented in the Essential Tag List.
6.2.1.3 Disturbance variables
The disturbance variables are defined as combustion variables that have an affect on
combustion and are independent of all other MVs and DVs. Variables which are functions
of MVs or other DVs will be considered as SVs. The associated costs with MVs all apply
to DV except for the operator manipulation costs. In addition to true process disturbances,
the DVs will contain MVs that we have chosen not to manipulate. The initial DVs are
listed in the Essential Tag List.
6.2.1.4 State variables
As mentioned above, CO will be considered a state variable. This will allow the conÂ¬
troller to constrain its allowable levels. This will not, however, be our only SV. The initial
set of SVs considered for modeling is presented in the Essential Tag List.
6.2.1.5 Variable representation
You probable noticed that many of the variables chosen to represent initial MVs, DVs,
SVs or CVs represent the same underlying process variable. The process in Section 5.2
103
"Variable Selection" resulted in subsets of tags that represent or impact the same physical
variable, but they each represent the variable in a slightly different way.
For example consider the physical variable of gross airflow, the total amount of air
entering the boiler. It is clear from firstprinciples that this variable has a significant
impact on NOx (see Chapter 3). The forced draft (FD) fans deliver gross airflow to the
combustion process (see Section 2.5 "FossilFired Power Generation"). The output of the
FD fans is derived from the boiler master signal. Forced draft output is specified along
with fuel flow by the fuelair curve of the boiler. The DCS has five different representative
tags for the variable of gross airflow:
1) Fan vane position: The FD fans are constant speed fans, meaning that
the fan shaft turns at a constant speed while more or less air with more
or less initial spin can be dumped into the blades by opening or closing
the vanes.
2) Fan amps: If the inlet vanes are opened wider the flow is greater, and
more work is being done and the amperage must increase.
3) FD fan demand: The fuelair curve gives a total air flow requirement,
which is characterized within the DCS as a demand signal for the FD
fan controller.
4) 02 trim: Prior to presenting the FD fan demand signal to the FD fan
controller, the operator is provided with a trim signal that can shift this
demand +10% of its range. This trim allows the operators to add or
remove gross air at their discretion.
5) FD fan setpoint: This DCS signal, which is simply the addition of the
FD fan demand with the 02 trim, is presented the FD fan control logic
where a PID control loop maintains desired airflow.
Notice that all of these representations are in our essential tag list marked as MVs.
Clearly, these variables are not all independent. In addition to â€œTypeâ€ you will notice a
field in the Essential Tag List called â€œGroup.â€ This field will be used to identify tags
which represent the same physical process variables. The following sections present a
104
methodology for reducing the MVs, DVs and SVs down to a minimal set which have the
greatest impact on combustion. All of the variable from a group can be removed from conÂ¬
sideration, but at most one variable from each group will be allowed in our final variable
sets.
6.3 Datasets
Section 5.3 presented the sampling methodology for retrieving process data from the
data historian. This methodology was applied to produce a contiguous time series of 3
months of process data from 1/18/98 to 4/18/98 while the unit was operating continuously.
This data also included parametric testing of key MVs thought to affect the process outÂ¬
puts of interest.
The 3 months of available data had to be divided into datasets for training, crossvaliÂ¬
dation and testing of each of the models. Due to the temporal nature of the data and the
dynamic nature of the models being considered, the datasets would each have to be contigÂ¬
uous in time yet disjoint from one another. In order to accomplish this while trying to keep
both the crossvalidation and testing data as close as possible to the training data, the data
was divided into 3 contiguous and disjoint time regions as follows: the first 2 weeks were
used for crossvalidation, weeks 2 thru 10 were used for training, and the remaining 2
weeks were used as a blind test set.
Each time region was then sampled according to the method outlined in Section 5.3 to
produce training, crossvalidation and testing datasets. The crossvalidation and testing
regions were only sampled once, producing only one crossvalidation and one testing
dataset common to all models considered. The training region, however, was sampled
105
D = 30 times, producing 30 training datasets as 30 observations of the underlying proÂ¬
cess. These 30 observations were sampled by offsetting the start of each sampling by 2
seconds from the previous, where Â±2 seconds is the approximate resolution of the data
historian.
6.4 Performance Criteria
The objective of this section is to develop the â€œbestâ€ possible models based on the
model definitions presented in Section 6.2. To this end, many models will be developed
for each of the six model definitions; these models will then compete for being the â€œbestâ€
for a particular model definition. The â€œbestâ€ models will then be used in Chapter 7.
This section will formally present the criteria by which the models will be judged.
Clearly, these criteria should be closely related to the objective function used to train the
models.
6.4.1 Objective Function
All of the model definitions considered are for predictors, i.e., they solve the generalÂ¬
ize nonlinear regression problem. Model training will therefore fall into the category of
supervised learning, where there is a known desired response for the model d(t) e 91 .
Although there are many objective function which can be applied to this problem, the
ordinary meansquarederror objective function is by far the most common and successful.
The MSE objective function will be given by
J =
Â¿1N t= j! = i
(77)
106
where y(t) e 91 is the output of the model, and T is the temporal length of the training
dataset. Note that both steadystate and dynamic models will be developed, therefore the
objective function has been written in a form that is applicable for both. When the model
is steadystate, T can be considered the number of samples in the training dataset.
The subsequent sections will be training multiple models for a given model definition,
and then comparing them to see which performed the best. The MSE could serve as a suitÂ¬
able candidate for our â€œbestâ€ metric. There are a few problems with comparing models
based on MSE, however:
1) It is difficult to derive meaning from the MSE value associated with
testing an individual model.
2) It cannot be used to compare the performance between individual outÂ¬
puts, because the MSE is significantly affected by the power of the indiÂ¬
vidual output variables.
3) It cannot be used to compare the performance of models tested on sepaÂ¬
rate datasets, since the power of each variable will be highly variable
across datasets.
Since all of the models will be tested against a common dataset, the third problem will
not be a factor here. The most compelling reason not to use MSE as a performance criteÂ¬
rion, is simply one of interpretation. It is very difficult to tell how well a model is performÂ¬
ing relative to the process. We will instead introduce two related performance criteria
which will not only serve as a metric with which to compare models against oneanother,
but whose values possess simple physical interpretation.
6.4.2 Normalized MeanSquared Error
The first metric is a close relative to the MSE, called the normalized meansquared
error (NMSE). This metric normalizes the MSE relative to what is sometimes called the
107
â€œtrivial predictor.â€ This predictor is simply an estimate for the process statistical mean,
and the NMSE is given by
T
NMSEy: = X
t = 1
(4(0MO)2
(!bT/(0)2
(78)
where pf is the / th element of the vector ft^, which is the statistical mean of the desired
response. Notice that the NMSE is based on individual output variables which allows us to
compare the model performance for specific model outputs.
The main advantages to the NMSE over the MSE are:
1) Differences in the power of individual outputs is normalized out.
2) Simple interpretation can be used to provide a feel for model perforÂ¬
mance based on the NMSE value alone. The best performance a model
can have would be to match the desired response precisely, which
would result in a NMSE = 0. Similarly, a NMSE = 1 indicates that
the model is doing no better than predicting the process mean. While
this is clearly not the worst performance a model can have, it is certainly
cause for speculation.
6.4.3 Correlation
Both the MSE and NMSE metrics represent the modelâ€™s error with respect to the
desired response. The error of a model describes how well it matches the actual value of
the desired response. It is very possible for the model to have a large error, and still conÂ¬
tain valuable information about the process. This is particularly true when using models
for optimization and control applications.
Consider the example illustrated in Figure 15. It is clear that the NMSE and MSE
between variable d and model m, will be lower than the respective errors between d and
m2 â– It is also clear that m2 has captured more of the dynamic information of d.
108
This example illustrates the need for an additional metric. We are building models
which are not going to be used for their ability to forecast the actual value of a variable,
but rather to explain the causeandeffect relationships between the modelâ€™s input and outÂ¬
put variables. For such applications, correlation provides a better metric. The correlation
of a model can be calculated as follows
where
(79)
T
?y = V). (80)
t= l
9
A R~ = 0 suggest that there is no linear relationship between the modelâ€™s output and
yÂ¡
2
the desired response, while R~ = 1 suggests that they are identical (in a linear sense).
Both NMSE and R provide useful metric with which to judge model performance. In
fact, used together they provide direct insight into the biasvariance dilemma of model
development. These will be the primary metric used to asses model performance in this
section.
109
6.5 Learning Algorithm
The learning algorithm, or algorithms for a neuralnetworkbased control application
must be able to deal with the following configurations:
1) Training: where the initial parameters of models and/or controllers are
determined for the first time.
2) Retuning: where the model and/or controller parameters are adjusted
online to allow the system to compensate for changes in its operating
environment.
3) Optimization: where the next MV setpoint, or setpoint trajectory must
be determined.
The proper choice of a learning algorithm needs to consider the specific requirements
for these three situations. The first decision is whether the learning algorithm should be
incremental or batch. It is the incremental learning schemes that are usually seen in close
relationship with online adaptation as seen in retuning and optimization. Indeed, the field
of adaptive controls applies incremental schemes almost exclusively.
By contrast, batch learning schemes are usually considered to be committed to offline,
nonadaptive operation, as found in training. However, it is equally possible to apply batch
learning in adaptive configurations running parallel to the plant in operation [31]. In fact,
there are many reasons to consider batch schemes when dealing with complex nonlinear
control systems, most importantly batch schemes are required for:
1) secondorder approximation methods requiring line searches. This conÂ¬
cerns particularly the conjugate gradient methods and Powellâ€™s algoÂ¬
rithm. These algorithms are not able to keep the search directions
conjugate without line searches.
2) most global optimization methods. Except for simulated annealing and
evolutionary algorithms, global optimization algorithms require the cost
function to be evaluated at various points of the state space with the
results compared.
110
The computational complexity lost by not using secondorder approximations is by
itself reason enough to restrict our learning to batch schemes. For firstorder methods,
there are no estimates of convergence even for exact quadratic functions. In other words,
they can converge arbitrarily slowly. Secondorder methods can be compared by the numÂ¬
ber of cost function evaluations that are necessary to reach the minimum of a quadratic
function. Hryces [31] demonstrates how the conjugate gradients algorithm can be applied
to speed up both the backpropagation and backpropagation through time algorithms.
These accelerations use only n(K+ 1) cost function evaluations, where n is the number
of free parameters and K is the number of evaluations required by the line search.
It should be noted that the firstorder gradient descent is not the last resort for increÂ¬
mental learning. The Kalman training algorithm proposed by Singhal and Wu [57] and
extended and applied to several neurocontrol problems by Puskorius and Feldkamp
[51 ][52] exploits secondorder information by building a parameter covariance matrix, an
analogy to the Hessian matrix in variable metric local optimization methods. Even with
sophisticated methods such as the Kalman training algorithms, incremental learning has
many practical limitations. For the identification of strongly nonlinear plants of realistic
size and difficulty, like those considered in this work, these methods can require tens of
thousands of sampled measurements [52], Furthermore, they do no support global optimiÂ¬
zation.
The robustness of a neurocontroller solutions will be predominately a function of its
performance during training, retuning and optimization. Consider the consequences of:
1) Showing up and trying to win operator confidence with poorly trained
models.
Ill
2) Having models which had worked properly being placed in closedloop
control of a power plant, which suddenly stop working following a
retraining.
3) A controller not able to see an optimal control trajectory because of the
nonconvex nature to the performance surface which exist between two
distinct operating modes of the unit.
Since the learning schemes described above will be applied to highly complex nonlinÂ¬
ear systems, and thus highly nonconvex performance surfaces, global optimization will
be considered a requirement. The notation ArgMin^{J} shall be used to refer to the gloÂ¬
bal optimization of cost functional J with respect to the variables . The global optimizaÂ¬
tion algorithms used throughout the rest of this work are detailed next.
6.5.1 Global Optimization Algorithm
An extremum of a completely arbitrary function of realvalued arguments is not comÂ¬
putable because it requires a combinatorial search in infinite spaces. Global optimization
approaches are typically forces to make explicit or implicit assumptions about the function
or are satisfied with trying to allocate the available resources so that the probability of
finding the optimum is as high as possible.
If an exhaustive search is not feasible, it is logical to try to cover the cost function
domain as far as possible with the given resources and then to use some local optimization
method. Then it is sufficient if every interesting local basin of attraction is hit by a starting
point. For ndimensional domains, the trivial coverage by all combinations of only two
values per dimension leads to generating 2n starting vectors. These correspond the corÂ¬
ners of a hypercube. Yet sparser coverage is by 2n vectors corresponding to starting
points in the centers of the hypercube sides.
112
For training and retuning learning tasks, these methods are still far too computationÂ¬
ally burdensome. For these configurations we will have to be satisfied with simply generÂ¬
ating IVs starting vectors uniformly. For optimization learning tasks, however, there are
far fewer free parameters, one for each MV. For this configuration, global optimization
shall be determined by running 2N1"' local optimizations from starting vectors x with
elements
1 i = 2/â€”1
1 i  2j â€¢
0 otherwise
(81)
6.5.2 Local Optimization Algorithm
The conjugate gradients algorithms used for all local optimization is given by:
I: Given a set of weights We, determine a search direction using the
PolkRibere algorithm presented in Section 3
II: Optimize the objective function J( W + IW^,) with respect to the
scaÂ¬
lar variable /, using the line search algorithm presented in Section
2.1.2.5
III: Update the weights according to JV(
e + 1
iWl
We + .,re
IV: Repeat until either e > e"'ax or J( W )  J( JPe + /) < A f"
where e is the index of iteration, e'ax is the maximum number of epochs, and Aj"â€œ" is
the minimum performance improvement for early stopping.
113
6.6 Variable Pruning
Appendix identified the initial tag sets to be used as MVs, DVs and SVs. The goal of
this section is to develop a methodology for pruning these tag sets into the smallest set of
variables required to produce accurate models for the model definitions presented in SecÂ¬
tion 6.2. At this point, we have exhausted our firstprinciples understanding of the process,
which was used to create the Essential Tag List. The variable pruning methodology will
therefore consist of training multiple models with different combinations of inputs, and
selecting the best models based on the performance criteria above. The methodology is an
educated direct search approach.
There are a few rules and assumptions that will be used to limit the search:
1) Only one variable from each group will be allowed for each type.
2) Operators do not want to manipulate more than 8
3) Variables for the best steadystate model will be the best variables for
dynamic modeling.
4) Variables for the best MLP(15,5) model will be the best variables for all
model architectures considered.
Here, MLP(15,5) is defined as a MLP with 2 hidden layers, with 15 processing eleÂ¬
ments in the first layer and 5 in the second. These assumptions are inherently flawed.
Given limited computational resources, however, they are reasonable assumptions.
The variable pruning methodology will consist of two algorithms. Both algorithms use
a MLP(15,5) using the steadystate CV model definitions with one modification; the CO
variable has been moved from the SV set to the CV set, i.e., from the models input to its
output. This modification was made to force the pruning algorithms to consider each variÂ¬
ables effect on both NOx and CO.
114
6.6.1 Group Pruning Algorithm
The goal of the group pruning algorithm is to identify which variable groups in the
Essential Tag List have the greatest impact on the performance of the model. The group
pruning algorithm was implemented as follows:
I: Train the MLP(15,5) 1^â€œ"' times for each unique group in the EssenÂ¬
tial Tag List
II: For each group model, select the â€œbest groupâ€ model as that with the
lowest crossvalidation MSE
III: Calculate the correlation of each â€œbest groupâ€ model across the blind
testing dataset, and define the â€œremoval groupâ€ model as the â€œbest
groupâ€ model with the highest correlation
IV: Remove all variables from the Essential Tag List that are in the group
associated with the â€œremoval groupâ€ model
V: Repeat this steps IV until there are only Ng'oups unique groups
remaining
This algorithm was run with N1u"s = 10 and ]\fg'Â°,ips = 10. Notice that there are a
total of 15 unique group models (ignoring NOx and CO), resulting in 150 model trainings
for the first iteration. This results in 15 â€œbest groupâ€ models. The total number of model
trainings required by this algorithm for all iterations is therefore
10(15 + 14+ 13 + 12+ 11) = 650. (82)
The average training run requires 30 minutes on a 300MHZ intel based workstation,
requiring approximately 14 dedicated days of CPU time. It is easy to see the limitations to
direct search methodologies. There is a lot of work recently in the literature applying
genetic algorithms to discrete optimization problems like variable and model architecture
selection. The author believes that these techniques have a great deal of merit, and will
result in significant improvements to model building methodologies. This is still a
115
research topic, however, and not the research topic chosen for this work. The â€œbruit forceâ€
method has been selected because of its robustness to parameterization.
The computational complexities of this algorithm and the algorithms discussed in subÂ¬
sequent sections are managed using a distributed computing environment. Distributing the
training of a neural network is yet another research project. Rather that distributing the
training of a single training run, the multiple training runs were distributed across a netÂ¬
work of computers. The author had evening access to more than 50 intel based workstaÂ¬
tions connected through a local area network. These machines were used in parallel to run
this and subsequent algorithms. This algorithm, for example, was run on a single SaturÂ¬
day, when the machines were not in use.
The results of this algorithm are provided in Figure 16. The algorithm was able to
remove 3 groups from the model without degrading its performance. In fact, there is an
increase in the generalization ability of the model. The 3 groups removed are listed in
Table 2 in the order they were removed.
116
Table 2: Types in order they were removed.
Order
Group
1
Fuel
2
SAS
3
PAS
There could be concern that the PAS group was removed using this algorithm. There is
an unbalanced number of variables in this group. It is not surprising that removing these
variables might increase the models ability to generalize. Given our restrictions on the
number of MVs, however, the PASs are not a good candidate for control. They were
included primarily to see if the models required them as DVs. Since they can be removed
with only minimal impact on modeling results, there is little reason at this time to investiÂ¬
gate their impact further. They are a good candidate for future expansion of our controlÂ¬
lers, however.
6.6.2 Representation Pruning Algorithm
The goal of the representation pruning algorithm is to identify the best variable repreÂ¬
sentation within each remaining group. The representation pruning algorithm was impleÂ¬
mented as follows:
I: Select the first group in the reduced Essential Tag List
II: If there is only one unique representation in the group then select the
next group and goto step II
III: For each unique representation in the selected group, train the
MLP(15,5) 10 times with the variables for this representation as the
only input variables from the selected group; The resulting models will
be called â€œrepresentationâ€ models
117
IV: For each â€œrepresentationâ€ model, the training result with the lowest
crossvalidation MSE is selected as the â€œbest representationâ€ model
V: Calculate the correlation of each â€œbest representationâ€ model across
the blind testing dataset, and remove all variables associated with
every â€œrepresentation modelâ€ except for the â€œbest representationâ€
model with the highest correlation
VI: Select the next group and goto step II
Notice that there are 6 remaining groups with more than one unique representation,
containing a total of 12 unique representations. This algorithm, therefore, requires 120
model trainings. The results after pruning the variable representations for each of the 4
groups are illustrated in Figure 17. As illustrated in this figure, each group was pruned
down to a single variable representation without loss of fidelity.
Figure 17: Representation pruning algorithm results.
6.6.3 Final Variable Sets
The final variable selections after the group and representation pruning algorithms are
provided in Table 3. These selections are of interest from a firstprinciples perspective.
Notice that the MVs contain the primary controls over gross air, i.e., the FD fans, and
118
recall from Chapter 3 that excess gross airflow is a primary cause of NOx formation. The
MVs also contain overfire air (OFA), and gas recirculation (GR) air, which recall from
Section 2.5 "FossilFired Power Generation" that the primary reason for installing the
OFA and GR subsystems is to better control NOx.
Table 3: Final variable selections after pruning.
Manipulated Variables
Disturbance Variables
State Variables
Control Variables
1/3 OFA Damper Pcs
Ambient Air Press
Sec Air Temp Side A
CEMNOx
2/3 OFA Damper Pcs
Ambient Air Temp
Sec Air Temp Side B
FD Fan 2A Inlet Vane
Bnr Atm Stm Press
OEM CO
FD Fan 2B Inlet Vane
Bnr Atm Stm Temp
Generated MW
GR Fan 2A Inlet Dmpr Pcs
OEM Barometric Pressure
Windbox Pressure
GR Fan 2B Inlet Qnpr Pcs
Cond Back Pres  Side A
GR Fan Hppr Dmpr A Pcs
Cond Back Pres  Side B
GR Fan Hppr Dmpr B Pcs
Fuel Gas Flew Indication
Fuel Oil Flow Indication
Fuel Temp Fired
Furnace Pressure
There are also some surprises. Consider, for instance, burner atomizing steam pressure
and temperature. During the initial operator interviews it was thought that the atomizing
steam would have a nominal affect on combustion, but their impact on the models develÂ¬
oped has been significant. We latter verified this impact by manipulating these variables at
the plant.
6.7 Architecture Selection
The final variable selections, identified through the pruning algorithms in the last secÂ¬
tion, will be considered the best variables for all model architectures. The author recogÂ¬
nizes that the best variables could be a function of the model architecture. The
computational complexity involved in variable pruning, however, makes finding the
topology specific variables impractical.
119
Their are a number of architecture specific parameters that will require selection for
each architecture, however. Identifying the best architecture parameters will employ a
similar methodology to that used for variable pruning, i.e., using a direct search of the feaÂ¬
sible parameter space. The following presents a detailed design for the model architectures
investigated in the study, along with the results used to identify their best parameterizaÂ¬
tion.
6.7.1 ARMA Model
The ARMA model considered in this study is given by
N"+]^T Nm
p(t) = Y an Y i + (83)
n = 1 m = 0
^ Nu N
where Ã¼ = {mv, dv} e 9? "are the model inputs, an g 91 y, Vn are the autoregressive
â€”^ N
coefficients, bm e 91 Vm are the moving average coefficients, Nn is the number of
N
autoregressive taps, Nm is the number of moving average taps, and t e 91 v is a constant
vector.
6.7.1.1 Parameter selection
Given the best set of MVs and DVs from variable pruning and only a single CV, there
are only two parameters to be identified for the ARMA model, Nn and Nm. The algorithm
used to determine N was as follows:
I: Set Nn = 0
II: Train ARMA(Nn, 1)
III: Increment Nn by 1 and repeat II until Nn = 10
120
IV: Calculate the correlation of each model in {ARMA(Nn, 1 )}VA, across
*
the blind testing dataset, and assign Nn, the optimum number autoÂ¬
regressive taps according to the model with the highest correlation
Figure 18: Results of autoregressive taps search algorithm for the NOx ARMA Model.
Figure 20 illustrates the results of this algorithm for the dCVModel. For the dCV
Model definition, ARMA(5, 1) was chosen as the model with the highest correlation, takÂ¬
ing into consideration our desire to minimize the number of taps. This algorithm was
repeated for the dSVModel definition, and the combined results are summarized in Table
6. Notice that only the dynamic model definitions have been considered here, since the
ARMA is a dynamic model.
121
Table 4: Results of autoregressive tap search algorithm for all dynamic models.
Model
PEs
Train NMSE
Train R
dCVModel
5
0.903
0.194
dSVModel
4
0.932
0.178
Given the optimum number of autoregressive taps, Nn, for each model definition, a
*
similar algorithm was used to determine the optimal number of moving average taps, Nm .
This algorithm can formally be stated as follows:
I: Set Nm = 1
II: Train the MLP(Nn, NJ
III: Increment Nm by 1 and repeat II until Nm = 10
*
IV: Calculate the correlation of each model in {ARMA(Nn, Nm)}yNm
*
across the blind testing dataset, and assign Nm , the optimum number
of moving average taps, according to the model with the highest correÂ¬
lation
Figure 19: Results of the moving average tap search algorithm for NOx ARMA Model.
122
Figure 21 illustrates the results of this algorithm for the dCVModel. For the dCV
Model definition, ARMA(5, 3) produced the model with the highest correlation. Once
again, the algorithm was repeated for each of the dynamic model definitions, and the
results are summarized in Table 7. The models summarized in this table will be taken as
the best ARMA models for their corresponding model definitions.
Table 5: Results of moving average tap search algorithm for all dynamic models.
Model
PEs
Train NMSE
Train R
dCVModel
3
0.888
0.225
dSVModel
4
0.902
0.219
6.7.2 Multilayer Perceptron
The multilayer perceptron (MLP) architecture considered in this study will consist of
2 hidden layers with tanh processing elements (PEs), and an output layer with a linear PE.
Formally, the architecture is given by
y
= #,a(#/'2a(#/,1Ã“ + S''1) + bhl) + b\
(84)
v Nu sb 1 Nh x N
where u = {mv, dv} e 91 " are the model inputs, W e 91 11 " is the matrix of
â–º /it N,
weights for the first hidden layer, b e 91 ' are the bias values for the first hidden layer,
 Â£V> Al2 X All â€¢
h2 Nâ€ž
â‚¬ 5H 'is the matrix of weights for the second hidden layer, b e 5H are the
bias values for the second hidden layer, e 91 y 1,2 is the matrix of weights for the
M
second hidden layer, b e 91 are the bias values for the second hidden layer, Nu is the
total number of MVs and DVs in the input layer, Nh] is the number of PEs in the first hid
123
den layer, Nh2 is the number of PEs in the second hidden layer, N is the number of PEs
in the output layer, and a is the tanh logistic function.
6.7.2.1 Parameter selection
Given the best set of MVs and DVs from variable pruning and only a single CV, there
are only two parameters to be identified for the MLP, Nh, and Nhl â– The algorithm used to
determine ./V/, ] was as follows:
I: Set Nhl = 5
II: Train the MLP(Nhx, 5) 10 times
III: Select MLP*(Nhx, 5) as the model with the lowest crossvalidation
MSE from the 10 training results
IV: Increment Nhx by 2 and repeat II until Nhx = 25
V: Calculate the correlation of each model in {MLP*(Nhx, 5)}VA,
*
across the blind testing dataset, and assign Nh Â¡, the optimum number
of hidden PEs in the first layer, according to the model with the highÂ¬
est correlation
124
1.000
0.900
0.800
0.700
0.600
0.500
0.400
0.300
0.200
0.100
0.000
5
7
9
11
13
15
17
19
21
23
25
â€”aâ€”Train R
0.589
0.643
0.666
0.757
0.793
0.817
0.875
0.876
0.894
0.884
0.887
â€”â– â€”Test R
0.475
0.512
0.563
0.635
0.668
0.750
0.774
0.793
0.786
0.791
0.752
PEs
Figure 20: Results of hidden layer #1 PE search algorithm for the NOx MLP Model.
Figure 20 illustrates the results of this algorithm for the ssCVModel. For the ssCV
Model definition, MLP( 19, 5) produced the model with the highest correlation on the
blind test set. This algorithm was repeated for each of the steadystate model definitions,
and the results are summarized in Table 6. Notice that only the steadysteady model defiÂ¬
nitions have been considered here, since the MLP is a static model.
Table 6: Results of hidden layer #1 PE search algorithm for all steadystate models.
Model
PEs
Train NMSE
Train R
Test NMSE
Test R
ssCVModel
19
0.130
0.876
0.182
0.793
ssSVModel
15
0.252
0.766
0.331
0.712
ssISVModel
17
0.087
0.892
0.143
0.844
ssIMVModel
23
0.293
0.698
0.354
0.637
*
Given the optimum number of hidden processing elements in the first layer, Nh ,, for
each model definition, a similar algorithm was used to determine the optimal number of
*
hidden PEs in the second hidden layer, Nh2 . This algorithm can formally be stated as folÂ¬
lows:
125
I: Set Nh2 = 2
II: Train the MLP(Nhx, Nhl) 10 times
*
III: Select MLP*(Nhx, Nhl) as the model with the lowest crossvalidation
MSE from the 10 training results
IV: Increment Nhl by 1 and repeat II until Nh2 =10
*
V: Calculate the correlation of each model in {MLP* (Nh j, Nh2)} vtv,!2
*
across the blind testing dataset, and assign Nhl, the optimum number
of hidden PEs in the second layer, according to the model with the
highest correlation
Figure 21: Results of the hidden layer #2 PE search algorithm for NOx MLP Model.
Figure 21 illustrates the results of this algorithm for the ssCVModel. For the ssCV
Model definition, MLP* (19, 7) produced the model with the highest correlation on the
blind test set. Once again, the algorithm was repeated for each of the model definitions,
and the results are summarized in Table 7. The models summarized in this table will be
taken as the best MLP models for their corresponding model definitions.
126
Table 7: Results of hidden layer #2 PE search algorithm for all steadystate models.
Model
PEs
Train NMSE
Train R
Test NMSE
Test R
ssCVModel
7
0.135
0.895
0.298
0.801
ssSVModel
4
0.265
0.786
0.401
0.723
ssISVModel
3
0.219
0.791
0.365
0.732
ssIMVModel
9
0.296
0.705
0.336
0.641
6.7.3 TimeDelayed Neural Network
As presented in Chapter 2, the most commonly applied timedelayed neural network
(TDNN) is simply a MLP with a tappeddelayline (TDL) preprocessor at its input. This
architecture can be formally presented as follows
p = JpyG(Jph2a(jthlTDL(Ã±) + S7'1) + b1'2) + by, (85)
Nu Nu x Nt
where TDL'.yi â€œ â€”Â» 91 is the TDL mapping, and NT is number of taps.
6.7.3.1 Parameter selection
The TDNN architecture, therefore, adds a single additional parameter to the MLP
architecture, NT. Clearly, the MLP parameters A^, and Nhl will be function of Nr. Once
again, however, NT will also be function of these parameters, and this study will simplify
this problem by assuming that the best parameter for the MLP will also be optimal for the
TDNN.
*
Given the optimum number of hidden processing elements for the MLP, Nh, and
*
N/i2 , for each model definition, the algorithm used to determine NT was implemented as
follows:
I: Set NT= 2
II: Train the TDNN(Nh j, Nhl, NT) 10 times
127
III: Select TDNN*(Nh], N/l2, NT) as the model with the lowest crossvalÂ¬
idation MSE from the 10 training results
IV: Increment NT by 2 and repeat II until Nr = 20
V: Calculate the correlation of each model in
* *
{TDNN* (Nh!, N/a, AV)}wvr across the blind testing dataset, and
*
assign Nt, the optimum number of taps, according to the model with
the highest correlation
Figure 22: Results of tap search algorithm for NOx TDNN Model.
Figure 22 illustrates the results of this algorithm. For the dCVModel definition,
TDNN* (19, 7, 8) produced the model with the highest correlation on the blind test set.
The algorithm was repeated for each of the dynamic model definitions, and the results are
summarized in Table 8. The models summarized in this table will be taken as the best
TDNN models for their corresponding model definitions.
128
Table 8: Results of tap search algorithm for all dynamic models.
Model
Taps
Train NMSE
Test NMSE
Train R
Test R
dCVModel
8
0.067
0.124
0.928
0.878
dSVModel
4
0.116
0.207
0.889
0.796
6.7.4 Gamma Neural Network
The Gamma Neural Network (GNN) is similar to the TDNN considered above, with
two fundamental differences: 1) the TDL feedforward memory mechanism is replaced
with an HR Gamma Filter (GF), and 2) the memory is embedded within the hidden layers
of the architecture. This study will consider an architecture containing two GFs, one in the
input layer, and the second in the first hidden layer. This architecture can be formally preÂ¬
sented as follows
p = JPyoClPh2Gh(cy(TP,''Gu(Ãº) + bhX)) + bh2) + by, (86)
where Gu\\R^" > iR^'1 X Nah is the hidden
layer GF, and these memories have NGu and NGh memory taps, respectively.
6.7.4.1 Parameter selection
The GNN, therefore, adds two additional parameter to the MLP architecture, NGu and
Nch . We again simplify the parameter selection problem by assuming that the best
parameter for the MLP will also be optimal for the GNN.
*
Given the optimum number of hidden PEs for the MLP for each model definition, Nh Â¡
*
and N,i2 , the algorithm used to determine NCu was implemented as follows:
I: Set NGu = 2
129
II: Train a GM(iV/;], Nh2, NGu, 4) 10 times
* *
III: Select GM*(Nhx, Nh2, NCu, 4) as the model with the lowest cross
validation MSE from the 10 training results
IV: Increment NGu by 1 and repeat II until NCu =10
V: Calculate the correlation of each model in
* *
{GM*(NÂ¡Ã±, N/i2, NGu, 4)}VCw across the blind testing dataset, and
*
assign NGu , the optimum number of taps, according to the model with
the highest correlation
1.000
0.900
A A * A A A A A *
0.800
0.000
0
1
2
3
4
5
6
7
8
9
10
â€”aâ€” Train R
0.893
0.916
0.933
0.936
0.931
0.930
0.936
0.934
0.936
0.930
0.940
â– TestR
0.804
0.832
0.834
0.846
0.835
0.835
0.831
0.830
0.819
0.824
0.819
PEs
Figure 23: Results of taps search algorithm for NOx GNN model.
Figure 23 illustrates the results of this algorithm. For the dCVModel definition,
GM*( 19, 7, 3, 4) produced the model with the highest correlation on the blind test set.
This algorithm was repeated for each of the dynamic model definitions, and the results are
summarized in Table 9.
130
Table 9: Results of taps search algorithm for all dynamic models.
Model
Taps
Train NMSE
Test NMSE
Train R
Test R
dCVModel
3
0.065
0.156
0.936
0.846
dSVModel
5
0.129
0.202
0.875
0.795
Given the optimum number of memory taps in the input layer, NCu, the optimal numÂ¬
ber of memory taps for the first hidden layer were determined as follows:
I: Set Nch = 2
II: Train the GM(Nhl, Nhl, NCu, Nch) 10 times
* * *
III: Select GM*(Nhx, N/l2, NGu, NGh) as the model with the lowest cross
validation MSE from the 10 training results
IV: Increment NCh by 1 and repeat II until NGh =10
V: Calculate the correlation of each model in
* * *
{GM* (Nh j, N/i2, NGu, NGh)} v/vc/1 across the blind testing dataset, and
*
assign NGh, the optimum number of memory taps in the first layer,
according to the model with the highest correlation
Figure 24: Results of hidden taps search algorithm for NOx GNN model.
131
Figure 24 illustrates the results of this algorithm for the dCVModel. For the dCV
* *
Model definition, GM*(Nhl, Nh2, 3, 4) produced the model with the highest correlation
on the blind test set. Once again, the algorithm was repeated for each of the model definiÂ¬
tions, and the results are summarized in Table 7. The models summarized in this table will
be taken as the best GNN models for their corresponding model definitions.
Table 10: Results of hidden taps search algorithm for all dynamic models.
Model
Taps
Train NMSE
Test NMSE
Train R
Test R
dCVModel
4
0.034
0.086
0.960
0.908
dSVModel
5
0.045
0.095
0.952
0.903
6.7.5 Nonlinear StateSpace Model
As presented in Chapter 2, the nonlinear statespace (NLSS) model actually consists of
two separate models; a state evolution model and a output observation model. Here we
consider the case where both of these models are MLPs, each with a single hidden layer.
Formally, the NLSS model considered for this study is given by
*(0 = JpxG(JphxK0 + ^hx) + ^
(87)
j>(0 = Tfa(^,yKt) + bhy) + by,
(88)
where r(t) = {u(t),x(t 1)} e 91 1 and s(t) = {u(t), x(t)} e 91 " Nx are state
hx ^N,r x (Nâ€ž + Nr) Â±hx ^Nhr x , Â±x â€žjVr
representation vectors; IFeiR ,SeiR , ir e iR and b e 91
are the weights of the state evolution model; W e 9?
V _ mNhyX(Nu+Nx) e ^Nhy
#6^X^and %ye*t y are the weights of the output observation model; N is the
total number of MVs and DVs in the input layer; Nhx is the number of hidden PEs in the
132
state evolution model; Nhy is the number of hidden PEs in the output observation; N is
the number of CVs in the output, and ct is the tanh logistic function.
There are therefore 3 parameters which need to be determined for the NLSS model: 1)
the number of hidden states Nx, 2) the number of hidden PEs in the state evolution model
Nhx, and 3) the number of hidden PEs in the output observation model Nhy. Once again,
these parameters will be determined using an exhaustive search methodology. We begin
by fixing the number of hidden PEs in both models to 4, and determine the number of hidÂ¬
den states. Defining NLSS(NX, Nhx, Nhy) as a NLSS model as described above, the numÂ¬
ber of hidden state can be determined as follows:
I: Set Nx = 2
II: Train the NLSS(NX, 4, 4) 10 times
III: Select NLSS*(NX, 4, 4) as the model with the lowest crossvalidation
MSE from the 10 training results
IV: Increment Nx by 1 and repeat II until Nx = 12
V: Calculate the correlation of each model in {NLSS* (Nx, 4, 4)} VN
*
across the blind testing dataset, and assign N , the optimum number of
hidden states, according to the model with the highest correlation
133
1.000
^
0.800
0.700
0.600
0.300
0.000
2
3
4
5
6
7
8
9
10
11
12
a Train R
0.911
0.943
0.906
0.911
0.931
0.922
0.945
0.948
0.955
0.921
0.949
â– Test R
0.704
0.738
0.783
0.772
0.806
0.773
0.781
0.777
0.725
0.752
0.730
States
Figure 25: Results of hidden states search algorithm for NOx NLSS model.
Figure 25 illustrates the results of this algorithm. For the dCVModel definition,
NLSS*(6, 4, 4) produced the model with the highest correlation on the blind test set. The
algorithm was repeated for each dynamic model definition, and the results are summarized
in Table 11.
Table 11: Results of hidden states search algorithm for all dynamic models.
Model
States
Train NMSE
Test NMSE
Train R
Test R
dCVModel
6
0.090
0.221
0.906
0.783
dSVModel
9
0.120
0.300
0.879
0.695
*
Given the optimal number of hidden states Nx, the optimal number of hidden PEs in
the state evolution model can be determined as follows:
I: Set Nhx = 2
*
II: Train the NLSS(NX, Nhx, 4) 10 times
134
III: Select NLSS*(NX, Nhx, 4) as the model with the lowest crossvalidaÂ¬
tion MSE from the 10 training results
IV: Increment Nhx by 1 and repeat II until Nln =12
V: Calculate the correlation of each model in {NLSS* (Nx, Nhx, 4)} Vv,lt
*
across the blind testing dataset, and assign Nhx according to the model
with the highest correlation
1.000
Zâ€”â€”'* 'â€”* *
0.800
2
3
4
5
6
7
8
9
10
11
12
a Train R
0.871
0.898
0.894
0.901
0.931
0.958
0.977
0.988
0.975
0.966
0.984
â€”â€¢â€”Test R
0.685
0.716
0.701
0.763
0.808
0.848
0.827
0.868
0.838
0.809
0.804
State Hidden PEs
Figure 26: Results of state hidden PEs search algorithm for NOx NLSS model.
Figure 26 illustrates the results of this algorithm. For the dCVModel definition,
NLSS*(6, 9, 4) produced the model with the highest correlation on the blind test set. The
algorithm was repeated for each of the model definitions, and the results are summarized
in Table 12.
Table 12: Results of state hidden PEs search algorithm for all dynamic models.
Model
States PEs
Train NMSE
Test NMSE
Train R
Test R
dCVModel
9
0.017
0.129
0.988
0.868
dSVModel
11
0.047
0.191
0.950
0.806
135
Finally, the optimal number of hidden states Nx and hidden PEs in the state evolution
*
model Nhx, the optimal number of hidden PEs in the output observation model can be
determined as follows:
I: Set Nhy = 2
II: Train the NLSS(N*X, Nhx, Nby) 10 times
III: Select NLSS*(NX, Nhx, Nhy) as the model with the lowest crossvaliÂ¬
dation MSE from the 10 training results
IV: Increment Nh by 1 and repeat II until Nh =12
V: Calculate the correlation of each model in
* *
{NLSS*(NX, Nhx, across the blind testing dataset, and
*
assign Nhy according to the model with the highest correlation
1.000
A A * * A * A A A
' " " "
'
2
3
4
5
6
7
8
9
10
11
12
â€”aâ€”Train R
0.896
0.939
0.959
0.965
0.976
0.975
0.963
0.974
0.971
0.963
0.966
â€”â– â€”Test R
0.735
0.845
0.884
0.896
0.891
0.895
0.854
0.845
0.773
0.785
0.770
State Output PEs
Figure 27: Results of output hidden PEs search algorithm for NOx NLSS model.
Figure 27 illustrates the results of this algorithm. For the dCVModel definition,
NLSS*(6, 9, 5) produced the model with the highest correlation on the blind test set. The
136
algorithm was repeated for each of the model definitions, and the results are summarized
in Table 13.
Table 13: Results of output hidden PEs search algorithm for all dynamic models.
Model
Output PEs
Train NMSE
Test NMSE
Train R
Test R
dCVModel
5
0.030
0.094
0.965
0.896
dSVModel
4
0.070
0.135
0.926
0.864
6.8 Analysis
Figure 28 summarizes the final modeling results after architecture selection. Recall
that the objective of this section was to find the most accurate model for the six model defÂ¬
initions required to implement our four control designs. The best models identified for
each definition are:
1) SteadyState SV Model: MLP*( 15, 4)
2) SteadyState CV Model: MLP*{ 19, 7)
3) SteadyState ISV Model: MLP*(\1, 3)
4) SteadyState IMV Model: MLP*(23, 9)
5) Dynamic SV Model: GM*( 15, 4, 3, 4)
6)Dynamic CV Model: GM*( 19, 7, 5, 5)
Figure 28: Best models for all model definitions by architecture.
The next chapter will implement each control design using these models.
CHAPTER 7
CONTROLLER IMPLEMENTATIONS
This section will implement the four control designs presented in Chapter 4, using the
â€œbestâ€ reference models developed in Chapter 6, and identify the â€œbestâ€ control design for
meeting the objectives outlined in Section 4.4 "Performance Criteria." The â€œbestâ€ controlÂ¬
ler will be determined as follows:
1) Using dynamic models as simulators for the plant, controller perforÂ¬
mance is quantified offline.
2) Allowing the controllers to manipulate actual plant values, controller
performance is quantified online.
7.1 Offline Quantification
To quantify the performance of the four control designs offline, a common dataset and
plant simulator need to be selected. The dataset should not include data that was used durÂ¬
ing training of the controllers or their underlying reference models. Recall that a blind 1
week dataset was set aside during modeling for testing and that none of the models have
ever seen this dataset during training. In addition, the dynamic process models chosen for
offline quantification should not be models that were used as reference models for the any
of the control designs. Since the best models developed in Chapter 6 were used as referÂ¬
ence models, i.e., the models with the lowest crossvalidation MSE from 10 separate trainÂ¬
ings, the offline quantification will use the second best models as the dynamic process
models.
138
139
Offline quantification will apply each controller to the dynamic process models across
the test dataset, and calculate the average NOx reduction along with the average CO proÂ¬
duction above the maximum CO constraint. Formally, the average NOx percent reduction
will be reported according to
A NOx = â€”â€” Â¿ NOx*(t)NOx(t), (89)
Tf~ T't = t,
where TÂ¡ is the start of the test dataset, Tj is its end, NOx(t) is the value of NOx at time t
predicted by the dynamic process models in response to the actual MV setpoints mv(t),
and NOx*(t) is the predicted value of NOx in response to the controllers optimal MV set
points inv (t) .
Similarly, the average CO above the maximum constraint will be reported according
to
rco]
i Tf r
Tf T Â¿
J â€˜t=T:
CO*(t)CO
v 0
CO*(t) > CO
else
(90)
where CO*(t) is the value of CO at time t predicted by the dynamic process models in
response to the controllers optimal MV setpoints mv*(t), and CO"'ax is the max limit set
for CO. All of the results presented are for a CO maximum of 500ppm.
The optimal MV, SV and CV trajectories across the test dataset produced by each conÂ¬
troller are estimated by:
140
I: Set optimal MV, SV and CV trajectories to their actual values
(i) = mv(t)
sv*(i) = sv(t)
cv* (t) = cv(t)
(91)
II: For t = TÂ¡ to Tjâ€” 1
^ â– T'
i: Get the optimal MVs from the controller, mv (t)
N 5Â¡â‚¬
ii: Use the SV model to calculate the resulting SVs, sv (t + 1)
 â– A ^
iii: Use the CV model to calculate the resulting CVs, cv (t + 1)
iv: Copy forward the MVs to carry forward the process state to initialize the
next step
(92)
V: Next t
The results of applying this algorithm to each controller are illustrated below. Figure
29 presents the average NOx reduction ANOx, while Figure 30 illustrates the average CO
above its max f CO~\. These are not particularly encouraging results. While each controlÂ¬
ler did manage to reduce NOx, the reductions were quite small. Furthermore, the controlÂ¬
lers seemed to have even less effect on the CO above 500ppm. The key question at this
stage is to determine whether these results represent:
1) all of the potential NOx reductions inherent to the process,
2) a problem with the control design, or
3) a problem with the reference models for the underlying process.
141
0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
0.016
0.018
Baseline
SteadyState
MPC
SteadyState
MIC
Dynamic MPC
Dynamic
MRAC
â–¡ Avg NOx Reduction
0.000
0.013
0.016
0.009
0.010
Figure 29: Average NOx reduction over testing dataset.
To help answer this question, a second run of the offline quantification algorithm was
run. This time, however, the test models were replaced with the reference models used to
develop each controller. These results are presented in Figures 31 and 32.
Figure 30: Average CO above max over testing dataset.
142
Figure 31: Average NOx reduction over testing dataset using train and test models.
Clearly, there is a problem. The following observation can be made:
1) the controllers appear to be working fine, and
2) the reference models and the test models are providing inconsistent
knowledge about the process.
Avg CO Above Max
Figure 32: Average CO above max over testing dataset using train and test models.
143
Chapter 8 addresses these issues in more detail, but for now the quantification of the
current controllers continues.
7.2 Online Quantification
Online quantification will be restricted to measuring the controllers ability to affect the
steadystate performance of the unit. Quantifying online performance will require running
online experiments, where MVs are moved and the resulting CVs measured. Each experiÂ¬
ment will have to be of relatively short duration, since the longer an experiment takes the
less likely it is that steadystate conditions will be maintained. In order to compare the perÂ¬
formance of different controllers, which will invariable have to perform their actions
under different conditions, baseline conditions will be established prior to each experiÂ¬
ment. These baseline conditions will be stated in terms of MVs since we have no direct
control over DVs. To account for differences between the DVs between individual experiÂ¬
ments, the MVs will be returned to the baseline conditions prior to each experiment.
Each online experiment will follow the following protocol: The unit operator is asked
to bring the unit to steadystate conditions, i.e., holding all MV setpoints constant. After
the unit has reached steadystate, measurements are taken to establish baseline conditions,
2k.bciS6 ^base
mv and cv . The controller is then queried for MV setpoints, mv (t), which will
be applied by the unit operator.
New setpoints will be repeatedly queried and applied until the unit has once again
returned to steadystate. The frequency with which an operator carries out this process,
will be at their discretion. If the operator feels that individual MV setpoints can not be
made, then these setpoints are constrained and the controller is queried for another set of
144
MVs. When none of the MV setpoints can be made, or the controller has no new advice,
i.e., it has saturated against its constraints, then the unit is at steadystate and the experiÂ¬
ment is terminated.
^cxp
The CVs are once again measured to establish experiment conditions, cv . The difÂ¬
ference between the experiment and baseline conditions will be called the experiment
delta, A^p. The operators are then asked to return the MV setpoints back to their baseline
cv
Â±t)ClS6
conditions mv , and a third measurement is taken to establish validation conditions
Â±vcil
cv . The difference between the experiment and validation conditions will be called the
validation delta, A^f /. Notice that if the validation conditions match the baseline condi
CV
tions, then the experiment and validation deltas will have identical magnitude and sign.
Controllers are then compared based on their ability to affect the CVs relative to these
baseline values across multiple experiments. Key to the accuracy of these experiments
will be how these measurements are reported. The next sections outline the measurement
methodology applied during offline quantification.
7.2.1 Measurement Methodology
One of the most critical aspect to quantifying the affect that different control strategies
have on a real process, will be in determining the significance of our results relative to
process noise and changing steadystate conditions. The following outlines the statistical
measurement methodology followed in this study.
145
â€”Â±base
Each measurement, e.g. cv
, taken of the units steadystate condition for a variable
at time ta, e.g. cv(ta) , will implement the following:
I: Collect T observations, {cv(t)}tÂ°= to
II: Calculate the sample mean
tn+T
\ Z ^(0
III: Calculate the sample standard deviation
(93)
L+T
,base
 Â£ (cv(0A )
(94)
t = tn
Each delta calculated, e.g. A^p, of changes to the units steadystate condition will
implement the following:
â€”Â±base â€”*â– val
I: Given measurements for cv and cv
II: Calculate the lower bound of a (1  a) 100% confidence interval for
the difference between two sample means according to
(fÂ£aseav_?')â€” hb"e + ^' [66],
vrcv 'ey T V cv cv L J
(95)
7.2.2 Results
Given the results obtained in Section 7.1 "Offline Quantification," there seems little
hope of conducting successful online experiments. While online tests might be useful in
confirming the offline results, there are considerable costs associated with conducting
them. In addition to the time and resource requirements, there is the invaluable capital of
buyin from operations and engineering to be considered. Most of these operators and
146
engineers have spent 5 to 20 years learning how to drive and maintain the unit. They are
the experts. Bringing in a new technology that is going to show them how to better operate
their unit, needs to be managed carefully. The operators and engineers have seen many
technologies come and go and have never seen a technology able to model unit emissions
much less control them.
With all of this said, 10 experiments were conducted with the steadystate optimizer.
This was the first controller implemented, and these experiments were conducted before
all of the above results we analyzed. The results were about as encouraging as they were in
our offline analysis, and are presented in Figures 33 and 34.
0.015
0.010
0.005
0.000
0.005
0.010
0.015
0.020
0.025
0.030
1
2
3
4
5
6
7
8
9
10
â€”â™¦â€” Experiment
0.005
0.006
0.002
0.022
0.023
0.008
0.010
0.003
0.027
0.003
â€”â€¢â€” Validation
0.006
0.029
0.016
0.022
0.019
0.027
0.003
0.032
0.015
0.030
 A  Average
0.001
0.018
0.007
0.022
0.021
0.018
0.003
0.014
0.021
0.017
Experiment
Figure 33: Change in NOx for steadystate controller experiments.
Figures 33 shows the measured NOx change between the baseline and experiment,
along with the corresponding change between the experiment and validation regions. We
can see that there seems to be a slight decrease in NOx. The average NOx percent change
147
for all experiments was 4.8%. The changes were so small with respect to the steadystate
NOx variance, however, that few of the experiments proved significant.
Figures 34 shows the corresponding CO changes. Here, its hard to see any trend. The
average CO percent change for all experiments was 2.7%. Once again, these changes are
so small that few of the experiments proved significant.
Figure 34: Final CO level for steadystate controller experiments.
CHAPTER 8
PARAMETERIZATION PROBLEM
The work thus far has resulted in some unexpected results. Applying accepted neural
network modeling techniques has produced several models for a process that contain drasÂ¬
tically different â€œknowledgeâ€ about the process. Here, knowledge refers to the causeand
effect relationships between MVs and CVs, which is what each of the control designs
depend upon. Contrast this with our modeling results from Section 6.8 "Analysis," where
these same models demonstrated consistent and robust â€œknowledgeâ€ of the process, where
knowledge was considered to be the models ability to predict or forecast a blind dataset.
Also notice, that this same blind test dataset was used to perform the offline quantification
of the controllers.
This problem is analogous to the parameterization problem in classical adaptive conÂ¬
trol theory. An adaptive control system centers around the idea that a process is described
as a mathematical function with parameters. It might therefore be expected that the way
parameters are estimated is essential to the success of an adaptive controller. It is useful to
view parameter estimation in the broader context of system identification. The key eleÂ¬
ments of system identification are the selection of model architecture, experiment design,
parameter estimation, and validation. Since system identification is executed automatiÂ¬
cally in adaptive systems, it is essential to have a good understanding of all aspects of the
problem. The elements of system identification are known to be fundamental issues in
adaptive control theory.
148
149
Even though this is a problem from classical controls, there are no classical solutions
which apply to neurocontrol designs. The primary reason for this is the use of neural netÂ¬
works as process models. Neural networks are fundamentally different than linear or first
principlesbased process models in that they are nonparametric. The individual parameters
have no physical interpretation, i.e., they are meaningless coefficients in a black box. We
are therefore seeking a method to validate the parameterization of a nonparametric system.
In classical adaptive control theory, parameterization is a designtime issue that is typiÂ¬
cally dealt with analytically. We can expect that the solution for neurocontrol theory will,
like most other aspects of neuralnetworkbased system design, have to be dealt with
empirically.
8.1 Search for a Validation Metric
Letâ€™s begin our search for a methodology to validate the parameterization of neural
network models by reviewing the metrics used to assert the performance of these models.
Figure 35 presents three of these metrics for the steadystate CV MLP model. The primary
metric of model quality used in this study has been correlation. The correlation between
the actual unit NOx emission and the MLPâ€™s prediction of NOx was 0.801. This metric
indicates that the MLP model understands a considerable amount about the variation in
NOx production. Furthermore, notice that the MLP produced this prediction by observing
the MVs, DVs and SVs only, implying that the MLP understands a considerable amount
about the relationship between these variables and NOx formation.
The normalized meansquared error (NMSE) was also used to asses the worthiness of
each model. Recall that a NMSE greater or equal to 1 indicates that the model is performÂ¬
ing no better than a trivial prediction, which simply predicts the statistical mean. Here the
150
MLP predicted the blind test dataset with a NMSE of 0.298, indicating that the model is
doing considerably better than simply predicting the mean.
ssCVModel MLP
Figure 35: Summary of validation metrics for MLP CV model.
The correlation and NMSE results provided in Figure 35 are for the model with the
lowest crossvalidation MSE from a set of 10 models trained from different random initial
conditions. Also recall that the offline quantification of the controllers was performed
using the models with the second lowest crossvalidation MSE. Furthermore, remember
that we determined that the controllers preformed very well when their reference models
were used to quantify their performance. We are therefore looking for a validation metric
which would provide some insight into the difference between these two models. The corÂ¬
relation and NMSE results presented in Figure 35 cannot differentiate between these two
models, because they relate to only one of these models.
It is clear that the validation metric that we are seeking should provide an indication
about how a particular model compares with the ensemble of possible models which could
151
be developed for the process. For nonlinear models trained with a gradient descent learnÂ¬
ing algorithm, this must include some understanding about the structure of the perforÂ¬
mance surface. Figure 36 presents the correlation and NMSE over the blind test dataset for
each of the 10 training runs of the steadystate CV MLP model. Recall that each training
run starts from random initial conditions. With the exception of two training runs, which
appear to have been trapped in local minima, there is relatively consistent performance
across the 10 sample models from the ensemble of possible models. Furthermore, there is
virtually no difference between the best model and the second best model. Clearly, the
correlation and NMSE metrics are not going to provide insight into why two parameter
izations for the same controller provide significantly different results.
0.900
0.800
0.700
0.000
L
0.769  0.763 0.791  0.447 0.727
0.711 0.801 I 0.483 0.780
â–¡ NMSE  0.331  0.407 0.304 0.544 j 0.420 ; 0.371 0.298  0.575 : 0.313
0.760
0.340
Figure 36: NMSE and R for all 10 training results for MLP CV model.
In addition to correlation and NMSE, Figure 35 presented a metric labeled â€œstd err/
var.â€ The realization that we are looking for a metric capable of differentiating between a
single model and the ensemble of possible models, leads naturally to considering standard
152
errors. Recall that Tibshirani [61] proposed the â€œbootstrapâ€ method for estimating the
standard errors of a MLPâ€™s predictions. Having estimates for the standard errors will proÂ¬
vide insight into variability in the predictions between models drawn from the ensemble.
The â€œstd err/varâ€ column in Figure 35 presents the average standard error calculated using
the 10 model trainings and the â€œbootstrapâ€ method across the test dataset, divided by the
measured NOx variance across this dataset. The reason for dividing by the variance is to
provide an intuitive feel for the reported value of this metric. A standard error equal to this
variance indicates almost no confidence in the model predictions, while a standard error
significantly smaller than the variance indicates confidence that the models understand
more that the natural variation of the variable. The â€œstd err/varâ€ metric value of 0.358
reported in Figure 35 fails to differentiate our MLP models.
The problem with all of these metrics is that they assess the models ability to predict
future outputs of the process. The models are able to make accurate predictions, because
although the test dataset is blind it must contain similar process relationships, causeand
effect, to the training dataset. When our control algorithm is run across this dataset, it is
changing these relationships. There are three likely mechanisms by which this could hapÂ¬
pen:
1) The MVs and DVs are not independent. When the controller moves a
MV it assumes that the other MVs and DVs remain constant.
2) The SV models are not accurately modeling the impact of MV moves.
3) The MVs are highly correlated. Thus changing a MV in the test dataset
breaks our assumption that the correlation relationships within the trainÂ¬
ing and test datasets are the same.
It is unlikely that case 1) is the cause of our troubles. For if the MVs and DVs are not
independent, then there is no way for any of the models to have determined this. Given
153
that all of the models have been developed from the same MV and DV variable sets. If in
fact these variables are not independent, then the impact of this on our control designs
would be the same as if these variables are correlated which will be considered in case 3).
If there is a problem with the SV models, this would definitely have an impact on the
results of our controllers. When examining the validation metric for the steadystate CV
model above, we were providing it with the actual SVs as input rather than the predicted
SVs from the SV model. Recall that the later configuration is how the controller utilizes
these models. We have seen, however, that the correlation and NMSE metrics for the SV
models demonstrate that these models are able to predict the SVs over the blind test
dataset. To verify that this is not the cause of our problems, Figure 37 presents the results
for the combined SV/CV configuration for the same metrics considered in Figure 35. Here
although using the SV/CV combined model did reduce the fidelity of our ability to predict
the CVs, the degradation in performance does not justify the poor performance of our conÂ¬
troller.
ssSVModel / ssCVModel
Figure 37: Summary of validation metrics for combined SV/CV model.
154
Case 3) is a valid cause for our parameterization problem. When modeling an indusÂ¬
trial process, correlated variables are going to be a fact of life. Unfortunately, the available
tools in the literature for dealing with such situations are very limited. The following secÂ¬
tions will demonstrate that correlation has a tremendous impact on the design of model
based controllers.
8.2 Correlation Paradox
Correlated variables are a natural phenomenon, as two variables are correlated when
they are related through some physical laws or process. In fact, the mission statement for
the empirical investigator is to infer, or learn, these physical laws from observations of
process data. Hence, correlation is a doubleedged sword. Without it learning is not possiÂ¬
ble as there is nothing to infer, while unanticipated correlation makes it difficult, if not
impossible, to understand what has been learned or inferred.
Industrial process plants, through centralized control, systematically correlate their
process variables. The plantâ€™s distributed control system (DCS) maintains a large number
(thousands to tens of thousands) of feedforward and feedback control loops, from a relaÂ¬
tively small number (tens to hundreds) of operator setpoints. The DCS is designed to batch
control over as many subsystems as possible, from the smallest possible number of operaÂ¬
tor setpoints. If it were feasible the operator would only have a single setpoint called
demand. As a rule, most variables within an industrial plant will be highly correlated to
plant demand.
The effects of correlation have been well document for modeling applications like sysÂ¬
tem identification [56] and regression [53], The most significant attribute that these appliÂ¬
cations have in common is that they deal with systems that are either linear, or have a
155
relatively simple parametric nonlinear form. This feature allows an investigator to apply
analytically assess the significance of model parameters. In fact, the correlation effects are
automatically accounted for in significance testing. Thus the validation required to assess
the parameterization of a model is accomplished through significance testing.
To illustrate the effect of correlation, consider the physical processing plant that proÂ¬
duces output y from inputs u x and u2. An investigator building a model for this plant is
presented with data for y, u Â¡ and u2, without any information about the physical system.
The investigator hypothesizes that the data was generated according the regression model
y = PjÂ«i + P2w2 + rlâ€™ (96)
where P, and P2 are unknown coefficients, and r is a zeromean uncorrelated disturÂ¬
bance term. Applying the method of least squares (LS), Ramanathan [53] shows that the
corresponding normal equations are given by
PlZ"l + PlZWlW2 = 2>!
(97)
Pll>lâ€œ2 + Pll>2 = YayUl
(98)
with solutions
= Yyu\Yu2'Zyu2Yu\u2
XmÃZm2(X"im2)2
p = Yyu2TuZ\Yyu\Yu\u2
5>iXM2~(I>1W2)2
(99)
(100)
156
and variances
Var(P,) = f (101)
2"l(* ~r)
2
Var( p2) = f (102)
Xm2(! ~r)
2
Cov(\3â€ž p2) =  7"a '' , (103)
vZMiZw2(1~r)
2
where aâ€œ = Far(r) and r is the correlation coefficient between ux and u2.
Suppose ux and u2 are highly correlated, r is near Â±1. It is evident from equations
(101) and (102) that the variances, and hence the standard errors of p, and p? will be
2
very large when r is close to 1. A large variance means poor precision and a low student
tstatistic, which results in insignificance. In addition, we can see from equation (103) that
the covariance between the regression coefficients will be very large, in absolute value. If
the estimates are correlated, each coefficient is capturing part of the effect of the other
variable and hence it is difficult to obtain the separate effects of w, and u2 on y. In other
words, we cannot hold u2 constant and increase u, alone, because u2 being correlated
with u j, will also change as a result. Or viseversa, since correlation is not cause.
Ramanathan [53] offers the following properties of models derived from correlated
input variables:
1) If two or more explanatory variables in the multiple input (MI) model
are exactly linearly related, then the model cannot be estimated.
157
2) If some explanatory variables are nearly linearly related, then OLS and
MLE and hence are unbiased, efficient, and consistent.
3) The effect of correlation among input variables is to increase the stanÂ¬
dard errors of the parameters and reduce the tstatistics, thus making
these parameters less significant (and possibly even insignificant). The
tests of hypotheses are, however, valid.
4) The covariance between the parameters of a pair of highly correlated
variables will be very high, in absolute value, thus making it difficult to
interpret individual coefficients.
5) Correlation may not affect the forecasting performance of a model and
may possibly even improve it.
8.2.1 Effects of Correlation on Neural Network Modeling
The neural network literature is not without reference to the issues of correlated inputs,
but the number of references are disproportionately low. These issues have not received
the attention within the neural network community that they have in related modeling
communities like system identification. The easy answer to why is that neural networks
are highly nonlinear structures making investigations into relevant statistics difficult if not
impossible. A more thorough understanding, however, lies in the way in which neural netÂ¬
works have been applied to date. The author offers the following observations:
1) Descent based learning will always arrive at a solutions, regardless of
the degree of correlation within the input space. The solution, however,
is rarely unique nor globally optimal.
2) The vast majority of applications for neural networks, rely purely on
their ability to forecast. For this reason, standard errors for a networks
predictions has been a recent topic of research.
3) If the correlation relationships in the input space of all testing datasets
are identical to those in the training dataset, then the correlated inputs
will not degrade the networkâ€™s predictions. This situation, which will
often be the case for prediction applications, is unlikely in control appliÂ¬
cations since the controller will independently move these inputs.
158
4) The network coefficients, its weights, have no parametric interpretation.
Their has been little reason, even if we knew how to, to calculate their
standard errors and significance.
Because of the way in which neural networks have been applied, the vast majority of
work done on the issue of correlated inputs deals almost entirely with its effect on the
dynamics of learning. Correlation in the input space drastically reduces the rate of converÂ¬
gence for descentbased learning algorithms [22], To counter this effect, the most comÂ¬
mon neural network solution to correlation is to projection the input spaces to a lower
dimensional subspace with orthogonal bases. Most of these projection operators are linear
and based on the energy or information content of the variables. The most common such
projection is principle component analysis. The result of using such a projection is to preÂ¬
cede the neural network with a simple matrix multiply. Thus the actual inputs to the neural
network are completely, or nearly completely uncorrelated features and thus accelerate the
convergence of the learning algorithm.
Although very useful, these techniques do not provide a solution to the parameterizaÂ¬
tion problem. These techniques can be used to improve the consistency of the neural netÂ¬
work portion of the model, but do not improve the consistency of the combined models
containing the transform stage with the neural network. This point may be confusing now,
but the following sections should help to clarify it.
In summary, neural networks have found an applications niche where this robust preÂ¬
dictor has demonstrated the ability to out forecast more traditional methods. One of the
problems with moving neural networks from an academic interest to an accepted modelÂ¬
ing methodology is the lack of standardized reporting. The vast majority of neural network
applications to date apply the inferences of a model without knowledge of their statistical
159
significance. Recent work in standard errors for these predictions have taken an important
step towards solving part of this problem [21].
8.2.2 Effects of Multicollinearity on ModelBased Control
Each of the controllers presented in Chapter 4 belong to the general family of model
based control, i.e., they each explicitly use a reference model during offline training or
online control. The process knowledge provided by the model in each of these control
designs is
dev
dmv
(104)
where the vectors cv and mv are the CVs and MVs, respectively. Recall our example in
Section 8.2 "Correlation Paradox," where the model was given by
y = Mi + P2M2 + rl
(105)
If this model was used as a reference model for a modelbased control scheme,
cv = {y} and mv = {ux,u2} could be considered to be the CVs and MVs, respecÂ¬
tively. Hence the process knowledge provided by our model is the set {p Â¡, P2}. But as we
have seen when the MVs and u2 are highly correlated, the standard errors for P, and
P? will be very large. A situation that will quickly render our process knowledge insignifÂ¬
icant. Clearly, this will have a tremendous impact on the performance and robustness of
our controller.
160
8.3 Validation Metric
Neural networks do not have the same convenient interpretation for their coefficients.
They do, however, infer relationships between the MVs and CVs, which is used directly
by the controllers. The method of extracting this relationship from a neural network
model, is commonly referred to as sensitivity analysis. The MV sensitivities dcv/dmv can
be calculated directly using backpropagation. These sensitivities will be a function of the
unitâ€™s operating state {mv, dv}. Recall that we are trying to get a feel for how these sensiÂ¬
tivities vary across the ensemble of possible models. Figure 38 illustrates the MV sensitivÂ¬
ities for a typical operating state across the 10 steadystate MLP models, representing a
sampling from this ensemble.
Figure 38: Sensitivity results for all 10 training results for NOx CV model.
For the first time, we can clearly see our parameterization problem! Each model is proÂ¬
viding significantly different process knowledge to the controller. More explicitly, our
best and secondbest models with respect to NMSE, R and SE, represented by training
161
runs number 3 and 7, look as if they are modeling two completely different processes.
Alas thereâ€™s hope.
Figure 38 provides the average sensitivity reported from the ensemble of models,
along with the corresponding 95% confidence intervals. This result demonstrates that we
are not able to assert the directional sensitivity with any degree of confidence. It is easy to
see why the controllers had so much trouble.
Figure 39: NOx CV model sensitivity with 95% confidence intervals.
8.4 Revised Representation Pruning Algorithm
Recall that the goal of the representation pruning algorithm is to identify the best variÂ¬
able representations within each group. In Chapter 6, the â€œbestâ€ variable representation
was determined by selecting the representation which produced a model with the highest
correlation on a blind test dataset. In hindsight, it is clear that the definition of "best" must
be applicationspecific. If the models were being applied to a forecasting application, then
the representation pruning algorithm would have suited our purposes. For control applica
162
tions, however, it is clear that the definition of "best" will have to be modified. Note that
the robustness of the predictors produced by this algorithm should still be in doubt. If the
forecasts are always made for a dataset with identical correlations characteristics to the
training dataset, then the modelâ€™s forecast would be fine. If, however, the correlation is
less physical, then there is still cause for concern.
8.4.1 Input Sensitivity Standard Errors
Recall the â€œbootstrapâ€ approach to calculating the prediction standard errors for a neuÂ¬
ral network presented in Section 2.2.4.0.1 "Bootstrap methods." The validation metric to
differentiate models to be used for our modelbased control designs will similarly need to
evaluate the standard error of the MV sensitivities. The following algorithm has been
developed to approximate the MV standard errors using a â€œbootstrapâ€ methodology:
I: Generate N8 datasets, each one of size NÂ° drawn with replacement
II: For each bootstrap dataset b e [ 1, iv ], find
ArgMin$h{J(dbp(ub, TvÂ¿))}
(106)
III: Estimate the standard error of the i th input sensitivity as
(107)
where
(108)
163
Conceptually, this algorithm simply calculates the variance of the input sensitivities
across multiple models, all trained using different random initial conditions and indepenÂ¬
dently sampled â€œbootstrapâ€ datasets. Clearly, the variance of the sensitivity calculations
presented in Figure 38 will be high. If an algorithm can be found that is capable of reducÂ¬
ing these standard errors, the offline controller quantification results should improve.
8.4.2 Algorithm
The revised representation pruning algorithm uses the standard error estimates for the
modelâ€™s input sensitivity across a dataset, as presented in Section 8.4.1 "Input Sensitivity
Standard Errors," to determine â€œbestâ€ as follows:
I: Perform the group selection pruning algorithm to the Essential Tag
List in the appendix
II: Select the first MV group in the reduced Essential Tag List
III: If there is only one unique â€œrepresentationâ€ in the group then select the
next MV group and goto step III
IV: For each unique representation in the selected group, train a
MLP(15,5) 30 times with the variables for this representation as the
only input variables from the selected group; the 30 models resulting
from each training will be called â€œrepresentationâ€ model set
V: Calculate the standard error for the input sensitivities for each MV,
VI: Calculate the average normalized input sensitivity standard error for
each â€œrepresentationâ€ model set according to
(109)
164
VII: Select the best â€œrepresentationâ€ model set as the set with the lowest
input sensitivity normalized average error, and remove all variables
associated with every â€œrepresentationâ€ model set except for the best
â€œrepresentationâ€ model set
VIITSelect the next group and goto step III
Figure 40: Results of revised representation pruning algorithm.
The results after running the revised pruning algorithm for each of the 4 groups are
illustrated in Figure 40. Notice that unlike the initial pruning algorithm, this algorithm sigÂ¬
nificantly degraded the fidelity of the model as it pruned each representation group. The
variable selections after the revised representation pruning algorithms are provided in
Table 14.
165
Table 14: Final variable selections after revised pruning.
Manipulated Variables
Disturbance Variables
State Variables
Control Variables
1/3 OFA Damper Bias
Ambient Air Press
Sec AirTemp Side A
CEM NOx
2/3 OFA Damper Bias
Ambient Air Ternp
Sec Air Temp Side B
FD Fan Bias
Bnr Atm Stm Press
CEM CO
" 1
02 Trim
Bnr Atm Stm Tern p
Generated M W
GR Fan 2A Inlet Dmpr Bias
CEM Barometric Pressure
W indbox Pressure
GR Fan 2B Inlet Dmpr Bias
Cond Back Pres  Side A
GR Fan HpprDmprA Bias
Cond Back Pres  Side B
GR Fan HpprDmprB Bias
Fuel Gas Flow Indication
Fuel Oil Flow Indication
Fuel Temp Fired
Furnace Pressure
8.4.3 Variable Representation
The results of the revised representation pruning algorithm are very interesting, and
provide some lessons about variable representation choices. For each MV group, the
revised algorithm chose the operator bias, rather than the positions selected by the initial
algorithm. Recall the example given in Section 6.2.1.5 "Variable representation," where 5
tags were identified that represented the single process variable of gross airflow. As we
have seen, this manytoone mapping between tags and variables is very common in
industrial plants. The tags selected for variable representation should have the following
characteristics:
â€¢ Uncorrelated: The tag or tags chosen to represent the variable should be uncorreÂ¬
lated with each other and with tags chosen to represent the other input variables.
â€¢ Representative: The tags chosen should provide a complete representation of the
dynamics of the variable, i.e., they should be as correlated as possible to the proÂ¬
cess variable.
â€¢ Dynamic: The tags chosen should contain as much of the real variation in the
process variable as possible. If the variable is changing for any reason, even if
the variation if not intended or not desirable, this information should be repre
166
sented in the chosen tags.
From the discussion above, it is easy to see that all 5 tags, except FD fan trim, will be
highly correlated to plant demand. As noted previously, most tags will be correlated with
plant demand. This would make these four tags a poor choice for representing FD fan airÂ¬
flow. FD fan trim, on the other hand, is not correlated to demand. The trim is a bias tag
that only moves when the operator wishes to alter the fuelair mixture, represented by FD
fan demand, that was designed into the DCS. The bias is also appealing from an optimizaÂ¬
tion perspective, since it provides a way to tune control over the process without having to
control the process.
Turning our attention to the second and third criteria for variable representation above.
The FD fan trim is certainly correlated to gross airflow, since moving it will alter the FD
fan setpoint. Dynamic, however, the FD fan trim is not. The only time the trim moves is
when the operator touches it. For the unit considered in this study, the trim was rarely
touched1. In addition, the trim does not contain any of the natural or unintended variability
in airflow, e.g. the slack in the PID controller. This variability is real, in the sense that airÂ¬
flow actually changed, and has an impact on the combustion process. Although not
intended, this variation provides rich data for learning.
So we can either choose a representation rich with dynamic information about the proÂ¬
cess but highly correlated to demand, or a representation completely uncorrelated to
demand that is rarely moved. Pruning the variable representations based on the fidelity of
the resulting model will select tags rich in dynamic information regardless of correlation:
1. This is an interesting fact, since FD fan trim turned out to be one of the signifiÂ¬
cant levers over optimizing NOx.
167
pruning based on standard errors of the knowledge extracted about the causeandeffect
relationships will prioritize independent tags. The answer to this dilemma identified in this
work requires two solutions: 1) with sufficient parametric testing, structured movement of
the MVs, the models will be able to extract enough knowledge about the process for the
controller to function properly; and 2) when the controllers are connected to the plant, they
will move the MVs and model retuning will continue to extract more knowledge about the
process.
Figure 41 illustrates the MV sensitivities for the same operating state illustrated in FigÂ¬
ure 38, across the first 10 of the 30 revised steadystate MLP models.
1.200
1.000
0.800
0.600
0.400
0.200
0.000
0.200
0.400
0.600
0.800
Figure 41: Sensitivity results for all 10 training results for revised NOx CV model.
There is a visible difference in the significance, i.e., our confidence, in these models as
compared to original models. These models have extracted consistent process knowledge
with respect to the relationships between the MVs and CVs. A comparison between the
168
standard error calculations between these models and the initial models is presented in
Figure 42. This result clearly validates our observations about Figure 41.
1.000
0.800
0.600
0.400
0.200
0.000
0.200
0.400
0.600
1/3 OFA
Damper
Bias
2/3 OFA
Damper
Bias
FD Fan
Bias
02 Trim
GR Fan
2A Inlet
Dmpr Bias
GR Fan
2B Inlet
Dmpr Bias
GR Fan
Hppr Dmpr
A Bias
GR Fan
Hppr Dmpr
B Bias
â–¡ Before
0.229
0.398
0.084
0.824
0.030
0.276
0.123
0.126
â– After
0.053
0.025
0.031
0.071
0.014
0.018
0.040
0.092
Figure 42: Revised NOx CV model sensitivity with 95% confidence intervals
8.5 Modeling
Figure 43 presents the results of training each of the best model architectures from
Chapter 6 with the new variable definitions presented in Table 14. Here, each model archiÂ¬
tecture was trained 10 times, the training result with the lowest crossvalidation error was
selected, and the selected result was tested against the blind test set. Notice that this proÂ¬
cess is equivalent to the training process for the previous models.
Comparing Figure 43 with Figure 28, the new models have lost some fidelity with
respect to their ability to predict a blind test dataset. The degradation in fidelity, however,
is quite small when compared with the increase in confidence with respect to the MV senÂ¬
sitivities that the new models have (Figure 42).
169
0.000
MLP
TDNN
GM
NLSS
â–¡ CV Model
0.753
0.818
0.880
0.851
â– SV Model
0.709
0.733
0.845
0.820
â–¡ ISV Model
0.687
â–¡ IMV Model
0.607
Figure 43: Best revised models for all model definitions by architecture
8.6 Control Implementation
The assumption is that models with higher confidence in the MV sensitivities will proÂ¬
vide better reference models for our modelbased control designs. Each control design will
now be implemented with these new models and their performance requantified.
8.6.1 Offline Quantification
The offline performance for the new controller will follow the same methodology preÂ¬
sented in Section 7.1 "Offline Quantification." Once again, the dynamic process models
chosen for offline quantification will not be the reference models used to implement any
of the control designs. In particular, the offline quantification will use the second best
model, with respect to crossvalidation MSE, as the dynamic process models.
Once again, offline quantification will apply each controller to the dynamic process
models across the test dataset and calculate the average NOx reduction along with the
amount of CO production above the maximum CO constraint. The NOx and CO results
are presented in Figures 44 and 44, respectively.
170
Avg NOx Reduction
Baseline
SteadyState
M PC
SteadyS tate
MIC
Dynamic MPC
Dynamic MRAC
â–¡ Before
0
0.01339
0.01556
0.009
0.01035
â– After
0
0.111
0.054
0.116
0.107
Figure 44: Average NOx reduction over testing dataset using old and revised models.
Avg CO Above Max
140
120
100
80
60
40
20
0
â–¡ Before
â– After
Figure 45: Average CO above max over testing dataset using old and revised models.
Recall that it was at this point in developing the controllers the first time that the probÂ¬
lem first surfaced. Clearly, the new controllers are performing much better than the origiÂ¬
nal ones, whose results have been included in Figures 44 and 45 for reference.
171
8.6.2 Online Quantification
The offline quantification first identified a problem with the control designs, and now
indicates that it has been solved. The proof, however, is always in the pudding, or at least
in the response of the actual plant. It stands to reason, that if we were not able to develop
models with consistent knowledge of the process, then there is little hope that the resulting
controllers would be able to improve the process. Consistent models of the process, howÂ¬
ever, do not necessarily imply that the resulting controllers will be able to improve the
process.
To quantify the ability of the controllers to improve the process, the methodology outÂ¬
lined in Section 7.2 "Online Quantification" is applied. Once again, 10 experiments were
conducted with the steadystate optimizer. The results this time, however, are very encourÂ¬
aging.
Figure 46 shows the measured NOx change between the baseline and experiment,
along with the corresponding change between the experiment and validation regions.
There is a significant decrease in NOx. The average NOx percent change for all experiÂ¬
ments is 24.83%.
Figure 47 shows the corresponding CO changes. Here, the controller was able to have
a significant impact on CO. The average CO for both baseline and experiment is approxiÂ¬
mately 500ppm, but the variance of the baseline data is 121.576ppm while the experiment
standard deviation is 13.473.
172
Figure 46: Change in NOx for revised steadystate controller experiments.
800
700
600
500
400
300
200
100
0
1
2
3
4
5
6
7
8
9
10
* Baseline
570
559
301
583
656
508
566
423
653
349
â™¦ Experiment
482
511
494
482
509
513
507
483
497
513
â€”â€¢â€”Validation
487
583
212
640
648
470
534
438
715
407
Experiment
Figure 47: Final CO level for revised steadystate controller experiments.
This offline quantification procedure was repeated for each of the four control designs,
and the NOx and CO results are summarized in Figures 48 and 49. Included in this figures
are the estimated results that each controller thought it would achieve across the same
173
dataset, this was accomplished by using a simulator for the plant as described in Section
7.1 "Offline Quantification."
Figure 48: Average percent NOx reduction for 10 online experiments.
There are some interesting observations about these results:
1) The performance of the MIC controller was substantially poorer than
the other control designs in both offline and online analysis.
2) There online performance of the steadystate optimizer was better than
either of the dynamic controllers.
3) The dynamic controllers performed significantly better in offline analyÂ¬
sis, that they did for online analysis.
The poor performance of the MIC controller can be explained by its treatment of conÂ¬
straints. Recall that the MV constraints were treated as penalty functions for the online
optimizer, and that the CO maximum constraint was addressed by placing CO in the CV
set and fixing it to 500ppm. This leaves only a single degree of freedom for the online
optimizer, which is the single CV of NOx. The problem is that when the online optimizer
starts moving NOx to meet its objective function, it will have to stop as soon as one of the
174
MV constraints exercise their penalty functions. There are no other degree of freedom for
the optimizer to explore, other than to simply stop moving NOx.
Notice that although penalty function are also employed in the steadystate optimizer,
this problem does not exist. Here, the online optimizer has N1"' degrees of freedom, the
number of MVs, so that when a MV constraintâ€™s penalty function is exercised it can simÂ¬
ply stop moving that MV. Furthermore, when a SV or CV constraintâ€™s penalty function is
exercised, the optimizer can explore other combinations of MVs to maintain this conÂ¬
straint while still trying to lower NOx.
Figure 49: Average percent CO reduction above 500ppm for 10 online experiments.
The observation that the steadystate optimizer outperformed the dynamic controllers
is most likely a function of the online test procedure. Recall that the controllers were
deployed online in an advisory mode. Here, the controllers optimal MV setpoints are forÂ¬
warded to an operator, who is then responsible for manipulating the actual plant setpoints.
One has to remember that the operators are responsible for the unit, which is worth several
175
hundred million dollars. Operators operate the unit based on years of experience and
dogma. Sorting out which is which is not an easy task, and invariably the operator will and
should make the final call about whether the requested setpoints are safe. The result was
that MV setpoints were not made on regular intervals, and what the optimizer recomÂ¬
mended as simultaneous MV moves might have been implemented one at a time over a
period as long as 20 minutes. Furthermore, recall that the measurement methodology, preÂ¬
sented in Section 7.2.1 "Measurement Methodology," required operating the unit under
steadystate conditions. Clearly, any dynamic advantage that these controllers might have
had was probably lost.
The observation that "the dynamic controllers performed significantly better in offline
analysis, that they did for online analysis" can also be chalked up to the online test proceÂ¬
dure. In fact, this result reinforces this conclusion. The fact that the offline analysis demÂ¬
onstrated a significant benefit to using dynamic controllers, provides incentive to continue
work toward a closedloop configuration.
CHAPTER 9
CONCLUSION
This project developed four neurocontrollers for the complex industrial process of
NOx formation. All the these neurocontrollers demonstrated benefit in an applications
area where traditional control designs have proven ineffective [39]. The first conclusion of
this study is that neurocontrollers are able to deal with highly complex industrial process
applications where more traditional methods have not been successful.
The objectives of this study went beyond demonstrating that a neurocontroller could
be developed, however. This work demonstrated neurocontrol designs that
1) are straightforward to implement,
2) account for dependent internal process states, and
3) can deal with correlated process variables.
9.1 Contributions
9.1.1 ApplicationBased Neurocontrol Implementation Methodology
A neurocontrol implementation methodology was developed whereby a process engiÂ¬
neer with reasonable knowledge about the process variables can develop advanced neuroÂ¬
controllers. The process engineer was able to simply classify process variables as MVs,
DVs, SVs and CVs, and the methodology was able to automate the development of each
stage of the controller. A set of supporting algorithms was proposed, implemented and
validated.
176
177
9.1.2 StateSpace Neurocontrol Designs
Several neurocontrol designs were implemented ranging from purely input/output to
full statespace. Consistent with the literature on neural network modeling [31], the full
statespace architecture was difficult to train and was thus not the best performer. The best
performer, however, was a partial statespace model. This work demonstrated that availÂ¬
able process knowledge can be used to create partial statespace models which can signifÂ¬
icantly improve the overall controller performance.
9.1.3 Methods for Dealing with Correlation
The primary limitations of neurocontrol identified during this study are directly related
to problems that arise when modeling with correlated input variables. In fact, the overÂ¬
whelming conclusion of this work is that correlation in the input space is the single most
important factor when designing a inductive modelbased control system. Decisions over
static vs. dynamic modeling, singlestage vs. multistage optimization, model topology,
controller type and every other seemingly monumental decision will prove irrelevant if
correlation issues are not properly addressed.
A new metric and methodology for variable selection were proposed, developed and
validated as a viable solution to correlation issues for industrial applications. This soluÂ¬
tions is a data mining approach for applications where many representations are available
of the same underlying process variables. This approach will not work for applications
where multiple choices for a process variable are not available, and the author proposes
using this new metric as an objective for the learning rule as an extension to this work.
178
9.1.4 Accurate Combustion Models
Neural networks have found an applications niche, where this robust predictor has
demonstrated the ability to out forecast more traditional methods. One of the problems
with moving neural networks from an academic interest to an accepted modeling methodÂ¬
ology is the lack of standardized reporting. The vast majority of neural network applicaÂ¬
tions to date, apply the inferences of a model without knowledge of their statistical
significance. The lack of standardized reporting has not clouded the success of neural netÂ¬
work because most of these applications rely solely on the modelâ€™s ability to forecast,
without regard for what relationships the model has inferred from the underlying process.
As clearly demonstrated in this work, the same cannot be said for neurocontrol, howÂ¬
ever. The results presented here raise significant questions about the metrics being used to
evaluate model performance in the field of neural networks. Metrics which evaluate the
predictions made by a model do not imply that the model has learning the fundamental
causeandeffect relationships within the process. A new metric was proposed and demonÂ¬
strated to overcome this limitation.
9.1.5 Novel Combustion Controller
An online NOx optimizer was developed and evaluated. This optimizer demonstrated
a consistent 45% reduction in the overall NOx emissions from the power plant.
9.2 Afterword
The benefits demonstrated in this study at the Canal Electric generating station have
been maintained for more than two years. Since these initial results, neurocontrollers have
179
been deployed on 12 generating units throughout the United States. In addition to being
installed on more units, the neurocontrollers have been extended with
1) closedloop implementations,
2) a software program that automates the neurocontrol design methodoloÂ¬
gies,
3) online automation of the variable and architecture pruning methodoloÂ¬
gies using evolutionary computing techniques, and
9.3 Future Direction
The author offers three important directions for extending this work:
1) development of learning algorithms that explicitly account for MV senÂ¬
sitivity standard errors,
2) algorithms to support automatic determination of variable type, and
3) online adaptation of the modelreference adaptive controller using reenÂ¬
forcement learning strategies.
APPENDIX
Table 15: Essensial tag list.
Descriptor
Type
Unit 2 1/3 OFA Master Bias
MV
Unit 2 1/3 OFA Master Out
MV
Unit 2 2/3 OFA Master Bias
MV
Unit 2 2/3 OFA Master Out
MV
APFI 2A Air In Temp
MV
APH 2B Air In Temp
MV
Fuel Temp Fired
MV
Boiler Efficiency (abbr. Losses)
CV
Bnr Atm Stm Press
MV
Exit 02 BSide
SV
Exit 02 ASide
SV
Flue Gas CO BSide
SV
Flue Gas CO ASide
SV
Generated MW
SV
PAS3A1 Pos
MV
PAS2B1 Pos
MV
PAS2A1 Pos
MV
PAS 1B1 Pos
MV
PAS 1A1 Pos
MV
PAS6B1 Pos
MV
PAS 6A1 Pos
MV
PAS5B1 Pos
MV
PAS 5A1 Pos
MV
PAS4B1 Pos
MV
PAS 4A1 Pos
MV
PAS3B1 Pos
MV
PAS8B1 Pos
MV
PAS 8A1 Pos
MV
PAS7B1 Pos
MV
PAS 7A1 Pos
MV
U2 PAS Mast Gas Bias
MV
U2CEM CO
SV
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IDF 2AINL VANE
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Descriptor
Type
PAS 3A1 Bias
MV
PAS2B1 Bias
MV
PAS 2A1 Bias
MV
PAS 1 B1 Bias
MV
PAS 1 A1 Bias
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PAS 6B1 Bias
MV
PAS 6A1 Bias
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PAS5B1 Bias
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PAS 5A1 Bias
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PAS4B1 Bias
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PAS 4A1 Bias
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PAS 3B1 Bias
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PAS 8B1 Bias
MV
PAS 8A1 Bias
MV
PAS7B1 Bias
MV
PAS 7A1 Bias
MV
Heat Rate
CV
GR Fan Hppr Dmpr B Pos
MV
GR Fan 2B Inlet Dmpr Pos
MV
GR Fan 2A Inlet Dmpr Pos
MV
Relative Efficiency
CV
Windbox Pressure
SV
CEM Barometric Pressure
SV
Unit 2 FD Fan Bias
MV
Unit 2 Heat Rate
CV
CEM Nox
CV
Boiler 02 Trim Unit 2
MV
CEM Opacity Unit 2
CV
CEM Stack Temp
SV
AIR PREHTR 2A GAS OUT TEMP
SV
AIR PREHTR 2B GAS OUT TEMP
SV
Unit 2 SAS Master
SV
Unit 2 SAS Master Bias
SV
Unit 2 PA Shroud Mast 0
MV
Unit 2 PAS Master Bias
MV
U2 PA Shroud Mast Gas
MV
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BIOGRAPHICAL SKETCH
Wesley Curtis Lefebvre was bom in Athens, Georgia, August 28, 1965. Curt gradÂ¬
uated from Grandville High School in Grandville, MI, and continued his education at
Michigan State University. During his freshman year at Michigan State, Curt accepted a
cooperative education position with International Business Machines (IBM) in Owego,
NY. Curt continued his education, splitting each year evenly between Michigan State and
IBM. After several years Curt left Michigan State to finish his undergraduate education at
the State University of New York at Binghamton year round while maintaining fulltime
employment with IBM.
After receiving a bachelor's degree in electrical engineering, Curt accepted a posiÂ¬
tion at IBMâ€™s research center in Yorktown Heights, NY where he was first exposed to the
field of neural networks. Curt decided to pursue graduate studies in neural networks, and
chose to attend the University of Florida in order to study under Dr. Jose Principe, one of
the first investigators of dynamic neural network theory. He graduated with a master of
engineering in electrical engineering, and his master's thesis was titled "An ObjectOriÂ¬
ented Implementation of Artificial Neural Networks."
Curt began working on his doctoral degree while founding NeuroDimension, Inc.,
a software company based on the work from his masterâ€™s thesis. The case study for his
doctoral work led to the founding of a second company in the Boston area called NeuCo,
Inc. Curt is currently the President and CEO of NeuCo where he will continue to work in
the field of neural networks and neurocontrol after receiving his Ph.D. in May 2000.
187
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