1 Schedulng of head-dependent cascaded hydro systems: mxed-nteger quadratc programmng approach J.P.S. Catalão a, *, H.M.I. Pousnho a, V.M.F. Mendes b a Department of Electromechancal Engneerng, Unversty of Bera Interor, R. Fonte do Lamero, 6201-001 Covlha, Portugal b Department of Electrcal Engneerng and Automaton, Insttuto Superor de Engenhara de Lsboa, R. Conselhero Emído Navarro, 1950-062 Lsbon, Portugal Receved 17 March 2009 Abstract Ths paper s on the problem of short-term hydro schedulng, partcularly concernng head-dependent cascaded hydro systems. We propose a novel mxed-nteger quadratc programmng approach, consderng not only headdependency, but also dscontnuous operatng regons and dscharge rampng constrants. Thus, an enhanced short-term hydro schedulng s provded due to the more realstc modelng presented n ths paper. Numercal results from two case studes, based on Portuguese cascaded hydro systems, llustrate the profcency of the proposed approach. 2009 Elsever Ltd. All rghts reserved. Keywords: Hydro schedulng; mxed-nteger quadratc programmng; head-dependency 1. Introducton In ths paper, the short-term hydro schedulng (STHS) problem of head-dependent cascaded hydro systems s consdered. In hydro plants wth a large storage capacty avalable, head varaton has neglgble nfluence on operatng effcency n the short-term [1]. In hydro plants wth a small storage capacty avalable, also known as run-of-the-rver hydro plants, operatng effcency s senstve to the head: head change effect [2]. For nstance, n the Portuguese system there are several cascaded hydro systems formed by several but small reservors. Hence, t s necessary to consder head-dependency on STHS. In a cascaded hydraulc confguraton, where hydro plants can be connected n both seres and parallel, the release of an upstream plant contrbutes to the nflow of the next downstream plants, mplyng spatal-temporal couplng among reservors. Head-dependency coupled wth the cascaded hydraulc confguraton augments the problem complexty and dmenson. * Correspondng author. Tel.: +351 275 329914; fax: +351 275 329972. E-mal address: catalao@ub.pt (J.P.S. Catalão).
2 Hydro plants partcularly run-of-the-rver hydro plants are consdered to provde an envronmentally frendly energy opton, whle fossl-fuelled plants are consdered to provde an envronmentally aggressve energy opton, but nevertheless stll n nowadays a necessary opton [3]. However, the rsng demand for electrcty, lkely ncreases n fossl-fuel prces, and the need for clean emsson-free generaton sources, are trends n favor of ncreasng generaton from renewable sources. The Portuguese fossl fuels energy dependence s among the hghest n the European Unon. Portugal does not have endogenous thermal resources, whch has a negatve nfluence on Portuguese economy. Moreover, the Portuguese greenhouse emssons are already out of Kyoto target and must be reduced n the near future. Hence, promotng effcency mprovements n the explotaton of the Portuguese hydro resources reduces the relance on fossl fuels and decreases greenhouse emssons. In a deregulated proft-based envronment, such as the Norwegan case [4] or concernng Portugal and Span gven the Iberan Electrcty Market, a hydroelectrc utlty s usually an entty ownng generaton resources and partcpatng n the electrcty market wth the ultmate goal of mzng profts, wthout concern of the system, unless there s an ncentve for t [5]. The optmal management of the water avalable n the reservors for power generaton, wthout affectng future operaton use, represents a major advantage for the hydroelectrc utltes to face competton [6]. STHS models provde decson support for the operatonal task of bddng n the energy and system servces markets [7]. In the STHS problem a tme horzon of one to seven days s consdered, usually dvded n hourly ntervals. Hence, the STHS problem s treated as a determnstc one. Where the problem ncludes stochastc quanttes, such as nflows to reservors or electrcty prces, the correspondng forecasts are used [8]. Dynamc programmng (DP) s among the earlest methods appled to the STHS problem [9,10]. Although DP can handle the nonconvex, nonlnear characterstcs present n the hydro model, drect applcaton of DP methods for hydro systems wth cascaded reservors s mpractcal due to the wellknown DP curse of dmensonalty, more dffcult to avod n short-term than n long-term optmzaton wthout losng the accuracy needed n the model [11].
3 Artfcal ntellgence technques have also been appled to the STHS problem [12 15]. However, a sgnfcant computatonal effort s necessary to solve the problem for cascaded hydro systems. Also, due to the heurstcs used n the search process only sub-optmal solutons can be reached. A natural approach to STHS s to model the system as a network flow model, because of the underlyng network structure subjacent n cascaded reservors [16]. Ths network flow model s often smplfed to a lnear or pecewse lnear one [17]. Lnear programmng (LP) s a well-known optmzaton method and standard software can be found commercally. Mxed-nteger lnear programmng (MILP) s becomng often used for STHS [18 21], where nteger varables allow modelng of dscrete hydro unt-commtment constrants. However, LP typcally consders that hydroelectrc power generaton s lnearly dependent on water dscharge, thus gnorng head-dependency to avod nonlneartes. The dscretzaton of the nonlnear dependence between power generaton, water dscharge and head, used n MILP to model head varatons, augment the computatonal burden requred to solve the STHS problem. Furthermore, methods based on successve lnearzaton n an teratve scheme depend on the expertse of the operator to properly calbrate the parameters. For nstance, the selecton of the best under-relaxaton factor n [21] s emprc and case-dependent, renderng some ambguty to these methods. A nonlnear model has advantages compared wth a lnear one. A nonlnear model expresses hydroelectrc power generaton characterstcs more accurately and head-dependency on STHS can be taken nto account [2,6,22]. Although there were consderable computatonal dffcultes n the past to drectly use nonlnear programmng (NLP) methods to ths sort of problem, wth the drastc advancement n computng power and the development of more effectve nonlnear solvers n recent years ths dsadvantage seems to be elmnated. In earler works [2,6,22], the use of the nonlnear model n some case studes leads to a result that exceeds by at least 3 percent what s obtaned by a lnear model, requrng a neglgble extra computaton tme. However, the nonlnear model cannot avod water dscharges at forbdden areas, and may gve schedules unacceptable from an operaton pont of vew. Moreover, t s mportant to notce that a mnor change n the electrcty prce may gve a sgnfcant change n the water dscharge, and consequently n the power generaton of plants. Therefore, ramp rate of water dscharge s ncluded n the constrants to
4 keep a lesser and steady head varaton, whch s partcularly mportant for reservors wth a task of navgaton. Hence, n ths paper we propose a novel mxed-nteger quadratc programmng (MIQP) approach to solve the STHS problem, where nteger varables are used to model the on-off behavor of the hydro plants. The proposed approach consders head-dependency, dscontnuous operatng regons, and dscharge rampng constrants, n order to obtan more realstc and feasble results. Ths paper s organzed as follows. In Secton 2, the mathematcal formulaton of the STHS problem s provded. Secton 3 presents the proposed MIQP approach to solve the STHS problem. In Secton 4, the proposed MIQP approach s appled on two case studes, based on Portuguese cascaded hydro systems, to demonstrate ts effectveness. Fnally, concludng remarks are gven n Secton 5. 2. Problem formulaton The notaton used throughout the paper s stated as follows. I, Set and ndex of reservors. K, k Set and ndex of hours n the tme horzon. k Forecasted electrcty prce n hour k. p k Power generaton of plant n hour k. Future value of the water stored n reservor. v k Water storage of reservor at end of hour k. a k Inflow to reservor n hour k. M Set of upstream reservors of plant. q k Water dscharge of plant n hour k. s k Water spllage by reservor n hour k. k Power effcency of plant n hour k. h k Head of plant n hour k. l k Water level n reservor n hour k. mn, v v Water storage lmts of reservor.
5 mn, q q Water dscharge lmts of plant. u k Commtment decson of plant n hour k. R Dscharge rampng lmt of plant. H f x A Hessan matrx. Vector of coeffcents for the lnear term. Vector of decson varables. Constrant matrx. mn, b b Lower and upper bound vectors on constrants. mn, x x Lower and upper bound vectors on varables. mn, Power effcency lmts of plant. mn, h h Head lmts of plant. mn, l l Water level lmts of reservor. The STHS problem can be stated as to fnd out the perodc water dscharges, q k, the water storages, v k, and the water spllages, k s, for each reservor, 1,, I, at all hours of the tme horzon, k 1,, K, that optmze an objectve functon subject to constrants. The water storages at the end of the tme horzon, decson, u k v K, s ascertaned., must be decded accordng wth future operatons. Addtonally, the commtment In the STHS problem under consderaton, the objectve functon s a measure of the proft attaned by the converson of potental energy nto electrc energy, wthout affectng future operatons. Thus, the objectve functon to be mzed can be expressed as: F I K 1 k 1 k p k I 1 v K (1) The objectve functon n (1) s composed of two terms. The frst term represents the proft wth the hydro system durng the short-term tme horzon, where and p k s the power generaton of plant n hour k. k s the forecasted electrcty prce n hour k
6 The second term expresses the value of the water stored n the reservors for future operatons. Ths second term s only needed f no fnal water storage requrement s specfed. An approprate representaton when ths term s explctly taken nto account can be seen for nstance n [23]. The storage targets for the short-term tme horzon can be establshed by medum-term plannng studes. The optmal value of the objectve functon s determned subject to constrants of two knds: equalty constrants and nequalty constrants or smple bounds on the varables. The constrants are ndcated as follows: v k v k a k ( q mk s mk) q k s (2), 1 k m M p k q ( h k) (3) k k h k l v ) l ( v ) (4) f ( ) k ( f ( ) k t( ) k t( ) k v mn k v v (5) u k q mn k k q u q (6) q k R q, k 1 q k R (7) s 0 (8) Equaton (2) corresponds to the water balance equaton for each reservor, assumng that the tme requred for water to travel from a reservor to a reservor drectly downstream s less than the one hour perod, ndependently of water dscharge, due to the small dstance between consecutve reservors. In (2) v k s the water storage of reservor at end of hour k, k k a s the nflow to reservor n hour k, q k s the water dscharge of plant n hour k, s k s the water spllage by reservor n hour k, and M s the set of upstream reservors of plant. Tme-delay s a dffcult ssue, dependng on the dstance between the reservors and on the water dscharge, deservng partcular attenton and research. Tme-delay can be accounted for by consderng a dfferent model structure for dfferent flow levels n an teratve procedure, whch s outsde the scope of ths paper. In (3) hydroelectrc power generaton, p k, s consdered a functon of water dscharge and effcency,. We consder effcency gven by the outputnput rato, dependng on the head, h k. k
7 In (4) the head s consdered a functon of the water levels n the upstream reservor, denoted by f () n subscrpt, and downstream reservor, denoted by t () n subscrpt, dependng on the water storages n the respectvely reservors. Typcally for a powerhouse wth a reacton turbne, where the tal water elevaton s not constant, the head s modeled as n (4), and for a powerhouse wth an mpulse turbne, where the tal water elevaton remans constant, the head depends only on the upstream reservor water level as n [21]. Hence, talrace effects can be consdered by ncludng a correcton n the data regardng reservor water levels. In (5) water storage has lower and upper bounds. Here for each reservor, the mnmum storage, and bounds. Here for each reservor, mn v s v s the mum storage. In (6) water dscharge has lower and upper mn q s the mnmum dscharge, and q s the mum dscharge. The mum dscharge may be consdered a functon of the head, as n [6,22]. As a new contrbuton to earler studes, we consder the commtment decson of each hydro plant. Hence, the bnary varable, u k, s equal to 1 f plant s on-lne n hour k, otherwse s equal to 0. Also, we consder dscharge rampng constrants, n (7), whch may be mposed due to requrements of navgaton, envronment, and recreaton [24]. In (8) a null lower bound s consdered for water spllage. Normally, water spllage by the reservors occurs when wthout t the water storage exceeds ts upper bound, so spllng s necessary to avod damage. The ntal water storages, v 0, and the nflows to reservors are known nput data. 3 Soluton methodology The STHS problem can be formulated as a mxed-nteger quadratc problem, gven by: T T Max F( x) f x 1/ 2 x H x (9) subject to b mn A x b (10) mn x x x (11) x nteger, j J (12) j In (9) the functon F (.) s a quadratc objectve functon of decson varables, where f s the vector of coeffcents for the lnear term and H s the Hessan matrx. In (10) A s the constrant matrx, mn b and b are the lower and upper bound vectors on constrants. Equalty constrants are defned by
8 settng the lower bound equal to the upper bound,.e. and upper bound vectors on varables. The varables mn b b. In (11) mn x and x are the lower x J are restrcted to be ntegers. The lower and upper bounds for water dscharge mply new nequalty constrants that wll be rewrtten nto (10). In (3) the effcency depends on the head. We consder t gven by: 0 k h k (13) where the parameters and 0 are gven by: mn mn ( ) / ( h h ) (14) 0 h (15) In (4) the water level depends on the water storage. We consder t gven by: l k k 0 v l (16) where the parameters and 0 l are gven by: mn mn ( l l ) / ( v v ) (17) Substtutng (13) nto (3) we have: l 0 l v (18) p k 0 q ( h ) (19) k k Therefore, substtutng (4) and (16) nto (19), hydroelectrc power generaton becomes a nonlnear functon of water dscharge and water storage, gven by: p k f ( ) q k v f ( ) k t ( ) q k v t ( ) k q k (20) where the parameter s gven by: 0 0 0 ( l f ( ) l t ( ) ) (21) The parameters gven by the product of ' s by ' s are of crucal mportance for the behavor of head-dependent reservors n a cascaded hydro system, settng optmal reservors storage trajectores n accordance to ther relatve poston n the cascade. It should be noted that these parameters are not related to the soluton procedure. Instead, they are determned only by physcal data defnng the hydro system [2].
9 Equaton (20) can be converted n the format of (9), wth the parameter multpled by the forecasted electrcty prce k appearng n the vector f, and the parameters f () and t ( ) also multpled by the forecasted electrcty prce k appearng n the matrx H. A major advantage of our novel MIQP approach s to consder the head change effect n a sngle functon (20) of water dscharge and water storage that can be used n a straghtforward way, nstead of dervng several curves for dfferent heads. As a new contrbuton to earler studes, we model the on-off behavor of the hydro plants usng nteger varables. Thus, the unt commtment s consdered n (6), allowng for multple operatng regons. Snce we can acheve a soluton faster for a MILP approach than for the proposed MIQP approach, the MILP approach s used to fnd a startng pont for the MIQP approach. Afterwards, we check for an enhanced objectve functon value usng our novel MIQP approach. In our case studes we always arrve at convergence to a superor soluton. 4. Case studes The proposed MIQP approach, consderng head-dependency, dscontnuous operatng regons, and dscharge rampng constrants, has been appled on two case studes based on Portuguese cascaded hydro systems: a) hydro system wth three cascaded reservors; b) hydro system wth seven cascaded reservors. A comparson wth a MILP approach s carred out, makng clear the advantages of the proposed MIQP approach. Our novel MIQP approach has been developed and mplemented n MATLAB and solved usng the optmzaton solver package Xpress-MP. The numercal testng has been performed on a 600-MHz-based processor wth 256 MB of RAM. The compettve envronment comng from the deregulaton of the electrcty markets brngs electrcty prces uncertanty, placng hgher requrements on forecastng. A good forecastng tool reduces the rsk of under/over estmatng the proft of the utltes and provdes better rsk management. Several forecastng procedures are avalable for forecastng electrcty prces [25 27], but for the STHS problem the prces are consdered as determnstc nput data.
10 4.1 Case 1 The hydro system has three cascaded reservors and s shown n Fg. 1. "See Fg. 1 at the end of the manuscrpt". Inflow s consdered only on reservor 1. The fnal water storage n the reservors s constraned to be equal to the ntal water storage, chosen as 60% of mum storage. The tme horzon consdered s one week, dvded nto 168 hourly ntervals. The electrcty prce profle consdered over the tme horzon s shown n Fg. 2 ($ s a symbolc economc quantty). "See Fg. 2 at the end of the manuscrpt". The optmal storage trajectores for the reservors are shown n Fg. 3. The dash-dot lnes denote the results obtaned by a MILP approach whle the sold lnes denote the results obtaned by the proposed MIQP approach. "See Fg. 3 at the end of the manuscrpt". The comparson between MILP and MIQP approaches reveals the nfluence of consderng headdependency on the behavor of the reservors. The upstream reservor should operate at a sutable hgh storage level n order to beneft the operatng effcency of ts assocated plants, due to the head change effect. The storage level n the last downstream reservor s lower wth the proposed MIQP approach than wth the MILP approach, thereby mprovng the head for the mmedately upstream reservor. Hence, a hgher effcency of the last downstream plant s not mportant for the overall proft n ths realstc hydro system. The optmal dscharge profles for the reservors are shown n Fg. 4. Agan, the dash-dot lnes denote the results obtaned by a MILP approach whle the sold lnes denote the results obtaned by the proposed MIQP approach. "See Fg. 4 at the end of the manuscrpt". The comparson between MILP and MIQP approaches reveals that the water dscharge changes more quckly from the lower value to the upper value wth the MILP approach than wth the proposed MIQP approach, thus gnorng the head-dependency. The water dscharge and consequently the hydroelectrc power generaton tend to follow the shape of the electrcty prce profle shown n Fg. 2. As a new
11 contrbuton to earler studes [2,6,22], the water dscharges at forbdden ntervals are avoded, namely between 0 and mn q. Also, ramp rate of water dscharge s ncluded n the constrants. Thus, an enhanced STHS s provded due to the more realstc modelng presented n ths paper. 4.2 Case 2 The hydro system has seven cascaded reservors and s shown n Fg. 5. "See Fg. 5 at the end of the manuscrpt". The hydro plants numbered n Fg. 5 as 1, 2, 4, 5 and 7 are run-of-the-rver hydro plants. The hydro plants numbered as 3 and 6 are storage hydro plants. Hence, for the storage hydro plants headdependency may be neglected, due to the small head varaton durng the short-term tme horzon. Inflow s consdered only on reservors 1 to 6. The fnal water storage n the reservors s constraned to be equal to the ntal water storage, chosen as 80% of mum storage. Also, the mnmum storage s constraned to be equal to 30% of mum storage. The tme horzon consdered s one day, dvded nto 24 hourly ntervals. The electrcty prce profle consdered over the tme horzon s shown n Fg. 6. "See Fg. 6 at the end of the manuscrpt". The optmal storage trajectores for the reservors are shown n Fg. 7. The dash-dot lnes denote the results obtaned by a MILP approach whle the sold lnes denote the results obtaned by the proposed MIQP approach. "See Fg. 7 at the end of the manuscrpt". The optmal dscharge profles for the reservors are shown n Fg. 8. Agan, the dash-dot lnes denote the results obtaned by a MILP approach whle the sold lnes denote the results obtaned by the proposed MIQP approach. "See Fg. 8 at the end of the manuscrpt". Table 1 summarzes an overall comparson between the numercal results obtaned by both optmzaton methods. "See Table 1 at the end of the manuscrpt".
12 Although the average water dscharge s as expected the same for both optmzaton methods, the average storage s superor wth the proposed MIQP approach due to the consderaton of headdependency. Thus, regardless of the prce scenaro consdered, wth the proposed MIQP approach we have a hgher total proft for the hydroelectrc utlty, about 4.4%. Moreover, the extra computaton tme requred s neglgble, convergng rapdly to the optmal soluton. The benefts of consderng head-dependency are shown by provdng a MILP approach that does not consder the mpact of varable head. In order to model head varatons n MILP, the dscretzaton of the nonlnear dependence between power generaton, water dscharge and head s requred. However, such dscretzaton augments the computatonal burden requred to solve the STHS problem. For nstance, the optmal soluton reported n [19] requred 22 mnutes of CPU tme, on a 400-MHz-based processor wth 500 MB of RAM. A major advantage of our novel MIQP approach s to consder the head change effect n a sngle functon of water dscharge and water storage that can be used n a straghtforward way, nstead of dervng several curves for dfferent heads. Hence, the proposed MIQP approach provdes better results for cascaded hydro systems, where head-dependency plays a major role on the behavor of the reservors. 5. Conclusons A novel MIQP approach s proposed n ths paper to solve the STHS problem. Our approach allows an effcent consderaton of the nonlnear dependence between power generaton, water dscharge and head. As a new contrbuton to earler studes, nteger varables are used to model the on-off behavor of the hydro plants. Also, dscharge rampng constrants are ncluded to keep a lesser and steady head varaton. Numercal testng results show that the proposed approach s computatonally adequate for hydro systems wth run-of-the-rver hydro plants, consderng head-dependency, dscontnuous operatng regons, and dscharge rampng constrants, n order to obtan more realstc and feasble results. The addtonal computaton tme requred s neglgble, convergng rapdly to the optmal soluton. Hence, the proposed approach s both accurate and computatonally acceptable, provdng an enhanced STHS. Acknowledgements The authors gratefully acknowledge the contrbutons of Professor L.A.F.M. Ferrera and Dr. S.J.P.S. Marano.
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15 Fgure captons a 1k 1 v 1k s 1k q 1k Reservor 2 v 2k s 2 k q 2 k Plant 3 v 3k s 3k q 3k Fg. 1. Hydro system wth three cascaded reservor.
16 Fg. 2. Electrcty prce profle consderng a one week tme horzon.
17 Fg. 3. Optmal storage trajectores for the reservors case 1.
18 Fg. 4. Optmal dscharge profles for the reservors case 1.
19 a 1 k s 1 k 1 v 1 k q 1 k a 2 k a 3 k 2 v 2 k s 2 k s q 3 k 2 k 3 v 3 k q 3 k a 4 k s 4 k 4 v 4 k q 4 k a 6 k a 5 k 6 v 6 k 5 v 5 k s 6 k q 6 k s 5 k q 5 k s 7 k 7 v 7 k q 7 k Fg. 5. Hydro system wth seven cascaded reservors.
20 Fg. 6. Electrcty prce profle consderng a one day tme horzon.
21 Fg. 7. Optmal storage trajectores for the reservors case 2.
22 Fg. 8. Optmal dscharge profles for the reservors case 2.
23 Tables Table 1 Comparson of results Case 1 Case 2 Optmzaton method Average dscharge (%) Average storage (%) Total proft ($ 10 3 ) MILP 41.58 33.69 5258.37 2.16 MIQP 41.58 46.18 5477.53 6.22 MILP 25.00 83.08 716.64 1.59 CPU (s) MIQP 25.00 84.06 749.59 5.17