A Multi-Objective Stochastic Approach to Hydroelectric Power Generation Scheduling

Transcription

1 A Mult-Objectve Stochastc Approach to Hydroelectrc Power Geerato Schedulg Atas Sauhats, Roma Petrcheko, Karls Baltputs, Zae Broka, Reata Varfolomejeva Isttute of Power Egeerg Rga Techcal Uversty Rga, Latva Abstract I ths paper, we propose a ovel stochastc approach to mult-objectve optmzato of hydroelectrc power geerato short-term schedulg. Maxmzato of proft s chose as the ma objectve wth addtoal sub-objectve to reduce the umber of startups ad shutdows of geeratg uts. The radom ature of future electrcty prces ad rver water flow s take to accout. We use a artfcal eural etwork-based algorthm to forecast market prces ad water flow. Ucertaty modelg s troduced to represet the stochastc ature of parameters ad to solve the short-term optmzato problem of proft-based ut commtmet. A case study s coducted o a real-world hydropower plat to demostrate the feasblty of the proposed algorthm by provdg the power geerato compay wth the day-ahead bddg strategy uder market codtos ad a Pareto optmal hourly dspatch schedule of the geeratg uts. Idex Terms-- dyamc programmg, hydropower schedulg, mult-objectve optmzato, stochastc optmzato, ut commtmet. I. INTRODUCTION Optmzato problems of power system operato ad short-term schedulg are topcal for varous stakeholders, cludg power geerato compaes, wholesalers ad system operators. Depedg o the specfcs of the terested party, t may have dfferet objectves such as maxmzato of proft, relablty or socal welfare; mmzato of producto cost, emssos etc. Hydropower schedulg s a large, tme-coupled, stochastc, space-coupled ad olear optmzato problem []. Uder the codtos of a deregulated electrcty market, plag ad schedulg becomes eve more complex due to may ucertates volved. Recet challeges have forced a large wave of research targeted to mprove the ut commtmet (UC) algorthms ad tools ad tackle the ucertates by mplemetg stochastc methods. Whle there are well-developed tradtoal applcatos of stochastc programmg to power systems (mostly used for log-term plag), the most promsg drectos of curret studes are focused o the mplemetato of stochastc approaches for short-term plag wth the ew evromet of decetralzed operato, deregulated markets, ad competto [2]. Besdes that, most of the real-world problems volve several objectves (ofte coflctg) that eed to be cosdered, thus leadg to mult-objectve optmzato. For example, a geeratg compay terested maxmzg ts proft mght also wat to mmze the amout of emssos (called ecoomc emsso load dspatch [3] or ecoomc evrometal dspatch [4]) or, aother case, mmze rsk ad maxmze relablty at the same tme. I such a case, the soluto should be provded as a set of optmal solutos stead of oe optmum, because o sgle soluto ca be cosdered to be better tha ay others wth respect to all objectve fuctos [4]. A feasble soluto to a mult-objectve problem s effcet (also called o-feror or Pareto optmal) f t s ot possble to mprove oe of the objectves wthout depravg other oes. The effcet set (also kow as Pareto frot or trade-off curve) represets the values of the objectves for effcet solutos [5]. Oe of the most wdely used methods for geeratg effcet solutos s the weghted-sums approach [5], where the trade-off curve s obtaed by chagg the weght cotrbuto of each sgle objectve to the geeral objectve. The weght factors ca be adjusted depedg o the mportace of each objectve [6]. For example, [3] proposes weghted mmax method ad employs a stochastc approach (treatg ucertates as radom varables) for ecoomc emsso load dspatch. I [4], soluto for a smlar problem a hydrothermal system s preseted by usg mult-objectve dfferetal evoluto Dyamc programmg (DP) s employed [5], but the mult-objectve problem s formulated as weghted sum of objectves. I ths paper, we propose a stochastc optmzato approach for short-term schedulg of power plat operato from the pot of vew of a hydro geeratg compay (H-GENCO). Optmal maagemet of the avalable hydro resources provdes a major advatage for the power producer to face the compettveess [7]. The short-term hydro schedulg soluto preseted ths paper s valdated based o real-world data to serve as decso support for the H-GENCO developg ts bddg strategy for the day-ahead market. I cotrast wth tradtoal proft-based optmzato, we have cluded a addtoal sub-objectve, amely, the mmzato of the umber of startups ad shutdows. The work preseted ths paper has bee co-faced by the Natoal Research Program LATENERGI ( ).

2 H-GENCO ams to reduce the umber of startups sce t volves varous costs due to loss of water durg startup, wear ad tear of equpmet (geerator wdgs as well as mechacal equpmet), possble malfuctos of the cotrol equpmet durg the startup ad the resultg eed of mateace ad loss of water durg the mateace [8]. It s eve more mportat whe operatg cascade HPPs, sce malfucto of cotrol equpmet oe of the plats ca requre reschedulg of the etre cascade ad decrease eergy producto of the cascade. Mmzato of the umber of startups s also cosdered [9] by employg a two-step geetc algorthm. The frst objectve cosdered [9] s the maxmzato of hourly plat effcecy accordg to effcecy curves of each hydro ut. We assume the market prce ad water flow as stochastc varables. To predct the radom varables, a artfcal eural etwork (ANN) s used ad a ose s added for ucertaty represetato extracted from hstorcal data. I a large part of other studes o hydro schedulg, water flow ucertaty s eglected the most short-term optmzato methods [0]. For example, [] t s dcated that water flows for the ext 24 hours ca be forecasted wth rather good precso, so ucertaty s restrcted to electrcty market prces. However, t s mportat to esure that the water value curves provde a correct pcture of the future value of the water at each reservor because of ther drect fluece o the shape of the bd curves [2]. Ths paper s a further developmet of our research o optmal hydro schedulg part of whch has bee publshed prevously [3]. I the curret paper, we have expaded our study by cludg a addtoal crtero of optmzato (to mmze the umber of startups), thus troducg a two-stage mult-objectve approach. I addto to that, we have advaced the optmzato algorthm by troducg the last stage of optmal UC ad dspatch mplemeted by meas of DP. Soluto of the mult-objectve problem s provded as a Pareto optmal set, leavg the fal choce up to the power plat operator. The remader of ths paper s orgazed as follows. The statemet of the problem s descrbed secto II where the objectve fucto for the cetral part of optmzato procedure, stochastc olear optmzato, s preseted. I secto III, we troduce ucertaty modelg whle secto IV descrbes the optmzato algorthm by addressg more detal the UC ad dspatch optmzato mplemeted by meas of DP. Secto IV presets the case study wth results of multobjectve optmzato for a real-world HPP. Secto V cocludes the paper ad dscusses the future work. II. STATEMENT OF THE PROBLEM A. Ma Assumptos Sce our study presets a model to buld the geerato bds of cascade HPPs, the optmzato problem s formulated from the H-GENCO s pot of vew. We presume that the electrcty market s orgazed accordg to the day-ahead tradg rules as the Elspot market of the Nord Pool (NP) whch s also used for the case study. The H-GENCO s assumed to be a prce-taker ad electrcty prces are cosdered as exogeous varables because the producer uder cosderato operates oly a small part of the pool capacty ad caot fluece the market clearg prces. Aother mportat assumpto s that the sze of reservors uder study s relatvely small ad costrats o the maxmum ad mmum water levels, both upstream ad dowstream, eed to be cosdered. Cosequetly, a geeral case the fluctuatos of water level affect the effectve head avalable for power producto. Uder market codtos, the goal of producer s to maxmze ts proft, hece t forms the ma objectve of the problem. At the same tme, the H-GENCO ams to reduce the umber of startups ad shutdows of geerators for cost-effectve operato of the HPP. The operato of HPPs must be strct complace wth the evrometal ad safety requremets. To take to accout the aforemetoed goals, we employ multobjectve optmzato stead of the tradtoally used sgleobjectve optmzato. The proft of the H-GENCO ca be expressed as: T I ( t v ) t, () t= = PF = c om p where c t electrcty market prce at hour t (EUR/MWh); pt power geerato of ut at hour t (MW); T, t set ad dex of hours the plag horzo; I, set ad dex of geeratg uts; om v varable producto costs. Gve that the varable producto costs for a HPP are close to zero, we assume them to be eglgble ad dsregard further calculatos. B. Objectve Fucto For the partcular HPP cascade uder study (preseted secto V), the objectve fucto for stochastc olear optmzato (Fg. ) of daly bddg strategy s expressed as follows: where for = : for = { 2, 3} : 3 = max, (2) [ ] E[ f ] E f = 2 24 = ηturb. ηge., r, t, t r, t r= t= f g H v c ; (3) v = S Δ L τ ; (4) t, t, H = L L Δ L + k w ; (5) up dow rt,, rt,, rt,, t, rt,, H = L L Δ L + k w + b Δ L ; (6) up dow lateral rt,, rt,, rt,, t,, t

3 where: dex of HPP the cascade (Fg. 3); r realzato umber of prce ad water dscharge forecast (descrbed more detaled secto III); t hour; g accelerato of gravty (9.8 m/s 2 ); η turb. turbe (mechacal) effcecy; η geerator (electrcal) effcecy; ge. H water head (m) at the ed of hour t ; v, t rt,, rt, water dscharge (m 3 /s); c electrcty prce (EUR/MWh); S surface area of the reservor (m 2 ); τ expermetal costat to determe dscharge from decrease reservor level (s ); up dow Lrt,,, L rt,, water level of upstream ad dowstream reservors at the begg of hour t (m); Δ L, t chage of the water level of upstream reservor as a result of geerato durg hour t (m); w rt,, water flow the reservor durg hour t (m 3 /s); lateral w lateral water flow reservors 2 ad 3 (assumed to be costat 6 m 3 /s); coeffcet to express crease water level from k b flow w rt,,(s/m 2 ); dmesoless coeffcet to express crease water level of dowstream reservors from the dscharged water through the respectve upstream HPP durg the prevous hour ΔL, t. The varable of optmzato s the chage of water level of each reservor, Δ L, t. The output of optmzato provdes the H-GENCO wth the day-ahead bddg strategy whch cludes the total hourly power geerato ad bddg prce for the HPP cascade to maxmze ts proft. The optmzato problem (2) (6) s subject to several techcal, evrometal (Table I) ad safety costrats, such as the mmum ad maxmum power geerato ad rampg rate of the hydro uts, mmum ad maxmum water head of the plat ad water level of reservors, permssble rate of water level chages, ad others [4]. Besdes that, allowable operato zoes ad effcecy curves of the hydro uts, whch are subject to the power geerated, water head ad water dscharge through the turbes, eed to be take to accout. The power geerated by the HPP cascade s olearly depedet o several ucerta radom varables (water flow, water dscharge, water head) ad, addtoally, the proft of H- GENCO s subject to the market prce of electrcty havg a stochastc ature. The formulato of the optmzato problem (2) (6) ad ucertaty modelg approach descrbed the ext secto allows to properly cosder the radom ature of the problem wth a acceptable computato tme. III. UNCERTAINTY MODELING Ths secto descrbes the forecastg of electrcty market prce ad water flow ad the subsequet samplg of umerous forecast realzatos c rt, ad w rt,, whch are used as put data for the optmzato procedure. It s suggested that the approaches most ofte used for electrcty spot prce modelg are statstcal tme seres, computatoal tellgece models (cludg ANNs) ad hybrd combatos thereof [5]. The same study also cocludes that statstcal methods for market prce predcto perform rather poorly the presece of spkes, whereas computatoal tellgece models are flexble ad ca hadle complexty ad olearty whch makes them promsg for short-term predctos. However, the ablty to adapt to o-lear, spky behavors may ot ecessarly result better pot forecasts. ANNs are employed for prce forecastg [4], [6], ad [5] cotas a overvew of usg ANNs for electrcty prce forecastg. Varous models of water flow are compared [7] where t s cocluded that a dyamc autoregressve ANN model wth sgmod actvty fucto s superor to the autoregressve movg average (ARMA) ad autoregressve tegrated movg average (ARIMA) models forecastg perod, especally peak pots. I our study, we employ a three-layer ANN whch s traed o hstorcal data of market prces the Latva bddg area of NP, water flow the Rver Daugava ad ambet temperature. Day of week ad hour for whch the predcto s performed s also take to accout durg the trag ad forecastg. Besdes that, the algorthm of ANN mples automatc adaptato of some of the ANN parameters before performg each ew forecast for the ext day. The parameters of ANN are adjusted to sut the best forecast performace the day before. The output of ANN provdes pot forecasts of the hourly day-ahead electrcty prce ad water flow. To take to accout ucertates, we use hstorcal forecast resduals to geerate addtoal realzatos of the forecast. By assumg that the forecast errors reta geerally the same characterstcs the medum-term, we use the hourly relatve errors from the forecasts sce 0 days before. Each realzato s obtaed by addg or subtractg the hstorcal error to the ew forecast at the respectve hour. I such a way, the 0-day old hstorcal data provdes 20 realzatos addto to the oe tal pot forecast. We assume that all the realzatos have equal probabltes. Cosequetly, for the optmzato 2 realzatos of electrcty market prces, c rt,, ad water flow, w rt,,, are used as put data. Ths approach allows to model the stochastcty of electrcty prces ad water flow whle ot creasg computatoal burde too much for a practcal applcato to H-GENCO s daly operato optmzato. A more detaled descrpto of the forecastg approach s beyod the scope of ths paper.

4 IV. OPTIMIZATION PROCEDURE A. Geeral Algorthm of Optmzato We have decomposed the hydro schedulg problem several sub-problems (Fg. ). Frst, a determstc lear optmzato s carred out for dspatch of the water resources for a 4- day log plag horzo. Ths step s eeded oly to obta the water reservor level at the ed of the frst day to be used for the ext stage whch s stochastc olear optmzato (2) (6). For that, we use the forecasted tme seres of electrcty spot prces ad water flow ad employ the Quas-Newto method of olear programmg. As a result, the H-GENCO s bddg strategy for the ext day s bult based o the proft maxmzato task ad the bds are submtted to the market operator. As prevously stated, the optmzato procedure s developed to be used by a H-GENCO operatg the NP power exchage. The market rules requre that GENCOs bd ther etre geerato fleet at oce stead of bddg each ut separately. Cosequetly, the compay ca decde o ts UC schedule after the market has cleared ad the amout of power to be sold at each hour s kow. Optmal UC ad dspatch schedule comprses the ext step of the optmzato procedure for whch we employ determstc DP (troduced more detal the ext subsecto). Determstc lear optmzato wth a plag horzo of 4 days (dspatch of water resources) Stochastc olear optmzato of the day-ahead bddg strategy Clearg of the market by the market operator Determstc dyamc programmg for optmal hourly ut commtmet ad dspatch (sgle-objectve soluto) Mult-objectve optmzato to maxmze the proft ad reduce the umber of startups (Pareto optmal set of solutos) Fgure. A smplfed dagram of the optmzato procedure To assess the objectve of mmzato of startups, a addtoal stage of the optmzato procedure s troduced. By costrag the mmum operatg tme of the uts to respectvely 2 ad 3 hours, the dspatch of hydro uts s rescheduled retag the objectve of proft maxmzato. Ths stage provdes the H-GENCO wth a set of Pareto optmal solutos accordg to whch the compay may choose ts operatg strategy that maxmzes ts proft or allows more cost-effectve schedulg of the hydro uts. Power plat operators try to dmsh the umber of startups ad shutdows of hydro uts because of extra costs volved due to the lost water, wear of equpmet, addtoal mateace ad other factors [8]. B. Optmal Ut Dspatch After the market s cleared ad the hourly amout of power geerato for the ext day has bee determed, t s ecessary to establsh the optmal dspatch schedule of the HPP geeratg uts. At ths stage, the characterstc of each hydro ut has to be cosdered. These characterstcs llustrate the relatoshp betwee effectve water head, power ad water dscharge through the hydro ut. To eable usg these relatoshp curves calculatos, they have to be descrbed mathematcally. I ths study, we approxmate the characterstcs by a thrd-order polyomal such that for every value of water head the water dscharge ca be expressed as: 3 2 v = a p + a2 p + a3 p + a0, (7) where v water dscharge rate (m 3 /s) for ut, p geerator power (MW), ad a0, a, a2, a 3 polyomal coeffcets Water dscharge [m 3 /s] worst combato gve Power [MW] 60 power Fgure 2. Example of block characterstcs of hydro uts best combato As llustrated by curves for several hydro uts Fg. 2, varous combatos of turbes requre dfferet amout of water to produce the same power. Evdetly, the objectve fucto (8) of the UC sub-task s addtve ature. It provdes the opto to solve the problem by usg DP as opposed to performg exhaustve eumerato. Tradtoally, whe DP s appled for optmzato of HPP operato, t s doe for loger plag horzos ad decsos are made at dfferet tme stages as, for stace, [8] ad [9]. I ths paper, however, we employ DP to solve the hourly UC problem, whch s statc tme. There are two ways how to employ DP to fd the optmal HPP ut dspatch schedule. If the put varable for each hour s the total amout of water to be dscharged through the partcular HPP, the DP solves the problem of power geerato maxmzato. O the other had, f the put varable for each hour s the total power geerated by the HPP, water dscharge mmzato s performed. Both approaches essetally strve to crease the effcecy of operato ad thus hgher water

5 value, but we have chose the hourly water dscharge as the objectve fucto of DP: subject to: where p ad o the power of ut, I = v = v m, (8) tσ t I t t Σ = p p ; (9) p pt p t T, (0) p are, respectvely, the lower ad upper bouds p t Σ s the total power to be geerated the HPP at hour t. The amout of power geerato s ormally establshed after clearg of the market, but for the purposes of ths study, we obta t from the results of the stochastc olear optmzato descrbed before. Sce at ths stage the characterstcs of hydro uts are modeled wth greater accuracy, troducg addtoal restrctos o ther operato, a stuato mght arse where the amout of power outputted by olear optmzato caot be acheved by ay combato of the uts wth ther operatoal zoe, hece (8) caot be a equalty. For the DP, a recursve equato s formulated to descrbe the total dscharge of the HPP depedg o the power of ut ad the uts optmzed before t: { } ( ) max ( ) ( ) rec p = rec p p + v p. () k k k k k Recurso s used to obta termedate results whch are stored a array wth dmesos k I, where k s the umber of steps (value of the costrat (8) dvded by the cremet betwee the steps). Oce the array s flled, trace-back procedure s talzed startg from the last etry. The optmal trajectory s thereby acqured, whch, ths stace, s a vector cotag the power geerated by each hydro ut. V. CASE STUDY A. The Object of Optmzato Plavas HPP (the secod largest HPP the Europea Uo), whch s the frst power plat a cascade of three reservor HPPs o the Rver Daugava (Fg. 3), s used for valdato of the proposed mult-objectve optmzato algorthm. Sgleobjectve optmzato results for the etre cascade of HPPs have bee publshed our prevous papers, e.g. [3]. Operato of the HPP s subject to several lmtatos due to the evrometal ad safety cocers. The costrats mposed by bak eroso, reservor capacty, tegrty of dam facltes ad varous other factors alog wth the ma techcal parameters of the HPP are summarzed Table I. All the costrats are take to accout durg the optmzato. TABLE I. TECHNICAL PARAMETERS AND ENVIRONMENTAL CONSTRAINTS OF THE PLAVINAS HPP Type of Costrat Value Istalled capacty (MW) Surface area of the reservor (km 2 ) 35.0 Useful volume of the reservor (mll. m 3 ) 43.0 Maxmum effcecy (turbe/geerator) 0.95/0.977 Permssble upstream level (m) Permssble dowstream level (m) Maxmum dscharge (m 3 /s) 3030 Permssble hourly decrease reservor level (m/h) 0.3 Permssble daly decrease reservor level (m/day), depeds o the seaso Ma flow, m 3 /s S HPP, Pavas (0 uts) S 2 HPP2, Kegums (7 uts) HPP3, Rga (6 uts) Fgure 3. The cascade of HPPs o the Rver Daugava S 3 Fgure 4. Dspatch schedule of the Plavas HPP wth the mmum up tme of the hydro uts of oe hour

6 (a) (b) Fgure 5. Dspatch schedule of the Plavas HPP wth the mmum up tme of the hydro uts of (a) two hours ad (b) three hours B. Results of the Mult-Objectve Optmzato By troducg a addtoal costrat of mmum up tme of uts, three dspatch schedules for the Plavas HPP were obtaed wth mmum up tme of oe hour (Fg. 4) ad two or three hours (Fg. 5). The charts o the left preset the hourly power geerato ad cumulatve proft, whle the charts o the rght dcate whch uts are ole at each hour (marked by X). The hydro uts are operatg oly a part of the day gve the amout of water avalable. (The Plavas HPP operates at ts full capacty durg the aual sprg floodg oly. Durg the rest of the year t s scheduled mostly durg the peak prce perod order to maxmze the proft gve the lmted water resources.) Comparg all the three dspatch schedules, the maxmum dfferece of the proft s 7000 euros, whle the umber of startups vares from 0 to 7. The gve results allow us to costruct a Pareto frot (Fg. 6). Pots A ad B represet the odomat solutos ad belog to the Pareto frot sce oe of them s better tha the other oe wth respect to both objectve fuctos. However, pot C s ot o the Pareto frot because t s etrely domated by A regard to both the proft ad the umber of startups. The set of Pareto optmal solutos allows the HPP operator to make the fal decso o the operatg strategy to maxmze ts proft by also cosderg the umber of startups. Proft [M ] Number of startups Fgure 6. Pareto optmal set of solutos

7 C. Computatoal Tools ad Resources All the procedures of forecastg ad optmzato were carred out MATLAB R203a usg Optmzato Toolbox wth solvers for lear ad olear programmg ad Neural Network Toolbox. The Quas-Newto method was used for olear optmzato. A specal computg evromet has bee desged, based o a hgh-performace mult-processor server wth a 2-core 2.27 GHz, 6 GB of RAM ad 64- bt W Server 2008 operatg system. The total computato tme of forecastg ad optmzato procedures vares from to 228 secods mostly depedg o the parameters of the ANN (amely, the umber of euros the hdde layer whch s selected automatcally from 0 to 36 euros durg the trag stage). VI. CONCLUSIONS AND THE FUTURE WORK Ths study presets a practcal stochastc approach to optmal proft-based daly ad hourly hydro schedulg. The optmzato problem addressed here s a requste for short-term schedulg of ay H-GENCO operatg uder market codtos. The stochastc olear optmzato algorthm employed for buldg the daly bddg strategy allows to take to accout electrcty prce ad water flow forecast ucertates whle the forecasts are obtaed usg a self-adaptve ANN. The two-step mult-objectve optmzato allows the producer to mmze the umber of startups of the geeratg uts addto to the proft maxmzato. The soluto s provded as a Pareto optmal set leavg the fal decso up to the power plat operator. The curret study ca be exteded by cludg assessmet of the cost of startup of the uts the objectve fucto. Besdes, the optmzato based o the proposed approach wll be expaded to corporate thermal geerato of the same GENCO. I that case, a mult-objectve problem statemet ca also be used to maxmze the proft of the hydrothermal power geerato ad, addtoally, mmze the umber of startups of the uts ad mmze the emssos from thermal power plats. REFERENCES [] P. C. B. Rampazzo, A. Yamakam, ad F. 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