Benders Decomposition for Capacity Expansion Planning with Network Constraints and Uncertain Demand: the Spanish Case

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1 Benders Decomposition for Cpcity Expnsion Pnning with Network Constrints nd Uncertin Demnd: the Spnish Cse Fco. Aberto Cmpos #1 José Vir # # Institute for Reserch in Technoogy Technic Schoo of Engineering Comis Pontific University C/ Snt Cruz de Mrcendo, 26, Mdrid-Spin 1 cmpos@iit.upcomis.es Cristin A. Díz * * Opertion Pnning Deprtment XM, Coombin System Opertor Ce 12 Sur No Boque 2, Medeín, Coombi cdiz@xm.com.co Abstrct This pper presents stochstic mode for genertion expnsion pnning in power system with network constrints nd uncertin demnd. A DC representtion of the network is ssumed. To mke the probem computtiony fesibe when ppied to the Spnish power system, bi-eve Benders decomposition hs been deveoped. The min resuts of this ppiction prove the robustness nd the efficiency of the pnning in comprison with other deterministic soutions. Outputs re the cpcity to inst of ech technoogy (incuding renewbe ssets) in ech yer nd in ech Spnish Autonomous Community. The convergence of the Benders gorithm hs so been tested. Keywords Genertion expnsion pnning, Benders decomposition, stochstic optimiztion, network constrints, RES integrtion. JEL codes O21, Q55, L94, C61, D81. Indices t NOMENCLATURE Technoogies Yers n, m Nodes s,j Mode prmeters Stochstic scenrios Benders itertions ϕ Discount rte (%) CI t, CP t, CNQ Investment cost ( /MW) Production cost ( /MWh) Cost of non-suppy demnd ( /MWh) D n,, D n,s, S n,, S n,s, C U t, Q t,n, Q n,m, Q I t,n Di t,n, Ci t,n, X n,m, Inestic demnd t node n (MWh) Interntion exchnges t node n. Positive vues re imports (MWh) Cpcity of ech unit of technoogy t (MW) Mximum cpcity to inst of technoogy t by yer (MW) Cpcity of the trnsmission ines (MW) Initi inst cpcity of technoogy t (MW) Outge rte of ech technoogy t. It so modes the utiiztion fctor for renewbe technoogies (p.u.) Annu cosure rte of pnts of ech technoogy t tht were pnned before the first pnning yer (p.u.) Rectnce of ine n to m (p.u.) P n,s, Probbiity of ech demnd scenrio (%) BD gorithm prmeters j BD cut s type for ech itertion (0=fesibiity cuts,1=optimity cuts) RF j Sub-probem objective vue ( ) j t,n,s, QA j t,n, NG j t,n, Du vribe of the mximum production constrint ( /MWh) Cumutive insted cpcity (MW) Number of units to inst (1,..,) L, U Lower nd upper BD bounds ( ) ε Benders toernce (%) IT Continuous vribes Benders mximum itertions (1,..,)

2 2 c, mc, sc Probem, mster probem nd sub-probem tot costs ( ) rf() Benders resource function ( ) f n,m,, f n,m,s, n,, n,s, qp t,n,, qp t,n,s, qi t,n,, qi t,n,s, q t,n,, q t,n,s, nq n,, nq n,s, cm n,, cm n,s, Integer vribes ng t,n, Power fow from node n to m. Positive vues re imports (MW) Phse nge t node n (rd) Production of technoogy t (MWh) New insted cpcity (MW) Cumutive insted cpcity (MW) Non-suppy demnd t node n (MWh) Mrgin cost t node n (MWh) Number of units to inst of technoogy t (1,..,) I. INTRODUCTION The im of ong-term genertion expnsion probems, in the frmework of power systems, consists in seecting the most dequte technoogies to invest in, the mount of new genertion cpcity to inst, nd the sites for buiding the new genertion pnts. To do so, physic, geogrphic nd economic criteri must be ppied whie ensuring tht the tot insted cpcity dequtey meets the expected demnd in ong-term horizon. Sever pproches nd techniques for the genertion expnsion probem re proposed in the iterture, but cn be cssified under two min groups: modes for iberized systems (see [1] for review), nd modes tht ssume centrized systems, sometimes just to void the compexity of oigopoistic modeing (see [2] nd [3]). Within the first group (see [4] for deeper cssifiction of this group), sever modes consider the optimiztion of ony one compny, eding to MPEC structures ([5]). For exmpe, [6] obtins n MPEC tht represents the sttic bi-eve cpcity expnsion probem of compny when the demnd is uncertin. Investments nd productions re decided in the upper eve, whie the ower eve cers the mrket. The probem of one compny, compring Cournot nd Stckeberg pproches ([7]), is nyzed in [8] tking into ccount hydro nd pumpedhydro constrints. Reference [9] describes bi-eve mode for the investment decisions of one compny ssuming perfecty competitive mrket, nd considering uncertinty not ony in the demnd but so in the competitors genertion cpcities. A genertion pnning mode is proposed in [10] when sever gents re competing. To find soution, [10] ssumes centrized entity in chrge of the evution of ech compny pnning in n itertive process. Ech evution provides informtion regrding the competitors behvior, which is used to mximize the profit functions in the sense of Nsh. Reference [11] presents genertion pnning invoving decisions on new units construction by ppying Cournot mode ([12]) t singe point in time. EPEC structures ([13]) ppers so in expnsion modes under competition. Reference [14] proposes n EPEC for the genertion expnsion considering one future yer, which is inerized nd soved using Mixed Integer Liner Progrmming ([15]). In [16] two bi-eves expnsion probems re proposed under perfect nd Cournot competitions, discussing existence nd uniqueness issues. An EPEC tht tkes into ccount hydro power, demnd nd competitors investments uncertinty is described in [1]. Other reistic detis such s the introduction of cpcity mechnisms for finnci hedging re so considered in [1]. The second group (centrized modes) is very we studied in the iterture (see [2] for metheuristic techniques to sove the invoved modes, encompssing Genetic Agorithms, Expert Systems, Fuzzy Progrmming, Artifici Neur Networks, Anytic Hierrchy Processes, Network Fows, nd Simuted Anneing). Unike the gme-bsed modes of the first group (for iberized systems), these centrized modes cn esiy consider trnsmission network constrints (equiibrium existence is typicy ost in gme-bsed modes incuding these constrints, see [17]). In [18] robust genertion expnsion pnning is obtined considering set of probbe demnd scenrios. A MPEC mode is presented in [19] to obtin the genertion investments in the upper eve by minimizing tot costs, subject to ower eve for the mrket cering under different od nd wind conditions. References [3], [20] nd [21] describe sever mutiobjective modes for the expnsion pnning, obtining Preto soutions when cost, environment impct nd sever types of risks re optimized simutneousy by centrized entity. In gener, the min imittions in cpcity expnsion modes re the size nd compexity of the probems to be soved. These two spects depend on both the horizon nd the degree of deti with which the system is represented 1. To sove this, different resoution methods consisting in spitting the probem in subprobems soved by stges, hve been proposed. The method most used is Benders Decomposition (BD; see [22] nd the nnex for review), tht cn be ppied when the probem to be soved hs speci bock structure. Under this structure, BD is be to divide very dense probem into different inked subprobems, nd obtin the optim soution itertivey. The mode structure required to ppy BD is often presented in ppictions tht ed to stochstic progrmming, s it is the cse of this pper. Different ppictions of BD to eectric power systems cn be found in the iterture. In [23] mode tht combines the ppiction of Genetic Agorithms nd BD is proposed for the cpcity expnsion probem. Athough this mode tkes into ccount the vibiity of the genertion units, it does not consider uncertinty in the inputs. In [24] BD is ppied to sove trnsmission expnsion pnning. Genertion expnsion is not optimized in this cse. BD is so ppied in [25] to optimize the cpcity expnsion when considering some probbiistic reibiity constrints. The opertion for fixed cpcity is optimized in the subprobem, whie optim 1 Modes for iberized system so hve n ddition drwbck reted with the compexity of the equiibrium conditions to be soved.

3 3 cpcity investments with yery trnsvers constrints resut from the mster probem. In this pper, mode for pnning cpcity expnsion considering stochstic demnd is proposed. A robust expnsion pn ginst this uncertinty is obtined, simiry to [18] but considering the effect of network constrints. Ony the expnsion in genertion technoogies is modeed. Therefore, trnsmission ines cpcities re considered inputs (see [24] for possibe extension to trnsmission expnsion). As in the centrized modes reviewed (see for exmpe [8]), the mode minimizes the expected tot system cost, tking into ccount investment nd expected opertion costs. The min resuts re the tot cpcity to inst, which is given by technoogy nd node, the fows ong ech trnsmission ine, the production of ech technoogy nd the eectricity mrgin cost t ech node. The min contribution of this pper is the ppiction of the proposed mode to the Spnish eectric system, considering ech Spnish Autonomous Community (CA) 2 s node of the network, estimting the therm imits of ech ine bsed on historic dt from different pubic web pges, nd their rectnce bsed on [26]. Though Spnish mrket does not hs nod prices, our mode cn serve s vid representtion of the diy-hed Spnish mrket when combined with the impct of the technic constrints mrket ([27]). For computtion resons, bi-eve BD is used to sove the proposed ppiction, where investments decisions re optimized in the mster probem whie genertion units opertion is soved in the sub-probem. This pper is orgnized s foows. Section II presents the formution of the proposed genertion expnsion mode without considering demnd uncertinty, nd when this uncertinty is incuded in the mode. Section III exposes the ppiction of the bi-eve BD technique to the mode. Section IV nyses reistic cse study. Finy, some concusions re presented in the st section. The nnex describes the mthemtic formution nd n overview of the bi-eve BD technique ppied. II. THE GENERATION EXPANSION PROBLEM WITH NETWORK CONSTRAINTS A. The deterministic cse If the demnd is deterministic nd unitry durtions for ech yer re ssumed (hereinfter by simpicity), the objective function of the proposed mode consists in the minimiztion of the present vue of the tot system cost: U CIt, ngt, n, C t, t 1 Min c CP qp CNQ nq n, t, t, n, (1) 1 n t where: The first term is the tot investment cost, computed s the product of the unitry investment cost CI t,n, nd the insted cpcity, which is the number ng t,n, of units to be insted mutipied by the cpcity C U t, of ech unit. The second term represents the tot production cost computed s the product of the unitry production cost CP t, nd the production qp t,n,. Finy, the third term represents the cost of the nonsuppied demnd nq n,. The optimiztion probem must be soved subject to the foowing constrints: q t, n, I Qt, n Cit, n, 0 U qt, n, 1 ngt, n, Ct, Cit, n, 1 0 qpt, n, qt, n, Dit, n, t (2) (3) f qp f S nq D cm n t, n, m, n, n, n, n, :, m n, m, n, m, (5) X n, m, Q f Q (6) n, m, n, m, n, m, where: (2) ccutes the ccumuted insted cpcity q t,n,, tking into ccount Ci t,n,, the nnu cosure rte of pnts tht were pnned before the first pnning yer. (3) is the mximum production constrint of the genertion units, tking into ccount the vibiity rte Di t,n, of ech technoogy. This rte so modes the utiiztion fctor for renewbe technoogies. Constrint (3) is essenti since it retes investment nd opertion decisions, which in turns compictes significnty the resoution. (4) is the bnce between genertion nd demnd considering the interntion power exchnges nd the power fows modeed. The du vribes of these constrints provide the mrgin cost cm n, t ech node n. (5) ccutes the trnsmission ines power fow f n,m, ong the ine (n,m), s function of the rectnce x n,m, nd the phse nges n, nd m,. A reference phse nge ref, must be chosen such tht ref, =0. Ech nge must be so upper bounded by 2 rd. Finy (6) modes the therm imits of the power fows f n,m,. (4) 2 The Autonomous Communities (CCAA) re the first-eve poitic division of the Kingdom of Spin, estbished in ccordnce with the Spnish Constitution.

4 4 B. Stochstic cse: uncertin demnd The bove minimiztion probem cn be extended to the cse of uncertin demnd by minimizing the present vue of the foowing expected objective function: U CIt, ngt, n, C t, t 1 Min c CPt, qpt, n, s, 1 n P t n, s, s CNQ nqn, s, Indeed, to obtin robust cpcities to inst, different demnd scenrios D n,s, with different probbiities P n,s, re considered, though the number ng t,n, of units to be insted must not depend on index s. On the contrry, the rest of vribes depend on s since they re referred to the system opertion, which is different for ech demnd scenrio. Aprt from (2), the foowing constrints must be tken into ccount in the stochstic mode (insted of (3), (4), (5) nd (6)): 0 qpt, n, s, qt, n, Dit, n, t (8) qp f S nq D cm n s (9) f t, n, s, m, n, s, n, s, n, s, n, s, :,, m n, s, m, s, n, m, s, (10) X n, m, Q f Q (11) n, m, n, m, s, n, m, III. BD APPLIED TO THE CAPACITY EXPANSION PROBLEM WITH NETWORK CONSTRAINTS Bsed on (8), decomposition mode tht seprtes the investment nd opertion decisions is described in this section. The min dvntge of this pproch is tht two seprted smer probems cn be soved insted of the origin one, which is sometimes too rge to be soved with convention toos. A. Formution This subsection presents the formution of bi-eve BD for the mode described in II.B (see the nnex for the mthemtic detis). To do so, the mode hs been decomposed s foows: Investment decisions ng t,n, nd q t,n, re optimized in the mster probem. Productions qp t,n,s,, fows f n,m,s, nd phse nges n,s, re optimized in the sub-probem. In prticur, the mster probem objective is: (7) 1 Min mc CI ng C rf ng, q 1 t, n U t, n, t, n, t, n, (12) where rf(ng,q) is the we-known resources function in BD iterture, which depends on the mster probem decisions (ng,q)=(ng t,n,,q t,n,, t,n,). Function rf(ng,q) is pproximted by vribe rf, n outer pproximtion using optimity or fesibiity cuts (see the nnex for more detis). If j is booen prmeter tht identifies if BD cut is n optimity cut ( j =1) or fesibiity cut ( j =0), t ech itertion j, then rf is obtined using the cuts unti itertion, from the foowing iner equtions, tht must be embedded in the mster probem: j j j j t, n, s, t, n, t, n, t, n, s, rf RF q QA j 1,.., -1 (13) Aprt from these cut constrints, ddition restrictions tht must be tken into ccount in the mster probem re (2). The objective of the sub-probem is the resource function rf(ng,q) expressed s: 1 CPt, n, qp t, n, s, t n, s, (14) 1 n, s rf ng, q Min sc P CNQ nqn, s, At ech itertion, the sub-probem constrints re (8), (9), (10) nd (11) fixing vribes (ng,q) t their optim vues from the mster probem. Note tht since the subprobem constrints do not ink vribes from different yers, ech yer cn be soved independenty. The foowing subsection describes with more deti this itertive process. B. Agorithm The BD s itertive process to sove the proposed expnsion mode cn be described with the foowing pseudo-code 3 : 1. Prmeters initiiztion: =1. L=-, U=. 2. Sove the mster probem incuding (2) nd the cuts of (13). Do: NG t,n,=ng.l t,n, QA t,n,=q.l t,n, L=rf.L 3. Sove the sub-probem incuding (8), (9), (10) nd (11), nd fixing vector (ng,q) s foows: ng t,n, =NG t,n, q t,n, =QA t,n, 4. New cut buiding: 3 X.L hs been used to refer to the optim vue of vribe X.

5 Nucer Coi Fue Combined Cyce Wind Sor Anduci Argon Asturis Csti-L mnch Csti y Leon Ctuñ C. Vencin Extremdur Gici L Rioj Murci Nvrr Pis Vsco Cntbri Mdrid 5 If the sub-probem is unfesibe: o Set =0 o Sove the foowing sub-probem to nyze the infctibiity of the subprobem: Min sp t, n, s, t, n, s, s. t. qp q Di t, n, s, t, n, s, t, n, t, n, (9), (10) nd (11) (15) o RF =sp.l o t,n,s,=du vribes of the first constrints of (15). ese o Set =1 o RF =sc.l o U=Min(U, RF ) o t,n,s,=du vribes of (8). 5. Proof of convergence: if 1-(L/U) ε or =IT then end. Otherwise, go to step 2. A. System nd cse description IV. CASE STUDY The cse study consists in the ppiction of the proposed mode to the Spnish eectric system in 2011, considering time horizon of 9 yers (from 2012 to 2020). The ifespn of the pnts to be insted hs been negected since it is typicy higher thn the ength of the horizon. It hs been ssumed tht the demnd must be competey stisfied, i.e., vribes nq n, nd nq n,s, hve been fixed to zero in the executions. The Spnish system in 2011 ws composed by 82 therm units which, for this cse, hve been ggregted in 4 therm technoogies (Nucer, Coi, Fue nd Combined Cyce). Two ddition renewbe technoogies hve been so considered (Wind nd Sor; Sor incuding photovotic nd therm). These technoogies hve been octed in 15 nodes of the considered power grid, ech node representing Spnish peninsur CA. The vues of the initi inst cpcity Q I t,n re shown in Tbe I for ech technoogy nd hve been obtined from reports of the Spnish system opertor. TABLE I: INITIAL INSTALLED CAPACITY (MW) The demnd scenrios D n,s, to be stisfied using the considered technoogies (tht is, D n,s, is the tot demnd minus the productions with hydro, pumping nd cogenertion technoogies) re bsed on the historic demnd of 2011, incresed t n nnu rtes of 1.6% nd 2.6% for 2014 nd 2015 respectivey nd of 6% from 2016 onwrd. Historic nnu rte for 2012 nd 2013 due the economic crisis were nd -2.3% respectivey. Tbe II shows the three demnd scenrios considered bsed on the demnd of They hve been beed s MED, MAX, MIN to identify the cses of medium (MED, demnd of 2011), high (MAX, 1.35 times MED) nd ow demnd (MIN, 0.65 times MED) vues, respectivey. TABLE II: DEMAND SCENARIOS (MWH/H) MAX MED MIN Anduci Argon Asturis Csti-L mnch Csti y Leon Ctuñ C. Vencin Extremdur Gici L Rioj Murci Nvrr Pis Vsco Cntbri Mdrid The probbiity P n,s, of ech demnd scenrio hs been supposed constnt over the time horizon, nd independent of ech CA, being 25%, 50% nd 25% for the MAX, MED nd MIN demnd scenrios respectivey. The vues for the mximum cpcity of the trnsmission network ines hve been estimted bsed on historic dt from different pubic web pges (see Tbe IV). The network is depicted in Fig. 1. TABLE IV: MAXIMUM CAPACITY OF TRANSMISSION NETWORK (MW) Anduci Argon Asturis Csti-L mnch Csti y Leon Ctuñ C. Vencin Extremdur Gici L Rioj Murci Nvrr Pis Vsco Cntbri Mdrid Anduci Argon Asturis Csti-L mnch Csti Leon Ctuñ C. Vencin Extremdur Gici L Rioj Murci Nvrr Pis Vsco Cntbri Mdrid

6 Insted cpcity (MW) Anduci Argon Asturis Csti-L mnch Csti y Leon Ctuñ C. Vencin Extremdur Gici L Rioj Murci Nvrr Pis Vsco Cntbri Mdrid Insted cpcity (MW) 6 TABLE VII: INTERNATIONAL EXCHANGES OF 2011 (MWH/H) Anduci -513 rgon -37 Csti y Leon -28 Ctuñ 105 Extremdur 119 Gici -412 Pis Vsco 71 The utiiztion fctors of wind nd sor technoogies hve been fixed to 0.43 nd 0.4 p.u. respectivey, which cn be computed from historic dt. Fig. 1: Considered Spnish network A. Investment pnning: stochstic versus deterministic Fig. 2 shows the investment pnning obtined from the stochstic mode described in subsection II.B. Tbe V presents the rectnce of ech ine (upper digon mtrix), estimted bsed on the ir distnce (ower digon mtrix) mong the cpits of ech CA, nd using stndrd rectnce vue of pu/100km for 400kV ines (from [26]). Therefore, for simpicity, it hs been ssumed tht ech ine in the equivent eectric network hs simir rectnce to singe ine of 400kV. The reference nge is Csti y Leon in the Northwest of Spin, with rge number of CCAA in its neighborhood. TABLE V: REACTANCE AND DISTANCES (P.U. AND KM) ccyce coi fue nucer sor wind Fig. 2: Stochstic expnsion pnning by technoogies Anduci Argon Asturis Csti-L mnch Csti y Leon Ctuñ C. Vencin Extremdur Gici L Rioj Murci Nvrr Pis Vsco Cntbri Mdrid Investments nd opertion costs hve been estimted bsed on pubic sources, ssuming n nnu increse rte of 2%, equ to the discount rte ϕ (Tbe VI). TABLE VI: INVESTMENT ( /MW) AND PRODUCTION COSTS ( /MWH) Investment Production Nucer Coi Fue Combined Cyce Wind Sor Historic interntion exchnges in 2011 t ech node re shown in Tbe VII (obtined from the Spnish system opertor). Expected interntion exchnges for the yers of the pnning horizon hve been ssumed to be equ to those of Note tht decommissions for groups insted before 2012 hve been modeed (see the resuting fue evoution), coherent with the current genertion technoogies trend. Fig. 2 so shows the rge increment of the insted cpcity of wind technoogy due to its utiiztion fctor nd investment cost, in comprison with sor. To stisfy the demnd, combined cyce pnts re so insted t the end of the horizon becuse they recover more costs for the remining yers of the horizon thn wind frms, refecting very common probem of finite horizon pnning. To overcome this drwbck, ong-term pnning nysis with infinite periods is currenty being studied by the uthors. Fig. 3 shows the tot investment pnning obtined from the stochstic mode nd compred to the corresponding for the three scenrios optimized using the deterministic mode described in subsection II.A MAX MED MIN STOC Fig. 3: Comprison between the stochstic nd the deterministic tot expnsion pnning

7 Mrgin cost ( /MWh) 7 As cn be seen from Fig. 3, the expnsion pnning of the stochstic mode is very simir to the deterministic mode for MED, nticipting tht MED hs the higher probbiity to occur (50%). Pnning costs re , , M for MAX, MED nd MIN respectivey, nd M for the stochstic soution. This is quite sensibe since stochstic pnning is robust soution ginst the demnd uncertinty, nd therefore it hs higher cost thn MED (nd obviousy MIN), but esser thn MAX, since this st consider high demnd scenrio with probbiity 1, which requires too rge investments. An verge nod mrgin cost cn be computed by weighting the nod mrgin cost of ech scenrio (MAX, MED nd MIN) with the probbiity of ech scenrio. Fig. 4 compres those obtined from the deterministic nd stochstic resoution pproches by representing their difference. In ddition, bck ine represents the verge of these differences weighted by the demnd D n, of ech node. As cn be seen the stochstic mode eds to more efficient soution, since these verge vues trend is positive TABLE VI: NUMBER OF SOLAR PLANTS IN ANDALUCÍA Anduci sor Argon wind 300 Asturis wind 183 Cntbri wind 156 Csti y Leon ccyce wind Csti Mnch ccyce coi 1 wind Ctuñ ccyce wind C. Vencin ccyce 1 1 wind Gici ccyce 1 1 wind L Rioj wind 95 Mdrid ccyce 1 wind Murci wind 127 Nvrr wind 151 Pis Vsco ccyce 1 wind 300 From Tbe VI, ddition groups re insted in the rest of CCAA since imports from Anducí re bounded by the mximum therm imits of the ines deprting from Anducí to other CCAA. This mens tht enforcements in such ines shoud be pnned for more efficient pnning. Fig. 5 presents the corresponding mrgin costs cm n,s, in Anducí. As cn be seen, prices tend to zero due to the high penetrtion of the sor technoogy (even eding to spis) nduci rgon sturis cntbri cstieon cstimnc ctunn comvencin extremdur gici rioj mdrid murci nvrr pisvsco AVERAGE Fig. 4: Mrgin costs differences between the stochstic nd the deterministic cses MAX MED MIN Fig. 5: Anducí mrgin costs vs. new insted sor pnts B. Network investments This subsection shows tht network investments shoud be in ccordnce with genertion investments, to void non-suppied demnds, but so to tke dvntges from CCAA cimtoogy. To do so, rge sor penetrtion is ssumed in Anducí, consisting in insting 300 sor groups ech yer of the horizon. Under this scenrio, Tbe VI presents the number of insted groups pnned by the stochstic mode. C. BD gorithm convergence The executions were run on 64-bit Inter-Core CPU t 3.4 GHz, progrmmed in Gms ( nd soved using Cpex sover. The probem with the hours of ech yer without BD ws too rge nd coud not be soved (computer run out of memory). When ppying BD soution ws chieved in 99 itertions, with tot execution time of pproximtey 2 dys nd 3 hours. Fig. 6 shows the ower nd upper bounds of BD gorithm (L nd U respectivey) t ech itertion of the gorithm.

8 M % % 200% 150% 100% 50% where c(x) nd T(x) re functions of x, d nd h re vectors, W is mtrix, nd X nd Y sets of constrints on x nd y respectivey, being Y poyhedron. First bock of constrints correspond to the we-known couped constrints between x nd y. This probem cn be represented s bi-eve optimiztion probem s foows [28]: Upper bound (U) Lower Bound (L) Convergence 1-(L/U) Fig. 6: Evoution of BD gorithm V. CONCLUSIONS This pper proposes genertion expnsion pnning mode to obtin usbe signs for investors in genertion ssets when the eectricity demnd is uncertin nd DC network constrints without osses re considered. These signs provide the cpcity to inst nd the nodes to octe it for ech genertion technoogy. By considering uncertinty in the demnd of the Spnish system, the mode hs been ppied to simpified network where ech node is Spnish Autonomous Community. Equivent ines chrcteristics hve been inferred from pst historic dt. The portfoio incudes wind nd sor technoogies, which hve proved to be more competitive thn therm ones, though bck-up therm investments of renewbe genertion hve not been tken into ccount. Other resuts of this ppiction prove the robustness nd efficiency of the stochstic soution, nd so provide insights bout the necessry investments in the network when rge mount of renewbe genertion is insted. It hs been so tested tht the cse study cnnot be soved using convention techniques. BD decomposition is used to sove this drwbck. Future deveopments my be oriented to ppy this methodoogy for more recent nd onger time horizon, to consider network ines investments for combined genertion nd trnsmission pn, nd to improve the power network representtion. 0% Min x, c x s.t. x X x x Min dy y s.t. Wy h T x yy (17) (x) being the resources function, which is proved to be convex ([28]). The optimiztion probem in the first eve is nmed mster probem, whie the second eve contins the sub-probem. Since (x) (second eve) is not known priori, Benders proposes n itertive pproximtion of (x) by using outer iner functions t ech itertive soution x of the mster probem (first eve). Ech iner function or optimity cut on (x) t x hs the foowing sope: x x h T x Tx x h Tx x xx xx xx Therefore, the optimity cuts re: (18) T x x x (19) The foowing scheme describes the itertive gorithm proposed by Benders: ANNEX: BENDERS DECOMPOSITION Bi-eve BD is used for soving rge-sce probem by mens of prtitioning it into two seprted probems: mster probem (iner, non-iner, nd continuous or integer probem) nd sub-probem (iner probem). This prtition cn be reized by tempor periods, spti units or scenrios [22]. The probems structure for which BD cn be ppied is the foowing: Min xy, s.t. c x dy T x Wy h x X yy (16) Mster probem Min xx, cx j j j j s.t. T x x x j 1,.., 1 Subprobem x x, Min dy yy s.t. Wy h T x : Fig. 7: Benders gorithm Optimity cuts This itertive process converges when the pproximtion of the recourse function (x) is sufficienty good (tht is, when

9 9 the optim vue of vribe in the mster probem is simir to j for some j obtined in the subprobem). Nevertheess, sometimes the subprobem is unfesibe when fixing prticur soution x of the mster probem (tipicy due to the sign of the right-hnd side of the subprobem constrints). In this cse, it is necessry to brodcst to the mster probem fesibiity cuts insted of optimity cuts. These new cuts re obtined by minimizing the infesibiities of the subprobem, i.e.: x Min e e yy,, 0 s.t. Wy I I h T x : (20) I being the identity mtrix nd e vector of unitry vues. Since x mkes the subprobem unfesibe then =(x )>0. To void x in the foowing itertion, iner pproximtion of (x) is itertivey buit (s with (x)) isoting x by imposing the foowing iner cut (the fesibiity cut): 0 T x x x (21) These cuts re very simir to the optimity cuts (see (19)) but when pproximting (x) insted of (x). 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