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1 European Journal of Operaional Research 224 (2013) Conens liss available a SciVerse ScienceDirec European Journal of Operaional Research journal homepage: Decision Suppor Risk neural and risk averse Sochasic Dual Dynamic Programming mehod Alexander Shapiro a,, Wajdi Tekaya a, Joari Paulo da Cosa b, Murilo Pereira Soares b a School of Indusrial and Sysems Engineering, Georgia Insiue of Technology, Alana, GA , USA b ONS Operador Nacional do Sisema Elérico, Rua da Quianda, 196, Cenro Rio de Janeiro, RJ , Brazil aricle info absrac Aricle hisory: Received 12 January 2012 Acceped 24 Augus 2012 Available online 5 Sepember 2012 Keywords: Mulisage sochasic programming Dynamic equaions Sochasic Dual Dynamic Programming Sample average approximaion Risk averse Average Value-a-Risk In his paper we discuss risk neural and risk averse approaches o mulisage (linear) sochasic programming problems based on he Sochasic Dual Dynamic Programming (SDDP) mehod. We give a general descripion of he algorihm and presen compuaional sudies relaed o planning of he Brazilian inerconneced power sysem. Ó 2012 Elsevier B.V. All righs reserved. 1. Inroducion In his paper we discuss risk neural and risk averse approaches o mulisage (linear) sochasic programming problems based on he Sochasic Dual Dynamic Programming (SDDP) mehod. We give a general descripion of he algorihm and presen compuaional sudies relaed o operaion planning of he Brazilian inerconneced power sysem. The SDDP algorihm was inroduced by Pereira and Pino [6,7] and became a popular mehod for scheduling of hydro-hermal elecriciy sysems. The SDDP mehod uilizes convexiy of linear mulisage sochasic programs and sagewise independence of he sochasic daa process. I is based on building piecewise linear ouer approximaions of he cos-o-go funcions and can be viewed as a varian of he approximae dynamic programming echniques. The disinguishing feaure of he SDDP approach is random sampling from he se of scenarios in he forward sep of he algorihm. Almos sure convergence of he SDDP algorihm was proved in Philpo and Guan [8] under mild regulariy condiions (see also [12, Proposiion 3.1]). However, lile is known abou raes of convergence and compuaional complexiy of he SDDP mehod. I is someimes made a claim ha he SDDP algorihm avoids curse of dimensionaliy of he radiional dynamical programming mehod. This claim should be aken carefully. An analysis of he SDDP algorihm applied o wo sage sochasic programming indicaes ha is compuaional complexiy grows fas wih increase of he number of sae variables (cf., [13, Secion 5.2.2]), and his seems Corresponding auhor. Tel.: address: ashapiro@isye.gaech.edu (A. Shapiro). o be confirmed by some numerical experimens. Therefore i appears ha he SDDP mehod works reasonably well when he number of sae variables is relaively small while he number of sages can be large. Up o his poin, he sandard risk neural approach was implemened for planning of he Brazilian power sysem. The energy raioning ha ook place in Brazil in he period 2001/2002 raised he quesion of wheher a policy ha is based on a crierion of minimizing he expeced cos is a valid one when i comes o mee he day-o-day supply requiremens. As a consequence, a shif owards a risk averse crierion is underway, so as o enforce he coninuiy of load supply. Several risk averse approaches o mulisage sochasic programming were suggesed in recen lieraure. Eichhorn and Römisch [2] developed echniques based on he so-called polyhedral risk measures. This approach was exended furher in Guigues and Römisch [3] o incorporae he SDDP mehod in order o approximae he corresponding risk averse recourse funcions. Theoreical foundaions for a risk averse approach based on condiional risk mappings were developed in Ruszczyński and Shapiro [10] (see also [11, Chaper 6]). For risk measures given by convex combinaions of he expecaion and Average Value-a-Risk, i was shown in [12] how o incorporae his approach ino he SDDP algorihm wih a lile addiional effor. This was implemened in an exensive numerical sudy in Philpo and de Maos [9]. This aricle is organized as follows. In he nex secion we discuss general mehodological aspecs of he sochasic programming approach wih focus on he SDDP algorihm. In Secion 3 we give a brief descripion of he SDDP algorihm. Risk averse approaches wih various risk measures are discussed in Secion 4. We design in ha secion a general and simplified (as compared wih [12]) mehodology /$ - see fron maer Ó 2012 Elsevier B.V. All righs reserved. hp://dx.doi.org/ /j.ejor

2 376 A. Shapiro e al. / European Journal of Operaional Research 224 (2013) for implemenaion of he risk averse approach combined wih he SDDP mehod. In Secion 5 he Brazilian inerconneced power sysem is inroduced. In paricular, a muliplicaive imes series model is suggesed o deal wih nonnegaiviy of he monhly inflows. The performance of he SDDP algorihm is illusraed wih he aid of some numerical experimens and wih emphasis on he Brazilian power sysem case in Secion 6. Compuaional aspecs of he algorihm s convergence and soluion sabiliy are presened and discussed. Finally, Secion 7 is devoed o conclusions. 2. Mahemaical formulaion and modeling Mahemaical algorihms compose he core of he Energy Operaion Planning Suppor Sysem. The objecive is o compue an operaion sraegy which conrols he operaion coss, over a planning period of ime, in a reasonably opimal way. This leads o formulaion of (linear) large scale mulisage sochasic programming problems which in a generic form (nesed formulaion) can be wrien as Min A 1 x 1 ¼b 1 x 1 P0 2 c T 1 x 1 þ E4 2 þe4 min B 2 x 1 þa 2 x 2 ¼b 2 x 2 P0 min B T x T 1 þa T x T ¼b T x T P0 c T 2 x 2 þ E½ 333 c T T x 55 T 5: ð2:1þ Componens of vecors c, b and marices A, B are modeled as random variables forming he sochasic daa process n =(c, A, B, b ), =2,...,T, wih n 1 =(c 1, A 1, b 1 ) being deerminisic (no random). By n [] =(n 1,...,n ) we denoe he hisory of he daa process up o ime. I is ofen assumed in numerical approaches o solving mulisage problems of he form (2.1) ha he number of realizaions (scenarios) of he daa process is finie, and his assumpion is essenial in he implemenaions and analysis of he applied algorihms. In many applicaions, however, his assumpion is quie unrealisic. In forecasing models (such as ARIMA) he errors are ypically modeled as having coninuous (say normal or log-normal) disribuions. So one of he relevan quesions is wha is he meaning of he inroduced discreizaions of he corresponding sochasic process. We do no make he assumpion of finie number of scenarios, insead he following assumpions will be made. These assumpions (below) are saisfied in he applicaions relevan for he Brazilian power sysem generaion. We make he basic assumpion ha he random daa process is sagewise independen, i.e., random vecor n +1 is independen of n [] =(n 1,...,n ) for =1,...,T 1. In some cases across sages dependence can be deal wih by adding sae variables o he model. In paricular, he following consrucion is relevan for he considered applicaions. Suppose ha only he righ hand side vecors b are across sage dependen, while oher parameers of he problem form a sagewise independen process (in paricular, hey could be deerminisic). We are ineresed in cases where, for physical reasons, componens of vecors b canno be negaive. Suppose ha random vecors b follow ph order auoregressive process wih muliplicaive error erms: b ¼ e ðl þ / 1 b 1 þþ/ p b p Þ; ¼ 2;...; T; ð2:2þ where vecor l and marices / 1,...,/ p are esimaed from he daa. Here e 2,...,e T are independen of each oher error vecors and such ha wih probabiliy one heir componens are nonnegaive and have expeced value one, and ab denoes he erm by erm (Hadamard) produc of wo vecors. The muliplicaive error erm model is considered o ensure ha realizaions of he random process b have nonnegaive values. The auoregressive process (2.2) can be formulaed as a firs order auoregressive process b b 1 b 2 b pþ e l / 1 / 2 / p 1 / p 1 0 I ¼ 1 0 þ 0 I B I 0 b 1 b 2 b 3 b p 31 ; ð2:3þ 7C 5A where 1 is vecor of ones and I is he ideniy marix of an appropriae dimension. Denoe by z he column vecor in he lef hand side of (2.3), and by, M and U he respecive erms in he righ hand side of (2.3). Then (2.3) can be wrien as z ¼ ðm þ Uz 1 Þ; ¼ 2;...; T: ð2:4þ Consequenly he feasibiliy equaions of problem (2.1) can be wrien as z Uz 1 ¼ M; B x 1 þ A x ¼ b ; x P 0; ¼ 2;...; T: ð2:5þ Therefore by replacing x wih (x,z ), and considering he corresponding daa process n formed by random elemens of c, A, B and error vecors e, =2,...,T, we ransform he problem o he sagewise independen case. The obained problem is sill linear wih respec o he decision variables x and z. We consider he following approach o solving he mulisage problem (2.1). Firs, a (finie) scenario ree is generaed by randomly sampling from he original disribuion and hen he consruced problem is solved by he Sochasic Dual Dynamic Programming (SDDP) algorihm. There are hree levels of approximaions in ha seing. The firs level is modeling. The inflows are viewed as seasonal ime series and modelled as a periodic auo-regressive (PAR(p)) process. Any such modeling involves inaccuracies auoregressive parameers should be esimaed, errors disribuions are no precise, ec. We will refer o an opimizaion problem based on a curren ime series model as he rue problem. The rue model involves daa process n, =1,...,T, having coninuous disribuions. Since he corresponding expecaions (mulidimensional inegrals) canno be compued in a closed form, one needs o make a discreizaion of he daa process n. So a sample ~ n 1 ;...; ~ n N, of size N, from he disribuion of he random vecor n, =2,...,T, is generaed. These samples generae a scenario ree wih he oal number of scenarios N ¼ Q T ¼2 N, each wih equal probabiliy 1/N. Consequenly he rue problem is approximaed by he so-called Sample Average Approximaion (SAA) problem associaed wih his scenario ree. This corresponds o he second level of approximaion in he curren sysem. Noe ha in such a sampling approach he sagewise independence of he daa process is preserved in he consruced SAA problem. If we measure he compuaional complexiy, of he rue problem, in erms of he number of scenarios required o approximae he rue disribuion of he random daa process wih a reasonable accuracy, he conclusion is raher pessimisic. In order for he opimal value and soluions of he SAA problem o converge o heir rue counerpars all sample sizes N 2,...,N T should end o infiniy. Furhermore, available esimaes of he sample sizes N required for a firs sage soluion of he SAA problem o be e-opimal for he rue problem, wih a given confidence (probabiliy), sums up o a oal number of scenarios N which grows as O(e 2(T 1) ) wih decrease of he error level e > 0 (cf., [11, Secion 5.8.2]). This indicaes ha from he poin of view of he number of scenarios, complexiy of mulisage programming problems grows exponenially

3 A. Shapiro e al. / European Journal of Operaional Research 224 (2013) wih increase of he number of sages. In oher words even wih a moderae number of scenarios per sage, say each N = 100, he oal number of scenarios N quickly becomes asronomically large wih increase of he number of sages. Therefore a consruced SAA problem can be solved only approximaely. The SDDP mehod suggess a compuaionally racable approach o solving SAA, and hence he rue, problems, and can be viewed as he hird level of approximaion in he curren sysem. A heoreical analysis (cf., [13, Secion 5.2.2]) and numerical experimens indicae ha he SDDP mehod can be a reasonable approach o solving mulisage sochasic programming problems when he number of sae variables is small even if he number of sages is relaively large. 3. Generic descripion of he SDDP algorihm In his secion we give a general descripion of he SDDP algorihm applied o he SAA problem. Suppose ha N, =2,...,T, poins generaed a every sage of he process. Le n j ¼ðc j; A j ; B j ; b j Þ, j =1,...,N, =2,...,T, be he generaed poins. As i was already menioned he oal number of scenarios of he SAA problem is N ¼ Q T ¼1 N and can be very large Backward sep of he SDDP algorihm Le x, =1,...,T 1, be rial poins (we can, and evenually will, use more han one rial poin a every sage of he backward sep, an exension o ha will be sraighforward), and Q ðx 1 Þ be he (expeced value) cos-o-go funcions of dynamic programming equaions associaed wih he mulisage problem (2.1) (see, e.g., [11, Secion 3.1.2]). Noe ha because of he sagewise independence assumpion, he cos-o-go funcions Q ðx 1 Þ do no depend on he daa process. Furhermore, le Q ðþ be a curren approximaion of Q ðþ given by he maximum of a collecion of cuing planes Q ðx 1 Þ¼max k2i a k þ b T k x 1 ; ¼ 1;...; T 1: ð3:1þ A sage = T we solve N T problems Min c T x T 2R n Tj x T s:: B Tj x T 1 þ A Tj x T ¼ b Tj ; x T P 0; j ¼ 1;...; N T : T ð3:2þ Recall ha Q Tj ðx T 1 Þ is equal o he opimal value of problem (3.2) and ha subgradiens of Q Tj () ax T 1 are given by B T Tjp Tj, where p Tj is a soluion of he dual of (3.2). Therefore for he cos-o-go funcion Q T ðx T 1 Þ we can compue is value and a subgradien a he poin x T 1 by averaging he opimal values of (3.2) and he corresponding subgradiens. Consequenly we can consruc a supporing plane o Q T ðþ and add i o collecion of supporing planes of Q T ðþ. Noe ha if we have several rial poins a sage T 1, hen his procedure should be repeaed for each rial poin and we add each consruced supporing plane. Now going one sage back le us recall ha Q T 1;j ðx T 2 Þ is equal o he opimal value of problem Min c T x T 1 2R n T 1;j x T 1 þq T ðx T 1 Þ s:: B T 1;j x T 2 þ A T 1;j x T 1 T 1 ¼ b T 1;j ; x T 1 P 0: ð3:3þ However, funcion Q T ðþ is no available. Therefore we replace i by Q T ðþ and hence consider problem Min c T x T 1 2R n T 1;j x T 1 þ Q T ðx T 1 Þ s:: B T 1;j x T 2 þ A T 1;j x T 1 T 1 ¼ b T 1;j ; x T 1 P 0: ð3:4þ Recall ha Q T ðþ is given by he maximum of affine funcions (see (3.1)). Therefore we can wrie problem (3.4) in he form Min x T 1 2R n T 1 ;h2r c T T 1;j x T 1 þ h s:: B T 1;j x T 2 þ A T 1;j x T 1 ¼ b T 1;j ; x T 1 P 0 h P a Tk þ b T Tk x T 1; k 2 I T : ð3:5þ Consider he opimal value, denoed Q T 1;j ðx T 2 Þ, of problem (3.5), and le p T 1,j be he parial vecor of an opimal soluion of he dual of problem (3.5) corresponding o he consrain B T 1;j x T 2 þ A T 1;j x T 1 ¼ b T 1;j, and le Q T 1 ðx T 2 Þ :¼ 1 NT 1 X Q T 1;j ðx T 2 Þ N T 1 and g T 1 ¼ 1 NT 1 X B T N T 1 T 1;jp T 1;j : Consequenly we add he corresponding affine funcion o collecion of Q T 1 ðþ. And so on going backward in Forward sep of he SDDP algorihm The compued approximaions Q 2 ðþ;...; Q T ðþ (wih Q Tþ1 ðþ 0 by definiion) and a feasible firs sage soluion x 1 can be used for consrucing an implemenable policy as follows. For a realizaion n = (c,a,b,b ), =2,...,T, of he daa process, decisions x, =1,...,T, are compued recursively going forward wih x 1 being he chosen feasible soluion of he firs sage problem, and x being an opimal soluion of Min x c T x þ Q þ1 ðx Þ s:: A x ¼ b B x 1 ; x P 0; ð3:6þ for = 2,..., T. These opimal soluions can be used as rial decisions in he backward sep of he algorihm. Noe ha x is a funcion of x 1 and n, i.e., x is a funcion of n [] = (n 1,...,n ), for =2,...,T. Tha is, policy x ¼ x ðn ½Š Þ is nonanicipaive and by he consrucion saisfies he feasibiliy consrains for every realizaion of he daa process. Thus his policy is implemenable and feasible. If we resric he daa process o he generaed sample, i.e., we consider only realizaions n 2,...,n T of he daa process drawn from scenarios of he SAA problem, hen x ¼ x ðn ½Š Þ becomes an implemenable and feasible policy for he corresponding SAA problem. On he oher hand, if we draw samples from he rue (original) disribuion, his becomes an implemenable and feasible policy for he rue problem. Since he policy x ¼ x ðn ½Š Þ is feasible, he expecaion " # E XT c Tx ðn ½Š Þ ð3:7þ ¼1 gives an upper bound for he opimal value of he corresponding mulisage problem. Tha is, if we ake his expecaion over he rue probabiliy disribuion of he random daa process, hen he above expecaion (3.7) gives an upper bound for he opimal value of he rue problem. On he oher hand, if we resric he daa process o scenarios of he SAA problem, each wih equal probabiliy 1/N, hen he expecaion (3.7) gives an upper bound for he opimal value of he SAA problem condiional on he sample used in consrucion of he SAA problem. Of course, if he consruced policy is opimal, hen he expecaion (3.7) is equal o he opimal value of he corresponding problem. The forward sep of he SDDP algorihm consiss in generaing M random realizaions (scenarios) of he daa process and compuing he respecive opimal values # j :¼ XT ¼1 c T j x j; j ¼ 1;...; M: ð3:8þ Tha is, # j is he value of he corresponding policy for he realizaion n 1 ; n j 2 ;...; nj T of he daa process. As such, # j is an unbiased esimae

4 378 A. Shapiro e al. / European Journal of Operaional Research 224 (2013) h of he expeced value of ha policy, i.e., E½# j Š¼E P i T ¼1 ctx ðn ½Š Þ. The forward sep has wo funcions. Firs, some (all) of compued soluions x j can be used as rial poins in he nex ieraion of he backward sep of he algorihm. Second, hese soluions can be employed for consrucing a saisical upper bound for he opimal value of he corresponding mulisage program (rue or SAA depending on from wha disribuion he sample scenarios were generaed). Le ~# M :¼ 1 M X M # j and ~r 2 M ¼ 1 X M ð# j # M 1 ~ M Þ 2 be he respecive sample mean and sample variance of he compued values # j. Since # j is an unbiased esimae of he expeced value of he consruced policy, we have ha ~ # M is also an unbiased esimae of he expeced value of ha policy. By invoking he Cenral Limi Theorem we can say ha ~ # M has an approximaely normal disribuion provided ha M is reasonably large. This leads o he following (approximae) (1 a)-confidence upper bound for he value of ha policy u a;m :¼ # ~ ~r M M þ z a p ffiffiffiffi : ð3:9þ M Here 1 a 2 (0,1) is a chosen confidence level and z a = U 1 (1 a), where U() is he cdf of sandard normal disribuion. Tha is, wih probabiliy approximaely 1 a he expeced value of he consruced policy is less han he upper bound u a;m. Since he expeced value (3.7) of he consruced policy is bigger han or equal o he opimal value for he considered mulisage problem, we have ha u a;m also gives an upper bound for he opimal value of he mulisage problem wih confidence a leas 1 a. Noe ha he upper bound u a;m can be used for he SAA or he rue problem depending on from wha disribuion he sampled scenarios were generaed. Since Q ðþ is he maximum of cuing planes of he cos-o-go funcion Q ðþ we have ha Q ðþ P Q ðþ; ¼ 2;...; T: ð3:10þ Therefore he opimal value of he problem compued a a backward sep of he algorihm, gives a lower bound for he opimal value of he considered SAA problem. This lower bound is deerminisic (i.e., is no based on sampling) if applied o he corresponding SAA problem. As far as he rue problem is concerned, recall ha he opimal value of he rue problem is greaer han or equal o he expecaion of he opimal value of he SAA problem. Therefore on average his is also a lower bound for he opimal value of he rue problem. On he oher hand, he upper bound u a;m is a funcion of generaed scenarios and hus is sochasic even for considered (fixed) SAA problem. This upper bound may vary for differen ses of random samples, in paricular from one ieraion o he nex of he forward sep of he algorihm. The SDDP algorihm wihin he risk neural framework is summarized in Fig Adapive risk averse approach h In formulaion (2.1) he expeced value E P i T ¼1 ct x of he oal cos is minimized subjec o he feasibiliy consrains. Tha is, he oal cos is opimized (minimized) on average. Since he coss are funcions of he random daa process, hey are random and hence are subjec o random perurbaions. For a paricular realizaion of he daa process hese coss could be much bigger han heir average (i.e., expecaion) values. We will refer o he formulaion (2.1) as risk neural as opposed o risk averse approaches which we will discuss below. The goal of a risk averse approach is o avoid large values of he coss for some possible realizaions of he daa process a every sage of he considered ime horizon. One such approach will be o mainain consrains c T x 6 h, =1,...,T, for chosen upper levels h and all possible realizaions of he daa process. However, rying o enforce hese upper limis under any circumsances could be unrealisic and infeasible. One may ry o relax hese consrains by enforcing hem wih a high (close o one) probabiliy. However, inroducing such so-called chance consrains can sill resul in infeasibiliy and moreover is very difficul o handle numerically. So we consider here penalizaion approaches. Tha is, a every sage he cos is penalized while exceeding a specified upper limi. In a simple form his leads o a risk averse formulaion where coss c T x are penalized by / c T x h, wih h þ and / P 0, =2,...,T, being chosen consans. Tha is, he coss c T x are replaced by funcions f ðx Þ¼c T x þ / c T x h in he objecive þ of he problem (2.1). The addiional penaly erms represen he penaly for exceeding he upper limis h. An immediae quesion is how o choose consans h and /. I could be noed ha in ha approach he upper limis h are fixed (chosen a priori) and no adaped o a curren realizaion of he random process. Le us observe ha opimal soluions of he corresponding risk averse problem will be unchanged if he penaly erm a h sage is changed o h þ / c T x h by adding he consan h þ. Now if we adap he upper limis h o a realizaion of he daa process by aking hese upper limis o be (1 a )-quaniles of c T x condiional on observed hisory n [ 1] =(n 1,...,n 1 ) of he daa process, we end up wih penaly erms given by AV@R a wih a =1//. Recall ha he Average Value-a-Risk of a random variable 1 Z is defined as AV@R a ½ZŠ ¼V@R a ðzþþa 1 E½Z V@R a ðzþš þ ; ð4:1þ wih V@R a (Z) being he (say lef side) (1 a)-quanile of he disribuion of Z, i.e., V@R a ðzþ ¼inff : FðÞ P 1 ag; where F() is he cumulaive disribuion funcion (cdf) of he random variable Z. This leads o he following nesed risk averse formulaion of he corresponding mulisage problem (cf., [10]) Min A 1 x 1 ¼b 1 x 1 P0 c T 1 x 1 þq 2jn1 6 4 min B 2 x 1 þ A 2 x 2 ¼ b 2 x 2 P 0 c T 2 x 2 þþq Tjn½T 1Š 6 4 min B T x T 1 þ A T x T ¼ b T x T P 0 c T T x T77 55 : ð4:2þ Here n 2,...,n T is he random process (formed from he random elemens of he daa c, A, B, b ), E½Zjn ½ 1Š Š denoes he condiional expecaion of Z given n [ 1], AV@R a ½Zjn ½ 1Š Š is he condiional analogue of AV@R a ½ZŠ given n [ 1], and q jn½ 1Š ½ZŠ ¼ð1 k ÞE½Zjn ½ 1Š Šþk AV@R a ½Zjn ½ 1Š Š; ð4:3þ wih k 2 [0,1] and a 2 (0,1) being chosen parameers. In formulaion (4.2) he penaly erms a 1 c T x V@R a c T x are condiional, i.e., are adaped o he random process by he opimizaion þ procedure. Therefore we refer o he risk averse formulaion (4.2) as adapive. I is also possible o give he following inerpreaion of he adapive risk averse formulaion (4.2). I is clear from he definiion (4.1) ha AV@R a [Z] P V@R a (Z). Therefore q jn½ 1Š ½ZŠ P. jn½ 1Š ½ZŠ, where. jn½ 1Š ½ZŠ ¼ð1 k ÞE½Zjn ½ 1Š Šþk V@R a ½Zjn ½ 1Š Š: ð4:4þ If we replace q jn½ 1Š ½ZŠ in he risk averse formulaion (4.2) by. jn½ 1Š ½ZŠ, we will be minimizing he weighed average of means 1 In some publicaions he Average Value-a-Risk is called he Condiional Value-a- Risk and denoed CV@R a. Since we deal here wih condiional AV@R a, i will be awkward o call i condiional CV@R a.

5 A. Shapiro e al. / European Journal of Operaional Research 224 (2013) Fig. 1. Risk neural SDDP. and (1 a)-quaniles, which will be a naural way of dealing wih he involved risk. Unforunaely such formulaion will lead o a nonconvex and compuaionally inracable problem. This is one of he main reasons for using AV@R a insead of V@R a in he corresponding risk averse formulaions. I is possible o show ha in a cerain sense AV@R a () gives a bes possible upper convex bound for V@R a (), [5]. I also could be menioned ha he (condiional) risk measures of he form (4.4) are no he only ones possible. We will discuss below some oher examples of risk measures Risk averse SDDP mehod Wih a relaively simple addiional effor he SDDP algorihm can be applied o risk averse problems of he form (4.2). Le q, =2,...,T, be a sequence of chosen (law invarian coheren) risk measures and q jn½ 1Š be heir condiional analogues (see, e.g., [11, Chaper 6] for a discussion of opimizaion problems involving coheren risk measures). Specifically, apar from risk measures of he form (4.3), we also consider mean-upper semideviaion risk measures of order p: q ðzþ ¼E½ZŠþj E ½Z E½ZŠŠ p 1=p þ ; ð4:5þ where p 2 [1,1) and j 2 [0,1]. In paricular, for p = 1 his becomes q ðzþ ¼E½ZŠþj E½Z E½ZŠŠ þ ¼ E½ZŠþ 1 2 j EjZ E½ZŠj: ð4:6þ Assuming ha he sagewise independence condiion holds, he corresponding dynamic programming equaions are Q ðx 1 ; n Þ¼ inf c T x 2R n x þq þ1 ðx Þ : B x 1 þ A x ¼ b ; x P 0 ; ð4:7þ wih Q þ1 ðx Þ :¼ q þ1 ½Q þ1 ðx ; n þ1 ÞŠ: A he firs sage problem ð4:8þ Min c T 1 x 1 þq 2 ðx 1 Þ s:: A 1 x 1 ¼ b 1 ; x 1 P 0; ð4:9þ x 1 2R n 1 should be solved. Noe ha because of he sagewise independence, he cos-o-go funcions Q þ1 ðx Þ and he risk measures q +1 in (4.8) do no depend on he daa process. Noe also ha since he considered risk measures are convex and monoone, he cos-o-go funcions Q þ1 ðx Þ are convex (cf., [11, Secion 6.7.3]) Backward sep for mean-upper semideviaion risk measures The corresponding SAA problem is obained by replacing he expecaions wih heir sample average esimaes. For risk measures of he form (4.5), he dynamic programming equaions of he SAA problem ake he form Q j ðx 1 Þ¼ inf x 2R n for j =1,...,N 1, wih Q þ1 ðx Þ¼ b Q þ1 ðx Þþj 1 N X N n o c T j x þq þ1 ðx Þ : B j x 1 þ A j x ¼ b j ; x P 0 ; ð4:10þ ½Q þ1;j ðx Þ b Q þ1 ðx ÞŠ p þ! 1=p ; ð4:11þ

6 380 A. Shapiro e al. / European Journal of Operaional Research 224 (2013) = T,...,2 and Q T+1 () 0, where bq þ1 ðx Þ¼ 1 X N Q N þ1;j ðx Þ: The opimal value of he SAA problem is given by he opimal value of he firs sage problem Min c T x 1 2R n 1 x 1 þq 2 ðx 1 Þ s:: A 1 x 1 ¼ b 1 ; x 1 P 0: 1 ð4:12þ In order o apply he backward sep of he SDDP algorihm we need o know how o compue subgradiens of he righ hand side of (4.11). Le us consider firs he case of p = 1. Then (4.11) becomes Q þ1 ðx Þ¼Q b þ1 ðx Þþ j X N h Q N þ1;j ðx Þ Q b i þ1 ðx Þ þ : ð4:13þ Le c +1,j be a subgradien of Q +1,j (x ), j =1,...,N, a he considered poin x. In principle i could happen ha Q +1,j () is no differeniable a x, in which case i will have more han one subgradien a ha poin. Forunaely we need jus one (any one) of is subgradiens. Then he corresponding subgradien of b Q þ1 ðx Þ is ^c þ1 ¼ 1 X N c N þ1;j ; and he subgradien of ½Q þ1;j ðx Þ Q b þ1 ðx ÞŠ þ is ( m þ1;j ¼ 0 if Q þ1;jðx Þ b Q þ1 ðx Þ < 0; c þ1;j ^c þ1 if Q þ1;j ðx Þ b Q þ1 ðx Þ > 0; and hence he subgradien of Q þ1 ðx Þ is g þ1 ¼ ^c þ1 þ j N X N m þ1;j : ð4:14þ ð4:15þ ð4:16þ In he backward sep of he SDDP algorihm he above formulas are applied o he piecewise linear lower approximaions Q þ1 ðþ exacly in he same way as in he risk neural case (discussed in Secion 3). Le us consider now he case of p > 1. Noe ha hen he cos-ogo funcions of he SAA problem are no longer piecewise linear. Neverheless he lower approximaions Q þ1 ðþ are sill consruced by using cuing planes and are convex piecewise linear. Similar o (4.16) he corresponding subgradien of Q þ1 ðx Þ is (by he chain rule) g þ1 ¼ ^c þ1 þ p 1 j q p X N g N þ1;j ; ð4:17þ P where q ¼ 1 N N ½Q þ1;jðx Þ Q b þ1 ðx ÞŠ p þ and ( g þ1;j ¼ 0 if Q þ1;jðx Þ Q b þ1 ðx Þ < 0; p½q þ1;j ðx Þ Q b þ1 ðx ÞŠ p 1 ðc þ1;j bc þ1 Þ if Q þ1;j ðx Þ Q b þ1 ðx Þ > 0: Backward sep for mean-av@r risk measures Le us consider risk measures of he form (4.3), i.e., q ðzþ ¼ð1 k ÞE½ZŠþk AV@R a ½ZŠ: The cos-o-go funcions of he corresponding SAA problem are Q þ1 ðx Þ¼ð1 k Þ b Q þ1 ðx Þ þ k Q þ1;i ðx Þþ 1 a N X N ½Q þ1;j ðx Þ Q þ1;i ðx ÞŠ þ!; ð4:18þ ð4:19þ ð4:20þ where i 2 {1,...,N } corresponds o he (1 a ) sample quanile, i.e., numbers Q +1,j (x ), j =1...,N, are arranged in he increasing order Q þ1;pð1þ ðx Þ 6 6 Q þ1;pðnþðx Þ and i ¼ ^j such ha pð^jþ is he smalles ineger such ha pð^jþ P ð1 a ÞN. Noe ha if (1 a )N is no an ineger, hen i remains he same for small perurbaions of x. The corresponding subgradien of Q þ1 ðx Þ is g þ1 ¼ð1 k Þbc þ1 þ k c þ1;i þ 1 a N X N where ( f þ1;j!; ð4:21þ f þ1;j ¼ 0 if Q þ1;jðx Þ Q þ1;i ðx Þ < 0; c þ1;j c þ1;i if Q þ1;j ðx Þ Q þ1;i ðx Þ P 0: ð4:22þ The above approach is simpler han he one suggesed in [12], and seems o be working as well Forward sep The consruced lower approximaions Q ðþ of he cos-o-go funcions define a feasible policy and hence can be used in he forward sep procedure in he same way as i was discussed in Secion 3.2. Tha is, for a given scenario (sample pah), saring wih a feasible firs sage soluion x 1, decisions x, =2,...,T, are compued recursively going forward wih x being an opimal soluion of Min x c T x þ Q þ1 ðx Þ s:: A x ¼ b B x 1 ; x P 0; ð4:23þ for = 2,..., T. These opimal soluions can be used as rial decisions in he backward sep of he algorihm. Unforunaely here is no easy way o evaluae he risk-adjused cos c Tx h 1 1 þ q 2jn1 c Tx 2 2ðn ½2Š Þþþq Tjn½T 1Š c Tx i T Tðn ½TŠ Þ ð4:24þ of he obained policy, and hence o consruc an upper bound for he opimal value of he corresponding risk-averse problem (4.2). Therefore a sopping crierion based on sabilizaion of he lower bound was used in numerical experimens. Of course, he expeced value (3.7) of he consruced policy can be esimaed in he same way as in he risk neural case by he averaging procedure. 5. Case sudy descripion The Brazilian inerconneced power sysem is a large scale sysem planned and consruced considering he inegraed uilizaion of he generaion and ransmission resources of all agens and he use of iner-regional energy inerchanges, in order o achieve cos reducion and reliabiliy in power supply. The power generaion faciliies as of December 2010 are composed of more han 200 power plans wih insalled capaciy greaer han 30 megawa, owned by 108 public and privae companies, called Agens 57 Agens own 141 hydro power plans locaed in 14 large basins, 69 wih large reservoirs (monhly regulaion or above), 68 run-of-river plans and four pumping saions. Considering only he Naional Inerconneced Sysem (SIN), wihou capive self-producers, he insalled capaciy reaches gigawa in 2020, wih an incremen of 61.6 gigawa over he 2010 insalled capaciy of gigawa. Hydropower accouns for 56% of he expansion (34.8 gigawa), while biomass and wind power accoun for 25% of he expansion (15.4 gigawa). The main ransmission grid is operaed and expanded in order o achieve safey of supply and sysem opimizaion. The iner-regional and iner-basin ransmission links allow inerchanges of large blocks of energy beween areas making i possible o ake advanage of he hydrological diversiy beween river basins. The main ransmission grid sysem has km of lines above

7 A. Shapiro e al. / European Journal of Operaional Research 224 (2013) kv, owned by 66 Agens, and is planned o reach kilomeer in 2020, an incremen of 41.2%, mainly wih he inerconnecion of he projecs in he Amazonian region. I is worh noing ha he insalled capaciy of hydro plans corresponds o 79.1% of he December 2010 oal insalled capaciy, bu is relaive posiion should diminish o 71.0% in Neverheless, considering he domesic elecriciy supply, he hydro power supremacy will coninue in 2020, sanding for 73.4% of he oal, as compared o 80.6% in December, 2010 (including impors). This relaive reducion is offse by a srong peneraion of biomass and wind generaion. In his conex, renewable sources mainain a high paricipaion in elecriciy supply marix (87.7%) (see [4]) Operaion planning problem The purpose of hydrohermal sysem operaion planning is o define an operaion sraegy which, for each sage of he planning period, given he sysem sae a he beginning of he sage, produces generaion arges for each plan. The usual objecive is o minimize he expeced value of he oal cos along he planning period, so as o mee requiremens on he coninuiy of energy supply subjec o feasibiliy consrains. The operaion coss comprise fuel coss, purchases from neighboring sysems and penalies for failure in load supply. This is referred o as he risk neural approach: he oal cos is opimized on average, and for a paricular realizaion of he random daa process he coss could be much higher han heir average values. Risk averse approaches, on he oher hand, aim a finding a compromise beween minimizing he average cos and rying o conrol he upper limi of he coss for some possible realizaions of he daa se a every sage of he process. The risk averse approach will be discussed laer in his aricle. The hydrohermal operaing planning can be seen as a decision problem under uncerainy because of unknown variables such as fuure inflows, demand, fuel coss and equipmen availabiliy. The exisence of large muli-year regulaing reservoirs makes he operaion planning a mulisage opimizaion problem; in he Brazilian case i is usual o consider a planning horizon of 5 years on a monhly basis. The exisence of muliple inerconneced hydro plans and ransmission consrains characerizes he problem as large scale. Moreover, because he value of energy generaed in a hydro plan canno be measured direcly as a funcion of he plan sae alone bu raher in erms of expeced fuel savings from avoided hermal generaion, he objecive funcion is also nonseparable [6]. In summary, he Brazilian hydro power operaion planning problem is a mulisage (60 sages), large scale (more han 200 power plans, of which 141 are hydro plans), nonlinear and nonseparable sochasic opimizaion problem. This seing far exceeds he compuer capaciy o solve i wih adequae accuracy in reasonable ime frame. The sandard approach o solve his problem is o resor o a chain of models considering long, mid and shor erm planning horizon in order o be able o ackle he problem in a reasonable ime. For he long-erm problem, i is usual o consider an approximae represenaion of he sysem, he so-called aggregae sysem model, a composie represenaion of a mulireservoir hydroelecric power sysem, proposed by Arvanidiis and Rosing [1], ha aggregaes all hydro power plans belonging o a homogeneous hydrological region ino a single equivalen energy reservoir, and solves he resuling much smaller problem. The major componens of he aggregae sysem model are: he equivalen energy reservoir model and he oal energy inflow (conrollable and unconrollable), see Fig. 2. The energy sorage capaciy of he equivalen energy reservoir can be esimaed as he energy ha can be produced by he depleion of he reservoirs of a sysem, provided a simplified operaing rule ha approximaes he acual depleion policy. Fig. 2. Aggregae sysem model. Fig. 3. Case-sudy inerconneced power sysem. For he Brazilian inerconneced power sysem i is usual o consider four energy equivalen reservoirs, one in each one of he four inerconneced main regions, SE, S, N and NE. For his simplified problem, one can use he SDDP approach and obain he coso-go funcions for each of he sages of he planning period. The resuling policy obained wih he aggregae represenaion can hen be furher be refined, so as o provide decisions for each of he hydro and hermal power plans. This can be done by solving he mid-erm problem, considering a planning horizon up o a few monhs and individual represenaion of he hydro plans wih boundary condiions (he expeced cos-o-go funcions) given by he soluion of he long-erm problem. This is he approach nowadays used for solving he long and mid erm hydrohermal power planning in he Brazilian inerconneced power sysem. The numerical experimens were carried ou considering insances of mulisage linear sochasic problems based on an aggregae represenaion of he Brazilian Inerconneced Power Sysem long-erm operaion planning problem, as of January 2010, which can be represened by a graph wih four generaion nodes comprising sub-sysems Souheas (SE), Souh (S), Norheas (NE) and Norh (N) and one (Imperariz, IM) ransshipmen node (see Fig. 3). The load of each area mus be supplied by local hydro and hermal plans or by power flows among he inerconneced areas. A

8 382 A. Shapiro e al. / European Journal of Operaional Research 224 (2013) Table 1 Defici coss and dephs. % of oal load curailmen Cos Table 2 Inerconnecion limis beween sysems. To SE S NE N IM From SE S NE N IM slack hermal generaor of high cos ha increases wih he amoun of load curailmen accouns for load shorage a each area (Table 1). Inerconnecion limis beween areas may differ depending of he flow direcion, see Table 2. The energy balance equaion for each sub-sysem has o be saisfied for each sage and scenario. There are bounds on sored and generaed energy for each sub-sysem aggregae reservoir and on hermal generaions. The long-erm planning horizon for he Brazilian case comprises 60 monhs, due o he exisence of muli-year regulaion capaciy of some large reservoirs. In order o obain a reasonable cos-ogo funcion ha represens he coninuiy of he energy supply afer hese firs 60 sages, a common pracice is o add 60 more sages o he problem and consider a zero cos-o-go funcion a he end of he 120h sage. There is no definiive answer for how many sages should be added o remedy he end of horizon effec. Empirically, we could observe ha wih 60 addiional sages his effec is dissipaed. Hence, he objecive funcion of he planning problem is o minimize he expeced cos of he operaion along he 120 monhs planning horizon, while supplying he area loads and obeying echnical consrains. In he case of he risk neural approach, he objecive funcion is he minimizaion of he expeced value along he planning horizon of hermal generaion coss plus a penaly erm ha reflecs energy shorage. The case s general daa, such as hydro and hermal plan daa and inerconnecion capaciies were aken as saic values hrough ime. The demand for each sysem and he energy inflows in each reservoir were aken as ime varying. The unis for he energy inflows are in Mega Was monh (MWm) and he coss are in Brazilian real per MWm. In order o se he hydrohermal operaing planning problem wihin he framework of Secion 2 one can proceed as follows. Considering he aggregae represenaion of he hydroplans, he energy conservaion equaion for each equivalen energy reservoir n can be wrien as SE ;n ¼ SE 1;n þ CE ;n GH ;n SP ;n : ð5:1þ Tha is, he sored energy (SE) a he end of each sage (sar of he nex sage) is equal o he iniial sored energy plus conrollable energy inflow (CE) minus oal hydro generaed energy (GH) and losses (SP) due o spillage, evaporaion, ec. A each sage, he ne subsysem load L, given by he remaining load afer discouning he unconrolled energy inflow from he oal load, has o be me by he oal hydro, he sum of all hermal generaion belonging o sysem n, given by he se NT n, and he ne inerconnecion energy flow (NF) o each subsysem. In oher words, he energy balance equaion for sysem n is GH ;n þ X GT ;j þ NF ;n ¼ L ;n : ð5:2þ j2nt n Fig. 4. Box plo of he inflows for each sysem.

9 A. Shapiro e al. / European Journal of Operaional Research 224 (2013) x ¼ðSE; GH; GT; SP; NFÞ > ; b ¼ðCE; LÞ > ; c ¼ð0; 0; CT;0; 0Þ > ; A ¼ I I 0 I 0 0 I D 0 I ; B ¼ I where D ={d n,j = 1 for all j 2 NT n and zero else}, I and 0 are ideniy and null marices, respecively, of appropriae dimensions and he componens of CT are he uni operaion cos of each hermal plan and penaly for failure in load supply. Noe ha hydroelecric generaion coss are assumed o be zero. Physical consrains on variables like limis on he capaciy of he equivalen reservoir, hydro and hermal generaion, ransmission capaciy and so on are aken ino accoun wih consrains on x. More deails can be found in [16,6] Time series model for he inflows ; Fig. 5. Box plo of he log-observaions of SE inflows. Fig. 6. Parial auocorrelaion of he residuals of he log-observaions of SE inflows. where (GT) denoes he hermal generaion. Noe ha his equaion is always feasible (i.e., he problem has complee recourse) due o he inclusion of a dummy hermal plan wih generaion capaciy equal o he demand and operaion cos ha reflecs he social cos of no meeing he energy demand (defici cos). Consrains B x 1 + A x = b are obained wriing The hisorical daa The hisorical daa are composed of 79 observaions of he naural monhly energy inflow (from year 1931 o 2009) for each of he four sysems. Le X, =1,...,948 denoe a ime series of monhly inflows for one of he regions. Hisograms for he hisorical observaions show posiive skew for each of he 4 sysems. This observaion moivaes considering Y = log(x ) for analysis. Afer aking he logarihm, he hisograms become more symmeric. Fig. 4 shows monhly box plos of regions inflows. I could be seen ha inflows of he N, NE and SE sysems have a clear seasonal behavior, while for he S sysem i is no obvious Time series analysis of SE As an example we give below he analysis of he ime series X of he SE daa poins. Analysis of he oher 3 regions were carried ou in a similar way. Fig. 5 shows box plos of monhly inflows of he log-observaions Y = log(x ) of SE inflows. One can clearly noe he seasonal behavior of he series, suggesing ha a periodic monhly model could be a reasonable framework for his series. Le ^l ¼ ^l þ12 be he monhly averages of Y and Z ¼ Y ^l be he corresponding residuals. Fig. 6 shows he parial auocorrelaion of he Z ime series. High value a lag 1 and insignifican values for larger lags sugges he firs order AR(1) auoregressive ime series model for Z : Z ¼ a þ /Z 1 þ : ð5:3þ Fig. 7. Bounds for risk neural SDDP wih ime series model.

10 384 A. Shapiro e al. / European Journal of Operaional Research 224 (2013) Fig. 8. Approximae gap for risk neural SDDP. Table 3 Toal CPU ime, bounds saus and gap a ieraion 2000 and Ieraion CPU ime (h) LB (10 6 ) UB average (10 6 ) UB upper end (10 6 ) Gap (%) For he adjused model he esimae for he consan erm a was highly insignifican and could be removed from he model. This is no surprising since values Z by hemselves are already residuals. Trying second order AR(2) model for Z did no give a significan improvemen of he fi. Similar resuls were obained for he oher hree subsysems. Therefore, we consider an AR(1) model for all subsysems in he subsequen analysis Model descripion The analysis of Secion suggess he following model for he ime series Y for a given monh Y ^l ¼ /ðy 1 ^l 1 Þþ ; ð5:4þ where is iid sequence having normal disribuion N(0,r 2 ). For he original imes series X his gives Fig. 9. Individual sage coss a ieraion 2000 and 3000.

11 A. Shapiro e al. / European Journal of Operaional Research 224 (2013) Fig. 10. CPU ime per ieraion. Denoe by R ¼ X e^l e^l. If he error erm is se o zero, i.e., he muliplicaive error erm e is se o one, (5.7) can be wrien as: R ¼ c R 1 ð5:8þ For each monh, we perform a leas square fi o he R sequence o obain he monhly values for c, assuming ha c = c +12. The errors are modelled as a componen of he mulivariae normal disribuion Nð0; R b Þ, where R b is he sample covariance marix for X log ½ e^l þ c e^l ^l 1 ðx 1 e^l 1 ÞŠ on a monhly basis, i.e., b R þ12 ¼ b R. Validaion of his model can be found in a working paper a: hp:// 6. Compuaional experimens Fig. 11. Toal policy value for 60 sages for a 2 {0.05, 0.1} as funcion of k. X ¼ e e^l /^l 1 X / 1 : ð5:5þ Unforunaely his model is no linear in X and would resul in a nonlinear mulisage program. Therefore we proceed by using he following (firs order) approximaion of he funcion y = x / a e^l 1 x / ðe^l 1 Þ / þ /ðe^l 1 Þ / 1 ðx e^l 1 Þ; which leads o he following approximaion of he model (5.5) X ¼ e e^l þ /e^l ^l 1 ðx 1 e^l 1 Þ : ð5:6þ We allow, furher, he consan / o depend on he monh, and hence o consider he following ime series model X ¼ e e^l þ c e^l ^l 1 ðx 1 e^l 1 Þ ð5:7þ wih c = c +12. We esimae he parameers of model (5.7) direcly from he daa. The numerical experimens are performed on an aggregaed represenaion of he Brazilian Inerconneced Power Sysem operaion planning problem wih hisorical daa as of January The sudy horizon is of 60 sages and he oal number of considered sages is 120. We use he high demand profile seing described in [15]. We implemen wo versions of he risk averse SDDP algorihm, one wih he mean-av@r and one wih he mean-upper semideviaion risk measures boh applied o solve he problem wih he model suggesed in Secion 5 (wih 8 sae variables a each sage). The SAA ree, generaed in boh cases, has 100 realizaions in every sage wih he oal number of scenarios = In he following experimens we run he SDDP algorihm wih 1 rial soluion per ieraion. The individual sage coss and policy value are evaluaed using 3000 randomly generaed scenarios. Boh implemenaions were wrien in C++ and using Gurobi 4.6. Deailed descripion of he algorihms can be found in [15]. The codes were run on 1 core of (2 quad-core Inel E5520 Xeons 2.26 gigaherz, and 24 gigabye RAM) machine. Dual simplex was used as a defaul mehod for he LP solver. In Secion 6.1 he resuls for he SDDP algorihm wih he ime series model (5.7) are discussed wihin he risk neural framework. The following Secion 6.2 discusses he compuaional experimens for he risk averse approach wih mean-av@r and mean-upper

12 386 A. Shapiro e al. / European Journal of Operaional Research 224 (2013) Fig. 12. Individual sage coss for k = 0.15 and a 2 {0.05,0.1}. Fig. 13. SE sysem quaniles of sored volumes for each sage. semideviaion risk measures. Finally, we conclude his par by discussing variabiliy of he SAA problems and sensiiviy o he iniial condiions in Secions 6.3 and Risk neural approach resuls In his secion we invesigae some compuaional issues relaed o he risk neural SDDP applied o he problem wih ime series model (5.7). Fig. 7 shows he bounds for risk neural SDDP for more han 7000 ieraions. In he legend, we have he following noaion: LB: he lower bound (i.e., he firs sage opimal value). UB: he upper end of he 95% confidence inerval of he upper bound compued approximaely using as observaions he pas 100 forward sep realizaions. UBMA: moving average of UB using he pas 100 values. Fig. 7 illusraes ypical behavior of he SDDP bounds fas increase in he lower bound for he firs ieraions and hen a slow increase in laer ieraions. The upper bound exhibis some variabiliy along wih a decreasing rend. We should noice he relaively slow convergence. Fig. 8 shows he evoluion of he approximae gap over ieraions. The coninuous line provides a smoohing of he observaions o ge he approximae rend. We consider an approximae gap defined by UBMA LB. This is jus LB an approximaion of he real gap defined as he difference beween he upper end of 95% confidence inerval and he lower bound. Due o he significan compuaional effor o evaluae adequaely his gap, we approximae he observaions by aking he pas 100 forward sep realizaions. We perform a ieraion 2000 and ieraion 3000 a proper forward sep wih 3000 scenarios (see Table 3 for he deails). We can see in Fig. 8 he fas decrease in he firs 1000 ieraions and hen a relaively slow decay for laer ones. Table 3 shows he lower bound, upper bound 95% confidence inerval (mean and upper end), he CPU ime along wih he gap (i.e., UBupper LB ) a ieraion 2000 and The confidence inerval LB for he upper bound was compued using 3000 randomly generaed scenarios. The approximae gap (of 35.84% and 29.24% a ieraions 2000 and 3000, respecively) gives a slighly higher value han he accurae gap. Also, o reach a gap of 22.11%, which is quie large, 3000 ieraions were needed. The experimen ook 136,740.7 seconds (i.e., approximaely 38 hours). Fig. 9 shows he individual sage coss a ieraion 2000 and ieraion 3000 for he risk neural SDDP. Some differences are perceivable beween he individual sage cos disribuions a ieraion 2000 and ieraion These differences are noiceable in he 95% quanile. However, hese differences don have a dramaic effec on he general shape of he disribuion. This observaion is expeced since running he algorihm furher will provide a more accurae represenaion of he opimal soluion. For his research purposes, i seems reasonable

13 A. Shapiro e al. / European Journal of Operaional Research 224 (2013) Fig. 14. CPU ime per ieraion for he risk neural and risk averse approaches. Fig sages oal coss for p 2 {1,2,3}. o sop he algorihm a a compuaionally accepable running ime wihou any significan impac on he general conclusions. Fig. 10 shows he CPU ime per ieraion for he risk neural SDDP for more han 7000 ieraions. We can see he linear rend of he CPU ime per ieraion. The discrepancy occurring a some ieraions is mos likely due o he shared resources feaure of he compuing environmen Risk averse approach resuls Mean-AV@R risk measures In his secion, we invesigae some compuaional issues relaed o he mean-av@r risk averse SDDP applied o he operaion planning problem wih he ime series model suggesed in Secion 5.2. Fig. 11 shows he oal policy value for he firs 60 sages a ieraion 3000 for a 2 {0.05, 0.1} and k 2 {0,0.05, 0.1,...,0.35}. The doed line corresponds o a = 0.1 and he coninuous line corresponds o a = The figure plos he average cos and he 90%, 95% and 99% quaniles for each a. Pracically here is no significan difference beween he 60 sages oal cos for a = 0.05 and for a = 0.1 when 0 6 k For k = 0.35, he oal 60 sages policy value is lower for a = 0.1. Furhermore, we can see ha as k increases (i.e. more imporance is given o he high quanile minimizaion) he average policy value increases and an improvemen in some quaniles is observed for 0<k The price of risk aversion is wha we lose on average compared o he risk neural case (i.e. k = 0). I is he price paid for some proecion agains exreme values. Fig. 12 shows he individual sage coss a ieraion 3000 for k = In his figure, we compare he individual sage coss for he risk neural case, a = 0.05 and a = 0.1. When we compare he risk averse approach and he risk neural approach, we can see he significan reducion in he 99% quanile and he loss in he average policy value ha occurs mosly in he firs sages. Mos of he reducion of he 90% and 95% quaniles happens in he las 15 sages. Furhermore, here is no significan difference beween he individual sage coss for a = 0.05 and a = 0.1. As an example of he impac of he risk averse approach in he decision variables, Fig. 13 shows he evoluion of SE sysem sored volumes along he planning period, for risk neural and risk averse for k = 0.15 and a = 0.05, where one can observe higher values of he sored volumes wih he risk averse approach, as is expeced. The availabiliy of higher sored volumes makes i possible, in case of droughs, o be able o avoid large deficis and coss spikes as shown in Fig. 12. Noice also ha wih he risk averse mehod, here is an increased likelihood of spillage. Among he ineresing quesions ha we can ask is: how much does he risk averse approach cos in erms of CPU ime compared o he risk neural approach? Fig. 14 shows he CPU ime per ieraion for he risk neural and he AV@R risk averse approach wih