REVISTA INVESTIGACIÓN OPERACIONAL Vol. 28, No.1, 4-16, 2007

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1 REVISTA INVESTIGACIÓN OPERACIONAL Vol. 28 No ALGORITHMS FOR MEAN-RISK STOCHASTIC INTEGER PROGRAMS IN ENERGY Rüdiger Shultz Frederike Neise Department of Mathematis University of Duisburg-Essen Campus Duisburg Lotharstr. 65 D Duisburg Germany ABSTRACT We introdue models and algorithms suitable for inluding risk aversion into stohasti programming problems in energy. For a system with dispersed generation of power and heat we present omputational results showing the superiority of our deomposition algorithm over a standard mixed-integer linear programming solver. RESUMEN: Introduimos modelos y algoritmos adeuados para inluir la aversión al riesgo en problemas de programaión estoástia en la energía. Para un sistema on una dispersión de la generaión de potenia y alor presentamos resultados omputaionales que muestran la superioridad de uatro algoritmos de desomposiión sobre los típios resolvedores de la programaión lineal entera mixta. KEY WORDS : Dispersed storage and generation Cogeneration Renewable Resoures Mathematial Programming Unertainty Deomposition methods Risk aversion MSC: 90C15 1. INTRODUCTION Inomplete information is beoming a more and more prevailing issue in energy optimization problems. Power market liberalization and the impending deentralization of eletriity supply the latter with substantial infeed from renewable resoures are just two phenomena that have fostered this development. In addition the appropriate handling of risk is of growing importane in the power industry. Stohasti programming is a proven tool for handling data unertainty in optimization problems [1]. The mentioned fats have motivated an extension of traditional stohasti programming methodology in various diretions of whih in the present paper we address the inlusion of integer deision variables and the transition from risk neutral models to those inorporating risk aversion. Our paper is organized as follows: In Setion 2 we introdue mean-risk extensions of risk neutral stohasti integer programs. In Setion 3 we study the blok strutures that arise in the equivalent mixed-integer linear programs if the underlying probability distributions are disrete. Setion 4 is devoted to deomposition and approximation algorithms exploiting these blok strutures. Computational tests for illustrative examples based on an energy system with dispersed generation of power and heat are reported in Setion 5. Finally we have a onlusions setion. 2. STOCHASTIC INTEGER PROGRAMS WITH RISK AVERSION To explain basi features of extending risk neutral into mean-risk stohasti integer programs let us start out from the following random optimization problem min{ T x + q(ω) T y : T(ω)x + W (ω)y = z(ω) x Xy Y} (1) together with the information onstraint that x must be seleted without antiipation of the realization of the random data (qtw z)(ω). This leads to a two-stage sheme of alternating deision and observation: The deision on x is followed by observing (qtw z)(ω) and then y 4 is taken thus

2 depending on x and ω. Aordingly x and y respetively. are alled first- and seond-stage deisions In energy optimization suh a setting is plausible in many ontexts. For example when planning a dispersed generation system the deisions on what types of and how many generators to install orrespond to x and operation deisions (under stohasti demand power pries or infeed from renewable resoures) orrespond to y [2]. Or as another example take day-ahead trading (with one round of bidding) for a utility. Here the first stage orresponds to the bids and the seond e.g. to prodution on the next day. The random input stems from the bids of the foreign utilities [3]. Coming bak to (1) we assume that X and Y are polyhedra possibly involving integer requirements to vetor omponents suh that (1) is a mixed-integer linear program under unertainty. The mentioned two-stage dynamis of (1) beomes more expliit by the following reformulation min T x + min x { q(ω) T y :W (ω)y = z(ω) T(ω)xy Y y { }: x X} = min{ T x +Φ(xω):x X}. x The problem of finding a best nonantiipative deision x in (1) thus turns into finding a best member in the indexed family of random variables ( T x +Φ(xω) ) x X. This leads to the topi of omparing or ranking random variables whih is studied in stohastis both oneptually and omputationally [4]. Ranking random variables aording to statistial parameters refleting mean values and/or dispersions is partiularly attrative from the omputational perspetive. In the risk neutral ase this parameter is the expetation ΙΕ whereas in the risk averse ase meanrisk models ome to the fore that are based on a weighted sum of ΙΕ and a later to be speified parameter R refleting risk. Denoting f (xω):= T x + Φ(xω) this paradigm leads to the following mean-risk stohasti integer program derived from (1) min{ Q MR (x):x X} (2) where Q MR (x):=( ΙΕ+ρ R) [ f (xω)] =ΙΕ[ f (xω)]+ ρ R[ f (xω)] = Q ΙΕ (x) + ρ Q R (x) with some fixed weight ρ > 0. In two respets this model goes beyond traditional linear stohasti programs: It allows for integer deision variables and by the term ρ Q R (x) it inludes risk aversion into the objetive. The fat that (2) is a nonlinear optimization problem arries only little weight algorithmially. Its objetive is a multivariate integral whose integrand is the optimal value of another (namely the seond-stage) optimization problem. This leads to insurmountable numerial diffiulties when omputing (with ompliated probability distributions) funtion values or gradients of Q MR. The latter by the way not even need to exist. Far worse Q MR is disontinuous in general and (2) may have a finite infimum that is not attained. This may happen when R is speified as variane see [5] for an (aademi) example. These observations indiate that the risk measure R beside being meaningful for the pratitioner should also fulfil requirements regarding the mathematial struture it indues in Q MR and the algorithmi possibilities for solving (2). Sine the initial random optimization problem (1) is mixedinteger and linear our aim is to find risk measures that enable (for disrete probability distributions) the solution of (2) by mixed-integer linear programming methodology. We will see that for dimensionality reasons it will not be suffiient to formulate just some mixed-integer linear program (MILP). Rather it will be important that the MILP obeys blok struture amenable to deomposition or approximation. Speifiations of R fulfilling these requirements are: 5

3 - Exess Probability whih reflets the probability of exeeding a presribed target level η IR s : Q IP η (x):= IP[ { ω : f (xω) > η} ]. - Conditional Value-at-Risk whih for a given probability level α ] 01[ reflets the expetation of the (1 α) 100% worst outomes. There are different ways to formalize this quantity see for instane [6]. One possibility is where Q αcvar (x):= min g(η x) (3) η IR g(η x):= η + 1 IE[ max f (xω) η0 1 α { } ]. - Expeted Exess whih reflets the expeted value of the exess over a given target η IR s : Q D η (x):= IE[ max{ f (xω) η0} ]. - Semideviation whih is similar in spirit to the Expeted Exess but with the pre-fixed target replaed by the mean: Q D + (x):= IE[ max{ f (xω) Q IE (x)0}]. Detailed expositions of mathematial strutures and algorithms assoiated with speifiations of (2) using these risk measures an be found in [578]. 3. BLOCK STRUCTURES OF MEAN-RISK STOCHASTIC INTEGER PROGRAMS Assume that (qtw z)(ω) is disretely distributed with finitely many senarios (q j T j W j z j ) and { } then probabilities π j j =1.... It is well-known that the risk neutral model min Q IE (x):x X admits the following equivalent representation as a mixed-integer linear program min T x + π j q T j : T j x + W j x X Y j =1.... (4) The onstraint matrix of (4) has the blok struture depited in Figure 1. The essential feature of this struture is that there is no onstraint with two seond stage variables 1 and 2 for different senarios j 1 j 2 { 1...} in ommon. 6

4 Figure 1: Blok struture of onstraint matrix Seond-stage variables are oupled only impliitly via x. This fat is ruial in deomposition shemes for the solution of (4). Considering the mean-risk model (2) the question arises on how MILP equivalents for different speifiations of R look like and whether the prinipal matrix struture of Figure 1 is preserved. Let R := Q IP η (Exess Probability) and ρ 0. If X exists a onstant M > 0 suh that (2) is equivalent to min T x + π j q T j + ρ π j θ j : T j x + W j is bounded it an be shown f. [5] that there T x + q T j η M θ j x X Yθ j { 01} j = 1...}. (5) So the struture of Figure 1 is maintained. The speifi nature of the risk measure however has led to additional Boolean variables θ j that an be seen as seond-stage variables. Let R := Q αcvar (Conditional Value-at-Risk) and ρ 0. Then f. [8] (2) is equivalent to min T x + T π j q j + ρ η α π j v j : T j x + W j T x + q T j η v j x Xη IR Yv j IR + j =1...}. (6) Here the matrix struture of Figure 1 is also maintained but a new ontinuous variable η appears that an be understood as a first-stage variable. Furthermore there are additional ontinuous seondstage variables stemming from a resolution of a max-expression. v j Let R := Q D η (Expeted Exess) and ρ 0. Then f. [8] (2) is equivalent to min T x + π j q T j + ρ π j v j : T j x + W j T x + q T j η v j x X Yv j IR + j =1...}. (7) 7

5 This model is very similar to (4). The prinipal matrix struture is maintained and only the resolution of a max-expression led to new onstraints and new seond-stage variables v j. Let R := Q D + (Semideviation) and ρ [ 01]. Then f. [7] (2) is equivalent to { T x + ( 1 ρ) π j q T j + ρ π j v j : T j x + W j min ( 1 ρ) Here the prinipal struture of Figure 1 is not maintained. The onstraints T x + q T j v j T x + π j q T j v j x X Yv j IR j =1...}. (8) T x + π j q T j v j j =1... (9) expliitly ouple seond-stage variables from different senarios. Summing up we see that the hoie of the risk measure in (2) is a sensitive issue regarding the blok strutures in the arising MILP equivalents. Consequenes may be mild as in (7) the risk measure may be the soure of additional integrality requirements as in (5) or it even may generate additional oupling in the model as in (8). 4. DECOMPOSITION AND APPROXIMATION ALGORITHMS The mean-risk models (2) for the different speifiations of R all are non-onvex non-linear optimization problems in general. Even when alleviating numerial integration problems by using disrete probability distributions the problem remains to find a global minimizer in a non-onvex optimization problem. Analytial properties of Q MR are partiularly poor when imposing a disrete μ see [578]. For lak of smoothness (even lak of ontinuity) hene loal (sub-) gradient based desent approahes to minimizing Q MR are not promising. The equivalent problem formulations of the previous setion provide an alternative. Although problem dimension was inreased onsiderably now there is the possibility of finally resorting to the welldeveloped algorithmi methodology of mixed-integer linear programming. Diret solution of (5) - (8) with general-purpose mixed-integer linear programming software however is prohibitive due to dimensionality reasons. Rather the mixed-integer linear programming tehniques are imbedded into lower bounding proedures of a branh-and-bound sheme Branh-and-Bound Framework To solve (2) by branh-and-bound the set X is partitioned with inreasing granularity. Linear inequalities are used for this partitioning to maintain the (mixed-integer) linear desription. On the urrent elements of the partition upper and lower bounds for the optimal objetive funtion value are sought. This is embedded into a oordination proedure to guide the partitioning and to prune elements due to infeasibility optimality or inferiority. For the following generi desription of the branh-and-bound method let Ρ denote a list of problems and ϕ LB (P) be a lower bound for the optimal value of P Ρ. By ϕ we denote the urrently best upper bound to the optimal value of (2). - Step 1 (Initialization): Let Ρ : = (2) and ϕ :=+. - Step 2 (Termination): 8

6 If Ρ := then the x that yielded ϕ = Q MR (x ) is optimal. - Step 3 (Bounding): Selet and delete a problem P from Ρ. Compute a lower bound ϕ LB (P) and find a feasible point x of P. - Step 4 (Pruning): If ϕ LB (P) = + (infeasibility of a subproblem) or ϕ LB (P) = Q MR (x ) (optimality of a subproblem) or ϕ LB (P) > ϕ (inferiority) then go to Step 2. If Q MR (x ) < ϕ then ϕ := Q MR (x ). - Step 5 (Branhing): Create two new subproblems by partitioning the feasible set of P. Add these subproblems to Ρ and go to Step 2. Step 4 is to be understood aordingly if no feasible point x was found in Step 3: Then none of the riteria applies and the algorithm goes to Step 5. The partitioning in Step 5 in priniple an be arried out by adding (in a proper way) arbitrary linear onstraints. The most popular way though is to branh along oordinates. This means to pik a omponent x (k ) of x and add the inequalities x (k) a and x (k ) a +1 with some integer a if x (k ) is an integer omponent or otherwise add x (k) a ε and x (k ) a + ε with some real number a where ε > 0 is some tolerane parameter to avoid endless branhing. When passing the lower bound ϕ LB (P) of P to its two hildren reated in Step 5 the differene or gap ϕ min P Ρ ϕ LB (P) provides information of the quality of the urrently best solution. Step 2 often is modified by terminating in ase the relative size of this gap drops below some threshold expressed in per ent Deomposition by Lagrangean Relaxation of Nonantiipativity Lower bounding is a ritial part in the above branh-and-bound framework. This is the plae where deomposition of the relevant stohasti programs into senario-speifi subproblems will beome ruial. Assume the stohasti program has the blok struture of Figure 2 introdue opies j =1... of x and reformulate the problem equivalently by adding the onstraints x 1 =... = x. This is nothing but formulating expliitly the nonantiipativity (NA) of x i.e. the requirement that x has to be independent of the senarios. The onstraint matrix of the reformulated problem has the prinipal struture shown in Figure 2. Figure 2: Blok struture of reformulation Relaxation of the NA-blok or in other words of the nonantiipativity of the first-stage variables hene deomposes the onstraints aording to senarios. In what follows we spell this out in detail in 9

7 terms of Lagrangean relaxation of NA for the Conditional-Value-at-Risk model (6). Reall that in (6) additionally to the deision variable x the auxiliary variable η is independent on j and hene an be understood as a first-stage variable that has to meet nonantiipativity. We express the nonantiipativity of x and η by the equations H j = 0 H j η j = 0 with suitable l m matries H j and l 1 vetors H j j =1.... The following Lagrangean funtion results L(x yvηλ):= where λ = ( λ λ ) and L j ( v j η j λ) L j ( v j η j λ λ ):= π j ( T + q T j + ρη j + ρ 1 1 α v ) + λ T j H j + λ T H j η j. The Lagrangean dual reads where { l + l } (10) max D(λ):λ IR D(λ) = min L j ( v j η j λ): T j + W j T + q T j η j v j Xη j IR Yv j IR + j =1...}. This optimization problem is separable with respet to the individual senarios i.e. D(λ) = where for j =1... D j (λ) D j (λ) = min{ L j ( v j η j λ) : T j + W j T + q T j η j v j X η j IR Yv j IR + }. (11) The Lagrangean dual (10) is a non-smooth onave maximization (or onvex minimization) problem with pieewise linear objetive for whose solution advaned bundle methods see for instane [910] an be applied. In this way solving the dual or in other words finding a desired lower bound redues to funtion value and subgradient omputations for D(λ) (when adopting a onvex minimization setting in (10)). A subgradient of D at λ is given by H j x λ λ j H j η j 10

8 where λ and η j λ are the orresponding omponents in an optimal solution vetor to the optimization problem defining D j (λ). As a onsequene the desired lower bound in Step 3 of the generi framework an be omputed by solving the single-senario mixed-integer linear programs in (11) instead of working with the full-size model (6). This deomposition is instrumental sine (11) may be tratable for MILP solvers while (6) may not. Results of the dual optimization also provide the basis for finding promising feasible points in Step 3 of the generi framework. Indeed starting from an optimal or nearly optimal λ the omponents x λ j η λ j j =1... of optimal solutions to (11) are seen as proposals for a nonantiipative firststage solution ( xη). A promising point then is seleted for instane by deiding for the most frequent one or by averaging and rounding to integers if neessary. The speifi nature of η as an auxiliary variable and argument in an optimization problem see (3) allows to improve this heuristi. Instead of seleting right away a andidate for ( xη) a andidate x for x is fixed first and then η is omputed as the best possible value namely as optimal solution to min{ g(ηx ):η IR} whih is equivalent to min η + 1 π j v j : T x +Φ(x ω j ) η v j 1 α η IR v j IR + j =1..}. The input quantities Φ(x ω j ) j =1... are readily omputed as optimal values of min{ q T j y :W j y T j x y Y}. If Φ(x ω j ) =+ (infeasibility of a seond-stage subproblem) then x is disarded. This onludes the onsideration of (6) as an illustration for how to ahieve deomposition in the lower bounding of (5) - (7) Approximation by Deomposable Lower Bounds Lagrangean relaxation of nonantiipativity of ourse is possible for (8) too. However the ounterpart to (11) then is no longer separable in the senarios due to the presene of the onstraints (9). An alternative are approximate lower bounds to (8) that deompose into senario speifi subproblems after relaxation of nonantiipativity. The expetation problem min Q IE (x):x X { } is an immediate suh bound although negleting risk effets ompletely. An improvement is the following relation established in [7]. For η Q IE (x) and all ρ [ 01] it holds that Q IE (x) (1 ρ)q IE (x) + ρq D η + ρη Q IE (x) + ρq D + (x). A possible speifiation of η fulfilling η Q IE (x) is given by the wait-and-see solution of the expetation model. This is the expeted value of Φ WS (ω):= min{ T x + q(ω) T y : T(ω)x + W (ω)y = z(ω)x X y Y}. Clearly Φ WS (ω) f (xω) for all x X and all ω suh that η = IEΦ WS (ω) Q IE (x) for all x X. 11

9 When wishing to solve (8) the bounding steps in the generi branh-and-bound framework from Subsetion 4.1 must be speified as follows. Lower bounds are obtained via Lagrangean relaxation of nonantiipativity applied to min{ ( 1 ρ)q IE (x) + ρq D η (x) + ρη : x X} (12) (with X of ourse replaed by partition elements at later stages of the algorithm). Sine Q D η leads to deoupled single-senario models after relaxation of nonantiipativity see (7) the ounterpart to (11) enjoys deomposition in the senarios. The results of the dual optimization again serve as inputs for the generation of upper bounds. Nonantiipative proposals however are not inserted into the objetive of (12) but into the original objetive Q IE (x) + ρq D + (x). 5. DISPERSED GENERATION - NUMERICAL EXPERIMENTS To illustrate models and algorithms of the previous setions let us onsider a system with dispersed generation (DG) of power and heat. The DG system run by a German utility onsists of 5 enginebased ogeneration stations eah equipped with a thermal storage and a ooling devie and altogether involving 8 gas boilers 9 gas motors and one gas turbine. The latter are ombined heat and power (CHP) units. The DG system is ompleted by 12 wind turbines and one hydroeletri power plant. Power is fed into a global grid enabling eletriity trade at energy markets. Heat is fed into loal networks around eah ogeneration station. Unertainty is present both at the input and the output sides of the system. At the input side this onerns the infeed from the wind turbines. At the output side power and heat demand as well as power pries typially are prone to unertainty. Taking into aount all relevant operational onstraints suh as minimum and maximum prodution levels for the DG units bounds for the fill of the storage demand fulfilment fuel onsumption and minimum up times of the units the DG system is modeled as a mixed-integer linear program with onoff deisions of units as the major soures of integrality. The objetive is to minimize fuel osts (minus revenues from trading). The planning horizon is 24 hours with a disretization into quarter-hourly subintervals. As a random optimization problem or in other words a speifiation of (1) the model has roughly variables (9.000 Boolean ontinuous) and onstraints. For a detailed desription of the model see [11]. With standard software for mixed-integer programs suh as ILOG-CPLEX [12] instanes of the random optimization problem with stohasti entities set to fixed values an be solved on a Linux-PC with a 30 GHz Pentium proessor and 20 GB RAM in less then 20 seonds and with gaps of less than 01 %. As a next step we impose the information onstraint that the random data of the model are known with ertainty in the first four hours of the time horizon and are unertain from then on. This leads to a twostage planning problem under unertainty where the first-stage variables are the deisions from the initial 16 quarter-hourly intervals and the seond-stage is given by the remaining variables. Table 1: Dimension of a risk neutral model The speifiation of the risk neutral expetation-based model (4) then no longer is tratable in aeptable time by ILOG-CPLEX. Table 1 shows problem dimensions and Table 2 omputational results. 12

10 The stopping riterion has been a gap of less than 001 %. While for the 50-senario instane the deomposition method from Subsetions 4.1 and 4.2 reahes this gap in roughly 10 minutes ILOG-CPLEX did not sueed in 24 hours. Table 2: Solution times for the risk neutral model The diffiulty of finding optimal nonantiipative deisions by means of senario analysis is illustrated in Table 3. For the heat prodution of the boilers in one ogeneration station we have listed an optimal nonantiipative solution (of a risk neutral stohasti program) against optimal solutions for different single-senario models. Clearly the optimal nonantiipative solution displayed in the last row of the table annot simply be dedued from the olletion of the antiipative first-stage deisions in the rows above. Table 3: Dispersion of first-stage deisions The following results for mean-risk models inorporating Expeted Exess Exess Probability or Conditional Value-at-Risk show some effets of inluding risk aversion and onfirm the superiority of deomposition over standard mixed-integer linear programming solvers. 13

11 Table 4 shows ontributions to the optimal objetive values of the individual senarios omputed with a mean-risk model involving Expeted Exess and varying risk weights ρ. The target value η was set to It beomes apparent that with inreasing weight optimal values are getting above or near target for an inreasing number of senarios. Table 4: Optimal solutions for different risk weights Computations for a model using the Conditional Value-at-Risk as the objetive are reported in Table 5. For instanes with 20 and more senarios the deomposition approah reahes the required gap of 0005 % in less omputational time than ILOG-CPLEX does. Appropriate solutions are found up to 8 times faster with the deomposition method. Table 5: Results for the Conditional Value-at-Risk Table 6 and Table 7 show results for a mean-risk model involving Exess Probability. Here time limits inreasing with the number of senarios were used as stopping riteria. The tables illustrate that for instanes with more than 10 senarios the deomposition method finds solutions with smaller gaps 14

12 than ILOG-CPLEX does. For instanes with 20 and more senarios ILOG-CPLEX fails to reah a feasible solution at all. Table 6: Results for the Exess Probability using the Deomposition Table 7: Results for the Exess Probability using ILOG-CPLEX 6. CONCLUSION Traditional risk neutral stohasti integer programs an be extended into mean-risk models by the help of risk measures suh as Exess Probability Conditional Value-at-Risk Expeted Exess or Semideviation. These risk measures indue sound mathematial strutures and allow for numerial treatment of the resulting mean-risk models in the framework of mixed-integer linear programming. 15

13 For the latter blok strutures arising within equivalent MILPs are essential. These strutures depend on the risk measures seleted and either enable a diret deomposition via Lagrangean relaxation of nonantiipativity or require approximation by deomposable models. The operation of energy systems with deentralized generation of power and heat and with infeed from renewable resoures is an ativity whose optimization is inherently infeted by unertainty. Mean-risk models of the mentioned types are proper tools for optimizing nonantiipative deisions in this ontext. The introdued deomposition algorithms are omputationally promising and outperform general purpose mixed-integer linear programming solvers at this lass of problems. 7. ACKNOWLEDGEMENT This researh has been supported by the German Federal Ministry for Eduation and Researh under grant 03SCNIVG. REFERENCES RUSZCZNYSKI A; SHAPIRO A. (EDS.) (2003): Handbooks in Operations Researh and Management Siene 10: Stohasti Programming Elsevier Amsterdam. VERHOEVEN M. (2005): Optimierung der Konfiguration von dezentralen Energieumwandlungsanlagen mit Kraft-Wärme-Kopplung Master Thesis Department of Mathematis University of Duisburg- Essen. NOWAK M.P.; SCHULTZ R.; WESTPHALEN M. (2005): A stohasti integer programming model for inorporating day-ahead trading of eletriity into hydro-thermal unit ommitment Optimization and Engineering MÜLLER A.; STOYAN D. (2002): Comparison Methods for Stohasti Models and Risks Wiley Chihester. SCHULTZ R.; TIEDEMANN S. (2003): Risk Aversion via Exess Probabilities in Stohasti Programs with Mixed-Integer Reourse SIAM ournal on Optimization ROCKAFELLAR R.T.; URYASEV S. (2003): Conditional Value-at-Risk for General Loss Distributions ournal of Banking & Finane MÄRKERT A.; SCHULTZ R. (2005): On deviation measures in stohasti integer programming Operations Researh Letters SCHULTZ R. TIEDEMANN S. (2006): Conditional value-at-risk in stohasti programs with mixedinteger reourse Mathematial Programming HELMBERG C.; KIWIEL K.C. (2002): A spetral bundle method with bounds Mathematial Programming KIWIEL K.C. (1990): Proximity ontrol in bundle methods for onvex nondifferentiable optimization Mathematial Programming HANDSCHIN E.; NEISE F.; NEUMANN H.; SCHULTZ R. (2005): Optimal operation of dispersed generation under unertainty using mathematial programming Preprint Dep. Of Mathematis University of Duisburg-Essen and presented at PSCC 05 Liege/Belgium CPLEX CALLABLE LIBRARY 8.1 (2003): ILOG Reeived une 2006 Revised anuary