LONG-TERM HYDROTHERMAL SCHEDULING VIA META-HEURISTICS

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1 LONG-TERM HYDROTHERMAL SCHEDULING VIA META-HEURISTICS Alexandre F. Amendola Energy Trading PETROBRAS Rio de Janeiro, Brazil Absrac This paper focuses on he long-erm hydrohermal coordinaion ask, considering a deerisic approach for inflows and individual represenaion of hydroelecric and hermal plans. This problem is non-convex and exposed o he curse-ofdimensionaliy. Therefore, i seems appropriae o invesigae he applicaion of mea-heurisics for opimizing is soluion. The operaional planning obecive is defined as he imizaion of he hermal unis coss, subec o load balance and hydraulic/hermal consrains. Three mea-heurisics have been exhausively compared. The es sysem is par of he Brazilian nework, wih 7 hydraulically coupled hydroelecric plans from he norheasern basin plus 6 hermal unis. Keywords: Geneic algorihms, Paricle swarm, Simulaed annealing, Operaional planning, Opimizaion mehods. INTRODUCTION Elecric power generaion has been produced based on differen energy sources. In counries where hydro plans are responsible for a grea percenage of he oal amoun of generaion, he hydrohermal scheduling problem becomes even more imporan. In Brazil, in order o explore waer resources more efficienly, he hydrohermal coordinaion problem has been solved in a cenralized way. This paper inends o conribue on esablishing he applicabiliy limis of mea-heurisics for such a non-convex opimizaion problem. Geneic Algorihms (GAs), Paricle Swarm (PS), and Simulaed Annealing (SA) have been seleced because hey have already passed hrough a necessary period of mauring. The possibiliy of combining heir search mechanisms wih convenional opimizaion echniques has also been invesigaed.. Thermal Plans Modeling The operaional coss for hermal plans are usually approximaed by a monoonically increasing funcion wih respec o he generaed power. For long-erm sudies, a linear relaionship is usually adoped, i.e.: ( g ) = a g + b ψ () where he parameer a is he incremenal cos of he generaion uni. Alexandre P. Alves da Silva PEE-COPPE Federal Universiy of Rio de Janeiro Rio de Janeiro, Brazil alex@coep.ufr.br When he generaion sysem is hermal based, for monhly discreized sages, and neglecing fuel limiaions on any h hermal uni, he opimizaion problem can be formulaed as follows: T J ψ ( g, ) (2) = = subec o D J = g (3), =, g g g (4) where J : number of hermal plans; ψ (.) : plan cos funcion; g, : generaion of plan during inerval ; D : load during inerval ; g : imum generaion a ; and g : imum generaion a. For a hermal generaion sysem wih linear cos unis, he scheduling problem for each inerval is solved by he prioriy lis crierion, i.e., he commimen is defined according o he incremenal coss..2 Hydro Plans Modeling Operaional characerisics of he ih hydro plan are represened by he following variables: x i, : sorage in he reservoir during inerval ; q i, : urbine flow during inerval ; v i, : spillage during inerval ; φ i ( x) : forebay volume waer head fourh-order polynomial relaionship; θ i ( u) : aferbay discharge head polynomial; pc i, : hydraulic loss during inerval ; x i, : imum sorage a he end of inerval ; x i, : imum sorage a he end of inerval. A hydrohermal generaion sysem has for operaional coss he ones associaed wih is hermal unis plus evenual defici coss. Therefore, he hydrohermal coordinaion problem can be formulaed for he horizon of ineres, in which an annual discoun rae r is applied, as follows, T J λ ψ ( g, ) = = (5) subec o D = G + P (6) 6h PSCC, Glasgow, Scoland, July 4-8, 2008 Page

2 G J = g (7) =, g g g (8), P = p i I i=, (9) Δ xi, = xi, + yi, + uk, ui, (0) 6 k Ω 0 i med ( ) θ( ) h = φ x u pc () i, i, i, i, x p med i, xi, xi, = + (2) 2 = k h q (3) i, i i, i, u = q + v (4) i, i, i, x x x (5) i, i, i, u u u (6) i, i, i, q q q ( h ) (7) i, i, i, i, vi, 0 (8) λ = ( + r) Where, for each hermal uni and each hydro uni i T : number of inervals; I : number of hydroelecric plans; J : number of hermal plans; λ : discoun rae for inerval ; k i : produciviy of plan i [MW/((m 3 /s).m)]; p i, : hydro uni i generaion during inerval ; G : oal hermal generaion during inerval ; P : oal hydro generaion during inerval ; D : load level during inerval ; x med i, : average volume during inerval ; h i, : average waer head for i during inerval ; u i, : oal discharge flow rae during inerval ; y i, : inflow rae during inerval ; Δ : lengh of inerval (one monh); and : se of upsream hydro plans. Ω i 2. META-HEURISTICS FOR OPTIMIZATION (9) Three mea-heurisics are used o solve such a nonconvex opimizaion problem for a Brazilian real sysem. Their implemenaion deails are described nex, along wih all daa specificaion. 2. Geneic Algorihms In large-scale opimizaion problems and in hydrohermal coordinaion in paricular [,2], real coding has been successfully adoped in GAs because of is simpliciy and he advanage of faciliaing he definiion of special purpose search operaors. Procedures for avoiding infeasible elemens in he maing pool may no be appropriae in case of opimum soluion on he boundaries of he feasible region, which is usually he case for he hydrohermal coordinaion problem. In his work, a penaly funcion approach [3] has been used o reduce he chances of producing infeasible elemens, bu sill allowing search raecories from he ouside of he feasible region, which usually faciliae he localizaion of opimum soluions. Therefore, consrains violaions have been included in he finess funcion wih penaly facors of he order of he expeced soluion. 2.2 Paricle Swarm Lieraure shows promising resuls from he applicaion of PS o operaional planning problems [4]. However, only he hermal uni commimen problem has been sudied in deail, and he poenial of PS for providing hydrohermal coordinaion soluions sill needs invesigaion. Reference [5] has esablished a general calibraion for he PS search (conrol) parameers. This calibraion has also been proved appropriae for power sysem problems [6]. In he presen work, a differen idea is employed. The PS mehod has a varian called Consricion Facor Approach [7], which has improved he robusness of he search. The consricion facor χ is defined as a funcion of k, ϕ and ϕ 2, where k [0;]. Then, k+ k k k vi = χ vi + crand( pbesi si ) + c2rand2( gbes si ) (20) 2k χ = (2) 2 2 ϕ ϕ 4ϕ where ϕ = c +c 2, for ϕ >4; k v + i : velociy of paricle i a ieraion k; c, c 2 : weigh facors; rand : uniformily disribued variable [0;]; k s i : ih paricle posiion a kh ieraion; pbes : bes so far obained by he ih paricle; gbes : bes so far obained by any paricle; and k : exploraion / inensificaion parameer. The consricion facor χ allows he uning of he exploraion capaciy by varying k. When k is close o zero, he swarm is no allowed o explore disan regions of he search space, which is convenien when promising valleys have already been found. On he oher hand, for k close o uni, he paricles are free o look for disan promising valleys. Finally, s = s + v (22) k+ k k+ i i i 6h PSCC, Glasgow, Scoland, July 4-8, 2008 Page 2

3 where k+ s i k s i : new posiion for paricle i; : previous posiion for paricle i. 2.3 Simulaed Annealing The search parameers for SA are: iniial emperaure T 0 ; annealing schedule given by T k+ = g(t k ); and number of ransiions N k for a T k. There are some proposals in he lieraure o deere a convenien iniial emperaure [8]. In his work, he following procedure has been successfully adoped: + ΔV T0 = ln X 0 (23) where Δ V + is he average degradaion on he obecive funcion values, wih respec o he curren soluions, for ransiions under he iniial emperaure ha do no improve heir curren soluions. X 0 denoes he fracion of accepance for hose ransiions, which is usually made equal o The number of ransiions N k for each emperaure level is defined in his paper as a consan value. The emperaure scheduling has been implemened as follows: T = k+ βt (24) k for β <. 3. TEST SYSTEM The hree mea-heurisics adoped in his work are applied o he Brazilian norheasern hydrohermal sysem, which has is hydro plans cascaded along he São Francisco river basin. No energy imporaion is assumed. The scheduling horizon is wo years wih a monhly inerval discreizaion. The average load level is assumed o be equal o 8500 MW. The monhly laeral (naural) inflows are made equal o he corresponding long erm means for he 24 monh horizon. The financial discoun rae is equal o % per monh. The hermal plans characerisics and opimized dispaches for differen ranges of oal hermal generaion are presened in Tables and 2, respecively. The hydro plans cascading scheme is shown in Figure. Their operaional feaures (Tables 3, 4, and 5) are described in deail o allow he reproducion of he resuls presened in his paper. As basic operaional policy, he ouflow from he hydro plan Iaparica is direced o Paulo Afonso 4 up o is imum urbine discharge (2400 m 3 /s). Beyond ha limi, he ouflow of Iaparica is direced o Moxoó. Plan Maximum Generaion [R$/MWh] [MW] () Pernabuco (2) Foraleza (3) Fafen (4) Ceará (5) Bahia (6) Camaçari Toal: Defici Cos Table : Thermal generaion capaciies and coss. Thermal Generaion [MW] Commimen Cos [R$/h] 0 G 638 () 60 G 638< G 985 (), (2) G < G 36 (), (2), (3) 7.29 G < G 356 (), (2), (3), (4) G < G 542 (), (2), (3), (4), (5) 87.2 G < G 889 All hermal unis G G > 889 All hermal unis plus load shedding G Table 2: Opimized hermal commimen. Paulo Afonso 4 Fig. : Hydro plans from he São Francisco river. Hydro Plan Três Marias Sobradinho Iaparica Moxoó Paulo Afonso, 2 and 3 Xingó Insalled Capaciy [MW] Ne Volume [hm 3 ] (I) Três Marias (II) Sobradinho (III) Iaparica (IV) Moxoó (V) Paulo Afonso ,2,3 (VI) Paulo Afonso (VII) Xingó Table 3: Elecric characerisics of hydro plans. Run-ofhe-river Reservoir 6h PSCC, Glasgow, Scoland, July 4-8, 2008 Page 3

4 Plan (I) (II) (III) (IV) (V) (VI) (VII) Max Vol hm 3 Min Vol. a a a a a a 0* a * a 2* a 3* a 4* Produciviy Turbine flow m 3 /s Max ouflow Min ouflow Table 4: Hydraulic feaures of hydro plans (a i and a i* sand for he forebay and aferbay volume-head polynomial coefficiens, respecively). Plan Monh (I) (II) (III) (IV) (V) (VI) (VII) May , June , July , Augus Sepember Ocober November December January February March April Table 5: Laeral inflows for he 24-monh horizon. rae has been kep consan (%) for all generaions, oo. The muaion operaor has been implemened as an uniformily disribued random perurbaion in he feasible inerval of he decision variable. Eliism has always saved he bes wo individuals from he previous generaion. Therefore, six combinaions of selecion and crossover procedures have been esed 30 imes each. Resuls are presened in Table 6, in which he symbols RU, RP, RI, TU, TP, and TI sand for he specific selecion and crossover operaors. Mean Deviaion Min. RU RP RI TU TP TI Table 6: Resuls from GA. As usual, proporional selecion (R) has lead o premaure convergence. The imum cos soluion (7 millions) has been found by ournamen selecion and uniform crossover. Figure 2 presens he corresponding sored volume of he reservoirs along he 24 monhs. Sorage May Jun Jul Aug Sep Oc Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oc Nov Dec Jan Feb Mar Apr Três Marias Sobradinho Iaparica Fig. 2: Percenage of he reservoirs sorage (TU). Figure 3 presens he bes soluion wih ournamen selecion and inermediae crossover. The operaing cos is R$79 millions. 4. RESULTS This work has chosen he iniial populaion size as wice he number of decision variables, i.e., 72 ouflows relaed o he hree hydro plans wih reservoirs during 24 monhs. The iniial populaion has been randomly creaed from uniform disribuions wih limis defined by he ouflow limis. Selecion has been ried via roulee wheel (R) and ournamen (T). Uniform (U), - poin (P), and inermediae (I; average values) crossovers have been compared. As sopping crierion, a imum number of generaions equal o 2000 has been employed. Addiionally, a search is considered sagnaed if he bes individual has no evolved for 200 generaions. Crossover rae has been equal o 85% allover he search process. Muaion Sorage May Jun Jul Aug Sep Oc Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oc Nov Dec Jan Feb Mar Apr Três Marias Sobradinho Iaparica Fig. 3: Percenage of he reservoirs sorage (TI). Afer several aemps wih differen PS variaions, he Gbes approach wih consricion facor has proved o be he mos reliable implemenaion. An 44 swarm 6h PSCC, Glasgow, Scoland, July 4-8, 2008 Page 4

5 size has been adoped, wih a imum number of 5000 ieraions per rial. The iniial posiions for he swarm are defined as for GAs. The sensiiviy wih respec o he search parameers c, c 2 and k has been invesigaed by running 30 rials for each of he following combinaions: Case (C): c =2.0, c 2 =2.0, k=.0 (χ=.0) Case 2 (C2): c =3.0, c 2 =2.0, k=.0 (χ=0.382) Case 3 (C3): c =2.0, c 2 =3.0, k=.0 (χ=0.382) Case 4 (C4): c =2.0, c 2 =2.0, k=0.5 (χ=0.5) Case 5 (C5): c =3.0, c 2 =2.0, k=0.5 (χ=0.9) Case 6 (C6): c =2.0, c 2 =3.0, k=0.5 (χ=0.9) Table 7 shows he corresponding resuls. Mean Deviaion Min. C C2 C3 C4 C5 C Table 7: Resuls from PS. Figure 4 shows he leas cos operaion policy wih PS, which has been obained using combinaion C5 for he conrol parameers. This imum cos is equal o R$ millions. The ohers provide beer resuls for random iniializaion, because he deerisic iniializaion has lead o premaure convergence. Again, 30 ess have been ried for each of he following six combinaions of conrol parameers: Case (C): N k = 500 and β=0,7; Case 2 (C2): N k = 2000 and β=0,7; Case 3 (C3): N k = 500 and β=0,8; Case 4 (C4): N k = 2000 and β=0,8; Case 5 (C5): N k = 500 and β=0,9; Case 6 (C6): N k = 2000 and β=0,9. Resuls from SA, for each of he above menioned cases, are presened in Table 8. Mean Deviaion Min. C C2 C3 C4 C5 C Table 8: Resuls from SA. Figure 5 shows he bes resul from he applicaion of SA, which has been obained wih combinaion C6 of search parameers. Minimum cos in his case is equal o R$ millions Sorage Sorage May Jun Jul Aug Sep Oc Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oc Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oc Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oc Nov Dec Jan Feb Mar Abr Três Marias Sobradinho Iaparica Fig. 4: Percenage of he reservoirs sorage (C5). The iniial emperaure T 0 has been seleced according o Equaion (23), for X 0 = 0,84, wih a corresponding value of T 8 0 = 2, Regarding he emperaure schedule, β values equal o 0.7, 0.8, and 0.9 have been esed. The number of ransiions wih he same emperaure value, N k, has been se o 500 and The mechanism for exploring he search space is based on a normally disribued addiive perurbaion, wih zero mean and variance proporional o he emperaure value. Two possibiliies have been used for sopping he algorihm: (i) imum emperaure (0-8 ) or (ii) ransiion reecions in a row. For he saring poin, a feasible soluion wih all reservoirs preserving heir iniial sored volume along he whole horizon of ineres (ouflow equal o inflow for all inervals) has been experimened. The cos for his iniial soluion is equal o R$ millions. I is imporan o emphasize ha SA is he only meaheurisic able o improve from such an iniializaion. Três Marias Sobradinho Iaparica Fig. 5: Percenage of he reservoirs sorage (C6). In order o give an idea of he difficuly relaed o he size of he seach space, he urbined flows can be discreized wih seps of 00 m 3 /s. Therefore, he number of saes in he discreized search space would be equal o (2 ) 0. The imum number of saes ha have been visied by GA, PS, and SA are , , and 50000, respecively. This paper avoids comparing CPU execuion imes because of he dependence on programg. An overall comparison among he hree meaheurisics is summarized in Table 9. Alhough depending on an appropriae saring poin, he superior performance of simulaed annealing wih respec o imum and average coss can be verified. However, GAs has been he mos robus, which can be noiced from he variabiliy coefficien (deviaion over mean). All es resuls in Table 9 have reached feasibiliy. 6h PSCC, Glasgow, Scoland, July 4-8, 2008 Page 5

6 GA - TU PS - C5 SA - C6 Mean Deviaion Min Dev/Mean % 7.23% Table 9: Global comparison. Even PS, alhough no having achieved he bes performances wih respec o he imum soluion, average cos, or robusness, has been he easies algorihm o adus. Regarding his paricular aspec, he geneic algorihm has been he hardes o se. As a benchmark, he Generalized Reduced Gradien (GRG) mehod implemened in he Fronline Solver Plaform ( has achieved a imum cos of R$ 6.68 millions afer he same iniializaion adoped for simulaed annealing. However, his resul has been obained by a manual ineracive process in which he decision variables relaed o each reservoir are iniially opimized considering one reservoir a a ime (wih he oher variables frozen). Aferwards, wih he values from he previous soluions, he decision variables are opimized considering pairs of reservoirs. Finally, he variables are opimized alogeher (Fig. 6). Tha has been he only way o allow GRG o converge o meaningful soluions, which is no pracical for a larger number of reservoirs. he ime (excep for he las inervals, which are no imporan because of he end of he opimizaion horizon). Because Iaparica has a small reservoir compared wih he oher wo hydro plans, i has been harder for he mea-heurisics o coordinae is operaion policy. Tha has been he reason why GRG has slighly improved he soluions provided by he mea-heurisics. 5. CONCLUSIONS This work has compared differen mea-heurisics when applied o he long-erm hydrohermal coordinaion problem. The seleced mea-heurisics have been chosen based on previous experience of heir applicaion o large-scale opimizaion problems. The resuls presened in his paper have confirmed heir convergence robusness and he soluions qualiy compared wih a classical opimizaion echnique. The uning of search parameers has been harder for he Generalized Reduced Gradien mehod. Alhough no invesigaed in his paper, he mea-heurisics have he possibiliy of self-adaping heir parameers as par of he search process. This exension is being developed along wih a pracical procedure for incorporaing hydrology, load, and fuel coss uncerainies based on muli-scenario opimizaion. 0 ACKNOWLEDGMENTS Sorage May Jun Jul Aug Sep Oc Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oc Nov Dec Jan Feb Mar Apr Três Marias Sobradinho Iaparica Fig. 6: Percenage of he reservoirs sorage (GRG). Besides, seing he GRG conrol parameers has no been easier. Similar soluions have been achieved by GRG wihou he need for he ineracive opimizaion process when GRG is iniialized wih soluions provided by he mea-heurisics. Because GRG needs a good saring poin, his is a very efficien way o hybridize hese opimizaion echniques, using he abiliy of mea-heurisics o explore differen regions of he search space and he local search capaciy of GRG. When comparing he soluions provided by GRG and by he mea-heurisics, i can be verified ha all mehods agree regarding he operaion policy for he reservoirs of Três Marias and Sobradinho. Noice ha he soluions for hese hydro plans follow he inflows seasonaliy, wih lile variaion on he urbined flows. On he oher hand, here is no general agreemen wih respec o he operaion of Iaparica. The GRG mehod has preserved is sored volume close o 0 mos of This work has been funded by Perobras and by he Brazilian Research Council (CNPq). REFERENCES [] ALVES DA SILVA, A.P. and FALCÃO, D.M., Fundamenals of Geneic Algorihms, in Modern Heurisic Opimizaion Techniques: Theory and Applicaions o Power Sysems, Wiley, [2] ZOUMAS, C.E., BAKIRTZIS, A.G., THEOCHARIS, J.B. e al., A Geneic Algorihm Soluion Approach o Hydrohermal Coordinaion Problem, IEEE Transacions on Power Sysems, Vol. 9, No. 2, May [3] GEN, M. and CHENG, R., A Survey of Penaly Techniques in Geneic Algorihms, Proc. 3rd. IEEE Conf. on Evoluionary Compuaion, pp , 996. [4] GAING, Z.L., Paricle Swarm Opimizaion o Solving he Economic Dispach Considering he Generaor Consrains, IEEE Trans. Power Sysems, Vol.8, No., Augus [5] SHI, Y. and EBERHART, R., Parameer Selecion in Paricle Swarm Opimizaion, Proc. Annual Conference on Evoluionary Programg, San Diego, 998. [6] FUKUYAMA, Y., Fundamenals of Paricle Swarm Opimizaion Techniques, in Modern Heurisic Opimizaion Techniques: Theory and Applicaions o Power Sysems, Wiley, [7] CLERC, M. and KENNEDY, J., The Paricle Swarm Explosion, Sabiliy and Convergence in a Mulidimensional Complex Space, IEEE Transacions on Evoluionary Compuaion, Vol. 6, No., February [8] WONG, K.P. and WONG, Y.W., Shor Term Hydrohermal Scheduling, Par I: Simulaed Annealing Approach, IEE Proc.-C, Vol. 4, pp , h PSCC, Glasgow, Scoland, July 4-8, 2008 Page 6