Philippe L. Toint
MOS Chair's Column
September 30, 2010. Starting a new chairmanship of the Mathe
matical Optimization Society is at the same time very motivating
and humbling. Very motivating because I believe the Society is in
an excellent position: the membership is strong and the importance
of our scientific subject is definitely growing in importance, both in
industry and academia. The role of the Society, as the main profes
sional body of researchers in optimization and associated subjects,
thus gains in importance and visibility.
My current position as a new Chair also makes me realize how
much is due to the past officers of the Society. Under the tireless
leadership of Steve Wright, the Society has achieved much in past
years, successfully organizing (with much help from others of course)
the major conferences which are part of its mission: IPCO, ISMP and
ICCOPT. The Society has also continued to organize, for the bene
fit of our worldwide community, the publication of its top journals
MPA, MPB and MPC as well as the Optima newsletter. Finally, Steve
has been the motor (and everything else) behind the Optimization
Online web service. In short Steve's contribution as Chair has been
central, active, and respectful, efficient and modest at the same time.
We are all indebted to him. The past MOS Treasurer David Gay has
also been as usual helpful, dedicated and responsive self, with no
hesitation to spend his time on our behalf in such tasks as financial
accounting, legal issues and more. He obviously also deserves our
best thanks. I also wish to stress the much appreciated positive role
of jon Lee, the past chair of the MOS Executive Committee, whose
interaction with the Council, willingness to exchange and continuous
stream of valuable proposals that have contributed greatly to the dy
namism of the Council. On behalf of all optimizers worldwide, many
thanks to all three.
It is also a pleasure to report on the success of the last ICCOPT
and associated winter school, organized by Alejandro jofr6 and his
local energetic team in Santiago, Chile, from August 24 to 29, 2010.
The many participants to this toplevel meeting will surely remember
the quality of the presentations and scientific interactions, but also
the most kind hospitality of our Chilean colleagues, the lively win
tery activity of Santiago, and the snowy Andes in the background.
Again think you dear Alejandro and colleagues for this remarkable
opportunity.
The Mathematical Optimization Society moves on, and I am
pleased to announce an important new development. After some
sleepy years, the Society Publications Committee has been rein
stated. This Committee is chaired by Alexander Shapiro and its
members are Nick Gould, Christoph Helmberg, Jie Sun and Robert
Weismantel. Its mission, as indicated in the Society's bylaws, includes
overseeing the publication process of our journals, improving their
effectiveness and making proposals to the Council regarding the suc
cession of the journal's Editors in Chief at the end of their terms.
I am very confident that this excellent group will work actively to
further promote the quality and attractiveness of the MOS publica
tions.
Finally, let me close this column by an appeal to all. For MOS
to continue to be, as we all wish, a true and efficient support for
our research activities, it must be strong and representative of our
community. In particular, the size of membership is a clear argument
when negotiating facilities at conferences among other things. The
current membership is faring reasonably well, but my goal is to in
crease significantly. This hope is justified because I am certain that
many more researchers (students and more senior staff) working in
Our scientific domain are not yet members of our Society. I wish,
therefore, to encourage them all to join us, so we can be stronger
together to promote our field, its meetings and journals. With this
in mind, I hope that you will promote MOS membership among your
colleagues.
I wish you all a most fruitful research time and hope to meet you
soon at one of our conferences.
Alberto Caprara and Andrea Lodi
Farewell
As announced in the MOS Chair's column of Optima 83, the current
one is the last issue for the Editorial Board of Optima in its present
form. It has been a nice journey for us over a period of four years in
which Optima was back in good place on the desk on many of our
colleagues! This would not have been possible without our coeditor
Katya Scheinberg, the enthusiasm and support of Steve Wright, the
competence and the patience of the authors of the scientific con
tributions, the dedication of Christina Loosli and Christoph Eyrich,
former and current designers, respectively. We warmly thank all of
them. Of course, the deepest gratitude goes to the Optima readers
whose feedback has shown us we were going in the right direction.
New challenges will be faced by the new, scientifically outstand
ing and energetic, Editorial Board of which Katya takes the lead as
new Optima editor together with Sam Burer and Volker Kaibel as
coeditors. We wish them all the best, and the same to Optima and
of course to MOS.
Contents of Issue 83 / July 20 10
IPhilippe L. Toint, MOS Chair's Column
IAlberto Caprara and Andrea Lodi, Farewell
2 Grace HechmeDoukopoulos, Sandrine BrignolCharousset,
J6r~me Malick, Claude Lemar~chal, The shortterm electricity
production management Problem at EDF
6 Antonio Frangioni, Unit Commitment problems: A tale in
Lagrangian optimization
7 Imprint
OP IMA
Mathematical Optimization Society Newsletter
OPTIMA 84
o On the short term (a few days to a few hours): to compute pro
duction schedules that satisfy the technical constraints and the
demand/production equilibrium, and provide marginal costs of the
system.
Aspects of shortterm management
We focus here on the shortterm management problem, commonly
called unitcommitment. The decisionmaking problem is expressed
as an optimization problem whose main characteristics are the fol
lowing:
o large size (106 variables, 106 constraints): all production plants are
modeled with a large number of technical constraints and on a
48 hours horizon discretized in halfhourly time steps;
ononconvex and noncontinuous nature: some production costs have
discontinuities and the production variables are discrete;
o strict computational limits, due to a very tense operational pro
cess: the latest data collection phase ends at I2:30 and the feasi
ble schedules have to be sent to the transport system operator
at I 6:30. Moreover postoptimization treatments involving human
expertise are required; altogether, about 15 minutes are left for
solving the optimization problem.
This article is mainly devoted to modeling aspects of the problem:
we describe the technical characteristics of the production units in
some details, and we outline the general solution methodology. For
more details on the latter, as well as numerical results, the reader is
referred to [4].
The structure of this paper is then as follows. Section 2 presents
technical aspects of the production units (thermal and hydraulic),
and their mathematical modeling. Section 3 sketches the methods
that are used in practice to tackle the production optimization prob
lem. Current issues and research directions are discussed in Sec
tion 4.
2 Description of the shortterm management problem
2. 1 Overall optimization problem
The goal of generation management is to compute technically fea
sible production schedules with a good supplydemand balance at a
minimal operating cost.
At a high level, this optimization problem is written as follows.
Let T be the discretized time horizon and DE R T the total pre
dicted demand. A production unit v has a production P, E RBT, an
Operating cost c,, and a technical functioning domain T, c RT. The
unitcommitment problem is
min Cc, (P,), s.t. P, E Tv, CP, = D. (1)
The production units are thus subject to two types of constraints:
the global linking demand constraint, and local structural constraints.
These local constraints are detailed in the next two sections. Then
section 3 explains how to deal with all of these constraints when
solving problem (1).
2.2 Constraints and cost for thermal units
TechnicGI GSpedtS
Thermal units consist of both classical (coal, oil, gas) and nuclear
UnitS Since their operating domains are similar. We give a simplified
description in the following.
When a thermal unit is turned on, the production level must re
main between a minimum and a maximum value, which can vary in
time. FOr inStance, the maximum level is equal to zero during refu
eling periOds. All levels between the maximum and minimum cannot
be used, and the production levels are discrete. Moreover, produc
tion variations must follow several rules, namely: a minimal duration
Grace HechmeDoukopoulos, Sandrine BrignolCharousset,
j6r~me Malick, Claude Lemar~chal
The shortterm electricity production
management problem at EDF
IIntroduction
Production management aims at meeting the demand of customers
at minimum cost. Since electricity cannot be stored, an electricity
producer mainly faces the permanent challenge of matching genera
tion and demand to avoid physical failures of the production system.
This article presents how EDF (French Electricity Board) solves this
optimization problem: we sketch some issues and the resolution
scheme, and we highlight parts of the numerical optimization pro
cess of the short term management.
From long to shortterm management
EDF manages a mix of generation units, composed of nearly 60 nu
clear plants, 100 classical thermal plants (coal, fuel, gas, combined
cycles), and 500 hydraulic plants dispatched in 50 valleys (a hydro
valley is a set of rivers, grouped in such a way that two different
valleys are geographically independent). The electricity generation
management at EDF consists in determining a strategy (illustrated
by Figure 1) which manages the chain of decisions.
The overall decisionmaking problem is highly complex and way
too difficult to be solved globally. Starting from longterm decisions
to the shortterm ones, a time horizon decomposition amenable to
dynamic programming is thus used. Each horizon computes Bellman
values for stocks (reservoirs, nuclear plants, ...) to be used by the
shorter horizon.
On the longerterm horizons, we take into account uncertainties
by using statistical forecast models within a stochastic optimization
framework; technical constraints are drastically simplified. By con
trast, a very fine vision of the generation mix is necessary for the
short term. Real technical constraints make the use of stochastics
prohibitive; rather, deterministic counterparts are solved.
The main decisions at each timehorizon are the following ones:
o On the long term (five to twenty years): to design the genera
tion mix: planning investments, choosing the right kind of plants,
forecasting polluting emissions, ...
o On the mid term (one to five years): to define the planning for nu
clear outages (refueling and overhaul), to calculate management
strategies for the main reservoirs (hydraulic reservoirs, demand
side management, polluting emission, fuel stocks, ...), to buy fos
sil fuel and to evaluate the failure risks and associated hedging
decisions.
Innstmentdecision Stck napment
Spte dexbilies Fuel supplies (120 tuclear, lrge
musearmentenanc pay nernton plans
scheduling
ml Fnne Intradayr os
Longterm Mediumterm Cnrln ak hr
vs, mamsement arbs
.S"U""If"ct Loadfoecas Lons h ding
Stateyfofrtrheuse
Longterrnl gas roadnc rd
Figure I. The EDF generation monogement decision chain
October 2010
between consecutive level changes, upper and lower bound values,
and variation prohibition for the rest of the time period after a level
decrease.
Switching on or off a thermal unit is not instantaneous. Specific
startup and shutdown curves must be followed, as well as mini
mum durations of shutdown. There are also some daily constraints:
the number of startups, shutdowns, and production level variations
are limited in a day.
The production cost of a thermal plant is composed by two parts:
a fuel cost depending on the generation level, plus some startup
costs depending on the duration of the previous shutdown.
Mathematical formulation
Denote by P[, the production level of unit v at time step t t T. Let
TnO [T "f] be the set of time steps when the unit is online (that is
with Pf, > 0) offlinee (with Pf, = 0)]. The constraints on the produc
tion levels can be expressed as follows:
o Static constraints: Ns, being the number of discrete generation
levels for v, and 0 < P( = Pqi and P = PV,Nw, the minimal and
maximal generation levels, the static constraints are just
where P,,,i <  < Ps,~, <  < Pt*,N .
o Dynamic constraints: the three constraints on the production
variations are: the minimal duration between level variations,
if P~t+1 Pt, then V7t [t + ; tdm'] +, dii Prgt+1,
the variation prohibition after a decrease
if P(+1P{, < then V7t [t +2; T], P, = Pt I
and the bound constraints:
gtF+1 _Pt
&, and A, being the minimum and maximum values, and dt the
duration of time step t.
o Startup or shutdown curves constraints: A startup [shutdown]
curve is a set of different generation levels at each timestep of the
startup [shutdown] period:
10 i Pistart,i < P ,, i t [1, dy~art] ,
and
0 5 Ptstop, srtop~l!,
where dy~art [dstop] is the duration of startup [shutdown]. Note
that the particular curve to follow depends on the duration of the
previous offlinelonline period. The constraints are then described
as follows:
o A plant has to follow a minimum duration for any off line period:
if P{, 0 and Pf, I 0 then:
V7tE [t + 1; t + dgtop], PV = 0.
o A plant has to follow the adequate startingup curve when go
ing online: if P{, and P{, fO then:
VT E [t + 1;t t +df~art] Pp r sarrt,7t
o A plant has to follow the adequate shutdown curve when going
offline: if P, fO and P{, I 0 then:
oThe aforementioned daily constraints turn out to make the prob
lem intractable. They are therefore incorporated as penalties
Daily in the objective function.
In view of the previous discussions, the total generation cost has
the form:
cv,(Ps,) =C Chad (propp~r Pdaily(tzPer1)~ Cstart, (2)
where estart is a startup cost depending on the previous off line pe
riod.
2.3 Constraints and cost for hydraulic units
Technical aspects
A hydroplant consists of a set of turbines that discharge water from
its upstream reservoir into its downstream one. The reverse is also
possible for some plants equipped with pumping units: pumping up
water at low demand hours allows one to reuse the water at higher
demand ones. Unlike thermal or nuclear units, the production of a
hydroplant is not computed individually. It is rather optimized in a
more global entity, a hydrovalley, that depicts the interaction be
tween a set of hydroplants and the reservoirs connecting them.
The power delivered by a hydroplant can take only a finite num
ber of values (designated in what follows as discrete production
points). These values correspond to the power produced by its
turbines that are switched on successively. Since the time period
is short, the considered turbines' rates are fixed, just because the
water level in the upstream reservoir is considered as constant.
The production of a hydrovalley is subject to a set of constraints
that deal with technical functioning aspects, aimed at preventing a
fast degradation of the units or simply at following some external
regulations. Of course, in addition to the flows induced by pumping
or turbining, a reservoir is subject to outer water inputs due to rain,
snow or spillage. Hence, through the time period, the volume of a
reservoir is governed by an equilibrium flow constraint that rules
these factors.
As for the power plants, their production variations are subject
to upper and lower bound constraints. A minimal delay of one hour
is also imposed between two production variations of opposite na
ture. Furthermore, when two reservoirs are connected with both
turbining and pumping plants, simultaneous pumping and turbining
is forbidden, and a minimal halt of thirty minutes before switching
from pumping into turbining (and vice versa) is imposed.
Mathematical formulation
Given a hydrovalley v, we denote by U its set of production plants
and by R its reservoirs. Each plant u E U is described by a set of
turbining/pumping units G(u). Each unit is characterized by its flow
capacity Fu,B and its power rate pu,8; units are ranked according to
decreasing p's. At time step t, the state of unit g is given by a binary
variable et,,B E0, 1) so that we have
F[, e~gFu, (the flow capacity of u)
geG(u)
PL= C e, ,~,aFu,, (the power of u).
geG(u)
For all u, the binary variables must follow a sequence constraint:
e(t 7 [,4
Vt g E G(u),
to have PL equal to a discrete production point at each time step t.
Note that it is then sufficient to apply the pumping/turbining techni
Cal COnStraintS for g 1 Only.
VT t [t + 1; t + d~] vtop stop,7t
OPTIMA 84
Denote by ur and up the turbining and pumping plants respec
tively. The constraints
prohibit simultaneous pumping and turbining, while the minimal halt
delay before a flow mode switch can be imposed by:
Vt t [1, T ], e,,i+e,, ,e,,,,i +eT e,,i < .
The production variation constraints are expressed as:
Vt t [2, T ], Vg t G(u), 1 i e[,, ef,, etl, < 0,
for the minimal delay between production variations, and
Vt t [1, T ], St,1 < (e(,) [,,F,, E
geG(u)
for the bound constraints, S, and St, being the lower and upper
bounds respectively.
Denote by V~t the volume of reservoir r tR at time step t. The
flow constraint has the following form:
(iv) However, the converse in (ii) is false: finding aAh maximizing
8 does not necessarily yield a primal optimal P(A). Usually,
~, Ps,(A) + D for any AE R T, even for A A.
(v) Nevertheless, the dual Problem (ii) does provide a certain pri
mal point P = Ps, which solves a certain convexified form of
(1); the Ps's need not lie in T, but Z,, Ps D.
(vi) Besides, a dual optimum 1 gives the marginal cost of the link
ing constraints 7, Ps, = D, associated with the convexification
alluded to in (v).
Accordingly, a twophase strategy is adopted.
o Phase 1 maximizing 8(A). Property (iii) makes (ii) possible and
common approaches use the popular subgradient algorithm. In
stead, the present operational software uses the bundle algo
rithm of [IO], which in turn uses a quadratic solver written by
K. C. Kiwiel. A first advantage is robustness: a reliable dual opti
mum 1 is computed which, according to (vi), provides useful in
formation on the marginal prices of the demand D in (1). Besides,
a primal point P as described in (v) is also obtained. We will see
in the forthcoming sections that hydraulic valleys cannot always
be optimized exactly; this results in a noisy 8, which is handled by
the technique of [7].
o Phase  producing schedules. Because of (iv), solving the dual
problem as above can only be viewed as a first step toward solving
(1). It is therefore followed by a second phase, aimed at comput
ing schedules that do lie in T,, while realizing a good compromise
between minimizing the cost and satisfying the balance equation.
The method currently used is based on augmented Lagrangian [3]
where a quadratic stabilization term is added to the dual function,
Since the quadratic term destroys the decomposability property,
a "partial linearization" as in [2] is applied. Altogether the local
problems of (3) are replaced by
Vt = Vt1 + C F~tpdte,r)
uthit(r)
C F~td(r'U)) g
utas(r)
where Nt (r) [NJ (r)] is the set of hydroplants up [down] reser
voir r, If is the outer water input, and d (u, r) is the travel time of
water between reservoir r and plant u and vice versa. Note finally
that the volume is also subject to bound constraints (resulting from
the hydraulicity, environment, or regulations due to the recreational
use of the reservoir).
The production cost of a hydrovalley is the global water loss
through the time horizon:
ct,(Ps,) =C rM(Vo V T).
The value wr, of water of reservoir r is estimated with marginal in
dicators resulting from midterm models, giving the future gain if the
water is not discharged (remember Figure 1).
3 Optimization methods
3. 1 Solving the overall Problem via decomposition
In our overall optimization problem (1), v indices the thermal plants
of Section 2. 1 and the hydrovalleys of Section 2.2. Then we see that
each T, depends on no other Ps, and that the balance constraint
~,Ps, = D is the only link between the "local agents" Ps,. The
problem is thus clearly decomposable and, as already seen in [1],
Lagrangian relaxation is an attractive approach. Thus, for given dual
variable AE R T, (1) is replaced by the decomposed problem
min (ct,(P,,) Ps,+ rPs, Qz,2
Pera~t
where the "stability center" Q,, is just the previous iterate P,91
(thus an initialization Po is required).
In order to assess schedules satisfying all technical constraints of
Section 2, while not matching the linking constraints, a "total cost"
C(P):= [cr,(Ps,) +n DP,) (5)
is introduced, where r penalizes the balance mismatch. Conver
gence of the model is measured as the gap between the value 8(A)
of the dual function computed in Phase I and the total cost of the
primal solution obtained in Phase II (from weak duality, this gap gives
a bound for the optimal cost C).
3.2 Solving the subproblems
Thermal units
A standard MIP formulation would be quite complex and require
considerably high computational time to obtain a satisfactory solu
tion. A specific dynamic programming approach has been developed
to solve the thermal subproblems, which can be outlined as follows.
A fourdimensional state (S1, S2,S3 S4~ is defined: S1 represents
the onlineloffline state (S1 1 if the plant is online, S1 0 if not),
S2 = ft, iS the discrete production level, S3 represents the sign of
the last production variation (S3 = 1 for an increase, O otherwise),
S2 = dt, repreSentS the duration of the current onlineloffline period.
At each time step t, the set of authorized states is calculated taking
into account all the production constraints (limits on the production
levels, halts ...). Authorized transitions between states at different
timesteps are then computed taking into account the timing con
straints (minimum durations, startup curves ...).
8(A) :=Pilli Sc,,(P,_) + D
c~t')
A D +C min(cz,(P,) Ps,)
and the issue becomes that of finding an adequate A, so as to re
produce a solution of (1). To this end, duality theory (see, e.g.,
[6, 8, 5, 9]) tells us that:
(i) If an optimal solution P(A) of (3) is feasible in (1), then it is also
optimal in (1).
(ii) To achieve this, A must maximize the dual function 8 of (3).
(iii) This function is concave, D Z, Ps,(A) t RBT being a subgradi
ent (of 8 at A).
October 2010
Costs associated to each transition are then calculated: a transi
tion to an off line state costs 0; a transition to a generation state is
associated to the cost
Cv ted + c "Ppf + daily (P~ ,pt1 _tP
in the Lagrangian (2) (3); a transition to a startup curve is associated
to the sum of the generation cost and the startup cost.
Dynamic programming is then applied, calculating backward the
Bellman value of each state. The graph is a set of nodes and a set
of transitions between nodes. Call Tri the set of transitions going to
node i; then
Vb," = mn ~[C(tr) + Vb'r i)
is the Bellman value of node i, where trI(i) is the node connected
to node i by transition tr, C(tr) is the cost associated to transi
tion tr.
Hydrovalleys
The hydraulic subproblems, modeled as MIP's described in Sec
tion 2.2, cannot be solved to optimality in a reasonable computing
time. This is the reason why three versions of the model are at
stake:
(a) the actual implementation currently in operation uses a continu
ous relaxation; combinatorial constraints are taken into account
only in Phase 11 with the help of various heuristics;
(b) a forthcoming version will keep the continuous relaxation in
Phase 1 only, while Phase 11 will solve inaccurately the actual
model 2.2;
(c) a third version is planned, where inaccurate solutions of the ac
tual model will be computed in both phases.
Effort is currently focused on version (b); Section 4 will explain
why version (c) has been postponed so far.
3.3 Numerical illustration
Let us illustrate the behaviour of the forthcoming version b) above.
The code is distributed and runs on a cluster of 32 processors. It
performs around 500 total iterations for each phase. An iteration
takes about 0.1 seconds for Phase 1, and 1.5 seconds for Phase II
(remember that Phase I solves "easy" hydraulic subproblems). The
total computational time of one run is therefore around 900 sec
onds.
On average over one year, Phase 1 optimizes 8 within 0.1 % and
produces a convexified P satisfying the balance constraint within
0.1 % as well. Phase II produces a schedule P, whose total cost C(P)
is optimal within 1.3 %.
Two figures below give an idea of the progression of the algorithm,
for both phases. The evolution of the dual function (3) through
Phase I is depicted in Figure 2. As for Phase II, Figure 3 displays
two curves: the balance mismatch  IP, D (averaged over the
time period [0, T]), and the total cost of (5). Both figures use a log
arithmic scale; for industrial privacy, all values have been normalized
as follows:
an a
for n =1,...,N, an an = with a = min(an)
Good convergence behavior is sometimes difficult to achieve (es
pecially in Phase II). For instance, we can remark in Figure 3 that the
best solution was found at iteration 414 (a414 = 0), and the method
carried on for almost 100 additional iterations without being able to
improve it.
Dual function
O 50 100 150 200 250 300 350 400 450 500
Iterations
Figure 2. Evolution of the (normalized) dual function along the iterations
Total production cost
Balance mismatch
ph
0 50 100 150 200 Itr 0n 300 350
400 450 500
Figure 3. Evolution of the (normalized) bolonce mismotch and total production
cost
4 Perspectives
Including all possible reallife constraints of an electricity production
management problem is pure dream (remember the human postpro
cessing mentioned in Section 1). The model is therefore regularly
improved, in order to better reflect reality and to achieve better
numerical performances.
For instance, Phase 1 of the actual implementation (a) "sees" a
less constrained hydraulic model; implementing version (c) should
be desirable. However, it turns out that inserting the true model in
Phase I results in chaotic 1. This undesired behaviour might be due
to stiff hydraulic constraints (which impacts the balance constraints),
or to inaccurate computations of the dual function (3). Further anal
ysis is needed to fix this question; an important question because
Phase I not only initializes Phase II but also provides marginal indica
tors about costs of the demand (A mentioned in (vi) of Section 3. 1).
Another point concerns Phase II, which uses a local search
method strongly dependent on the initial point. Moreover the al
gorithm does not have guaranteed convergence properties on this
II .
OPTIMA 84
nonconvex problem. Alternative methods are worth investigating to
improve this phase.
Grace HechmeDoukopoulos, EDR R&D, I avenue du General de Gaulle,
92141 Clamart, France. grace.doukopoulos@edf.fr
Sandrine BrignolCharousset, EDR R&D, I avenue du General de Gaulle'
92141 Clamart, France. sandrine.charousset@edf.fr
Jer~me Malick, CNRS, Lab.J.Kunztmann, INRIA, 655 avenue de I'Europe,
Montbonnot, 38334 Saint Ismier, France. jerome.malick@inria.fr
Claude Lemarechal, INRIA, 655 avenue de I'Europe, Montbonnot,
38334 Saint Ismier, France. claude.lemarechal@edf.fr
References
[I] D.P Bertsekas, G.S. Lauer, N.R. Sandell, and T.A. Posberg. Optimal short
term scheduling of largescale power systems. IEEE Transactions on Auto
motic Control, AC28:11 1, 1983
[2] G. Cohen. Auxiliary problem principle and decomposition of optimization
problems. journal of Optimization Theory ond Applications, 32:277305, 1 980.
[3] G. Cohen and D. L. Zhu. Decomposition coordination methods in large
scale optimization problems, the nondifferentiable case and the use of aug
mented Lagrangians. In J.B. Cruz, editor, Advances in Large Scale Systems, 1,
pages 203266. JAl Press Inc., Greenwich, Connecticut, 1984.
[4] L. Dubost, R. Gonzalez, and C. Lemarechal. A primalproximal heuristic
applied to the french unitcommitment problem. Mathematical Program
ming, 104(1):129152, 2005.
[5] A.M. Geoffrion. Duality in nonlinear programming: a simplified applicati
onsoriented development. SIAM Review, 13(1):137, 1971
[6] R.C. Grinold. Lagrangian subgradients. Management Science, 17(3):185
188, 1970.
[7] K.C. Kiwiel. A proximal bundle method with approximate subgradient
linearizations. SIAM journalon Optimization, 16(4):1007 1023, 2006.
[8] L. Lasdon. Optimization Theory for Large Systems. Macmillan Series in Oper
ations Research, 1970.
[9] C. Lemarechal. Lagrangian relaxation. In M. Junger and D. Naddef, editors,
Computational Combinatorial Optimization, pages I 12156. Springer Verlag,
2001.
[IO] C. Lemarechal and C. Sagastiz~ibal. Variable metric bundle methods: from
can ept~ual to implementable forms. Mathematical Programming, 76(3):393
Discussion Column
Antonio Frangioni
Unit Commitment problems: A tale in
Lagrangian optimization
The paper provides an account of the current status of a multiluster
collaboration between academia and industry about the application
of Lagrangian techniques for the solution of Unit Commitment (UC)
problems in electrical power production.
UC problems have played an important role, perhaps as significant
as that of multicommodity flows, to popularize solution techniques
based on Lagrangian relaxation both in the mathematical program
ming community and within practitioners (in particular, in this case
in the electrical engineering community). This is due to a number of
factors:
oUC is indeed a largescale problem with both nonlinear and dis
crete components. As such, it has been until recently firmly out
of reach of solution techniques based on generalpurpose mixed
integer solvers, and specialized approaches have been a neces
sity. However, singleunit scheduling problems, both for thermal
and hydro units, are relatively easy to solve with appropriate
approaches (typically dynamic programming and flowllinear pro
gramming techniques).
o As a consequence, UC, at least in some of its "easiest" versions, is
incredibly wellsuited for the technique. Lower bounds obtained
by Lagrangian relaxation can have a ludicrously small inherent gap,
as low as a small fraction of a percentage point, especially for
the largest hydrothermal instances that used to be the norm in
several relevant application environments (and still are in that de
scribed in the Scientific Contribution). Each Lagrangian iteration
is quite fast, which coupled with a good method to update the
multipliers produces these terrific bounds quickly enough.
o UC has really tight operational constraints, making the develop
ment of methods capable of quickly and reliably producing good
quality solution of utmost importance.
o UC is, or at least used to be, the almost perfect example of an
"easy sell" for advanced applied research. The problem is huge in
terms of costs involved and has to be solved daily, thus small say
ings rapidly add up to incredibly vast sums. Optimizing the sched
ules of production units has no noticeable negative effect, and
therefore no meaningful opposition by any of the parties involved.
The problem used to be of concern of a single monopolistic pro
ducer often stateowned with almost unlimited financial re
sources.
This is not to say that UC is an easy problem. Indeed, successful ap
plication of Lagrangian techniques to UC (as well as to other difficult
combinatorial problems) requires several nontrivial steps which have
motivated relevant theoretical contributions, to which the authors
of the Scientific Contribution are by no means unconnected:
o algorithmic recovery [7] of the continuous solutions of the "pri
mal counterpart" of the Lagrangian Dual [8];
o fast converging bundle methods using techniques like disaggrega
tion in the master problem and preconditioning [2];
o general yet efficient approaches for recovering primal feasible so
lutions out of a Lagrangian dual [3], comprised an entire new class
of Lagrangianbased heuristics [6].
All this illustrates a very nice instance of a central credence (hope?
wishful thinking?) in the mathematical programming community, and
in applied mathematics in general: important practical problems mo
tivate relevant theoretical developments, sophisticated mathematical
theory is necessary to solve crucial practical applications.
Research on UC problems is by no means over, due to several
factors:
o Most countries in the world have been transitioning from elec
trical systems based on monopolistic producers to those based
on free market, where competition between producers and con
sumers (regulated by a central authority, typically taking care of
the electrical network) is supposed to increase the overall ef
ficiency. In this setting, the UC problem takes different forms
for different actors, sometimes complicating matters consider
ably (most of the system is in the hand of other decision makers,
whose behavior is unknown), sometimes simplifying them some
what (each decision maker needs to model only his units, which
are a subset of the whole system, and the effects of the constraints
on the transmission network is somewhat less pronounced).
o It is possible (perhaps desirable) that the future will bring very rel
evant changes in the characteristics of the generating units. Other
than several hundreds of large and relatively reliable plants, many
thousands of smaller and less reliable units based on renewables
or even much more with fuel cells cars in the envisaged hydro
gen economy may have to be taken into account. Distribution
grids may also undergo substantial updates to the socalled "smart
grids". All this may clearly have very profound impacts on the UC
models to be solved.
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oFrom the mathematical programming viewpoint, two somewhat
opposite phenomena are manifesting themselves. On the one
hand, the continuous push to represent more and more closely
the reality of producing plants (valve points and startup ramps [ I]
in thermal units, nonlinear powertodischargedwater relation
ships [4], forbidden operating zones and pumping in hydro units
[5],...) in the mathematical model contributes to keep the prob
lem rather difficult to solve, and therefore justifies the develop
ment of specialized approaches. On the other hand, the impres
sive rise of solution power of generalpurpose MILP and MINLP
solvers renders Lagrangian approaches less competitive, and at
least not necessarily the unique choice as they used to be [9], es
pecially considering the superior flexibility of methods based on
generalpurpose tools.
Thus, UC (in its varied forms) is likely to remain an important prob
lem in practice for the foreseeable future, and a fine and worthy
playground for the mathematical programming community.
This discussion column would not be complete, however, without
mentioning how important the last author of the Scientific Contribu
tion has been in obtaining many of the above theoretical results, and
fostering their practical application in his own country and in many
Others. Not content of helping shaping generations of researchers
in convex analysis and applied mathematical programming with his
deep and broad theoretical contributions, his restless pursuit of clar
ity of presentation and higher educational value in his writings, and
his profound humanity, he has also substantially contributed to say
ing his own and many others countless millions in power generat
ing costs and the corresponding CO2 emissions. I think many of
us would be very happy to have even a fraction of the impact that
Claude Lemar~chal has had on his community.
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and not for the benefit of any library or institution.
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Antonio Frangioni, Dipartimento di Informatica, Universita di Pisa, Polo Uni
versitario della Spezia, Via dei Colli 90, 19121 La Spezia, 1taly.
frangio@di.unipi.it
References
[I] J.M. Arroyo and A.J. Conejo. Modeling of StartUp and ShutDown
Power Trajectories of Thermal Units. IEEE Transactions on Power Systems,
19(3):15621568, August 2004.
[2] L. Bacaud, C. Lemarechal, A. Renaud, and C. Sagastizibal. Bundle Methods
i~nrStochasitic Optima IPower Mana em nt:tiA Dinaggre aed A proac~h7Usin
2001.
[3] A. Belloni, A. Diniz, M.E. Maceira, and C. Sagastizibal. Bundle Relaxation and
Primal Recovery in Unit Commitment Problems. The Brazilian Case. Annals
of Operations Research, 1 20:2144, 2003.
[4] A. Borghetti, C. D'Ambrosio, A. Lodi, and S. Martello. An MILP Ap
proach for ShortTerm Hydro Scheduling and Unit Commitment With
HeadDependent Reservoir. IEEE Transactions on Power Systems, 23(2):1 I 15
I 124, 2008.
[5] J.PS. Catalso, H.M.I. Pousinho, and VM.E. Mendes. Scheduling of Head
dependent Cascaded Reservoirs Considering Discharge Ramping Con
straints and Start/stop of Units. International journal of Electrical Power &
Energy Systems, 32(8):9049 10, 20 10.
[6] A. Daniilidis and C. Lemarechal. On a primalproximal heuristic in discrete
optimization. Mathematica/ Programming, 1 04: 1051 28, 2005.
[7] S. Feltenmark and K.C. Kiwiel. Dual Applications of Proximal Bundle Meth
ods, Including Lagrangian Relaxation of NonConvex Problems. SIAM journal
on optimization, 10(3):697721, 2000.
[8] C. Lemarechal and A. Renaud. A geometric study of duality gaps, with ap
plications. Mathematical Programming, 90:399427, 2001i.
[9] T Li and M. Shahidehpour. PriceBased Unit Commitment: A Case of La
grangian Relaxation Versus Mixed Integer Programming. IEEE Transactions on
Power Systems, 20(4):20 152025, 2005.
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Editor: Andrea Lodi, DEIS University of Bologna, Viale Risorgimento 2, 40 136 Bologna, Italy. andrea.lodi@unibo.it a CoEditors: Alberto Caprara, DEIS
University of Bologna, Viale Risorgimento 2, 40 136 Bologna, Italy. alberto.caprara@unibo.it n Katya Scheinberg, Department of Industrial and Systems Engineering,
Lehigh University, Bethlehem, PA. katyascheinberg@gmail.com a Founding Editor: Donald W. Hearn a Published by the Mathematical Optimization Society.
m Design and typesetting by Christoph Eyrich, Berlin, Germany. optima@0x45.de
a Printed by Oktoberdruck AG, Berlin, Germany.
OPTIMA 84
ZIB Optimization Suite 2.0 released
The suite contains new versions of the constraint integer program
ming framework SCIP, the LP solver library SoPlex, and the mod
elling language ZIMPL. The package can solve LPs, MIPs and non
convex MIQCPs outofthebox. At its core lies SCIP, a hybrid solver
which integrates techniques from Integer Programming, Constraint
Programming, and Satisfiability Testing. It supports automated con
flict analysis/nogood learning and global constraints while retaining
the full strength of MIP solving.
Due to its pluginbased design SCIP can be used standalone or as a
very flexible branchcutandprice framework. New features of ver
sion 2.0 include full support for MIQCP, both in the solver and the
modelling language. First global CP constraints, new primal heuris
tics and separators have been implemented and the LP preprocessing
and solver interfaces have been improved. A couple of new examples
have been added to the distribution.
The complete source code and precompiled binaries can be
downloaded for academic use at http://zibopt.zib.de.
VII ALIO/E URO
Workshop on Applied Combinatorial Optimization
Porto, Portugal, May 46, 20 II
The main purpose of the ALIO/EURO Conferences of Combinato
rial Optimization is to bring together Latin American and European
researchers and to stimulate activities and discussions about meth
ods and applications in the field of combinatorial optimization. Re
searchers from other countries are obviously welcome too. Previous
editions of ALIOEURO were held in: Rio de Janeiro, Brazil (1989),
Valparaiso, Chile (1996), Erice, Italy (1999), Pucon, Chile (2002),
Paris, France (2005), and Buenos Aires, Argentina (2008). In this
meeting contributions dealing with any aspect of Applied Combina
torial Optimization are welcomed. This includes theoretical achieve
ments, algorithms development and realworld implementations.
Confirmed invited Speakers: Rolf Mojhring, D~bora P. Ronconi,
Andrea Lodi, and Miguel Constantino.
With a new introduction
by Stuart Dreyfus
Dynamic
Programming
Richard E Belbmna
This classic book is an introduction to dynamic progranining,
presented by the scientist who coined the term and developed
the theory in its early stages. In Dynamic P~ropniniing,
Richard Bellman introduces his groundbreaking theory
and fixrnishes a new and versatile nxathenatical tool for the
treatment of many complex problems, both within and outside
of the discipline.
Princeton Landmarks inMathematics
Paper $39.50 9780691146683
Submission deadline: March 14, 201 I
Information and contact: www.dcc.fc.up.pt/ALIOEURO201 I or
email alio.euro.201 l @gmail.com
IPCO XV will be held on June 15 17, 20 1 I at the IBM T.J. Watson
Research Center in Yorktown Heights, New York, USA.
Authors are invited to submit extended abstracts of their recent
work by November 15, 2010. Submission details and other infor
mation can be found at: http://ipco20 I I.uai.cl
The IPCO conference is under the auspices of the Mathematical
Optimization Society (formerly known as the Mathematical Pro
gramming Society). It is held every year, except for those years
in which the "International Symposium on Mathematical Program
ming" takes place. The conference is a forum for researchers and
practitioners working on various aspects of integer programming
and combinatorial optimization. The aim is to present recent devel
opments in theory, computation, and applications in these areas.
Program committee: Nikhil Bansal (IBM), Michele Conforti (Padova),
Bertrand Guenn (Waterloo), Oktay Giinliik (IBM), Tibor Jordain
(ELTE Budapest), jochen Koenemann (Waterloo), Andrea Lodi
(Bologna), Franz Rendl (Klagenfurt), Giovanni Rinaldi (Roma), Giin
ter Rote (FU Berlin), Cliff Stein (Columbia), Frank Vallentin (Delft),
Jens Vygen (Bonn), Gerhard Woeginger (Eindhoven, chair).
Organizing committee: Sanjeeb Dash, Oktay Giinliik (chair), Jon Lee,
Maxim Sviridenko.
important dates:
Abstract submission:
Notification:
Conference:
November 15, 2010
January 31, 20I I
June 1 5 17, 20 1 I
Kicbsnl Bsllmsn
IPCO 20 II
The I 5th Conference on Integer Programming and Combinatorial Optimization
Announcement and call for papers
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