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CGRASP APPLICATION TO THE ECONOMIC DISPATCH PROBLEM By INGRIDA RADZIUKYNIENE A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2010 @ 2010 Ingrida Radziukyniene I dedicate this to my wonderful son, Matas ACKNOWLEDGMENTS I am grateful to many people for supporting me throughout my graduate study in United States. First of all, I would like to express my earnest gratitude to my advisor, Dr. Panos M. Pardalos, for directing this study and reading previous drafts of this work. Without his guidance, inspiration, and support throughout the course of my research, this work would not be complete. Many thanks to Arturas who has been there for me, listening to me and supporting me. I am also thankful to my friends at the Center for Applied Optimization who mentally supported and made my student life more colorful. TABLE OF CONTENTS ACKNOW LEDGMENTS ............................ LIST O F TABLES . . LIST O F FIG URES . . ABSTRACT .................................. CHAPTER 1 INTRO DUCTIO N .. .. .. .. .. .. .. 1.1 M otivation . . 1.2 Literature Overview ........................ 2 ECONOMIC DISPATCH (ED) PROBLEM ............... 2.1 ED C onstraints . . 2.1.1 LoadGeneration Balance ................. 2.1.2 Generation Capacity Constraint .............. 2.1.3 Generating Unit Ramp Rate Limits ............ 2.1.4 Reserve Contribution ................ ... 2.1.5 System Spinning Reserve Requirement ......... 2.1.6 Tieline Limits .. ............... ..... 2.1.7 Prohibited Zone ................ ..... 2.2 Objective Functions .. ..................... 2.2.1 Smooth Cost Function .. ................ 2.2.2 Nonsmooth Cost Functions with Valvepoint Effects .. 2.2.3 Nonsmooth Cost Functions with Multiple Fuels ..... 2.2.4 Nonsmooth Cost Functions with ValvePoint Effects and Fuels ........................... 2.2.5 Em mission Function .. .................. page . 4 Multiple 3 SOLUTION METHODS .. ........................... 3.1 Continuous Greedy Randomized Adaptive Search Procedure (CGRASP) 3.2 Genetic Algorithms (GA) .......................... 3.3 Simulated Annealing (SA) ........................ 3.4 Constraints Handling .............................. 3.4.1 PenaltyBased Approach .. .................... 3.4.2 Heuristic Strategy ......................... 4 EXPERIMENTS AND RESULTS .. ...................... 4.1 Experim ents . . . 4.1.1 System 1 . 4.1.2 System 2 .................... ........... 33 4.1.3 System 3 .................... ........... 35 4.1.4 System 4 .................... ........... 35 4.1.5 System 5 .................... ........... 37 4.2 Results ........................ ............. 37 4.2.1 Case 1 ................... ............. 38 4.2.2 Case 2 ................... ............ 38 4.2.3 Case 3 ................... ............ 40 4.2.4 Case 4 .................. ............ 41 5 CO NCLUSIO N .. .. .. . .. .. 43 REFERENCES ...................................... 44 BIOGRAPHICAL SKETCH .................... ........... 50 LIST OF TABLES Table 41 42 43 44 45 46 47 48 49 410 411 412 413 414 415 Generation costs for 13unit system with demand 1800 MW Generation costs for 40unit system with demand 10500 MV Best solution for case 4 .................... Generating units characteristics of fiveunit system . Load demand ................ ......... Generating units characteristics of sixunit system . Rumpup limits and prohibited zones of sixunit system . Generating units characteristics of 13unit system . Generating units characteristics of 40unit system . Generating units characteristics of 10unit system . Load demand for 24 hours . . Generation costs for case 1 .................. Best solution for case 1 . . Best solutions for case 2 ... ... Best results, when demand is1263 MW . & page 33 34 34 35 35 3 6 37 37 38 38 39 39 4 0 4 0 V ........... 41 41 LIST OF FIGURES Figure page 21 Example of cost function with two prohibited operating zones ... 19 22 Cost function with valvepoint effects ..... .... .. 21 23 Cost function with multiple fuels ... 22 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science CGRASP APPLICATION TO THE ECONOMIC DISPATCH PROBLEM By Ingrida Radziukyniene August 2010 Chair: Panos M. Pardalos Major: Industrial and Systems Engineering Economic dispatch plays an important role in power system operations, which is a complicated nonlinear constrained optimization problem. It has nonsmooth and nonconvex characteristic when generation unit valvepoint effects are taken into account. This work adopts the CGRASP algorithm to solve differently formulated economic dispatch problems. The comparison of the feasibility and effectiveness of the CGRASP, SA and GA is given as well. CHAPTER 1 INTRODUCTION 1.1 Motivation The economic dispatch (ED) optimization problem is one of the fundamental issues in power systems to obtain optimal benefits with the stability, reliability and security [52]. Essentially, the ED problem is a constrained optimization problem in power systems that have the objective of dividing the total power demand among the online participating generators economically while satisfying the various constraints. ED problem have complex and nonlinear nonconvex characteristics with equality and inequality constraints. Therefore, good solutions of the ED problem would result in great economical benefits. Over the years, many efforts have been made to solve this problem, incorporating different kinds of constraints or multiple objectives, through various mathematical programming and optimization techniques [42]. In the conventional methods such as the lambdaiteration method, the base point and participation factors, and the gradient methods, an essential assumption is that the incremental cost curves of the units are monotonically increasing piece wise linear functions, but the practical systems are nonlinear [52]. Hence, global optimization techniques, such as the genetic algorithms (GAs), simulated annealing (SA), and particle swarm optimization (PSO) have been studied in the past decade and have been successfully used to solve the ED. However, the references with continuous greedy randomized adaptive search procedure (CGRASP) application to such type of problems hadn't appear yet. The aim of this work is to apply the CGRASP to the ED problem and compare its effectiveness and produced solution feasibility with ones of other heuristic methods like the GAs and SA. 1.2 Literature Overview Since Carpentier introduced a network constrained economic dispatch problem in 1962 [9] and the first paper in the area of dynamic dispatching was published by Bechert and Kwatny in 1972 [6], a lot of researches have employed various mathematical programming optimization methods for solving ED problems [30]. These optimization techniques can be classified into three main categories. The first category contains deterministic methods that include the linear programming algorithm [26, 57, 69], quadratic programming algorithm [18, 37], nonlinear programming algorithm [39], etc. The LP method application to the powersystem rescheduling problem with securityconstrained economic dispatch/control for multiplevalvedturbine units was given by Stott and Marinho [57]. Rosehart et al. [48] discovered that for the economic dispatch problem, SLP appears to be a better tool than SQP. An approach based on efficient SLP techniques to solve the multiobjective environmental/economic load dispatch problem was presented by Zehar and Sayah [69]. Granelli et al. [18] solved a security constrained economic dispatch problem using modified SQP techniques. A dual feasible starting point is found by relaxing transmission limits and then constraint violations are enforced applying the dual quadratic algorithm. In [59] and [35], a security constrained economic dispatch problem was solved by SLP and the interior point dualaffine scaling algorithm. Momoh et al. [37] proposed an IPM for ED problem formulated as linear and convex QP. However each of traditional methods has some defects: it would generate large errors to use the linear programming algorithm to linearize the ED model; for the quadratic programming and nonlinear programming algorithms, the objective function should be continuous and differentiable [30]. The second category contains the methods based on artificial intelligence. Artificial intelligence technology has been successfully used to solve the ED problem. A chaos optimization algorithm (CAO) has been proposed by Jiang et al. [29] to deal with the economic dispatch problem of a hydro power plant. Zhijiang et al. [71] also applied a COA and the simulation results verified that the proposed approach is effective and precise. A mutative scale COA was applied by Xu et al. [65] to the economic operation of power plants. However, the results showed that the method is timeconsuming. An improved mutative scale COA hes been developed by Han and Lu [19]. According to the authors, their algorithm is highly efficient and can be applied not only to ED but to many power system problems, such as economic operation, OPF, system identification and optimal control, as well. In [36], Mahdad et al. proposed an efficient decomposed parallel GA to solve the multiobjective environmental/economic dispatch problem. In the first stage, the original network is decomposed into multi subsystems and the problem is transformed to optimize the active power demand associated with each partitioned network. In the second stage, an active power dispatch strategy is proposed to enhance the final solution of the optimal power flow of the original network. The proposed approach was tested on the Algerian 59bus test system. The computational results showed the convergence at the near solution and obtain a competitive solution at a reduced time. GAs with fuzzy logic controllers to adjust its crossover and mutation probabilities was applied by Song et al. [56] to solve a combined environmental economic dispatch problem. SA techniques were used by RoaSepulveda and PavezLazo [47], however, long computational time to obtain an optimal solution was reported. Tabu search was applied by Altun and Yalcinoz [2]. Simulation results on power systems consisting of 6 and 20 generating units exhibited good performance. In [38], an application of TS for solving security constrained ED problem was given by Muthuselvan and Somasundaram. Base case and contingency case line flow constraints were considered. Tests on 66bus and 191bus Indian utility systems revealed the reliability, efficiency and suitability of the proposed algorithm for practical applications. The third category consists the hybrid methods, which combine two or more techniques in order to get best features in each algorithm. Typically, significant improvement with hybrid methods can be achieved over each of the individual methods. Hybrid methods gained increasing popularity in the last 10 years. For the ED problem, Wong and Wong [63] combined an incremental GA with SA techniques. Coelho and Mariani [12] proposed a method combining a DE algorithm with selfadaptive mutation factor in the global search stage and chaotic local search techniques in the local search to solve an ED problem associated with the valvepoint effect. The same authors report another successful application of chaotic PSO in combination with an implicit filtering local search method to solve economic dispatch problems [13]. The chaotic PSO approach is used to produce good potential solutions, while the implicit filtering is used to finetune the final solution of the PSO. The hybrid methodology is validated for a test system consisting of 13 thermal units whose incremental fuel cost function takes into account the valvepoint loading effects. In [11], Coelho and Lee improved PSO approaches for solving an ED problem taking into account nonlinear generator features such as ramprate limits. Prohibited operating zones in the power system operation are developed as well. Their algorithm combines the PSO, Gaussian probability distribution functions and/or chaotic sequences. The PSO and its variants are validated for two test systems consisting of 15 and 20 thermal generation units, respectively. A combination of chaotic and selforganization behavior of ants in the foraging process was presented by Cai et al. [8]. This algorithm was applied to ED problems with thermal generators. The thesis is organized as follows: In Section 2, we briefly discuss a general ED problem formulation. The methods applied to solve ED are shortly discussed in Section 3. Section 4 describes experimental cases and presents calculation results. We conclude with Section 5. CHAPTER 2 ECONOMIC DISPATCH (ED) PROBLEM ED is one of the important optimization problems in power system operations, which is used to determine the optimal combination of power outputs of all generating units to minimize the total fuel cost while satisfying various constraints over the entire dispatch periods [67]. The traditional or static ED problem assumes constant power to be supplied by a given set of units for a given time interval and attempts to minimize the cost of supplying this energy subject to constraints on the static behavior of the generating units like system load demand. Shortly, static ED determines the loads of generators in a system that will meet a power demand during a single scheduling period for the least cost. Therefore, it might fail to capture large variations of the load demand due to the ramp rate limits of the generators. Due to large variation of the customers load demand and the dynamic nature of the power systems, it became necessary to schedule the load beforehand so that the system can anticipate sudden changes in demand in the near future. Dynamic ED is an extension of static ED to determine the generation schedule of the committed units so that to meet the predicted load demand over the entire dispatch periods at minimum operating cost under ramp rate and other constraints [64]. The ramp rate constraint is a dynamic constraint which used to maintain the life of the generators, i.e. plant operators, to avoid shortening the life of the generator, try to keep thermal stress within the turbines safe limits [20]. Since the violations of the ramp rate constraints are assessed by examining the generators output over a given time interval, this problem cannot be solved for a single value of MW generation [20]. The objective function of dynamic ED is formulated as follows T N minC(P) =_ C,(Pf) (21) t=l i=1 where N is the set of committed units; Pi is the generation of unit i; C,(Pi) is the cost of producing Pi from unit i; T is the number of intervals in the study period. The fuel cost functions C,(.) is derived from the fuel consumption function that can be measured and are discussed in Section 2.2. The dynamic ED is not only the most accurate formulation of the economic dispatch problem but also the most difficult to solve because of its large dimensionality [3]. The DED problem is normally solved by discretization of the entire dispatch period into a number of small time intervals, over which the load demand is assumed to be constant and the system is considered to be in a temporal steady state. Over each time interval a static ED problem is solved under static constraints and the ramp rate constraints are enforced between the consecutive intervals [34]. In the DED problem the optimization is done with respect to the dispatchable powers of the units. Some researchers have considered the ramp rate constraints by solving SED problem interval by interval and enforcing the ramp rate constraints from one interval to the next. However, this approach can lead to suboptimal solutions [23]; moreover, it does not have the lookahead capability. Since dynamic ED was introduced, variuos methods have been used to solve this problem. However, all of those methods may not be able to provide an optimal solution and usually getting stuck at a local optimal. 2.1 ED Constraints The constrained ED problem is subjected to a variety of constraints depending upon assumptions and practical implications. Usually, formulation of ED problem includes such constraints as load generation balance, minimum and maximum capacity constraints. To maintain system reliability and security, spinning reserve constraints and security constraints can be added to the dynamic ED problem. The inclusion of the prohibited zones, ramprate limits and other practical constraints results in nonconvex ED of generating units. All these constraints are discussed bellow. 2.1.1 LoadGeneration Balance The generated power from all the running units must satisfy the load demand and the system losses given by (22) N Pf = D' + Losst, t=1,2,..., T (22) i= 1 where Dt is the demand and Losst is the system transmission loss. Their sum represents the effective load to be satisfied at the tth interval. The transmission line losses can be expressed in terms of the unit outputs: N N N Losst = PfBPj + P BoPf + Boo i=1 j=1 i=1 where B, is the ijth element of the loss coefficient square matrix, Bio is the ith element of the loss coefficient, and Boo is the constant loss coefficient. Sometimes the last two terms are omitted. In a competitive environment, the loadgeneration balance constraint is relaxed and each generating company schedules its production to maximize its profits given a forecast of electricity prices for the scheduling period. As a first approximation, each generating unit could be optimized separately in this problem because of the decoupling made possible by the availability of prices at each period. Dynamic constraints (such as ramp rates and minimum up and down time constraints) complicate the problem because a generating company that owns a portfolio of units must then decide whether to buy "flexibility" on the market or meet the dynamic constraints with its own resources [21]. 2.1.2 Generation Capacity Constraint For normal system operations, real power output of each generator is restricted by lower and upper bounds as follows: P + Sf < P ax = 1,2,... N, t= 1,2,..., T (23) min < p i= 1,2,...N, t = 1,2,..., T (24) where Pmin and pmax are the minimum and maximum power produced by generator i, Sf is the reserve contribution of unit during time interval t. 2.1.3 Generating Unit Ramp Rate Limits One of unpractical assumption that prevailed for simplifying the problem in many of the earlier research is that the adjustments of the power output are instantaneous [43]. Therefore, the power output of a practical generator cannot be adjusted instantaneously without limits. The operating range of all online units is restricted by their ramprate limits during each dispatch period. So, the subsequent dispatch output of a generator should be limited between the constraints of up and down ramprates [66] as follows pit+_ P < URi. At (25) Pf Pf+ < DR; At i = 12 ...N, t= 1, 2..., T 1 (26) where URi and DR, are the maximum ramp up/down rates for unit i and At is the duration of the time intervals into which the study period is divided. The inclusion of ramp rate limits modifies the generator operation constraints (23, 24) as follows max(P"i, Pf DR,) < Pi < min(Pmax, P,1 + UR,) (27) 2.1.4 Reserve Contribution The maximum reserve contribution has to satisfy following constraints: 0 < < S ax i = 1, 2, ... N, t = 1,2,..., T (28) where S"ax is the maximum contribution of unit i to the reserve capacity. Maximumramp spinning reserve contribution is defined as in (29) 0< Sf < UR At = 1,2,..., N, t= 1,2,..., T (29) where Sf is the spinning reserve of unit i. 2.1.5 System Spinning Reserve Requirement Sufficient spinning reserve is required from all running units to maximize and maintain system reliability [14]. There are many ways to determine the system spinning reserve requirement. It can be calculated as the size of the largest unit in operation or as a percentage of forecast load demand or even as a function of the probability of not having sufficient generation to meet the load [64]. The spinning rezerve can be defined by (210) N St > SRt t= 1,2,..., T (210) i=1 where SRt is the system spinning reserve requirement for time interval t. Also, the system spinning reserve requirement for interval t can sometimes be given by the following equation [20]: SRt = odDt +g g *max(Pmax scheduled at time t, = 1,2,...N) (211) where ad and ag are constants which depend on the system required reliability level [55]. Besides the determination of the system spinning reserve requirement, the issue of allocation the spinning reserve among the committed units is very important; however, it has received very little attention in the dynamic ED literature. 2.1.6 Tieline Limits The economic dispatch problem can be extended by importing additional constraint like transmission line capacity limit given by (212) PTjk,rn < PTk + Sjk < PTkmax (212) where PrTk,mn and PTjkmax specify the tieline trasnmission capability, i.e. the transfer from area to area k should not exceed the tieline transfer capacities for security consideration [28]. Each area has own special load and its spinning reserve [68]. 2.1.7 Prohibited Zone The generating units may have certain ranges where operation is restricted on the grounds of physical limitations of machine components or instability, e.g. due to steam valve or vibration in shaft bearings. So, there is a quest to avoid operation in these zones in order to economize the production [43]. These ranges are prohibited from operation and a generator with prohibited regions (zones) has discontinuous fuelcost characteristics (Fig. 2.1.7) [53]. The acceptable operating zones of a generating unit can be formulated as follows pmin < Pt < P' (213) P:j_ < P < Pj, ie O, j =2,3,..., ni, t = 1, 2,..., T (214) PiUn, < < < pmax (215) where ni is the number of the prohibited zones in unit i, 0 is the set of units that have prohibited zones, P/i, P P are the lower and upper bounds of thejth prohibited zone. PZ: Prohibited Zone 0 IPZ1I IPZ2, I I I J I II I I . I I I I I I I I I  Min Max Power output (MW) Figure 21. Example of cost function with two prohibited operating zones 2.2 Objective Functions The dynamic ED problem has been solved with many different forms of the cost function, such as the smooth quadratic cost function (216) or the nonsmooth cost function due to the valvepoint effects (217). Also, a linear cost function [20] and piecewise linear cost function [27, 41] have been employed. For smooth cost function it is usually assumed that its incremental cost function. In some power systems combined cycle units are used to supply the base load. For these units the cost function can be given as linear, piecewise or quadratic with decreasing incremental cost function [41]. For units with prohibited zones, the fuel cost function is discontinuous and nonconvex. An interesting departure from this standard formulation is the approach proposed by Wang and Shahidehpour [61] who include in the objective function a term representing the reduction in the life of the turbine caused by excessive ramping rates. This flexible technique makes possible a tradeoff between the system operating cost and the life cycle cost of the generating units [21]. 2.2.1 Smooth Cost Function The most simplified cost function of each generator can be represented as a quadratic function as given in (216) whose solution can be obtained by the conventional mathematical methods Ci(Pf,) = ai + bPf + c,(Pf)2 (216) where ai, bi,c, are cost coefficients of generator i. 2.2.2 Nonsmooth Cost Functions with Valvepoint Effects The generating units with multivalve steam turbines exhibit a greater variation in the fuel cost functions because in order to meet the increased demand a generator with multivalve steam turbines increase its output and various steam valves are to be opened [67]. This valveopening process produces ripple like effect in the heatrate curve of the generator. The inclusion of valvepoint loading effects makes the modeling of the incremental fuel cost function of the generators more practical [60]. Therefore, in reality, the objective function of ED problem has nondifferentiable property. Consequently, the objective function should be composed of a set of nonsmooth cost functions. Considering nonsmooth cost functions of generation units with valvepoint $'MlWh E D /. C A: Primary Vah B: Secondary Valve MW C : Tertiary Valv D : Quateramry Vahl E: Qukmary Valv Figure 22. Cost function with valvepoint effects effects, the objective function is generally described as the superposition of sinusoidal functions and quadratic functions [52] Ci(Pf) = ai + bPf + ci(Pf)2 + leisin(hi(P" Pft)) (217) where ei and hi are the coefficients of generator i reflecting valvepoint effects. As shown in Fig. 2.2.2, this increases the nonlinearity of curve as well as number of local optima in the solution space [60] compared with the smooth cost function due to the valvepoint effects. Also the solution procedure can easily trap in the local optima in the vicinity of optimal value. 2.2.3 Nonsmooth Cost Functions with Multiple Fuels Since the dispatching units are practically supplied with multifuel sources [49], each unit should be represented with several piecewise quadratic functions reflecting the effects of fuel type changes, and the generator must identify the most economic fuel to burn. The resulting cost function is called a "hybrid cost function." Each segment of the hybrid cost function implies some information about the fuel being burned or the PowerlMWI Min PI P2 Max Figure 23. Cost function with multiple fuels units operation. Thus, generally, the fuel cost function is a piecewise quadratic function described as follows ail + biPf + cil(Pf)2 if P/ mi < Pf < p ai2 + bi2Pf + Ci2(Pf)2 if Pf < Pf < pt 2 ci(P,) (218) ain + binPf + cn(Pft)2 if Pt1 < f where are a,p, bp, cp the cost coefficients of generator for the pth power level. The incremental cost functions are illustrated in Fig. (2.2.3) 2.2.4 Nonsmooth Cost Functions with ValvePoint Effects and Multiple Fuels To obtain an accurate and practical economic dispatch solution, the realistic operation of the ED problem should consider both valvepoint effects and multiple fuels. The reference [10] proposed an incorporated cost model, which combines the valvepoint loadings and the fuel changes into one frame. Therefore, the cost function should combine (217) with (218), and can be realistically represented as shown in (219) ail + bilPf + cil(Pt)2 + lei,1sin(hi,l(Pm n P )) ai2 + bi2Pf + i2(Pit)2 + ei,2sin(hi,2(Pin Pt,2)) ci(P,) = ain + binPf + Cin(Pf)2 + lei,nsin(hi,n(Pirn Pitn)) if Pt i < Pt < pt ,,mln  I i,1 f pt < ptf< pt 1,1 i,2 if i < pit < Pimax (219) 2.2.5 Emission Function Due to increasing concern over the environmental considerations, society demands adequate and secure electricity, i.e. not only at the cheapest possible price, but also at minimum level of pollution. In this case, two conflicting objectives, i.e., operational costs and pollutant emissions, should be minimized simultaneously [4, 5, 7, 62]. The atmospheric pollutants such as sulphur oxides (SO) and nitrogen oxides (NOx) caused by fossilfueled generating units can be modeled separately or as the total emission of them which is the sum of a quadratic [4] and an exponential function and can be expressed as (220) T N Z ai + p,Pf + 7i(Pf)2 + iexp(6iPft) t=l i=1 where a, iP, CHAPTER 3 SOLUTION METHODS 3.1 Continuous Greedy Randomized Adaptive Search Procedure (CGRASP) ContinuousGRASP (CGRASP) extends the greedy randomized adaptive search procedure (GRASP) that was introduced by Feo and Resende [16, 17] from the domain of discrete optimization to that of continuous global optimization in [24, 25]. It is described as a multistart local search procedure, where each CGRASP iteration consists of two phases, namely, a construction phase and a local search phase [24]. Construction combines greediness and randomization to produce a diverse set of goodquality starting solutions for local search. The local search phase attempts to improve the solutions found by construction. The best solution over all iterations is kept as the initial solution. The advantages of this method is simplicity to implement and no requirement for derivative information Pseudocode for CGRASP is shown in (3.1). CGRASP works by discretizing the domain into a uniform grid. Both the construction (see the high level pseudocode 3.2) and local improvement phases (see the high level pseudocode 3.3) move along points on the grid. As the algorithm progresses, the grid adaptively becomes more dense. The main difference between GRASP and CGRASP is that an iteration of CGRASP does not consist of a single greedy randomized construction followed by local improvement, but rather a series of constructionlocal improvement cycles with the output of construction serving as the input of the local improvement, as in GRASP, but unlike GRASP, the output of the local improvement serves as the input of the construction procedure [25]. Since CGRASP is essentially an unconstrained optimization algorithm, the constraints handling strategy needs to be incorporated into it in order to deal with the constrained ED problem. Approaches to manage these constraints are discussed in section 3.4. pseudocode 3.1 CGRASP (n, /, u, f(.),Maxlters, MaxNumlterNolmprov, NumTimesToRun, MaxDirToTry,a) 1: f*  c0 2: forj 1,..., NumTimesToRun do 3: x UnifRand(/, u); h 1; NumlterNolmprov 0; 4: for Iter 1,..., Maxlters do 5: x + ConstructGreedyRandomized(x, f(.), n, h, I, u, a); 6: x LocalSearch(x, f(.), n, h, I, u, MaxDirToTry); 7: if f(x) < f* then 8: x* + x; f* f(x); NumlterNolmprov 0; 9: else 10: NumlterNolmprovw NumlterNolmprov+1 11: end if 12: if NumlterNolmprov> MaxNumlterNolmprov then 13: h h/2; NumlterNolmprov 0; {/}*make grid more dense*/ 14: end if 15: end for 16: end for 17: return x* pseudocode 3.2 ConstructGreedyRandomizedSolution (Problem Instance) 1: Solution 0; 2: while Solution construction not done do 3: MakeRCL(RCL); 4: S + SelectRandomElement(RCL); 5: Solutions Solution U S; 6: AdaptGreedyFunction(S); 7: end while 8: return (Solution); pseudocode 3.3 LocalSearch(Solution,Neighborhood) 1: Solution* Solution 2: while Solution* not locally optimal do 3: Solution* SelectRandomElement(Neighborhood(Solution*)); 4: if Solution better than Solution* then 5: Solution* Solution; 6: end if 7: end while 8: return (Solution*) 3.2 Genetic Algorithms (GA) This section engages into the concept of genetic algorithms that reflects the nature of chromosomes in genetic engineering. GAs are a class of stochastic search algorithms that start with the generation of an initial population or set of random solutions for the problem at hand. Each individual solution in the population called a chromosome or string represents a feasible solution. The objective function is then evaluated for these individuals. If the best string (or strings) satisfies the termination criteria, the process terminates, assuming that this best string is the solution of the problem. If the termination criteria are not met, the creation of new generation starts, pairs, or individuals are selected randomly and subjected to crossover and mutation operations. The resulting individuals are selected according to their fitness for the production of the new offspring. Genetic algorithms combine the elements of directed and stochastic search while exploiting and exploring the search space [31]. More details about GA can be found in [22, 46, 58]. pseudocode 3.4 Genetic algorithm 1: initialize population( 2: while not converge do 3: assign population fitness( 4: for 1,..., npopsiz do 5: select parents(p1,p2) 6: reproduction(p ,p2,child) 7: end for 8: select next generation( 9: end while The advantages of GA over other traditional optimization techniques can be summarized as follows: *GA searches from a population of points, not a single point. The population can move over hills and across valleys. GA can therefore discover a globally optimal point, because the computation for each individual in the population is independent of others. GA has inherent parallel computation ability. * GA uses payoff (fitness or objective functions) information directly for the search direction, not derivatives or other auxiliary knowledge. GA therefore can deal with nonsmooth, noncontinuous and nondifferentiable functions that are the reallife optimization problems. This property also relieves GA of the approximate assumptions for a lot of practical optimization problems, which are quite often required in traditional optimization methods. GA uses probabilistic transition rules to select generations, not deterministic rules. They can search a complicated and uncertain area to find the global optimum. GA is more flexible and robust than the conventional methods [33]. The first attempt of the application of genetic algorithms in power systems is in the load flow problem [70]. It has been found that the simple genetic algorithm quickly finds the normal load flow solution for smallsize networks by specifying an additional term in the objective function. A number of approaches to improving convergence and global performance of GAs have been investigated [70]. 3.3 Simulated Annealing (SA) The SA is a generic probabilistic metaheuristic for the global optimization problem that was proposed by Kirkpatric et al. [32]. In the SA method, each point s of the search space is analogous to a state of some physical system, and the function E(s) to be minimized is analogous to the internal energy of the system in that state. The goal is to bring the system, from an arbitrary initial state, to a state with the minimum possible energy. In each step of the SA algorithm the current solution is replaced by a random "nearby" solution, chosen with a probability that depends on the difference between the corresponding function values and on a global parameter T (called the temperature), that is gradually decreased during the process. The dependency is such that the current solution changes almost randomly when T is large, but increasingly "downhill" as T goes to zero. The allowance for "uphill" moves saves the method from becoming stuck at local minima which are the bane of greedier methods. For certain problems, SA may be more effective than exhaustive enumeration. It has been shown that this technique converges asymptotically to the global optimal solution with probability one [1]. SA is an effective global optimization algorithm because of the following advantages [50]: * suitability to problem in wide area, * no restriction on the form of cost function, * high probability to find global optimization, * easy implementation by programming. The pseudocode implementing SA is given bellow. It starts from state sO and continue for kmax of steps or until a state with energy emax or less is found. The call neighbour (s) should generate a randomly chosen neighbour of a given state s; the call random() should return a random value in the range [0,1]. The annealing schedule is defined by the temp(r), which should yield the temperature to use, given the fraction r of the time budget that has been expended so far. pseudocode 3.5 Simmulated Annealing 1: s so; e E(s) 2: Sbest + S; ebest < e; 3: k 0; 4: while k < kmax and e > emax do 5: Snew neighbour(s) 6: enew E(snew) 7: if enew < best then 8: best Snew; best enew 9: end if 10: if P(e, enew, temp(k/max)) > random() then 11: S Snew; e enew 12: k k + 1 13: end if 14: end while 15: return Sbest Actually, the "pure" SA algorithm does not keep track of the best solution found so far: it does not use the variables Sbest and ebest, it lacks the first if inside the loop, and, at the end, it returns the current state s instead of Sbest. While saving the best state is a standard optimization, that can be used in any metaheuristic, it breaks the analogy with physical annealing since a physical system can "store" a single state only. In strict mathematical terms, saving the best state is not necessarily an improvement, since one may have to specify a smaller kmax in order to compensate for the higher cost per iteration. However, the step Sbest  Snew happens only on a small fraction of the moves. Therefore, the optimization is usually worthwhile, even when statecopying is an expensive operation. SA has the ability to avoid getting local solutions; then it can generate global or near global optimal solutions for optimization problems without any restriction on the shape of the objective functions [44]. SA is not memory intensive [45]. However, the setting of control parameters of the SA algorithm is a difficult task and the computation time is high [3]. The computational burden can be reduced by means of parallel processing [44]. 3.4 Constraints Handling Constraints lie at the hear to fall constrained engineering optimization applications. Practical constraints, which are often nonlinear and nontrivial,confine the feasible solutions to a small subset of the entire search space. There are several approaches which can be applied to handle constraints in heuristic approaches. These methods can be grouped into four categories: methods that preserve the feasibility of solutions, penaltybased methods, methods that clearly distinguish between feasible and unfeasible solutions, and hybrid methods [15, 62]. 3.4.1 PenaltyBased Approach The penalty function method is frequently applied to manage constraints in evolutionary algorithms. Such a technique converts the primal constrained problem into an unconstrained problem by penalizing constraint violations. The penalty function method is simple in concept and implementation. However, its primal limitation is the degree to which each constraint is penalized. These penalty terms have certain weaknesses that become fatal when penalty parameters are large. Such a penalty function tends to be ill conditioned near the boundary of the feasible domain where the optimum point is usually located [10]. The penalized fuel cost function in ED problem was employed in [51]. In [40] the ED problem was transformed into an unconstrained one by constructing an augmented objective function incorporating penalty factors for any value violating the constraints: Neq Nueq H(X) = J(X) + k, Z(hj(X))2 + k2 max[0, g(X)]2 (31) j=1 j=1 where J(X) is the objective function value of the ED problem. Neq and Nueq are the number of equality and inequality constraints, respectively; hj(X) and gj(X) are the equality and inequality constraints, respectively; kl and k2 are the penalty factors. Since the constraints should be met, the value of thek, and k2 parameters were chosen to have high value of 10,000. This approach was epmpoyed when applying SA method. The heuristic startegy that is discussed in nex section was used to get a feasible solution while applying CGRASP method. 3.4.2 Heuristic Strategy When the CGRASP is applied to solve ED problem, a key problem is how to handle constraints with efficiency. In this section we mainly focus on handling the real power limits and generators rampup constraints. Other than penalty based way to satisfy the real power balance equality constraints (22), is to specify the output of (N 1) generating units and to find the Nth from the equality constraint like in [4, 67]. In reference [67], authors employed a dependent generation power pt of randomly selected unit /. The heuristic strategy applied in LocalSearchO procedure in CGRASP algorithm can be formulated in a following way: Step 1. Set the dispatch period index t = 1 and iteration i = 1. Step 2. Calculate the violation of power balance constraint Pr,, at dispatch time t is calculated from 32 as follows N Pt = Dt + Losst Pf (32) i=1 If Prr = 0, then go to Step 5, otherwise to Step 3. Step 3. Randomly generate / the index of generating unit and calculate the real power of selected dependent generating unit pt from (33). N Pf =Dt Pft t = 1,2,..., T (33) i=1 i#/ However, considering transmission losses (2.1.1), these equality constraints become nonlinear and the output of dependent generating unit for every dispatch period t can be found from by solving a following equation N N N N N BII(Pf)2+(2Z BiiPf+B101)Pf+(Dt+ PfBiiPj +Boo+ BioPf, P) = 0 (34) i=1 i=1j=1 i=1 i=1 i/ i / jil i / i/ If it doesn't violate the generator operating limits and rampup constraints (if they are present), go to Step 5. Otherwise, the value has to be modified according to 35 P max if Pt > pmax pmin if Pf < pmin If ED incorporates rampup limits and dispatch period t > 1, then dependent unit output has to be calculated as 36 max(Pmin, P1 DR,) if Pf > max(Pmin, Pf1 DR) (36) P {(f= <(36) min(Pmax, Pf + URi) if Pf < min(P/ax, Pt + URI) After adjustment, go to Step 4. Step 4. Increase the iteration number by 1, i.e. I = / + 1. If I < /max go to Step 2, otherwise go to Step 5. Step 5. Increase the period number by 1, i.e. t = t + 1. If t < T go to Step 2, otherwise stop. The applied strategy for constraints handling will produce solutions satisfying real power limits constraint and generating unit ramp rate limits constraint, however not always the the real power balance constraint will be satisfied in dynamic ED due to rampup limits. The situation can be that in one dispatch period demand will meet generation, however in the next period the demand can be not because due to generating unit power increase or reduction limitation. In order not to consider such infeasible solution a large penalty is added to objective function value. CHAPTER 4 EXPERIMENTS AND RESULTS 4.1 Experiments In order to verify the feasibility and effectiveness of adopted CGRASP capabilities for solving ED problems, different ED problem formulations, i.e. static and dynamic ED and different systems were used. The CGRASP algotihm with heuristic strategy to deal with constraints was implemented in Matlab 7.5. For GA and SA algorithms, the standard Matlab functions form Genetic Algorithm and Direct Search Toolbox were employed. In standard GA function ga(), it is possible to include both linear and nonlinear equality and inequality constraints. However, SA function simulannealbnd( incorporates only lower and upper bound constraints, other constraints as a penalty function is added to objective function. Next, we will provide descriptions of systems used for our experiments. 4.1.1 System 1 The system consists of five generating units, whose the maximum total output is 925 MW. On this system dynamic ED problem was solved with the dispatch horizon one day with 12 intervals of one hour each. The demand of the system and generating unit data are given in Tables (41) and (42), respectively. Table 41. Generating units characteristics of fiveunit system ai, $ /h bi, $ /MWh ci, $ /(MW)2h pin, MW pmax, MW Unit 1 25.000 2.000 0.008 10.000 75.000 Unit 2 60.000 1.800 0.003 20.000 125.000 Unit 3 100.000 2.100 0.0012 30.000 175.000 Unit 4 120.000 2.000 0.001 40.000 250.000 Unit 5 40.000 1.800 0.0015 50.000 300.000 4.1.2 System 2 The system contains six thermal generating units. The total maximum output of generating units is 1470 MW. This system was used to solve static ED problem where load demand on the system is 1263 MW. Parameters of all the thermal units are Table 42. Load demand Time, h Load, MW Time, h Load, MW 1 410 7 626 2 435 8 654 3 475 9 690 4 530 10 704 5 558 11 720 6 608 12 740 reported in [30] and are given in Tables 43 and 44. In normal operation of the system, the loss coefficients B are as follows: 0.0017 0.0012 0.0007 0.0001 0.0005 0.0002 0.0012 0.0014 0.0009 0.0001 0.0006 0.0001 0.0007 0.0009 0.0031 0.0001 0.001 0.0006 0.0001 0.0001 0 0.0024 0.0006 0.0008 x 102 Table 0.0005 0.0006 0.001 0.0006 0.0129 0.0002 0.0002 0.0001 0.0006 0.0008 0.0002 0.015 Boi = [0.3908 0.12970.70470.05910.2161 0.6635] x 103 Boo = 0.056 43. Generating units characteristics of sixunit system Unit Pmin, MW Pmax, MW a, $ /h bi, $/MWh c,,$/(MW)2h Po, MW 1 100 500 0.007 7 240 440 2 50 200 0.0095 10 200 170 3 80 300 0.009 8.5 220 200 4 50 150 0.009 11 200 150 5 50 200 0.008 10.5 220 190 6 50 120 0.0075 12 190 110 Bi  44. Rumpup limits and prohibited zones of sixunit system Unit URi,MW DR,,MW Prohibited zone 1 80 120 [210 240] [350 380] 2 50 90 [90 110] [140 160] 3 65 100 [150 170] [210 240] 4 50 90 [80 90] [110 120] 5 50 90 [90 110] [140 150] 6 50 90 [75 85] [100 105] 4.1.3 System 3 This system consists of 13 generating units with valvepoint loading as given in Table (45). The parameters of this system showed is taken from [54]. The expected demand is 1800 MW and 2520 MW. Table 45. Generating units characteristics of 13unit system pmin, MW pmax, MW i i 680 360 360 180 180 180 180 180 180 120 120 120 120 ai bi ci ei f 0.00028 0.00056 0.00056 0.00324 0.00324 0.00324 0.00324 0.00324 0.00324 0.00284 0.00284 0.00284 0.00284 8.1 8.1 8.1 7.74 7.74 7.74 7.74 7.74 7.74 8.6 8.6 8.6 8.6 550 309 307 240 240 240 240 240 240 126 126 126 126 300 200 200 150 150 150 150 150 150 100 100 100 100 0.035 0.042 0.042 0.063 0.063 0.063 0.063 0.063 0.063 0.084 0.084 0.084 0.084 4.1.4 System 4 This system is composed of 40 generating units with valvepoint loading effects supplying a total demand of 10500 MW. Therefore, this system has nonconvex solution spaces and there are many local minima due to valvepoint effects and the global minimum is very difficult to determine. The parameters of this system showed in the Table (46) are available in [54] as well. Unit Table Table 46. Generating units characteristics of 40unit system Unit pmin, MW pmax, MW ai bi ci ei 1 36 114 0.0069 6.73 94.705 100 0.084 2 36 114 0.0069 6.73 94.705 100 0.084 3 60 120 0.02028 7.07 309.54 100 0.084 4 80 190 0.00942 8.18 369.03 150 0.063 5 47 97 0.0114 5.35 148.89 120 0.077 6 68 140 0.01142 8.05 222.33 100 0.084 7 110 300 0.00357 8.03 287.71 200 0.042 8 135 300 0.00492 6.99 391.98 200 0.042 9 135 300 0.00573 6.6 455.76 200 0.042 10 130 300 0.00605 12.9 722.82 200 0.042 11 94 375 0.00515 12.9 635.2 200 0.042 12 94 375 0.00569 12.8 654.69 200 0.042 13 125 500 0.00421 12.5 913.4 300 0.035 14 125 500 0.00752 8.84 1760.4 300 0.035 15 125 500 0.00708 9.15 1728.3 300 0.035 16 125 500 0.00708 9.15 1728.3 300 0.035 17 220 500 0.00313 7.97 647.85 300 0.035 18 220 500 0.00313 7.95 649.69 300 0.035 19 242 550 0.00313 7.97 647.83 300 0.035 20 242 550 0.00313 7.97 647.81 300 0.035 21 254 550 0.00298 6.63 785.96 300 0.035 22 254 550 0.00298 6.63 785.96 300 0.035 23 254 550 0.00284 6.66 794.53 300 0.035 24 254 550 0.00284 6.66 794.53 300 0.035 25 254 550 0.00277 7.1 801.32 300 0.035 26 254 550 0.00277 7.1 801.32 300 0.035 27 10 150 0.52124 3.33 1055.1 120 0.077 28 10 150 0.52124 3.33 1055.1 120 0.077 29 10 150 0.52124 3.33 1055.1 120 0.077 30 47 97 0.0114 5.35 148.89 120 0.077 31 60 190 0.0016 6.43 222.92 150 0.063 32 60 190 0.0016 6.43 222.92 150 0.063 33 60 190 0.0016 6.43 222.92 150 0.063 34 90 200 0.0001 8.95 107.87 200 0.042 35 90 200 0.0001 8.62 116.58 200 0.042 36 90 200 0.0001 8.62 116.58 200 0.042 37 25 110 0.0161 5.88 307.45 80 0.098 38 25 110 0.0161 5.88 307.45 80 0.098 39 25 110 0.0161 5.88 307.45 80 0.098 40 242 550 0.00313 7.97 647.83 300 0.035 4.1.5 System 5 This system has 10 generating units with valvepoint loading effects. Therefore, this system has nonconvex solution spaces and there are many local minima due to valvepoint effects. The parameters of this system are given in the Table 410 and are available in [4] as well. The forecasted demand with the dispatch horizon one day with 24 intervals of one hour each is shown in Table 48. Table 47. Generating units characteristics of 10unit system Unit pin", MW pmax, MW ai bi ci ei f URi URi 1 150 470 786.7988 38.5397 0.1524 450 0.041 80 80 2 135 470 451.3251 46.1591 0.1058 600 0.036 80 80 3 73 340 1049.9977 40.3965 0.028 320 0.028 80 80 4 60 300 1243.5311 38.3055 0.0354 260 0.052 50 50 5 73 243 1658.5696 36.3278 0.0211 280 0.063 50 50 6 57 160 1356.6592 38.2704 0.0179 310 0.048 50 50 7 20 130 1450.7045 36.5104 0.0121 300 0.086 30 30 8 47 120 1450.7045 36.5104 0.0121 340 0.082 30 30 9 20 80 1455.6056 39.5804 0.109 270 0.098 30 30 10 10 55 1469.4026 40.5407 0.1295 380 0.094 30 30 Table 48. Load demand for 24 hours Time, h Load, MW Time, h Load, MW Time, h Load, MW 1 1036 9 1924 17 1480 2 1110 10 2022 18 1628 3 1258 11 2106 19 1776 4 1406 12 2150 20 1972 5 1480 13 2072 21 1924 6 1628 14 1924 22 1628 7 1702 15 1776 23 1332 8 1776 16 1554 24 1184 4.2 Results One of the features that the heuristic algorithms possess is randomness. Therefore, their performances cannot be judged by the result of a single run and many trials with different initializations should be made to reach a valid conclusion about the performance of the algorithms. An algorithm is robust, if it can guarantee an acceptable performance level under different conditions. In this paper, 50 different runs of CGRASP have been carried out. 4.2.1 Case 1 In this case, the dynamic ED problem on system 1 is solved. It can be seen from Table (49) that the CGRASP provided the best solution compared to SA and GA. Table 49. Generation costs for case 1 Min 19645.87118 19817.50206 19675.35508 Avg 19725.6329 19819.44773 19777.37761 Max 19837.55210 19817.50206 19855.63880 The smallest total production cost is obtained by SA and it is $19645.87. Morever, we can notice that on the average CGRASP algorithm performs better than SA and GA. The lowest maximum value is provided by CGRASP as well, while the highest maximum value was produced by SA. shows that SA solutions are very sensitive to starting points and are more volatile. The best found solution satisfying demand and power limits is given in Table 410. Table 410. Best solution for case 1 Unit 1 10.00006975 10.00003632 26.00796462 26.75734182 49.04650601 19.63298415 36.5415225 13.70517701 41.8141424 21.88623928 28.88128653 25.81847079 Unit 2 20.00005644 52.60222864 87.4541663 95.20329384 56.01226378 103.7721597 92.86540909 92.67194012 102.4755387 102.6285523 58.20434065 113.8283152 Unit 3 80.00001477 30.00004416 80.84053733 170.8232815 91.33351254 51.19984471 61.43881398 138.6821921 123.5565313 129.2017437 161.8266865 174.999958 Unit 4 105.6819645 151.8901212 45.99580774 68.03647655 110.4114641 196.8271002 157.4967338 178.1269249 244.850131 249.9999302 222.1801306 147.8022832 Unit 5 194.3178945 190.5075697 234.701524 169.1796063 251.1962536 236.5679112 277.6575207 230.8137659 177.3036566 200.2835345 248.9075557 277.5509728 4.2.2 Case 2 Here, the static ED problem includes the nonlinear generationdemand equality constraints due to included transmission losses. The ramp up limits and prohibited Method CGRASP GA SA St.Dev. 26.8471 0.93349 36.1006 Hour 1 2 3 4 5 6 7 8 9 10 11 12 zones of generators are incorporated as well. The efficient of CGRASP is tested on sixunit systems that is described in Section 4.1.2. The same problem has been solved in [30] and their best solution and applied methods are presented in Table 411. The losses and total generation cost are given in Table 412. The best solutions among all solutions have been illustrated in the bold prints. From these data we can see that their provided objective function values are smaller that one obtained by CGRASP, but it should be noted that solutions gained by CPSO 1 and CPSO 2 violate the rampup limits of generator 3. When the solution of PSO has been pluged, it has been found the violation of generationdemand balance equality by 0.4661 MW, because according to given solution, the generation is equal to 1275.9571 MW and loss is 12.4910 MW. The minimum generation cost found by CGRASP is $15456.54469, while the average cost is $15507.10954 with standard deviation of value $28.10037477. According to these facts, it can be stated that CGRASP approach with applied heuristic strategy can produced feasible and good solutions. The results produced by SA and GA were not feasible or reasonably close to results presneted here, so they are not presneted here. Table 411. Best solutions for case 2 Pi P2 P3 P4 P5 P6 PSO 447.4970 173.3221 263.4745 139.0594 165.4761 87.1280 CPSO 1 434.4236 173.4385 274.2247 128.0183 179.7042 85.9082 CPSO 2 434.4295 173.3231 274.4735 128.0598 179.4759 85.9281 CGRASP 447.8181 200 253.5570 149.9999 150.3202 73.5022 Table 412. Best results, when demand is1263 MW Total output Loss Total generation cost PSO 1276.0 12.9584 15451 CPSO 1 1276.0 12.9583 15447 CPSO 2 1276.0 12.9582 15446 CGRASP 1275.1974 12.1974 15456.54 4.2.3 Case 3 In this case, the static ED problem with nonsmooth cost function due to the valvepoint effects is considered to check the ability of CGRASP to solve such type problems and its competitiveness with both GA and SA approaches. The experiment is performed on two different systems, namely, system 3 and system 4. The final fuel costs obtained using applied approaches are summarized in Table 413. It shows the minimum, average and maximum cost and standard deviation achieved by applied methods for 75 runs. From the computational results, the minimum cost achieved by CGRASP was the best, followed by SA and GA. The minimum cost, maximum cost and the mean cost values obtained by CGRASP are 18394.07 $/h, 18699.339 $/h, and 18550.105 $/h, respectively, which are lower than those obtained by SA and GA. The worst results are obtained by GA. It can be noticed that results produced by GA vary the least, this can be confirmed by the low standard deviation that is $12.562. In literature [54], the lowest reported generation cost for 1800 MW is $17994.07, however the solution is not presented. Table 413. Generation costs for 13unit system with demand 1800 MW Method Min Avg Max St. Dev. CGRASP 18394.070 18550.105 18699.339 65.729 GA 19384.229 19417.964 19438.914 12.562 SA 18950.174 19393.114 19782.516 181.920 The results on 40units are presented in table 414. CGRASP, SA and GA algorithms were run for 75 times and the minimum, maximum and average value of objective function are reported. Table 414. Generation costs for 40unit system with demand 10500 MW Method Min Avg Max St.Dev. CGRASP 128883.1965 130268.9796 132839.2181 972.757 GA 163401.9977 163534.9817 163623.3423 64.0606 SA 138975.7844 150757.5002 162578.6271 6118.65 4.2.4 Case 4 The last problem solved by CGRASP is dynamic ED problem including rampup limits, that makes this problem more difficult than in a case 1. For simplicity, the transmission losses are neglected. The minimum cost obtained by the CGRASP coupled with heuristic strategy is found to be $1,735,176.10, the best solution that satisfies demandbalance constraints as well as generators operation constraints including ones of rampup is given in Table 415. GA ans SA applied in this work couldn't produce the feasible solution. Table 415. Best solution for case 4 Hour 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 P1 197.5775 199.3353 279.3352 357.6864 341.5554 364.4919 291.8407 275.9255 355.9255 373.6142 453.6141 469.9999 405.7058 325.7058 308.0081 228.0081 252.7930 214.1486 253.8865 333.8865 334.2637 254.2637 174.2637 150.0001 P2 181.2115 142.0086 222.0086 196.9907 276.9907 356.9907 436.9906 452.7053 382.1891 395.9434 447.3342 469.9999 425.0176 367.8957 320.2882 369.5967 361.9796 441.9796 470.0000 448.3762 375.3046 295.3046 215.3046 171.8146 P3 163.4175 243.3161 163.3161 83.3162 104.5803 92.7176 162.9068 238.0856 318.0856 310.8613 339.9999 327.0836 281.3943 339.9999 284.4522 235.5079 165.9723 160.0032 240.0032 320.0032 329.5451 249.5451 218.8263 144.0519 P4 62.3581 112.3580 162.3580 212.3580 226.3765 205.6510 249.2158 230.3986 233.8956 235.4000 285.3999 235.4000 279.5788 229.5788 279.5788 230.6255 237.6272 187.6272 137.6272 149.1631 195.2469 245.2469 256.1493 259.4389 P5 127.0141 115.8294 80.9696 130.9696 180.9696 202.5198 210.4313 196.6733 197.8604 242.9999 197.8604 229.2473 226.4663 240.9742 190.9742 174.2616 169.1177 211.3444 242.9999 211.3444 242.9999 195.4727 161.5679 132.1630 Best solutions for case 4 continued Hour 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 P6 84.1156 87.3169 60.2349 107.8558 128.7563 140.9288 137.0179 89.7335 139.7335 159.9999 112.7650 112.5501 88.9431 65.2257 70.0661 108.1979 62.7680 112.7679 122.1729 159.9999 153.7373 127.9852 122.7771 72.7771 P7 77.5922 63.8052 93.8052 111.1225 81.1226 76.3086 46.3086 76.3086 106.3086 128.7672 98.7671 128.7671 129.9999 127.7126 97.7126 67.7126 70.2996 99.9863 97.8341 126.0825 100.4907 70.4907 40.4908 70.4908 Ps 83.0176 90.0000 119.9999 108.2973 78.2974 98.3418 68.3418 84.7578 93.1458 84.5977 114.5977 84.5977 114.5977 95.1739 119.9999 90.2077 97.6767 119.9999 112.1255 105.6771 119.9999 96.0201 66.0202 73.5756 P9 20.0001 46.0304 46.7507 59.4590 38.9946 56.9712 71.6510 80.0000 62.0940 73.0386 43.0386 73.0386 77.8105 76.7334 62.4338 32.4338 32.8367 48.3857 55.9241 78.8549 48.8549 70.0305 56.0837 60.6486 Pl0 39.6960 10.0001 29.2219 37.9445 22.3567 33.0785 27.2956 51.4117 34.7621 16.7777 12.6230 19.3156 42.4861 54.9999 42.4861 17.4482 28.9293 31.7572 43.4265 38.6122 23.5569 23.6405 20.5164 49.0393 CHAPTER 5 CONCLUSION * Economic disatch problem can be formulated in very different ways: as a simple linear programing problem to nonlinear nonconvex problem. * In this work, four different cases were analysed and three heuristic methods: CGRASP, GA and SA were applied to solve ED problem. * Since CGRASP is able to cope with optimization problem having box constraints, the heuristic strategy to deal with equality and inequality constraints for ED problem was incorporated. * The experimental results revealed, that CGRASP adopted to ED problem is able to provide good results and can outperfom SA and GA. 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PowerCon 4 23132317. BIOGRAPHICAL SKETCH Ingrida Radziukyniene got Bachelor of Science and Master of Science in computer science at Vytautas Magnus University, Lithuania in 2003 and 2005, respectively. In addition, she got a certificate of business management from Department of Business at Vytautas Magnus University. In 2010, she earned the Master of Science in industrial engineering from University of Florida. More information about her research interest can be found in her webpage http://plaza.ufl.edu/ingridar/. PAGE 1 CGRASPAPPLICATIONTOTHEECONOMICDISPATCHPROBLEMByINGRIDARADZIUKYNIENEATHESISPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFMASTEROFSCIENCEUNIVERSITYOFFLORIDA2010 PAGE 2 c2010IngridaRadziukyniene 2 PAGE 3 Idedicatethistomywonderfulson,Matas 3 PAGE 4 ACKNOWLEDGMENTS IamgratefultomanypeopleforsupportingmethroughoutmygraduatestudyinUnitedStates.Firstofall,Iwouldliketoexpressmyearnestgratitudetomyadvisor,Dr.PanosM.Pardalos,fordirectingthisstudyandreadingpreviousdraftsofthiswork.Withouthisguidance,inspiration,andsupportthroughoutthecourseofmyresearch,thisworkwouldnotbecomplete.ManythankstoArturaswhohasbeenthereforme,listeningtomeandsupportingme.IamalsothankfultomyfriendsattheCenterforAppliedOptimizationwhomentallysupportedandmademystudentlifemorecolorful. 4 PAGE 5 TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 9 CHAPTER 1INTRODUCTION ................................... 10 1.1Motivation .................................... 10 1.2LiteratureOverview .............................. 11 2ECONOMICDISPATCH(ED)PROBLEM ..................... 14 2.1EDConstraints ................................. 15 2.1.1LoadGenerationBalance ....................... 16 2.1.2GenerationCapacityConstraint .................... 16 2.1.3GeneratingUnitRampRateLimits .................. 17 2.1.4ReserveContribution .......................... 17 2.1.5SystemSpinningReserveRequirement ............... 18 2.1.6TielineLimits .............................. 18 2.1.7ProhibitedZone ............................. 19 2.2ObjectiveFunctions .............................. 19 2.2.1SmoothCostFunction ......................... 20 2.2.2NonsmoothCostFunctionswithValvepointEffects ........ 20 2.2.3NonsmoothCostFunctionswithMultipleFuels ........... 21 2.2.4NonsmoothCostFunctionswithValvePointEffectsandMultipleFuels ................................... 22 2.2.5EmissionFunction ........................... 23 3SOLUTIONMETHODS ............................... 24 3.1ContinuousGreedyRandomizedAdaptiveSearchProcedure(CGRASP) 24 3.2GeneticAlgorithms(GA) ............................ 26 3.3SimulatedAnnealing(SA) ........................... 27 3.4ConstraintsHandling .............................. 29 3.4.1PenaltyBasedApproach ........................ 29 3.4.2HeuristicStrategy ............................ 30 4EXPERIMENTSANDRESULTS .......................... 33 4.1Experiments .................................. 33 4.1.1System1 ................................ 33 5 PAGE 6 4.1.2System2 ................................ 33 4.1.3System3 ................................ 35 4.1.4System4 ................................ 35 4.1.5System5 ................................ 37 4.2Results ..................................... 37 4.2.1Case1 .................................. 38 4.2.2Case2 .................................. 38 4.2.3Case3 .................................. 40 4.2.4Case4 .................................. 41 5CONCLUSION .................................... 43 REFERENCES ....................................... 44 BIOGRAPHICALSKETCH ................................ 50 6 PAGE 7 LISTOFTABLES Table page 41Generatingunitscharacteristicsofveunitsystem ................ 33 42Loaddemand ..................................... 34 43Generatingunitscharacteristicsofsixunitsystem ................ 34 44Rumpuplimitsandprohibitedzonesofsixunitsystem ............. 35 45Generatingunitscharacteristicsof13unitsystem ................ 35 46Generatingunitscharacteristicsof40unitsystem ................ 36 47Generatingunitscharacteristicsof10unitsystem ................ 37 48Loaddemandfor24hours .............................. 37 49Generationcostsforcase1 ............................. 38 410Bestsolutionforcase1 ............................... 38 411Bestsolutionsforcase2 ............................... 39 412Bestresults,whendemandis1263MW ...................... 39 413Generationcostsfor13unitsystemwithdemand1800MW ........... 40 414Generationcostsfor40unitsystemwithdemand10500MW .......... 40 415Bestsolutionforcase4 ............................... 41 7 PAGE 8 LISTOFFIGURES Figure page 21Exampleofcostfunctionwithtwoprohibitedoperatingzones .......... 19 22Costfunctionwithvalvepointeffects ........................ 21 23Costfunctionwithmultiplefuels ........................... 22 8 PAGE 9 AbstractofThesisPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofMasterofScienceCGRASPAPPLICATIONTOTHEECONOMICDISPATCHPROBLEMByIngridaRadziukynieneAugust2010Chair:PanosM.PardalosMajor:IndustrialandSystemsEngineeringEconomicdispatchplaysanimportantroleinpowersystemoperations,whichisacomplicatednonlinearconstrainedoptimizationproblem.Ithasnonsmoothandnonconvexcharacteristicwhengenerationunitvalvepointeffectsaretakenintoaccount.ThisworkadoptstheCGRASPalgorithmtosolvedifferentlyformulatedeconomicdispatchproblems.ThecomparisonofthefeasibilityandeffectivenessoftheCGRASP,SAandGAisgivenaswell. 9 PAGE 10 CHAPTER1INTRODUCTION 1.1MotivationTheeconomicdispatch(ED)optimizationproblemisoneofthefundamentalissuesinpowersystemstoobtainoptimalbenetswiththestability,reliabilityandsecurity[ 52 ].Essentially,theEDproblemisaconstrainedoptimizationprobleminpowersystemsthathavetheobjectiveofdividingthetotalpowerdemandamongtheonlineparticipatinggeneratorseconomicallywhilesatisfyingthevariousconstraints.EDproblemhavecomplexandnonlinearnonconvexcharacteristicswithequalityandinequalityconstraints.Therefore,goodsolutionsoftheEDproblemwouldresultingreateconomicalbenets.Overtheyears,manyeffortshavebeenmadetosolvethisproblem,incorporatingdifferentkindsofconstraintsormultipleobjectives,throughvariousmathematicalprogrammingandoptimizationtechniques[ 42 ].Intheconventionalmethodssuchasthelambdaiterationmethod,thebasepointandparticipationfactors,andthegradientmethods,anessentialassumptionisthattheincrementalcostcurvesoftheunitsaremonotonicallyincreasingpiecewiselinearfunctions,butthepracticalsystemsarenonlinear[ 52 ].Hence,globaloptimizationtechniques,suchasthegeneticalgorithms(GAs),simulatedannealing(SA),andparticleswarmoptimization(PSO)havebeenstudiedinthepastdecadeandhavebeensuccessfullyusedtosolvetheED.However,thereferenceswithcontinuousgreedyrandomizedadaptivesearchprocedure(CGRASP)applicationtosuchtypeofproblemshadn'tappearyet.TheaimofthisworkistoapplytheCGRASPtotheEDproblemandcompareitseffectivenessandproducedsolutionfeasibilitywithonesofotherheuristicmethodsliketheGAsandSA. 10 PAGE 11 1.2LiteratureOverviewSinceCarpentierintroducedanetworkconstrainedeconomicdispatchproblemin1962[ 9 ]andtherstpaperintheareaofdynamicdispatchingwaspublishedbyBechertandKwatnyin1972[ 6 ],alotofresearcheshaveemployedvariousmathematicalprogrammingoptimizationmethodsforsolvingEDproblems[ 30 ].Theseoptimizationtechniquescanbeclassiedintothreemaincategories.Therstcategorycontainsdeterministicmethodsthatincludethelinearprogrammingalgorithm[ 26 57 69 ],quadraticprogrammingalgorithm[ 18 37 ],nonlinearprogrammingalgorithm[ 39 ],etc.TheLPmethodapplicationtothepowersystemreschedulingproblemwithsecurityconstrainedeconomicdispatch/controlformultiplevalvedturbineunitswasgivenbyStottandMarinho[ 57 ].Rosehartetal.[ 48 ]discoveredthatfortheeconomicdispatchproblem,SLPappearstobeabettertoolthanSQP.AnapproachbasedonefcientSLPtechniquestosolvethemultiobjectiveenvironmental/economicloaddispatchproblemwaspresentedbyZeharandSayah[ 69 ].Granellietal.[ 18 ]solvedasecurityconstrainedeconomicdispatchproblemusingmodiedSQPtechniques.Adualfeasiblestartingpointisfoundbyrelaxingtransmissionlimitsandthenconstraintviolationsareenforcedapplyingthedualquadraticalgorithm.In[ 59 ]and[ 35 ],asecurityconstrainedeconomicdispatchproblemwassolvedbySLPandtheinteriorpointdualafnescalingalgorithm.Momohetal.[ 37 ]proposedanIPMforEDproblemformulatedaslinearandconvexQP.Howevereachoftraditionalmethodshassomedefects:itwouldgeneratelargeerrorstousethelinearprogrammingalgorithmtolinearizetheEDmodel;forthequadraticprogrammingandnonlinearprogrammingalgorithms,theobjectivefunctionshouldbecontinuousanddifferentiable[ 30 ].Thesecondcategorycontainsthemethodsbasedonarticialintelligence.ArticialintelligencetechnologyhasbeensuccessfullyusedtosolvetheEDproblem.Achaosoptimizationalgorithm(CAO)hasbeenproposedbyJiangetal.[ 29 ]todealwiththeeconomicdispatchproblemofahydropowerplant.Zhijiangetal.[ 71 ]alsoapplieda 11 PAGE 12 COAandthesimulationresultsveriedthattheproposedapproachiseffectiveandprecise.AmutativescaleCOAwasappliedbyXuetal.[ 65 ]totheeconomicoperationofpowerplants.However,theresultsshowedthatthemethodistimeconsuming.AnimprovedmutativescaleCOAhesbeendevelopedbyHanandLu[ 19 ].Accordingtotheauthors,theiralgorithmishighlyefcientandcanbeappliednotonlytoEDbuttomanypowersystemproblems,suchaseconomicoperation,OPF,systemidenticationandoptimalcontro,aswelll.In[ 36 ],Mahdadetal.proposedanefcientdecomposedparallelGAtosolvethemultiobjectiveenvironmental/economicdispatchproblem.Intherststage,theoriginalnetworkisdecomposedintomultisubsystemsandtheproblemistransformedtooptimizetheactivepowerdemandassociatedwitheachpartitionednetwork.Inthesecondstage,anactivepowerdispatchstrategyisproposedtoenhancethenalsolutionoftheoptimalpowerowoftheoriginalnetwork.TheproposedapproachwastestedontheAlgerian59bustestsystem.Thecomputationalresultsshowedtheconvergenceatthenearsolutionandobtainacompetitivesolutionatareducedtime.GAswithfuzzylogiccontrollerstoadjustitscrossoverandmutationprobabilitieswasappliedbySongetal.[ 56 ]tosolveacombinedenvironmentaleconomicdispatchproblem.SAtechniqueswereusedbyRoaSepulvedaandPavezLazo[ 47 ],however,longcomputationaltimetoobtainanoptimalsolutionwasreported.TabusearchwasappliedbyAltunandYalcinoz[ 2 ].Simulationresultsonpowersystemsconsistingof6and20generatingunitsexhibitedgoodperformance.In[ 38 ],anapplicationofTSforsolvingsecurityconstrainedEDproblemwasgivenbyMuthuselvanandSomasundaram.Basecaseandcontingencycaselineowconstraintswereconsidered.Testson66busand191busIndianutilitysystemsrevealedthereliability,efciencyandsuitabilityoftheproposedalgorithmforpracticalapplications.Thethirdcategoryconsiststhehybridmethods,whichcombinetwoormoretechniquesinordertogetbestfeaturesineachalgorithm.Typically,signifcantimprovementwithhybridmethodscanbeachievedovereachoftheindividualmethods. 12 PAGE 13 Hybridmethodsgainedincreasingpopularityinthelast10years.FortheEDproblem,WongandWong[ 63 ]combinedanincrementalGAwithSAtechniques.CoelhoandMariani[ 12 ]proposedamethodcombiningaDEalgorithmwithselfadaptivemutationfactorintheglobalsearchstageandchaoticlocalsearchtechniquesinthelocalsearchtosolveanEDproblemassociatedwiththevalvepointeffect.ThesameauthorsreportanothersuccessfulapplicationofchaoticPSOincombinationwithanimplicitlteringlocalsearchmethodtosolveeconomicdispatchproblems[ 13 ].ThechaoticPSOapproachisusedtoproducegoodpotentialsolutions,whiletheimplicitlteringisusedtonetunethenalsolutionofthePSO.Thehybridmethodologyisvalidatedforatestsystemconsistingof13thermalunitswhoseincrementalfuelcostfunctiontakesintoaccountthevalvepointloadingeffects.In[ 11 ],CoelhoandLeeimprovedPSOapproachesforsolvinganEDproblemtakingintoaccountnonlineargeneratorfeaturessuchasrampratelimits.Prohibitedoperatingzonesinthepowersystemoperationaredevelopedaswell.TheiralgorithmcombinesthePSO,Gaussianprobabilitydistributionfunctionsand/orchaoticsequences.ThePSOanditsvariantsarevalidatedfortwotestsystemsconsistingof15and20thermalgenerationunits,respectively.AcombinationofchaoticandselforganizationbehaviorofantsintheforagingprocesswaspresentedbyCaietal.[ 8 ].ThisalgorithmwasappliedtoEDproblemswiththermalgenerators.Thethesisisorganizedasfollows:InSection 2 ,webrieydiscussageneralEDproblemformulation.ThemethodsappliedtosolveEDareshortlydiscussedinSection 3 .Section 4 describesexperimentalcasesandpresentscalculationresults.WeconcludewithSection 5 13 PAGE 14 CHAPTER2ECONOMICDISPATCH(ED)PROBLEMEDisoneoftheimportantoptimizationproblemsinpowersystemoperations,whichisusedtodeterminetheoptimalcombinationofpoweroutputsofallgeneratingunitstominimizethetotalfuelcostwhilesatisfyingvariousconstraintsovertheentiredispatchperiods[ 67 ].ThetraditionalorstaticEDproblemassumesconstantpowertobesuppliedbyagivensetofunitsforagiventimeintervalandattemptstominimizethecostofsupplyingthisenergysubjecttoconstraintsonthestaticbehaviorofthegeneratingunitslikesystemloaddemand.Shortly,staticEDdeterminestheloadsofgeneratorsinasystemthatwillmeetapowerdemandduringasingleschedulingperiodfortheleastcost.Therefore,itmightfailtocapturelargevariationsoftheloaddemandduetotherampratelimitsofthegenerators.Duetolargevariationofthecustomersloaddemandandthedynamicnatureofthepowersystems,itbecamenecessarytoscheduletheloadbeforehandsothatthesystemcananticipatesuddenchangesindemandinthenearfuture.DynamicEDisanextensionofstaticEDtodeterminethegenerationscheduleofthecommittedunitssothattomeetthepredictedloaddemandovertheentiredispatchperiodsatminimumoperatingcostunderramprateandotherconstraints[ 64 ].Theramprateconstraintisadynamicconstraintwhichusedtomaintainthelifeofthegenerators,i.e.plantoperators,toavoidshorteningthelifeofthegenerator,trytokeepthermalstresswithintheturbinessafelimits[ 20 ].Sincetheviolationsoftheramprateconstraintsareassessedbyexaminingthegeneratorsoutputoveragiventimeinterval,thisproblemcannotbesolvedforasinglevalueofMWgeneration[ 20 ].TheobjectivefunctionofdynamicEDisformulatedasfollows minC(P)=TXt=1NXi=1Ci(Pti)(2) 14 PAGE 15 whereNisthesetofcommittedunits;Piisthegenerationofuniti;Ci(Pi)isthecostofproducingPifromuniti;Tisthenumberofintervalsinthestudyperiod.ThefuelcostfunctionsCi()isderivedfromthefuelconsumptionfunctionthatcanbemeasuredandarediscussedinSection 2.2 .ThedynamicEDisnotonlythemostaccurateformulationoftheeconomicdispatchproblembutalsothemostdifculttosolvebecauseofitslargedimensionality[ 3 ].TheDEDproblemisnormallysolvedbydiscretizationoftheentiredispatchperiodintoanumberofsmalltimeintervals,overwhichtheloaddemandisassumedtobeconstantandthesystemisconsideredtobeinatemporalsteadystate.OvereachtimeintervalastaticEDproblemissolvedunderstaticconstraintsandtheramprateconstraintsareenforcedbetweentheconsecutiveintervals[ 34 ].IntheDEDproblemtheoptimizationisdonewithrespecttothedispatchablepowersoftheunits.SomeresearchershaveconsideredtheramprateconstraintsbysolvingSEDproblemintervalbyintervalandenforcingtheramprateconstraintsfromoneintervaltothenext.However,thisapproachcanleadtosuboptimalsolutions[ 23 ];moreover,itdoesnothavethelookaheadcapability.SincedynamicEDwasintroduced,variuosmethodshavebeenusedtosolvethisproblem.However,allofthosemethodsmaynotbeabletoprovideanoptimalsolutionandusuallygettingstuckatalocaloptimal. 2.1EDConstraintsTheconstrainedEDproblemissubjectedtoavarietyofconstraintsdependinguponassumptionsandpracticalimplications.Usually,formulationofEDproblemincludessuchconstraintsasloadgenerationbalance,minimumandmaximumcapacityconstraints.Tomaintainsystemreliabilityandsecurity,spinningreserveconstraintsandsecurityconstraintscanbeaddedtothedynamicEDproblem.Theinclusionoftheprohibitedzones,rampratelimitsandotherpracticalconstraintsresultsinnonconvexEDofgeneratingunits.Alltheseconstraintsarediscussedbellow. 15 PAGE 16 2.1.1LoadGenerationBalanceThegeneratedpowerfromalltherunningunitsmustsatisfytheloaddemandandthesystemlossesgivenby( 2 ) NXi=1Pti=Dt+Losst,t=1,2,...,T(2)whereDtisthedemandandLosstisthesystemtransmissionloss.Theirsumrepresentstheeffectiveloadtobesatisedatthetthinterval.Thetransmissionlinelossescanbeexpressedintermsoftheunitoutputs:Losst=NXi=1NXj=1PtiBijPtj+NXi=1Bi0Pti+B00whereBijistheijthelementofthelosscoefcientsquarematrix,Bi0istheithelementofthelosscoefcient,andB00istheconstantlosscoefcient.Sometimesthelasttwotermsareomitted.Inacompetitiveenvironment,theloadgenerationbalanceconstraintisrelaxedandeachgeneratingcompanyschedulesitsproductiontomaximizeitsprotsgivenaforecastofelectricitypricesfortheschedulingperiod.Asarstapproximation,eachgeneratingunitcouldbeoptimizedseparatelyinthisproblembecauseofthedecouplingmadepossiblebytheavailabilityofpricesateachperiod.Dynamicconstraints(suchasrampratesandminimumupanddowntimeconstraints)complicatetheproblembecauseageneratingcompanythatownsaportfolioofunitsmustthendecidewhethertobuyexibilityonthemarketormeetthedynamicconstraintswithitsownresources[ 21 ]. 2.1.2GenerationCapacityConstraintFornormalsystemoperations,realpoweroutputofeachgeneratorisrestrictedbylowerandupperboundsasfollows: Pti+StiPmaxii=1,2,...,N,t=1,2,...,T(2) 16 PAGE 17 PminiPtii=1,2,...,N,t=1,2,...,T(2)wherePminiandPmaxiaretheminimumandmaximumpowerproducedbygeneratori,Stiisthereservecontributionofunitduringtimeintervalt. 2.1.3GeneratingUnitRampRateLimitsOneofunpracticalassumptionthatprevailedforsimplifyingtheprobleminmanyoftheearlierresearchisthattheadjustmentsofthepoweroutputareinstantaneous[ 43 ].Therefore,thepoweroutputofapracticalgeneratorcannotbeadjustedinstantaneouslywithoutlimits.Theoperatingrangeofallonlineunitsisrestrictedbytheirrampratelimitsduringeachdispatchperiod.So,thesubsequentdispatchoutputofageneratorshouldbelimitedbetweentheconstraintsofupanddownramprates[ 66 ]asfollowsPt+1i)]TJ /F4 11.955 Tf 11.96 0 Td[(PtiURit (2)Pti)]TJ /F4 11.955 Tf 11.96 0 Td[(Pt+1iDRiti=1,2,...,N,t=1,2,...,T)]TJ /F5 11.955 Tf 11.96 0 Td[(1 (2)whereURiandDRiarethemaximumrampup/downratesforunitiandtisthedurationofthetimeintervalsintowhichthestudyperiodisdivided.Theinclusionoframpratelimitsmodiesthegeneratoroperationconstraints( 2 2 )asfollows max(Pmini,Pt)]TJ /F8 7.97 Tf 6.59 0 Td[(1i)]TJ /F4 11.955 Tf 11.95 0 Td[(DRi)Pimin(Pmaxi,Pt)]TJ /F8 7.97 Tf 6.59 0 Td[(1i+URi)(2) 2.1.4ReserveContributionThemaximumreservecontributionhastosatisfyfollowingconstraints: 0StiSmaxii=1,2,...,N,t=1,2,...,T(2)whereSmaxiisthemaximumcontributionofunititothereservecapacity.Maximumrampspinningreservecontributionisdenedasin( 2 ) 0StiURiti=1,2,...,N,t=1,2,...,T(2)whereStiisthespinningreserveofuniti. 17 PAGE 18 2.1.5SystemSpinningReserveRequirementSufcientspinningreserveisrequiredfromallrunningunitstomaximizeandmaintainsystemreliability[ 14 ].Therearemanywaystodeterminethesystemspinningreserverequirement.Itcanbecalculatedasthesizeofthelargestunitinoperationorasapercentageofforecastloaddemandorevenasafunctionoftheprobabilityofnothavingsufcientgenerationtomeettheload[ 64 ].Thespinningrezervecanbedenedby( 2 ) NXi=1StiSRtt=1,2,...,T(2)whereSRtisthesystemspinningreserverequirementfortimeintervalt.Also,thesystemspinningreserverequirementforintervaltcansometimesbegivenbythefollowingequation[ 20 ]: SRt=dDt+gmax(Pmaxischeduledattimet,i=1,2,...N)(2)wheredandgareconstantswhichdependonthesystemrequiredreliabilitylevel[ 55 ].Besidesthedeterminationofthesystemspinningreserverequirement,theissueofallocationthespinningreserveamongthecommittedunitsisveryimportant;however,ithasreceivedverylittleattentioninthedynamicEDliterature. 2.1.6TielineLimitsTheeconomicdispatchproblemcanbeextendedbyimportingadditionalconstraintliketransmissionlinecapacitylimitgivenby( 2 ) PTjk,minPTjk+SjkPTjk,max(2)wherePTjk,minandPTjk,maxspecifythetielinetrasnmissioncapability,i.e.thetransferfromareajtoareakshouldnotexceedthetielinetransfercapacitiesforsecurityconsideration[ 28 ].Eachareahasownspecialloadanditsspinningreserve[ 68 ]. 18 PAGE 19 2.1.7ProhibitedZoneThegeneratingunitsmayhavecertainrangeswhereoperationisrestrictedonthegroundsofphysicallimitationsofmachinecomponentsorinstability,e.g.duetosteamvalveorvibrationinshaftbearings.So,thereisaquesttoavoidoperationinthesezonesinordertoeconomizetheproduction[ 43 ].Theserangesareprohibitedfromoperationandageneratorwithprohibitedregions(zones)hasdiscontinuousfuelcostcharacteristics(Fig. 2.1.7 )[ 53 ].TheacceptableoperatingzonesofageneratingunitcanbeformulatedasfollowsPminiPtiPli,1 (2)Pui,j)]TJ /F8 7.97 Tf 6.59 0 Td[(1PtiPli,j,i2,j=2,3,...,ni,t=1,2,...,T (2)Pui,niPtiPmaxi (2)whereniisthenumberoftheprohibitedzonesinuniti,isthesetofunitsthathaveprohibitedzones,Pli,j,Pui,jarethelowerandupperboundsofthejthprohibitedzone. Figure21. Exampleofcostfunctionwithtwoprohibitedoperatingzones 2.2ObjectiveFunctionsThedynamicEDproblemhasbeensolvedwithmanydifferentformsofthecostfunction,suchasthesmoothquadraticcostfunction( 2 )orthenonsmoothcost 19 PAGE 20 functionduetothevalvepointeffects( 2 ).Also,alinearcostfunction[ 20 ]andpiecewiselinearcostfunction[ 27 41 ]havebeenemployed.Forsmoothcostfunctionitisusuallyassumedthatitsincrementalcostfunction.Insomepowersystemscombinedcycleunitsareusedtosupplythebaseload.Fortheseunitsthecostfunctioncanbegivenaslinear,piecewiseorquadraticwithdecreasingincrementalcostfunction[ 41 ].Forunitswithprohibitedzones,thefuelcostfunctionisdiscontinuousandnonconvex.AninterestingdeparturefromthisstandardformulationistheapproachproposedbyWangandShahidehpour[ 61 ]whoincludeintheobjectivefunctionatermrepresentingthereductioninthelifeoftheturbinecausedbyexcessiverampingrates.Thisexibletechniquemakespossibleatradeoffbetweenthesystemoperatingcostandthelifecyclecostofthegeneratingunits[ 21 ]. 2.2.1SmoothCostFunctionThemostsimpliedcostfunctionofeachgeneratorcanberepresentedasaquadraticfunctionasgivenin( 2 )whosesolutioncanbeobtainedbytheconventionalmathematicalmethods Ci(Pti)=ai+biPti+ci(Pti)2(2)whereai,bi,ciarecostcoefcientsofgeneratori. 2.2.2NonsmoothCostFunctionswithValvepointEffectsThegeneratingunitswithmultivalvesteamturbinesexhibitagreatervariationinthefuelcostfunctionsbecauseinordertomeettheincreaseddemandageneratorwithmultivalvesteamturbinesincreaseitsoutputandvarioussteamvalvesaretobeopened[ 67 ].Thisvalveopeningprocessproducesripplelikeeffectintheheatratecurveofthegenerator.Theinclusionofvalvepointloadingeffectsmakesthemodelingoftheincrementalfuelcostfunctionofthegeneratorsmorepractical[ 60 ].Therefore,inreality,theobjectivefunctionofEDproblemhasnondifferentiableproperty.Consequently,theobjectivefunctionshouldbecomposedofasetofnonsmoothcostfunctions.Consideringnonsmoothcostfunctionsofgenerationunitswithvalvepoint 20 PAGE 21 Figure22. Costfunctionwithvalvepointeffects effects,theobjectivefunctionisgenerallydescribedasthesuperpositionofsinusoidalfunctionsandquadraticfunctions[ 52 ] Ci(Pti)=ai+biPti+ci(Pti)2+jeisin(hi(Pmini)]TJ /F4 11.955 Tf 11.96 0 Td[(Pti))j(2)whereeiandhiarethecoefcientsofgeneratorireectingvalvepointeffects.AsshowninFig. 2.2.2 ,thisincreasesthenonlinearityofcurveaswellasnumberoflocaloptimainthesolutionspace[ 60 ]comparedwiththesmoothcostfunctionduetothevalvepointeffects.Alsothesolutionprocedurecaneasilytrapinthelocaloptimainthevicinityofoptimalvalue. 2.2.3NonsmoothCostFunctionswithMultipleFuelsSincethedispatchingunitsarepracticallysuppliedwithmultifuelsources[ 49 ],eachunitshouldberepresentedwithseveralpiecewisequadraticfunctionsreectingtheeffectsoffueltypechanges,andthegeneratormustidentifythemosteconomicfueltoburn.Theresultingcostfunctioniscalledahybridcostfunction.Eachsegmentofthehybridcostfunctionimpliessomeinformationaboutthefuelbeingburnedorthe 21 PAGE 22 Figure23. Costfunctionwithmultiplefuels unitsoperation.Thus,generally,thefuelcostfunctionisapiecewisequadraticfunctiondescribedasfollows ci(Pi)=8>>>>>>>>>><>>>>>>>>>>:ai1+bi1Pti+ci1(Pti)2ifPti,minPtiPti,1ai2+bi2Pti+ci2(Pti)2ifPti,1PtiPti,2......ain+binPti+cin(Pti)2ifPti,n)]TJ /F8 7.97 Tf 6.59 0 Td[(1PtiPti,max(2)whereareaip,bip,cipthecostcoefcientsofgeneratorforthepthpowerlevel.TheincrementalcostfunctionsareillustratedinFig.( 2.2.3 ) 2.2.4NonsmoothCostFunctionswithValvePointEffectsandMultipleFuelsToobtainanaccurateandpracticaleconomicdispatchsolution,therealisticoperationoftheEDproblemshouldconsiderbothvalvepointeffectsandmultiplefuels.Thereference[ 10 ]proposedanincorporatedcostmodel,whichcombinesthevalvepointloadingsandthefuelchangesintooneframe.Therefore,thecostfunctionshouldcombine( 2 )with( 2 ),andcanberealisticallyrepresentedasshownin 22 PAGE 23 ( 2 ) ci(Pi)=8>>>>>>>>>><>>>>>>>>>>:ai1+bi1Pti+ci1(Pti)2+jei,1sin(hi,1(Pmini,1)]TJ /F4 11.955 Tf 11.96 0 Td[(Pti,1))jifPti,minPtiPti,1ai2+bi2Pti+ci2(Pti)2+jei,2sin(hi,2(Pmini,2)]TJ /F4 11.955 Tf 11.96 0 Td[(Pti,2))jifPti,1PtiPti,2......ain+binPti+cin(Pti)2+jei,nsin(hi,n(Pmini,n)]TJ /F4 11.955 Tf 11.96 0 Td[(Pti,n))jifPti,n)]TJ /F8 7.97 Tf 6.59 0 Td[(1PtiPti,max(2) 2.2.5EmissionFunctionDuetoincreasingconcernovertheenvironmentalconsiderations,societydemandsadequateandsecureelectricity,i.e.notonlyatthecheapestpossibleprice,butalsoatminimumlevelofpollution.Inthiscase,twoconictingobjectives,i.e.,operationalcostsandpollutantemissions,shouldbeminimizedsimultaneously[ 4 5 7 62 ].Theatmosphericpollutantssuchassulphuroxides(SOx)andnitrogenoxides(NOx)causedbyfossilfueledgeneratingunitscanbemodeledseparatelyorasthetotalemissionofthemwhichisthesumofaquadratic[ 4 ]andanexponentialfunctionandcanbeexpressedas TXt=1NXi=1i+iPti+i(Pti)2+iexp(iPti)(2)wherei,i,i,i,andiareemissioncoefcientsofithgeneratingunit. 23 PAGE 24 CHAPTER3SOLUTIONMETHODS 3.1ContinuousGreedyRandomizedAdaptiveSearchProcedure(CGRASP)ContinuousGRASP(CGRASP)extendsthegreedyrandomizedadaptivesearchprocedure(GRASP)thatwasintroducedbyFeoandResende[ 16 17 ]fromthedomainofdiscreteoptimizationtothatofcontinuousglobaloptimizationin[ 24 25 ].Itisdescribedasamultistartlocalsearchprocedure,whereeachCGRASPiterationconsistsoftwophases,namely,aconstructionphaseandalocalsearchphase[ 24 ].Constructioncombinesgreedinessandrandomizationtoproduceadiversesetofgoodqualitystartingsolutionsforlocalsearch.Thelocalsearchphaseattemptstoimprovethesolutionsfoundbyconstruction.Thebestsolutionoveralliterationsiskeptastheinitialsolution.TheadvantagesofthismethodissimplicitytoimplementandnorequirementforderivativeinformationPseudocodeforCGRASPisshownin( 3.1 ).CGRASPworksbydiscretizingthedomainintoauniformgrid.Boththeconstruction(seethehighlevelpseudocode 3.2 )andlocalimprovementphases(seethehighlevelpseudocode 3.3 )movealongpointsonthegrid.Asthealgorithmprogresses,thegridadaptivelybecomesmoredense.ThemaindifferencebetweenGRASPandCGRASPisthataniterationofCGRASPdoesnotconsistofasinglegreedyrandomizedconstructionfollowedbylocalimprovement,butratheraseriesofconstructionlocalimprovementcycleswiththeoutputofconstructionservingastheinputofthelocalimprovement,asinGRASP,butunlikeGRASP,theoutputofthelocalimprovementservesastheinputoftheconstructionprocedure[ 25 ].SinceCGRASPisessentiallyanunconstrainedoptimizationalgorithm,theconstraintshandlingstrategyneedstobeincorporatedintoitinordertodealwiththeconstrainedEDproblem.Approachestomanagetheseconstraintsarediscussedinsection 3.4 24 PAGE 25 pseudocode3.1CGRASP(n,l,u,f(),MaxIters,MaxNumIterNoImprov,NumTimesToRun,MaxDirToTry,) 1: f 1 2: forj 1,...,NumTimesToRundo 3: x UnifRand(l,u);h 1;NumIterNoImprov 0; 4: forIter 1,...,MaxItersdo 5: x ConstructGreedyRandomized(x,f(),n,h,l,u,); 6: x LocalSearch(x,f(),n,h,l,u,MaxDirToTry); 7: iff(x) PAGE 26 3.2GeneticAlgorithms(GA)Thissectionengagesintotheconceptofgeneticalgorithmsthatreectsthenatureofchromosomesingeneticengineering.GAsareaclassofstochasticsearchalgorithmsthatstartwiththegenerationofaninitialpopulationorsetofrandomsolutionsfortheproblemathand.Eachindividualsolutioninthepopulationcalledachromosomeorstringrepresentsafeasiblesolution.Theobjectivefunctionisthenevaluatedfortheseindividuals.Ifthebeststring(orstrings)satisestheterminationcriteria,theprocessterminates,assumingthatthisbeststringisthesolutionoftheproblem.Iftheterminationcriteriaarenotmet,thecreationofnewgenerationstarts,pairs,orindividualsareselectedrandomlyandsubjectedtocrossoverandmutationoperations.Theresultingindividualsareselectedaccordingtotheirtnessfortheproductionofthenewoffspring.Geneticalgorithmscombinetheelementsofdirectedandstochasticsearchwhileexploitingandexploringthesearchspace[ 31 ].MoredetailsaboutGAcanbefoundin[ 22 46 58 ]. pseudocode3.4Geneticalgorithm 1: initializepopulation() 2: whilenotconvergedo 3: assignpopulationtness() 4: for1,...,npopsizdo 5: selectparents(p1,p2) 6: reproduction(p1,p2,child) 7: endfor 8: selectnextgeneration() 9: endwhile TheadvantagesofGAoverothertraditionaloptimizationtechniquescanbesummarizedasfollows: GAsearchesfromapopulationofpoints,notasinglepoint.Thepopulationcanmoveoverhillsandacrossvalleys.GAcanthereforediscoveragloballyoptimalpoint,becausethecomputationforeachindividualinthepopulationisindependentofothers.GAhasinherentparallelcomputationability. 26 PAGE 27 GAusespayoff(tnessorobjectivefunctions)informationdirectlyforthesearchdirection,notderivativesorotherauxiliaryknowledge.GAthereforecandealwithnonsmooth,noncontinuousandnondifferentiablefunctionsthatarethereallifeoptimizationproblems.ThispropertyalsorelievesGAoftheapproximateassumptionsforalotofpracticaloptimizationproblems,whicharequiteoftenrequiredintraditionaloptimizationmethods. GAusesprobabilistictransitionrulestoselectgenerations,notdeterministicrules.Theycansearchacomplicatedanduncertainareatondtheglobaloptimum.GAismoreexibleandrobustthantheconventionalmethods[ 33 ].Therstattemptoftheapplicationofgeneticalgorithmsinpowersystemsisintheloadowproblem[ 70 ].Ithasbeenfoundthatthesimplegeneticalgorithmquicklyndsthenormalloadowsolutionforsmallsizenetworksbyspecifyinganadditionaltermintheobjectivefunction.AnumberofapproachestoimprovingconvergenceandglobalperformanceofGAshavebeeninvestigated[ 70 ]. 3.3SimulatedAnnealing(SA)TheSAisagenericprobabilisticmetaheuristicfortheglobaloptimizationproblemthatwasproposedbyKirkpatricetal.[ 32 ].IntheSAmethod,eachpointsofthesearchspaceisanalogoustoastateofsomephysicalsystem,andthefunctionE(s)tobeminimizedisanalogoustotheinternalenergyofthesysteminthatstate.Thegoalistobringthesystem,fromanarbitraryinitialstate,toastatewiththeminimumpossibleenergy.IneachstepoftheSAalgorithmthecurrentsolutionisreplacedbyarandomnearbysolution,chosenwithaprobabilitythatdependsonthedifferencebetweenthecorrespondingfunctionvaluesandonaglobalparameterT(calledthetemperature),thatisgraduallydecreasedduringtheprocess.ThedependencyissuchthatthecurrentsolutionchangesalmostrandomlywhenTislarge,butincreasinglydownhillasTgoestozero.Theallowanceforuphillmovessavesthemethodfrombecomingstuckatlocalminimawhicharethebaneofgreediermethods.Forcertainproblems,SAmaybemoreeffectivethanexhaustiveenumeration.Ithasbeenshownthatthistechniqueconvergesasymptoticallytotheglobaloptimalsolutionwithprobabilityone[ 1 ]. 27 PAGE 28 SAisaneffectiveglobaloptimizationalgorithmbecauseofthefollowingadvantages[ 50 ]: suitabilitytoprobleminwidearea, norestrictionontheformofcostfunction, highprobabilitytondglobaloptimization, easyimplementationbyprogramming.ThepseudocodeimplementingSAisgivenbellow.Itstartsfromstates0andcontinueforkmaxofstepsoruntilastatewithenergyemaxorlessisfound.Thecallneighbour(s)shouldgeneratearandomlychosenneighbourofagivenstates;thecallrandom()shouldreturnarandomvalueintherange[0,1].Theannealingscheduleisdenedbythetemp(r),whichshouldyieldthetemperaturetouse,giventhefractionrofthetimebudgetthathasbeenexpendedsofar. pseudocode3.5SimmulatedAnnealing 1: s s0;e E(s) 2: sbest s;ebest e; 3: k 0; 4: whilek PAGE 29 standardoptimization,thatcanbeusedinanymetaheuristic,itbreakstheanalogywithphysicalannealingsinceaphysicalsystemcanstoreasinglestateonly.Instrictmathematicalterms,savingthebeststateisnotnecessarilyanimprovement,sinceonemayhavetospecifyasmallerkmaxinordertocompensateforthehighercostperiteration.However,thestepsbest snewhappensonlyonasmallfractionofthemoves.Therefore,theoptimizationisusuallyworthwhile,evenwhenstatecopyingisanexpensiveoperation.SAhastheabilitytoavoidgettinglocalsolutions;thenitcangenerateglobalornearglobaloptimalsolutionsforoptimizationproblemswithoutanyrestrictionontheshapeoftheobjectivefunctions[ 44 ].SAisnotmemoryintensive[ 45 ].However,thesettingofcontrolparametersoftheSAalgorithmisadifculttaskandthecomputationtimeishigh[ 3 ].Thecomputationalburdencanbereducedbymeansofparallelprocessing[ 44 ]. 3.4ConstraintsHandlingConstraintslieattheheartofallconstrainedengineeringoptimizationapplications.Practicalconstraints,whichareoftennonlinearandnontrivial,connethefeasiblesolutionstoasmallsubsetoftheentiresearchspace.Thereareseveralapproacheswhichcanbeappliedtohandleconstraintsinheuristicapproaches.Thesemethodscanbegroupedintofourcategories:methodsthatpreservethefeasibilityofsolutions,penaltybasedmethods,methodsthatclearlydistinguishbetweenfeasibleandunfeasiblesolutions,andhybridmethods[ 15 62 ]. 3.4.1PenaltyBasedApproachThepenaltyfunctionmethodisfrequentlyappliedtomanageconstraintsinevolutionaryalgorithms.Suchatechniqueconvertstheprimalconstrainedproblemintoanunconstrainedproblembypenalizingconstraintviolations.Thepenaltyfunctionmethodissimpleinconceptandimplementation.However,itsprimallimitationisthedegreetowhicheachconstraintispenalized.Thesepenaltytermshavecertainweaknessesthatbecomefatalwhenpenaltyparametersarelarge.Suchapenalty 29 PAGE 30 functiontendstobeillconditionedneartheboundaryofthefeasibledomainwheretheoptimumpointisusuallylocated[ 10 ].ThepenalizedfuelcostfunctioninEDproblemwasemployedin[ 51 ].In[ 40 ]theEDproblemwastransformedintoanunconstrainedonebyconstructinganaugmentedobjectivefunctionincorporatingpenaltyfactorsforanyvalueviolatingtheconstraints: H(X)=J(X)+k1NeqXj=1(hj(X))2+k2NueqXj=1max[0,)]TJ /F4 11.955 Tf 9.3 0 Td[(gj(X)]2(3)whereJ(X)istheobjectivefunctionvalueoftheEDproblem.NeqandNueqarethenumberofequalityandinequalityconstraints,respectivel;hj(X)andgj(X)aretheequalityandinequalityconstraints,respectively;k1andk2arethepenaltyfactors.Sincetheconstraintsshouldbemet,thevalueofthek1andk2parameterswerechosentohavehighvalueof10,000.ThisapproachwasepmpoyedwhenapplyingSAmethod.TheheuristicstartegythatisdiscussedinnexsectionwasusedtogetafeasiblesolutionwhileapplyingCGRASPmethod. 3.4.2HeuristicStrategyWhentheCGRASPisappliedtosolveEDproblem,akeyproblemishowtohandleconstraintswithefciency.Inthissectionwemainlyfocusonhandlingtherealpowerlimitsandgeneratorsrampupconstraints.Otherthanpenaltybasedwaytosatisfytherealpowerbalanceequalityconstraints( 2 ),istospecifytheoutputof(N)]TJ /F5 11.955 Tf 12.05 0 Td[(1)generatingunitsandtondtheNthfromtheequalityconstraintlikein[ 4 67 ].Inreference[ 67 ],authorsemployedadependentgenerationpowerptlofrandomlyselectedunitl.TheheuristicstrategyappliedinLocalSearch()procedureinCGRASPalgorithmcanbeformulatedinafollowingway:Step1.Setthedispatchperiodindext=1anditerationi=1. 30 PAGE 31 Step2.CalculatetheviolationofpowerblanceconstraintPterratdispatchtimetiscalculatedfrom 3 asfollows Pterr=Dt+Losst)]TJ /F6 7.97 Tf 17.3 14.95 Td[(NXi=1Pti(3)IfPterr=0,thengotoStep5,otherwisetoStep3.Step3.Randomlygenerateltheindexofgeneratingunitandcalculatetherealpowerofselecteddependentgeneratingunitptlfrom( 3 ). Ptl=Dt)]TJ /F6 7.97 Tf 17.3 14.94 Td[(NXi=1i6=lPtit=1,2,...,T(3)However,consideringtransmissionlosses( 2.1.1 ),theseequalityconstraintsbecomenonlinearandtheoutputofdependentgeneratingunitforeverydispatchperiodtcanbefoundfrombysolvingafollowingequation Bll(Ptl)2+(2NXi=1i6=lBliPti+Bl0)]TJ /F5 11.955 Tf 9.48 0 Td[(1)Ptl+(Dt+NXi=1i6=lNXj=1j6=lPtiBijPtj+B00+NXi=1i6=lBi0Pti)]TJ /F6 7.97 Tf 14.83 14.94 Td[(NXi=1i6=lPti)=0(3)Ifitdoesn'tviolatethegeneratoroperatinglimitsandrampupconstraints(iftheyarepresent),gotoStep5.Otherwise,thevaluehastobemodiedaccordingto 3 Ptl=8>><>>:PmaxlifPtl>PmaxlPminlifPtl PAGE 32 Step4.Increasetheiterationnumberby1,i.e.l=l+1.Ifl PAGE 33 CHAPTER4EXPERIMENTSANDRESULTS 4.1ExperimentsInordertoverifythefeasibilityandeffectivenessofadoptedCGRASPcapabilitiesforsolvingEDproblems,differentEDproblemformulations,i.e.staticanddynamicEDanddifferentsystemswereused.TheCGRASPalgotihmwithheuristicstrategytodealwithconstraintswasimplementedinMatlab7.5.ForGAandSAalgorithms,thestandardMatlabfunctionsformGeneticAlgorithmandDirectSearchToolboxwereemployed.InstandardGAfunctionga(),itispossibletoincludebothlinearandnonlinearequalityandinequalityconstraints.However,SAfunctionsimulannealbnd()incorporatesonlylowerandupperboundconstraints,otherconstraintsasapenaltyfunctionisaddedtoobjectivefunction.Next,wewillprovidedescriptionsofsystemsusedforourexperiments. 4.1.1System1Thesystemconsistsofvegeneratingunits,whosethemaximumtotaloutputis925MW.OnthissystemdynamicEDproblemwassolvedwiththedispatchhorizononedaywith12intervalsofonehoureach.ThedemandofthesystemandgeneratingunitdataaregiveninTables( 41 )and( 42 ),respectively. Table41. Generatingunitscharacteristicsofveunitsystem ai,$/hbi,$/MWhci,$/(MW)2hPmini,MWPmaxi,MW Unit1 25.0002.0000.00810.00075.000Unit2 60.0001.8000.00320.000125.000Unit3 100.0002.1000.001230.000175.000Unit4 120.0002.0000.00140.000250.000Unit5 40.0001.8000.001550.000300.000 4.1.2System2Thesystemcontainssixthermalgeneratingunits.Thetotalmaximumoutputofgeneratingunitsis1470MW.ThissystemwasusedtosolvestaticEDproblemwhereloaddemandonthesystemis1263MW.Parametersofallthethermalunitsare 33 PAGE 34 Table42. LoaddemandTime,h Load,MW Time,h Load,MW 1 410 7 6262 435 8 6543 475 9 6904 530 10 7045 558 11 7206 608 12 740 reportedin[ 30 ]andaregiveninTables 43 and 44 .Innormaloperationofthesystem,thelosscoefcientsBareasfollows:Bij=26666666666666640.00170.00120.0007)]TJ /F5 11.955 Tf 9.3 0 Td[(0.0001)]TJ /F5 11.955 Tf 9.3 0 Td[(0.0005)]TJ /F5 11.955 Tf 9.3 0 Td[(0.00020.00120.00140.00090.0001)]TJ /F5 11.955 Tf 9.3 0 Td[(0.0006)]TJ /F5 11.955 Tf 9.3 0 Td[(0.00010.00070.00090.00310.0001)]TJ /F5 11.955 Tf 9.3 0 Td[(0.001)]TJ /F5 11.955 Tf 9.3 0 Td[(0.0006)]TJ /F5 11.955 Tf 9.3 0 Td[(0.00010.000100.0024)]TJ /F5 11.955 Tf 9.3 0 Td[(0.0006)]TJ /F5 11.955 Tf 9.3 0 Td[(0.0008)]TJ /F5 11.955 Tf 9.3 0 Td[(0.0005)]TJ /F5 11.955 Tf 9.3 0 Td[(0.0006)]TJ /F5 11.955 Tf 9.3 0 Td[(0.001)]TJ /F5 11.955 Tf 9.3 0 Td[(0.00060.0129)]TJ /F5 11.955 Tf 9.3 0 Td[(0.0002)]TJ /F5 11.955 Tf 9.3 0 Td[(0.0002)]TJ /F5 11.955 Tf 9.3 0 Td[(0.0001)]TJ /F5 11.955 Tf 9.3 0 Td[(0.0006)]TJ /F5 11.955 Tf 9.3 0 Td[(0.0008)]TJ /F5 11.955 Tf 9.3 0 Td[(0.00020.015377777777777777510)]TJ /F8 7.97 Tf 6.59 0 Td[(2B0i=[)]TJ /F5 11.955 Tf 9.29 0 Td[(0.3908)]TJ /F5 11.955 Tf 11.95 0 Td[(0.12970.70470.05910.2161)]TJ /F5 11.955 Tf 11.95 0 Td[(0.6635]10)]TJ /F8 7.97 Tf 6.58 0 Td[(3B00=0.056 Table43. GeneratingunitscharacteristicsofsixunitsystemUnit Pmini,MWPmaxi,MWai,$/hbi,$/MWhci,$/(MW)2hP0i,MW 1 1005000.00772404402 502000.0095102001703 803000.0098.52202004 501500.009112001505 502000.00810.52201906 501200.007512190110 34 PAGE 35 Table44. RumpuplimitsandprohibitedzonesofsixunitsystemUnit URi,MWDRi,MWProhibitedzone 1 80120[210240][350380]2 5090[90110][140160]3 65100[150170][210240]4 5090[8090][110120]5 5090[90110][140150]6 5090[7585][100105] 4.1.3System3Thissystemconsistsof13generatingunitswithvalvepointloadingasgiveninTable( 45 ).Theparametersofthissystemshowedistakenfrom[ 54 ].Theexpecteddemandis1800MWand2520MW. Table45. Generatingunitscharacteristicsof13unitsystemUnit Pmini,MWPmaxi,MWaibicieifi 1 06800.000288.15503000.0352 03600.000568.13092000.0423 03600.000568.13072000.0424 601800.003247.742401500.0635 601800.003247.742401500.0636 601800.003247.742401500.0637 601800.003247.742401500.0638 601800.003247.742401500.0639 601800.003247.742401500.06310 401200.002848.61261000.08411 401200.002848.61261000.08412 551200.002848.61261000.08413 551200.002848.61261000.084 4.1.4System4Thissystemiscomposedof40generatingunitswithvalvepointloadingeffectssupplyingatotaldemandof10500MW.Therefore,thissystemhasnonconvexsolutionspacesandtherearemanylocalminimaduetovalvepointeffectsandtheglobalminimumisverydifculttodetermine.TheparametersofthissystemshowedintheTable( 46 )areavailablein[ 54 ]aswell. 35 PAGE 36 Table46. Generatingunitscharacteristicsof40unitsystemUnit Pmini,MWPmaxi,MWaibicieifi 1 361140.00696.7394.7051000.0842 361140.00696.7394.7051000.0843 601200.020287.07309.541000.0844 801900.009428.18369.031500.0635 47970.01145.35148.891200.0776 681400.011428.05222.331000.0847 1103000.003578.03287.712000.0428 1353000.004926.99391.982000.0429 1353000.005736.6455.762000.04210 1303000.0060512.9722.822000.04211 943750.0051512.9635.22000.04212 943750.0056912.8654.692000.04213 1255000.0042112.5913.43000.03514 1255000.007528.841760.43000.03515 1255000.007089.151728.33000.03516 1255000.007089.151728.33000.03517 2205000.003137.97647.853000.03518 2205000.003137.95649.693000.03519 2425500.003137.97647.833000.03520 2425500.003137.97647.813000.03521 2545500.002986.63785.963000.03522 2545500.002986.63785.963000.03523 2545500.002846.66794.533000.03524 2545500.002846.66794.533000.03525 2545500.002777.1801.323000.03526 2545500.002777.1801.323000.03527 101500.521243.331055.11200.07728 101500.521243.331055.11200.07729 101500.521243.331055.11200.07730 47970.01145.35148.891200.07731 601900.00166.43222.921500.06332 601900.00166.43222.921500.06333 601900.00166.43222.921500.06334 902000.00018.95107.872000.04235 902000.00018.62116.582000.04236 902000.00018.62116.582000.04237 251100.01615.88307.45800.09838 251100.01615.88307.45800.09839 251100.01615.88307.45800.09840 2425500.003137.97647.833000.035 36 PAGE 37 4.1.5System5Thissystemhas10generatingunitswithvalvepointloadingeffects.Therefore,thissystemhasnonconvexsolutionspacesandtherearemanylocalminimaduetovalvepointeffects.TheparametersofthissystemaregivenintheTable 410 andareavailablein[ 4 ]aswell.Theforecasteddemandwiththedispatchhorizononedaywith24intervalsofonehoureachisshowninTable 48 Table47. Generatingunitscharacteristicsof10unitsystemUnit Pmini,MWPmaxi,MWaibicieifiURiURi 1 150470786.798838.53970.15244500.04180802 135470451.325146.15910.10586000.03680803 733401049.997740.39650.0283200.02880804 603001243.531138.30550.03542600.05250505 732431658.569636.32780.02112800.06350506 571601356.659238.27040.01793100.04850507 201301450.704536.51040.01213000.08630308 471201450.704536.51040.01213400.08230309 20801455.605639.58040.1092700.098303010 10551469.402640.54070.12953800.0943030 Table48. Loaddemandfor24hoursTime,h Load,MW Time,h Load,MW Time,h Load,MW 1 1036 9 1924 17 14802 1110 10 2022 18 16283 1258 11 2106 19 17764 1406 12 2150 20 19725 1480 13 2072 21 19246 1628 14 1924 22 16287 1702 15 1776 23 13328 1776 16 1554 24 1184 4.2ResultsOneofthefeaturesthattheheuristicalgorithmspossessisrandomness.Therefore,theirperformancescannotbejudgedbytheresultofasinglerunandmanytrialswithdifferentinitializationsshouldbemadetoreachavalidconclusionabouttheperformanceofthealgorithms.Analgorithmisrobust,ifitcanguaranteeanacceptable 37 PAGE 38 performancelevelunderdifferentconditions.Inthispaper,50differentrunsofCGRASPhavebeencarriedout. 4.2.1Case1Inthiscase,thedynamicEDproblemonsystem1issolved.ItcanbeseenfromTable( 49 )thattheCGRASPprovidedthebestsolutioncomparedtoSAandGA. Table49. Generationcostsforcase1 MethodMinAvgMaxSt.Dev. CGRASP19645.8711819725.632919837.5521026.8471GA19817.5020619819.4477319817.502060.93349SA19675.3550819777.3776119855.6388036.1006 ThesmallesttotalproductioncostisobtainedbySAanditis$19645.87.Morever,wecannoticethatontheaverageCGRASPalgorithmperfomsbetterthanSAandGA.ThelowestmaximumvalueisprovidedbyCGRASPaswell,whilethehighestmaximumvaluewasproducedbySA.showsthatSAsolutionsareverysensitivetostartingpointsandaremorevolatile.ThebestfoundsolutionsatisfyingdemandandpowerlimitsisgiveninTable 410 Table410. Bestsolutionforcase1 HourUnit1Unit2Unit3Unit4Unit5 110.0000697520.0000564480.00001477105.6819645194.3178945210.0000363252.6022286430.00004416151.8901212190.5075697326.0079646287.454166380.8405373345.99580774234.701524426.7573418295.20329384170.823281568.03647655169.1796063549.0465060156.0122637891.33351254110.4114641251.1962536619.63298415103.772159751.19984471196.8271002236.5679112736.541522592.8654090961.43881398157.4967338277.6575207813.7051770192.67194012138.6821921178.1269249230.8137659941.8141424102.4755387123.5565313244.850131177.30365661021.88623928102.6285523129.2017437249.9999302200.28353451128.8812865358.20434065161.8266865222.1801306248.90755571225.81847079113.8283152174.999958147.8022832277.5509728 4.2.2Case2Here,thestaticEDproblemincludesthenonlineargenerationdemandequalityconstraintsduetoincludedtransmissionlosses.Therampuplimitsandprohibited 38 PAGE 39 zonesofgeneratorsareincorporatedaswell.TheefcientofCGRASPistestedonsixunitsystemsthatisdiscribedinSection 4.1.2 .Thesameproblemhasbeensolvedin[ 30 ]andtheirbestsolutionandappliedmethodsarepresentedinTable 411 .ThelossesandtotalgenerationcostaregiveninTable 412 .Thebestsolutionsamongallsolutionshavebeenillustratedintheboldprints.FromthesedatawecanseethattheirprovidedobjectivefunctionvaluesaresmallerthatoneobtainedbyCGRASP,butitshouldbenotedthatsolutionsgainedbyCPSO1andCPSO2violatetherampuplimitsofgenerator3.WhenthesolutionofPSOhasbeenpluged,ithasbeenfoundtheviolationofgenerationdemandbalanceequalityby0.4661MW,becauseaccordingtogivensolution,thegenerationisequalto1275.9571MWandlossis12.4910MW.TheminimumgenerationcostfoundbyCGRASPis$15456.54469,whiletheaveragecostis$15507.10954withstandarddeviationofvalue$28.10037477.Accordingtothesefacts,itcanbestatedthatCGRASPapprochwithappliedheuristicstrategycanproducedfeasibleandgoodsolutions.TheresultsproducedbySAandGAwerenotfeasibleorreasonablyclosetoresultspresnetedhere,sotheyarenotpresnetedhere. Table411. Bestsolutionsforcase2 P1P2P3P4P5P6 PSO447.4970173.3221263.4745139.0594165.476187.1280CPSO1434.4236173.4385274.2247128.0183179.704285.9082CPSO2434.4295173.3231274.4735128.0598179.475985.9281CGRASP447.8181200253.5570149.9999150.320273.5022 Table412. Bestresults,whendemandis1263MW TotaloutputLossTotalgenerationcost PSO1276.012.958415451CPSO11276.012.958315447CPSO21276.012.958215446CGRASP1275.197412.197415456.54 39 PAGE 40 4.2.3Case3Inthiscase,thestaticEDproblemwithnonsmoothcostfunctionduetothevalvepointeffectsisconsideredtochecktheabilityofCGRASPtosolvesuchtypeproblemsanditscompetitivenesswithbothGAandSAapproches.Theexperimentisperformedontwodifferentsystems,namely,system3andsystem4.ThenalfuelcostsobtainedusingappliedapprochesaresummarizedinTable 413 .Itshowstheminimum,averageandmaximumcostandstandarddeviationachievedbyappliedmethodsfor75runs.Fromthecomputationalresults,theminimumcostachievedbyCGRASPwasthebest,followedbySAandGA.Theminimumcost,maximumcostandthemeancostvaluesobtainedbyCGRASPare18394.07$/h,18699.339$/h,and18550.105$/h,respectively,whicharelowerthanthoseobtainedbySAandGA.TheworstresultsareobtainedbyGA.ItcanbenoticedthatresultsproducedbyGAvarytheleast,thiscanbeconrmedbythelowstandarddeviationthatis$12.562.Inliterature[ 54 ],thelowestreportedgenerationcostfor1800MWis$17994.07,howeverthesolutionisnotpresented. Table413. Generationcostsfor13unitsystemwithdemand1800MW MethodMinAvgMaxSt.Dev. CGRASP18394.07018550.10518699.33965.729GA19384.22919417.96419438.91412.562SA18950.17419393.11419782.516181.920 Theresultson40unitsarepresentedintable 414 .CGRASP,SAandGAalgorithmswererunfor75timesandtheminimum,maximumandaveragevalueofobjectivefunctionarereported. Table414. Generationcostsfor40unitsystemwithdemand10500MW MethodMinAvgMaxSt.Dev. CGRASP128883.1965130268.9796132839.2181972.757GA163401.9977163534.9817163623.342364.0606SA138975.7844150757.5002162578.62716118.65 40 PAGE 41 4.2.4Case4ThelastproblemsolvedbyCGRASPisdynamicEDproblemincludingrampuplimits,thatmakesthisproblemmoredifcultthaninacase1.Forsimplicity,thetransmissionlossesareneglected.TheminimumcostobtainedbytheCGRASPcoupledwithheuristicstrategyisfoundtobe$1,735,176.10,thebestsolutionthatsatisesdemandbalanceconstraintsaswellasgeneratorsoperationconstraintsincludingonesoframpupisgiveninTable 415 .GAansSAappliedinthisworkcouldn'tproducethefeasiblesolution. Table415. Bestsolutionforcase4 HourP1P2P3P4P5 1197.5775181.2115163.417562.3581127.01412199.3353142.0086243.3161112.3580115.82943279.3352222.0086163.3161162.358080.96964357.6864196.990783.3162212.3580130.96965341.5554276.9907104.5803226.3765180.96966364.4919356.990792.7176205.6510202.51987291.8407436.9906162.9068249.2158210.43138275.9255452.7053238.0856230.3986196.67339355.9255382.1891318.0856233.8956197.860410373.6142395.9434310.8613235.4000242.999911453.6141447.3342339.9999285.3999197.860412469.9999469.9999327.0836235.4000229.247313405.7058425.0176281.3943279.5788226.466314325.7058367.8957339.9999229.5788240.974215308.0081320.2882284.4522279.5788190.974216228.0081369.5967235.5079230.6255174.261617252.7930361.9796165.9723237.6272169.117718214.1486441.9796160.0032187.6272211.344419253.8865470.0000240.0032137.6272242.999920333.8865448.3762320.0032149.1631211.344421334.2637375.3046329.5451195.2469242.999922254.2637295.3046249.5451245.2469195.472723174.2637215.3046218.8263256.1493161.567924150.0001171.8146144.0519259.4389132.1630 41 PAGE 42 Bestsolutionsforcase4continued HourP6P7P8P9P10 184.115677.592283.017620.000139.6960287.316963.805290.000046.030410.0001360.234993.8052119.999946.750729.22194107.8558111.1225108.297359.459037.94455128.756381.122678.297438.994622.35676140.928876.308698.341856.971233.07857137.017946.308668.341871.651027.2956889.733576.308684.757880.000051.41179139.7335106.308693.145862.094034.762110159.9999128.767284.597773.038616.777711112.765098.7671114.597743.038612.623012112.5501128.767184.597773.038619.31561388.9431129.9999114.597777.810542.48611465.2257127.712695.173976.733454.99991570.066197.7126119.999962.433842.486116108.197967.712690.207732.433817.44821762.768070.299697.676732.836728.929318112.767999.9863119.999948.385731.757219122.172997.8341112.125555.924143.426520159.9999126.0825105.677178.854938.612221153.7373100.4907119.999948.854923.556922127.985270.490796.020170.030523.640523122.777140.490866.020256.083720.51642472.777170.490873.575660.648649.0393 42 PAGE 43 CHAPTER5CONCLUSION Economicdisatchproblemcanbeformulatedinverydifferentways:asasimplelinearprogramingproblemtononlinearnonconvexproblem. 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[71] Zhijiang,Y.,H.Zhijian,J.Chuanwen.2002.EconomicDispatchandOptimalPowerFlowBasedonChaoticOptimization.ProceedingsofInternationalConferenceonPowerSystemTechnology,2002.PowerCon42313. 49 PAGE 50 BIOGRAPHICALSKETCH IngridaRadziukynienegotBachelorofScienceandMasterofScienceincomputerscienceatVytautasMagnusUniversity,Lithuaniain2003and2005,respectively.Inaddition,shegotacerticateofbussinessmanagementfromDepartmentofBusinessatVytautasMagnusUniversity.In2010,sheearnedtheMasterofScienceinindustrialengineeringfromUniversityofFlorida.Moreinformationaboutherresearchinterestcanbefoundinherwebpagehttp://plaza.u.edu/ingridar/. 50 