Optimization in Short-Term Operation of Hydro Power Systems

Transcription

1 Proceedings of he 6h WSEAS/IASME In. Conf. on Elecric Power Sysems, High Volages, Elecric Machines, Tenerife, Spain, December 6-8, 6 9 Opimizaion in Shor-Term Operaion of Hydro Power Sysems Terje Gjengedal NTNU&Sakraf P.O.Box, Lilleaker, 6 Oslo, Norway ABSTRACT: The paper describes he use of an advanced opimizaion ool for shor erm scheduling of hydro power sysems. The ool akes ino accoun mixed-ineger formulaion and hence, sarup coss for he sar-up of unis can be accouned for. The goal has been o quanify he improvemens in cos reducion by using such a ool, and o see how his will influence upon he daily producion planning. The analyses are made from a posspo-view, meaning ha he spo marke has been cleared and he selling and buying volumes for he nex day are given. The problem formulaion will hen be o cover he load obligaions for he nex day, given a cos minimizing objecive, all relevan consrains aken ino accoun. The analyses show a poenial for significan cos reducion. This cos reducion is mainly due o improved oal efficiency, and a more opimal dispach and a reducion in sar and sop coss. Keywords: Shor-erm hydro scheduling, sar/sop coss, opimal use of resources, successive linear programming INTRODUCTION The New Norwegian Energy Ac came ino force on January, 99. This is he legal basis o open all neworks and creae compeiion in generaion and reail supply aiming a a more flexible and efficien elecriciy supply indusry. The Norwegian Energy Ac requires a separaion beween compeiive and monopoly aciviies []. This marked liberalizaion has increased he focus on profi and revenues. Each producer is making heir own plans based on commercial erms, and are making heir bids ino he marke depending on heir expeced prices in he shor and long erm. The power producer has in principle no obligaion o serve any paricular consumer. The objecive is o generae and sell elecriciy wih maximum profis. To obain his, he oal efficiency of he producion sysem will be of major significance. Bu also, a direc connecion beween he forecas of he marked prices and he marginal cos of producion will be imporan o make sure he resource is used o maximize he income. In such a scenario, he modeling of sar-up cos of unis will be imporan o obain a good balance beween cos due o reduced efficiency, and cos due o sar and sop of a uni. The generaion planning process for a hydro power producer is a very complex ask. The resource is limied over he year, and ypically, he inflow comes in imes when he consumpion is lowes. In he long erm he inflow can be described as a sochasic variable, while in he shor erm deailed scheduling issues including sar and sop of unis mus be considered. In addiion o his, he price varies no only over he year, bu also over he day and beween day and nigh. Due o he complexiy of he planning process ranging from shor erm o long erm consideraions, i is common o divide he overall ask ino hree differen sages; long erm generaion planning wih a ime horizon up o five years, medium erm generaion planning wih a ime horizon 3-8 monhs depending on he ime of he year, and he shor erm generaion planning which will be o schedule he generaion for he nex 4 hours o weeks ahead. All hese sages are conneced such ha he shor erm scheduling ges is boundary condiions from he medium erm planning, which again ges is boundary condiions from he long erm planning. The variables ha connec all hese sages are he waer-values. The waer-values, which migh be hough of as he fuel price for he differen reservoirs, represen in he long erm he expecaion for he fuure prices. In he economic equilibrium, he marginal cos is equal o he marginal demand, and

2 Proceedings of he 6h WSEAS/IASME In. Conf. on Elecric Power Sysems, High Volages, Elecric Machines, Tenerife, Spain, December 6-8, 6 9 hus, he waer-values can be used as he marginal cos for he hydro power producer. Since he resource is limied over he year, he waer-values give guiding signals for when o use he waer, by using he waer-values as he marginal cos. This paper presens he resuls of a sudy using advanced compuer programs for opimizing he shor erm operaion of complex hydro power generaion sysems ha is he deailed planning for he nex 4 hours ahead o weeks. Differen modelling and implemenaion issues are discussed, and experiences and benefis from pracical use are presened. OPTIMIZATION IN AN OPEN MARKET Sakraf is he larges power producer in Norway, wih a oal insalled capaciy of abou 87 MW in more han 55 power plans. Excep for 4 MW of wind power, all of he insalled capaciy is hydro power. Today, he shor erm generaion planning is done manually in wo seps; a cenral producion disribuion by he dispach cenre, and a local producion disribuion wihin each of he four regions. The dispach cenre makes a disribuion beween he hydro power sysems, while he regions make a disribuion beween he aggregaes wihin heir region. Boh of hese disribuions are made manually according o subjecive consideraions based on resuls from he long erm generaion planning. The basis for hese consideraions is he waer-values and he efficiency curves. I is expeced ha here exiss a poenial for economic profi by implemening an opimizaion ool for he shor erm generaion planning. The reason for his is he complexiy represened by he shor erm opimizaion problem, and how his complexiy increases by he number of plans and generaors. There are many variables influencing he oal efficiency of he producion, and a marginal improvemen of he efficiency can make a raher large cos reducion. Use of discree variables in shor erm hydropower scheduling has been invesigaed for some ime. Bu only recenly he use of discree variables has become an alernaive o heurisics and oher mehods like Dynamic Programming and Lagrange Relaxaion. There are several reasons o include he uni commimen in he hydro power scheduling: I is a rend for increasing variaion in spo price for elecric power in he Nordic marke. The power producers need ools for scheduling ha accoun for he coss of saring and sopping unis. The need arise when producers wan o opimize income and o obain coverage for coss from graned sysem services. A program used in a close o real-ime environmen needs o ake ino accoun all modeling opions he user feel relevan, unless he prepared plans will be of minor value. Shor-erm scheduling of hydropower is a challenging ask in cascaded reservoir sysems. The scheduling mus inerface wih he boundary condiions received from he mid-erm scheduling, fulfill discharge consrains wihin each ime inerval as well as mainain couplings as ramping consrains and reservoir balances beween he inervals. The oupu of he scheduling problem should be a proper uni commimen sequence ogeher wih a power dispach. This resuls in a large-scale opimizaion problem wih a mix of discree and coninuous decision variables. Appropriae sysem models and soluion sraegies are imporan o be able o solve such a problem. This paper describes our modeling approach, he implemenaion and discusses he performance on a hydro sysem. The program used is based on linear programming and branch and bound opimizaion mehods where discree variables are aken ino accoun. I has linear programming as kernel, where he nonlineariy s in he hydro sysem are aken ino accoun by successive linearizaions and by muliple ieraions wih model refinemen. This opens for he possibiliy o explicily accoun for sar-up cos of he unis. Plans suggesed by he program have been compared wih he equivalen plans made manually for he same scenarios, wih he same assumpions. Three hydro power sysems have been used, having a oal insalled capaciy of 333 MW divided on 9 power plans, and generaors. 3 SYSTEM MODELING 3. Sysem opology

3 Proceedings of he 6h WSEAS/IASME In. Conf. on Elecric Power Sysems, High Volages, Elecric Machines, Tenerife, Spain, December 6-8, 6 9 In a cascaded reservoir sysem, here is a need o model all he individual reservoirs as well as he differen discharge pahs beween he reservoirs. Such discharge elemens may be power producion plans, bypass gaes and spill pahs. This is illusraed in Figure. nu n d - Xi( - )+ Xi()- q u j ( -τ j )+ q d j ()= () j= j= where: X i (-) Reservoir conen of reservoir i in he end of ime inerval -. (Sar volume of ) X i () Reservoir conen in he end of inerval q u j (-τ j ) Inflow from upsream sources in ime inerval -τ j where τ j is he ime delay. These sources can be regular inflows, discharge hrough plans or hrough gaes. Q d j () Discharge from he reservoir. This can include a hydropower producion plan, and/or a number of bypass gaes and a reservoir overflow. N d Number of downsream elemens Number of upsream elemens n u Figure : Example opology Each reservoir will normally have several discharge opions and each pah may have is separae downsream desinaion. Several plans can be defined for each upsream reservoir. In Figure he configuraions denoed A and B complicaes he soluion process. In he unnel beween he wo reservoirs (A), he flow is highly a funcion of he reservoir levels in he wo reservoirs. How he reservoir level develops over he sudy period is unknown when he model in buil so his is source o srong non-lineariy. In he configuraion denoed B, he same plan discharges waer from wo reservoirs and he disribuion beween hem depends on he plan discharge and he difference in reservoir levels, This is a source o srong nonlineariy. 3. Reservoirs The reservoirs are he main connecing nodes of he waercourse. All plans and discharge gaes mus be associaed wih a reservoir. For each reservoir and ime incremen of he sudy period, a reservoir balance equaions as given below is se up: Reservoir endpoin descripion All reservoirs mus be given some guidelines abou feasible range of reservoir levels a end of he opimizaion period. Such a descripion is necessary since he opimizaion is based on a muli-sage formulaion where he use of he available resources over he sudy period will be opimized. These opions for defining he endpoin descripion can be given: Specified reservoir volume in he end of he sudy period Range of feasible reservoir volumes An incremenal cos of waer as a funcion of he endpoin reservoir volume (waer value) In he curren implemenaion we have used he waer values for each reservoir represening he fuel cos of he waer. Hence he decision variable is he fuel cos represened by he waer-value. Wha we achieve by using waer-values as he decision variable in he shor erm opimizaion is a close and direc connecion beween he planning periods o make sure of a direc connecion beween he forecas of he price and he marginal cos. The boundary condiions obained from he medium erm generaion planning are aken explicily ino he shor erm opimizaion. 3.3 Power Plans

4 Proceedings of he 6h WSEAS/IASME In. Conf. on Elecric Power Sysems, High Volages, Elecric Machines, Tenerife, Spain, December 6-8, 6 93 A hydro power plan is a main elemen in he formulaion of he opimizaion problem. The reason for using he plan as a main elemen raher han he individual unis is he hydraulic coupling wihin he plan. The producion on each uni can influence on each oher due o common unnels/pensocks or because of he ailrace effecs. Hs Pensocks αp Reservoir Tunnel αt 3 4 Unis Tailrace (Table) Oule line/downsream reservoir level Figure : Possible plan opology The figure above illusraes he inernal plan configuraion Waerways The plan is always conneced upsream o one or wo reservoirs. The waerways descripion consiss of he main unnel and/or a number of pressure shafs and is used o calculae he losses from he inake reservoir o he urbines. There can be several pressure shafs wih individual loss facors. Unis, like in figure, can share common pensocks or be conneced o individual pensocks. This deailed descripion of he waerways enables an accurae head loss calculaion. Figure also illusraes he problem of including explici modeling of sar/sop in hydropower opimizaion. The decisions abou which uni o run become more difficul when he unis influence on each oher, as is he case here. The uni would normally no be used ogeher wih uni if uni 3 or uni 4 were available because of he quadraic loss in he pensock unnel. In his implemenaion, hese aspecs are solved in a wo-sep approach. The descripion is simplified in he iniial ieraions when he uni commimens are made. The simplificaion is o assume individual unnels and pensock for all unis. Afer he uni commimen is made, he accurae modeling of he loss can be included Unis Each uni has i own se of efficiency curves for he urbine so ha he connecion beween head and efficiency can be modeled for each individual uni. In his way i is possible o obain an accurae modeling of he characerisic diagram for he hydropower urbine. Individual generaor efficiency curves can also be applied. Minimum and maximum of he urbine can eiher be fixed values, or be funcions of he plan head Head Calculaion of head includes he up- as well as downsream reservoir levels. Especially for run of river sysems, he ailrace is modeled. This means ha rising of he backwaer can be accouned for a high waer flows. Plan head opimizaion is imporan when he effecive plan head is low and he variaion of he reservoir level is high. This program uses a local approach o plan head opimizaion. P = k Q H () P = k Q H + k H Q (3) H = H H = V / A V / A (4) When he producion-discharge equaion is linearized, he conribuion from a perurbaion in plan head is included. Based on he discharge from he previous ieraion, a correcion erm is included in he load balance equaion. This mehod gives improved reservoir level profile on small reservoirs and improves he convergence since a more acive search for he bes soluion is used Consrains In hydro power scheduling i mus be possible o define differen ypes of consrains. Examples are: Schedules on discharges, power producion plans and unis Time dependen limis on reservoir levels

5 Proceedings of he 6h WSEAS/IASME In. Conf. on Elecric Power Sysems, High Volages, Elecric Machines, Tenerife, Spain, December 6-8, 6 94 Minimum and maximum power producion on unis and plans. Maximum rae of change on producion discharges and reservoir levels The complexiy of including hese consrains depends on he ype. Schedules and simple limis on variables are uncomplicaed o include. Special schemes for processing ramping consrains have been developed. This is discussed in he secion of solving sraegy. A major challenge wih many and possibly conflicing consrains is he possibiliy of non-exising soluion. Special care is needed in he design o idenify he source of infeasibiliy Modeling wih discree variables Using a discree model for he plan is opional. This make i is possible o model some hydropower plans wih coninuous and some wih discree variables in he same case. The use of discree variables is mos helpful in he case of srong hydraulic connecions in he waercourse. Hydropower plans conneced o reservoir wih some sorage capaciy can be handled by oher means. This makes i possible o model realisic sysems using discree variables only on he criical hydropower plans. Anoher consequence of his flexibiliy is he possibiliy o use ieraion logic where discree variables are added for one and one plan in each ieraion. The following equaions for he hydropower plans are used in he model. Similar modeling is used for saring and sopping pumps. The equaions are valid for he one uni case. Firs he sarup coss are subraced from he objecive funcion by adding he following erm: c su su (5) Calculae he producion on he plan: α + p = j j, q j, δ γ (6) Ensure δ = when p > : p δ, and m pmax (7) m Ensure p p min, when δ = : p pmin δ (8) Calculae sar variable su δ δ su (9) Consrains on discharge per segmen q () q j, max j, Definiion ofδ : δ {,} () su sar up variable C su is he sar up cos j is he segmen index is he ime index α j is he incline of he segmen δ is he plan running indicaor: δ =, plan is sanding δ =, plan is running γ is he consan erm on he PQ curve p is sum producion p min is minimum producion p max is maximum producion q j, is discharge per segmen is maximum discharge per segmen q max,j 3.4 Gaes/Tunnels Gaes can be defined connecing any wo reservoirs in he waercourse. The downsream desinaion for he gae does no have o be he same as he downsream desinaion for he plans conneced o he same reservoir. A special feaure for gaes is implemened o represen a unnel beween wo reservoirs. The flow in his unnel may depend on boh reservoir levels as indicaed in he equaions below. Q = f ( H, H, α) () Q = f ' ( H H ) (3) H = H H = V / A V / A (4) The reservoir volumes are explicily represened in he opimizaion problem. To ge an appropriae

6 Proceedings of he 6h WSEAS/IASME In. Conf. on Elecric Power Sysems, High Volages, Elecric Machines, Tenerife, Spain, December 6-8, 6 95 approximaion o he flow over he sudy period, he consrains mus be expressed as a funcion of he reservoir volumes. The relaion beween he change in flow for an incremenal change in he reservoir volumes is achieved by combining he equaions (3) and (4). The non-linear relaion of flow beween he reservoirs as a funcion of difference in reservoir level as is defined as a number of ables where each able is referred o a reference level of a reservoir. 3.5 Spinning Reserves Spinning reserves can be defined for each ime inerval. The principal formulaion is given by: P i R i ( P max, i,, ) (5) i All online unis We have chosen o le he maximum power producion be he upper producion limi in he equaion (5). The explici processing of sar/sop is imporan for he spinning reserve modeling. Wihou an explici represenaion of on-line unis, i is very difficul o model he reserves wihou using heurisics. 4 SOLVING STRATEGY The opimizaion approach is based on successive linear programming. A branch and bound echnique is used for handling ineger variables. The commercial opimizaion package CPLEX from he company ILOG CPLEX Opimizaion Inc. is used as a kernel in he calculaions. The overall soluion consiss of ieraions (main ieraions) using one of wo modeling modes. The goal for he firs mode (full descripion) is o find an iniial soluion ha can ac as a linearizaion poin for a deailed descripion. In he second mode (incremenal descripion) a more deailed modeling is used line arising around he efficiency curves for he commied unis. 4. Main ieraions The main ieraions make i possible o use he resuls from he las ieraion o perform a refinemen in modeling from ieraion o ieraion. The main moivaion for using an ieraive approach is ha some of he nonlineariies and consrains will depend on he power producion/discharge decisions. These decisions are unknown and canno be aken ino accoun in he firs main ieraion. An opimizaion model of he sysem is buil for he enire sudy period based on he available informaion. This model couples all ime inervals and akes ino accoun he endpoin reservoir consrains. The soluion of his problem is he opimal decisions based on he curren approximaion of he sysem descripion. The sequence of main ieraions refines he sysem descripion based on he discharge profiles and he reservoir rajecories from he previous ieraion. The refinemen involves new linearized descripions of: The uni efficiency curves The reservoir level - volume relaion The gae discharges Major nonlineariies are represened by more han one segmen in he linearizaion process. The consrain se is also updaed in he ouer loop. This involves consrains as ramping of reservoir volumes and releases. Afer each main ieraion ha consiss of he model building and an opimizaion, a feasible and close o opimal soluion will be available unless here are any consrains ha are excluded due o he sraegy of adding he consrains. Each main ieraion can eiher be based on a full or an incremenal descripion. 4.3 Full descripion In his mode he opimal decisions for producion level, gae discharge are compued aking head opimizaion and consrains ino accoun. Normally wo main full ieraions are used before i is changed o he incremenal. The second full ieraion is needed for he reservoir level/plan head opimizaion. When solving he full descripion he enire valid producion area of he hydropower plans is used. As linear programming require convexiy in he modeling he whole working area of he plan or uni canno be modeled in deail. Losses in he unnels and pensocks are accouned for in a simplified way. In he full descripion he discree naure of he sar/sop is accouned for by inroducing binary variables for he uni commi-

7 Proceedings of he 6h WSEAS/IASME In. Conf. on Elecric Power Sysems, High Volages, Elecric Machines, Tenerife, Spain, December 6-8, 6 96 men. This model needs o be solved by a Branch and Bound echnique. 4.4 Incremenal descripion Afer he uni commimen is made, an incremenal formulaion is used. Based on he online unis and he acual discharge, incremenal formulaions are esablished around he curren operaing poin. In hese incremenal descripions, he accurae loss models are included. This is possible because he commied unis and he enaive producion levels already are deermined. The number of incremenal ieraions is normally wo o hree o obain saisfacory convergence. 5 APPLICATIONS The program can in principle be used in wo major modes: Decision suppor for bidding in a spo marke. The challenge is hen o find appropriae amoun of power producion in he plans for forecased marke price profiles. To decide an opimal fulfillmen of a given load obligaion and o adap changed condiions. This is normally he case when he program is used in on-line environmens. The flexibiliy o combine schedules, load obligaions and marke prices wihin he same opimizaion model makes i possible o use he program wihin a wide range of shor-erm scheduling aciviies. 6 TEST CASE MARKET BIDDING In order o demonsrae he effec from including sar-sop coss in he marke bidding, he program was used o opimize he hydro producion in Figure versus a 4 hour price profile from he Nordic elecriciy marke as given in Figure 3. Figure 3: Price profile from he Nordic marke. There are wo high price periods during he day. The las peak has shorer duraion and makes he focus on sar/sop coss imporan. The highes price is 93 NOK/MWh (hour 9) and he lowes is 63 NOK/MWh (during he nigh). Iniially an opimizaion was performed where coss associaed wih sar/sop was ignored. Each reservoir was given an individual resource cos descripion (waer value). Using waer values adds flexibiliy o he opimizaion and he discharge from each reservoir will be a funcion of price level and profile. In order o compare he imporance of including he sar-sop cos, wo differen opimizaions were performed: - The firs opimizaion: sar-sop coss were no included. Opimal bidding of he plans was calculaed by using he ool. - Second opimizaion: sar-sop coss ( NOK 3) were included in he opimal bidding The resuls for power plans 6 and 7 for he wo opimizaions are shown in Figure 4 and 5 respecively. For power plan 6, he firs opimizaion resuled in a producion profile where i was in operaion from hour 8 o 3 and for a single hour in he afernoon when he price is high. Ignoring he coss involved wih sar/sop, shor periods of operaion for marginal changes in marke price can occur. When sar-sop coss were included, he operaion plan changed by removing he sarup a he las price spike. The revenue from he marke is no compensaing he sar-up cos. Figure 4: Producion plan power plan 6. Marke price profile NOK / MW Time

8 Proceedings of he 6h WSEAS/IASME In. Conf. on Elecric Power Sysems, High Volages, Elecric Machines, Tenerife, Spain, December 6-8, 6 97 Power producion - Vinje 3 Including sar cos Wihou sar cos 5 MW Power producion - Tokke Including sar cos Wihou sar cos Time cos. The resuling generaion schedule is given in Figure MW Time MW 6 Figure 5: Producion plan plan 7. The resuls for power plan 7 are given in Figure 5. A significan change is observed wih sar coss included. Now, he plan also runs from 3 pm o 5 pm. The margin beween he waer value and he marke price is making his profiable. For he periods wih high prices, all he unis run a maximum oupu. For he oher periods, he plans run according o is marginal cos. The marginal cos is defined by he incremenal uni efficiency and he waer values. This plan has four unis. There are wo unis on each pensock. In some cases when he difference beween waer value and price is low, he uni head loss may make i profiable o only run one uni on each pensock. In his case hough, all four unis were used in he same ime inervals. Number of unis Hours Figure 6: Load profile for 4 hour Hours Figure 7. Uni Schedule Plan Plan 4 Plan 6 Plan 7 7 TEST CASE LOAD COVERING In his es case we have changed from a marked bidding issue o a load covering case. The load profile is specified for a 4 hour period as illusraed in Figure 6. The hydro sysem performance is opimized in order o cover he load a minimum As we can see from Figure 7, The number of unis sared and sopped are following a nice sequencedue o he given load profile in Figure 6. 8 LARGE SCALE STUDY

9 Proceedings of he 6h WSEAS/IASME In. Conf. on Elecric Power Sysems, High Volages, Elecric Machines, Tenerife, Spain, December 6-8, 6 98 In order o demonsrae he benefis from using such a ool on large scale sysems, i was applied o hree differen hydro sysems locaed in hree differen waer courses. The opimizaion period is now increased o 68 hours. Each hydro sysem comprises of muliple reservoirs and power plans as illusraed in Figure, bu he srucure and complexiy varies from one sysem o he nex. Sysem : The hydro power sysem has five reservoirs in parallel wih a oal reservoir capaciy of 536 GWh. The mean annual inflow o he five reservoirs is 978 GWh. The generaion is performed in wo power plans wih a oal insalled capaciy of MW. The hydro sysem is a pure parallel sysem wih no serial coupling, meaning ha he waer from plan can no be uilized in plan and vice versa. Sysem : This hydro power sysem comprises of he nine reservoir sysem in Figure wih a oal reservoir capaciy of 837 GWh. The mean annual inflow is 53 GWh. The generaion sysem comprises of five power plans wih a oal insalled capaciy of 55 MW. The sysem is a combined parallel and series sysem as illusraed in Figure. Sysem 3: The sysem comprises of en reservoirs wih a oal reservoir capaciy of 4 GWh. The mean annual inflow is 897 GWh. The power is generaed by hree power plans wih a oal insalled capaciy of 36 MW. This is a pure series sysem, meaning ha he waer from plan can be uilized in plan, and waer from plan can be uilized in plan 3. The load profiles for he differen sysems are given in Figure 8. The oal load for all hree sysems, which is found by adding he hree local sysem loads, is given in Figure 9. The case represens a high load level wih a large volailiy from peak o off-peak hours. Figure 8: Local Sysem loads in MW as funcion of hours Toal Load Hour Figure 9: Toal load all hree sysems in MW as funcion of hours. In order o find he economic benefi, he opimizaion was carried ou in hree seps: - Firs sep: Opimal schedules were found by manual procedures by covering he local loads wihou using he ool. We can name his as manual local scheduling. - Second sep: each sysem was opimized separaely wih respec o he local sysem load by using he opimizaion ool. We can name his as local opimizaion. 5 5 Local Loads S S S3 - Third sep: all hree sysems were opimized simulaneously as one join large sysem covering he oal load in Figure 9. We can name his as oal opimizaion. The economic resuls were calculaed for all hree seps, and he benefis from using he opimizaion 5

10 Proceedings of he 6h WSEAS/IASME In. Conf. on Elecric Power Sysems, High Volages, Elecric Machines, Tenerife, Spain, December 6-8, 6 99 ool was found by using he economic crieria from sep as he reference. The benefi for sep compared o sep is found as he difference beween he respecive cos funcions, while he benefi beween sep 3 and sep is found as he difference beween he cos funcions of sep and sep 3 respecively. The benefi using he ool for opimizing hree small scale sysems compared o opimizing one large scale sysem was also found as he cos difference beween he sep and sep 3 opimizaion. The economic benefis are shown in Figure. $ $ $8 $6 $4 $ $ -$ -$4 -$6 -$8 Sysem Sysem Sysem3 Toal Local opimizaion $3 $7 $ $4 Toal opimizaion $97 -$569 -$47 $56 Figure : Economic benefi (cos reducion) from using he opimizaion ool compared o manual planning From Figure, i appears ha he local opimizaion gives a producion plan wih abou $4 in cos reducion, while he oal opimizaion gives a producion plan wih abou $56 in cos reducion - corresponding o,% and 5,6% of he oal cos respecively. I clearly demonsraes ha he larges benefi is found when opimizing he hree sysems as a whole for covering he lumped oal load. We can also see ha when performing he oal opimizaion, here has been a shif in generaion from sysem o sysem and 3 (seen as reduced operaion cos in sysem and increased generaion cos in sysem and 3) due o an improved disribuion in relaion o he waer-values in addiion o improved oal efficiency. The resuls reveal a large poenial for cos reducion by using an opimizaion ool in he shor erm generaion planning. The sar-up cos represens abou.5% of he oal cos. Alhough his cos is a raher small share of he oal cos, i is imporan o ake he sar-up cos ino accoun as he producion plan will be more realisic and beer adaped o he overall schedule. By using he opimizaion ool, he sar-up cos is reduced wih an average of abou $ 5 per week. The cos calculaions in SHOP are made by muliplying used waer wih he corresponding waervalue. This involves boh he waer-values, he efficiency curves for boh generaor and urbine, and loss in he waer courses. The efficiency for he generaors depends on he producion level, and he efficiency for he urbines depends on boh reservoir level, and he amoun of waer ino he urbine. Also, by opimizing a larger par of a sysem, we will obain a beer oal efficiency for he whole sysem, as well as a beer disribuion according o he waer-values. Finally, his will resul in a beer overall plan in relaion o he operaion coss. 9 COMPUTATIONAL ASPECTS This work has shown ha Branch and Bound echniques can be useful in shor-erm scheduling of hydropower sysems. However, such echniques can be ime consuming if he number of discree variables becomes oo large. The ime horizon of he sudy and he number of hydro unis wih explici modeling of sar/sop will be criical for he performance. Mos river sysems in Norway have less ha 5 producion unis. For such sysems i is no problem o sudy periods of -3 days wih hourly ime resoluion wihou more han a few minues compuaion ime. When using Branch and Bound, he algorihm may find he opimum raher fas, only o spend a long ime verifying ha he soluion is indeed he opimal one. Especially when here are many differen soluions wih almos he same value of he objecive funcion. This is he case when he price profile is fla, or almos fla. I is also found ha he soluion ime spen on a specified load profile is higher han o solve a case giving he same producion profile agains a marke price profile. The compuaion ime, on he sysem and wih a marke price profile used in his paper, is very low. DISCUSSION/CONCLUSION Inroducion of discree variables in he hydro scheduling is an imporan exension o he mehods based on radiional linear programming. This capabiliy will reduce he operaion cos and give uni

11 Proceedings of he 6h WSEAS/IASME In. Conf. on Elecric Power Sysems, High Volages, Elecric Machines, Tenerife, Spain, December 6-8, 6 3 biliy will reduce he operaion cos and give uni plans ha are more pracical o implemen. However, since he echniques a his sage are infeasible for really large sysem, i is imporan o apply a hierarchy for he decisions. For coordinaion of muliple river sysems wih a week s ime horizon, i is necessary o use he radiional echniques based on successive linear programming wih only coninuous variables. The main resul from his sage will be how much he differen river sysems should conribue o he overall plan. The nex sage is hen he deailed sudy of how o implemen each sub sysem requiremen. Oher principles have also been esed for he hydro uni commimen decisions. Dynamic programming (DP) has been used in combinaion wih he successive linear echniques. The DP was hen used for he posprocessing of he plan producions wih he sensiiviy signals/marginal coss from he linear programming as decision suppor. This echnique was fas and gave almos he same resuls as he Branch and Bound in cases wih some flexibiliy of he reservoir sorages. However, for small buffer reservoirs i was difficul o ge saisfacory resuls by DP. The Branch and Bound echnique was found o be superior on small/medium sysems due o he robusness and sysemaic approach for finding he opimal soluion while aking all flow consrains ino accoun. REFERENCES [] Bellman, R Dynamic Programming. Princeon, New Jersey: Princeon Universiy Press. [] Belsnes M., Fosso O.B., Røynsrand J., Gjengedal T., Valhovd E.: Uni Commimen in Hydro Power Operaion Scheduling, Hydropower Conference in Bergen, Norway in June. [3] Flaabø, N., A. Johannesen, E. Olaussen, S. Nyland, K. Hornnes & A. Haugsad 988. EFI s models for hydro scheduling. Technical Repor 3483, SINTEF Energy Research, Trondheim, Norway. [4] Gjelsvik, A. & S. W. Wallace 996. Mehods for sochasic medium-erm scheduling in hydrodominaed power sysems. Technical Repor A4438, SINTEF Energy Research, Trondheim, Norway. [5] Mo B., Haugsad A., Fosso O.B.: Inegraing long- and shor-erm models for hydro scheduling, Hydropower 997 Balkema, Roerdam, ISBN