SCHEDULING OF HEAD-SENSITIVE CASCADED HYDRO SYSTEMS: A COMPARISON BASED ON NUMERICAL SIMULATION RESULTS

Transcription

1 SCHEDULING OF HEAD-SENSITIVE CASCADED HYDRO SYSTEMS: A COMPARISON BASED ON NUMERICAL SIMULATION RESULTS J.P.S. Catalão,* S.J.P.S. Marano,* V.M.F. Mendes,** and L.A.F.M. Ferrera*** Abstract In the present day, wth the deregulaton of the electrc power sector, busness s recognzed as a bd to wn the best proft. In ths new and compettve envronment, a hydroelectrc power utlty has to decde the optmal management of the nflows and the water stored n ts reservors, maxmzng proft from sellng energy wthout compromsng future potental proft. Ths artcle s on the problem of short-term hydro schedulng, concernng head-senstve cascaded reservors, and the algorthmc aspects of ts soluton. We propose and compare optmzaton methods based on dynamc programmng, lnear and non-lnear network programmng. Fnally, based on numercal smulaton results, we report and llustrate our experence. Key Words Hydro schedulng, optmzaton methods, numercal smulaton, cascaded reservors, varable head * Department of Electromechancal Engneerng, Unversty of Bera Interor, Rua Fonte do Lamero, Covlha, Portugal; e-mal: catalao@dem.ub.pt, sm@dem.ub.pt ** Department of Electrcal Engneerng and Automaton, Insttuto Superor de Engenhara de Lsboa, Avenda Emdo Navarro, Lsbon, Portugal; e-mal: vfmendes@deea.sel.pl.pt *** Department of Electrcal Engneerng and Computers, Insttuto Superor Tecnco, Avenda Rovsco Pas, Lsbon, Portugal; e-mal: lmf@st.utl.pt

2 1. Introducton The electrc power ndustry has undergone sgnfcant transformatons n the past decades. Pror to 1973, successve advances n the domans of generaton, transmsson and dstrbuton of electrcal power lead to relatvely low producton costs. The redundancy of nstalled and avalable eupments kept the relablty levels hgh. The energy crss of 1973 and ts repercusson over the cost of the eupments ntroduced new economc concerns to the ssue. Snce then, the electrcty prce ncreased, becomng ncreasngly dffcult to mantan the relablty levels relyng on the former strategy, wthn a healthy economc prospectve. Nowadays, the deregulaton of the electrc power sector and, partcularly concernng Portugal and Span, the creaton of the Iberan Electrcty Market (MIBEL), poses new challenges to the electrc power utltes. The possblty of choosng the electrcty suppler by the consumers, presents the electrc power utltes a new concern: compettveness. The development of new methodologes carryng away an mproved plannng s absolutely crucal n a compettve envronment. In the electrc lberalzed energy framework, a utlty wth hydroelectrc facltes faces the optmal trade-off problem of how to make the present proft by the management of the water resources wthout compromsng future potental proft [1]. Ths problem s known as the hydro schedulng problem. Hydro schedulng s a very mportant actvty for hydroelectrc power utltes because of ts sgnfcant economc mpact [2, 3]. Short-term hydro schedulng s concerned wth the operaton durng a tme horzon of one to seven days, usually dscretzed n hourly ntervals. In ths case, the problem s treated as a determnstc one. Where the problem ncludes stochastc uanttes, such as nflows to reservors or energy prces, the correspondng forecasts are used [4]. Hydro schedulng s guded by prespecfed hourly weghtng factors, whch uantfy the energy prce n the correspondng hours [4]. The goal s to maxmze the value of total hydroelectrc generaton throughout the tme horzon consdered satsfyng all hydraulc constrants, and conseuently to maxmze the proft of the electrc utlty from sellng energy. A reservor has a dynamc behavour and constrants couplng the hourly generaton across tme. Furthermore, snce the reservors n a rver catchment are hydraulcally coupled, the generaton of an upstream reservor affects the water volumes of the downstream reservors [5].

3 To solve ths complex large-scale problem many algorthms have been developed, ncludng dynamc programmng, lnear network programmng, mxed-nteger lnear programmng and non-lnear network programmng. Dynamc programmng (DP) s flexble and can handle the prevous mentoned constrants n a straghtforward way [6-8]. However, drect applcaton of DP methods, for systems wth cascaded reservors and dscrete operatng states, s mpractcal due to the curse of dmensonalty, snce the computatonal burden ncreases exponentally wth problem sze. Modern codes and computers make lnear network programmng (LNP) a wdely used method for hydro schedulng [8-11]. Indeed, these algorthms accommodate easly constrants such as the water balance euaton for cascaded systems, reservors lmts of operaton, water flow through canals and spllways, and water draft through powerhouses, transt tmes n between reservors, and other constrants. In addton, LNP algorthms lead to extremely effcent codes, mplementatons of whch can be found commercally. However, ts major lmtaton s the nablty to deal wth head-senstve reservors, snce n ths case power generaton s a nonlnear functon of water dscharge and head. Also, mxed-nteger lnear programmng (MILP) s becomng freuently used for hydro schedulng [12-16], where bnary varables allow modellng of start-up costs to avod unnecessary start-ups and of dscrete hydro unt-commtment constrants. Even though the objectve functon of MILP s lnear, t can easly accommodate a convex, pece-wse lnear representaton of the power generaton versus water dscharge curve. However, the pecewse-lnear approxmatons augment the computatonal burden reured to solve ths problem. Ths artcle proposes a non-lnear network programmng (NLNP) method to handle such non-lnear functons as they relate to head changes. A NLNP approach [1-3, 17] s more realstc and justfed for mprovng the results, partcularly n reservors where the head greatly depends on the water volume stored. 2. Notaton The notaton used throughout the artcle s descrbed as follows.

4 A. Indces and sets k I K M Index of hydro resource Index of hour n schedulng perod Total number of hydro resources Total number of hours n schedulng perod Set of reservors upstream to reservor B. Varables and constants l Water level n reservor n hour k h Head of plant n hour k h Maxmum head of plant h Mnmum head of plant v Water volume of reservor at end of hour k v Maxmum water volume of reservor v Mnmum water volume of reservor Water dscharge of plant n hour k Maxmum water dscharge of plant Mnmum water dscharge of plant s Water spllage by reservor n hour k a Natural nflow to reservor n hour k p Power generaton of plant n hour k p Maxmum power generaton of plant p Mnmum power generaton of plant

5 λ k Forecasted energy prce n hour k Ψ Future value of water stored n reservor τ m Tme reured for the water dscharged from reservor m to reach reservor, n hours s τ m Tme reured for the water splled by reservor m to reach reservor, n hours C. Vectors and matrxes A x b x x f H Node-arc ncdent matrx Vector of the flux varables correspondng to the arcs of the network Rght hand sde vector Upper bound vector Lower bound vector Vector of coeffcents for the lnear term Hessan matrx 3. Formulaton The water balance euaton of reservor n hour k can be stated as: v s,k 1 a + ( m,k τ m + s m,k τ m) m M + = v + + s (1) In ths artcle, we assume that the dscharge from any upstream reservor flows drectly nto the succeedng downstream reservor wth no tme lag, thus (1) becomes: v = v + + s,k 1 + a + 1,k + s 1,k (2) Ths assumpton s only for theorcal smplfcaton and poses no dffcultes n the problem formulaton. The objectve functon consdered for the hydro schedulng problem s composed of two terms: the frst term represents the proft wth the watershed durng the short-term tme horzon and the last term expresses the economc value of the future use of the fnal stored water n the reservors.

6 Thus, the hydro schedulng problem can be formulated as: subject to: Water balance euaton: I K Max [ λ p (,h )] + Ψ ( v ) (3) k K = = 1 k 1 v = v + + s,k 1 + a + 1,k + s 1,k Power generaton euaton: Head euaton: p k ( 0) = α h + η, where α, R (4) η 0 h = l l + 1,k (5) Water level euaton: Reservor water volume constrants: l = β v + l, where, l R (6) 0 β 0 v v v (7) Plant water dscharge constrants: (8) Plant power generaton constrants: p p p (9) Reservor water spllage constrants: s 0 (10)

7 The optmal value of the objectve functon (3) s determned subject to constrants. The constrants are of two knds: eualty constrants and neualty constrants or smple bounds on the varables. In (4) power generaton s consdered a non-lnear functon of water dscharge and head. In (5) the head s consdered a functon of the water levels n the reservors upstream and downstream to the plant. In (6) the water level s consdered a lnear functon of the reservor water volume. Usng (5) and (6) n (4) power generaton becomes a non-lnear functon of water dscharge and reservor volume, gven by: p α β v α β v + δ = , k, where α, β,δ R (11) In (7), (8), (9), reservors water volume, plant water dscharge and power generaton have lower and upper bounds. In (10) s consdered that the lower bound of water spllage s zero. Water spllage by the reservors exsts when maxmum volume of the water stored exceeds ts upper bound or t s proftable to dscharge to the downstream reservor. The ntal water volumes and nflows to reservors are known, for ths problem. 4. Soluton Methodologes DP methods were among the earlest methods appled to the hydro schedulng problem. DP has the advantage that t can drectly handle non-convex, non-lnear, and even dscrete characterstcs present n the hydro model [9]. These methods, however, are not deally suted for hydro systems wth two or more cascaded reservors, due to excessve computaton tmes and the well-known curse of dmensonalty. For these knds of hydro systems, LNP and NLNP methods are, therefore, superor to DP methods. LNP can be stated as: Max f T x (12) subject to: A x = b (13) x x x (14)

8 In ths approach, we gnore the head change effect and non-lneartes thus power generaton s lnearly dependent on water dscharge, gven by: p σ, = where σ R (15) The prevous smplfcaton may lead to naccuraces. A more accurate model should nclude the hydro generaton characterstc descrbng the relatonshp between the head, the water dscharge and the power generaton. Ths relatonshp can be represented as a famly of non-lnear, non-convex curves (fg. 1), whch are also known as plant performance curves, each for a specfed value of the head. h Water dscharge (m 3 /s) h 0 p Power generaton (MW) p fgure 1. Plant performance curves. NLNP, namely uadratc programmng, can be stated as: Max 1 2 x T H x T + f x (16) subject to: A x = b x x x In ths approach, power generaton s a non-lnear functon of water dscharge and reservor volume: p α β v α β v + δ, where α, β = , k,δ R

9 The sparsty pattern of matrx H s gven below n fg. 2, consderng a case wth three cascaded reservors and three hours n the tme horzon. fgure 2. Structure of matrx H: sparse and symmetrc. 5. Numercal Smulaton Results The numercal smulaton, based on two test cases, was performed on a 1.6-GHz-based processor wth 512 MB of RAM. A tme horzon of 168 hours was used for both test cases. The forecasted energy prce consdered for the tme horzon s shown n fg. 3 ($ s a symbolc uantty). Energy prce ($/MWh) fgure 3. Forecasted energy prce. Tme (h) The frst test case, consstng of a sngle hydroelectrc plant, s dscussed for llustratng the use and the success of the proposed NLNP approach n comparson wth a DP approach that provdes an exact soluton for the problem.

10 The DP approach was developed usng a dscretzaton level of 0.01 hm 3, thus wth 2000 states on each hour gven a maxmum volume of 20 hm 3, consderng the head change effect usng the non-lnear objectve functon. In the next fgure (fg. 4) the sold lne denotes NLNP results whle the dashed lne denotes DP results. Water dscharge of reservor (hm 3 ) Water volume of reservor (hm 3 ) Tme (h) fgure 4. NLNP results denoted by the sold lne versus DP results denoted by the dashed lne. To compare DP wth NLNP, the man results are summarzed n table 1. Table 1 Comparson of DP wth NLNP Results Method Average volume 3 ( hm ) Average dscharge ( hm 3 / h) Total proft 3 ($ 10 ) CPU (s) DP NLNP The proposed NLNP approach acheves an optmal soluton wth nearly the same total proft obtaned wth the DP approach (less than 0.036%), but wth an nferor computaton tme (almost 1000 tmes less). Thus, NLNP represents a vald and better soluton for the problem n comparson wth DP for a sngle hydroelectrc plant.

11 The second test case, consstng of three hydroelectrc plants as shown n fg. 5, s based on a realstc cascaded hydro system. Only reservor 1 has natural nflow. Ths second test case s dscussed for llustratng the benefts of the NLNP approach n comparson wth the LNP approach that gnores the head change effect. s 1k s 2k s 3 k 3 2k 2 v 2 k 1k 1 v 1 k a 1 k 3k v 3k fgure 5. Hydro system watershed. The data of the hydro system s gven n table 2. Table 2 Hydro System Data Reservor Volume capacty 3 ( hm ) Maxmum dscharge ( hm 3 / h) Power capacty (M W) Ths test case consders that fnal water volume of reservors s eual to ts ntal volume. Intal and fnal reservor volumes can be obtaned by a medum-term plannng procedure. Conseuently, the future values of water stored n reservors are not consdered.

12 In the next fgures, the sold lne denotes NLNP results whle the dashed lne denotes LNP results. The results for the water volume are shown n fg. 6. Water volume of reservor 3 (hm 3 ) Water volume of reservor 2 (hm 3 ) Water volume of reservor 1 (hm 3 ) Tme (h) fgure 6. Water volume of reservors. The sold lne denotes NLNP results whle the dashed lne denotes LNP results. In fg. 6 we can see the major nfluence of the head change effect n the optmal behavour of the reservors. Consderng the head change effect, upstream reservors should operate at the hghest possble storage level n order to maxmze the generaton s effcency of ther assocated plants. The optmzaton process wth NLNP tres to mantan the maxmum storage levels n the two upstream reservors, pullng up the reservors trajectory, opposng to the change n the thrd reservor n order to maxmze electrc generaton effcency. Ths mples that reservors play a completely dfferent role n the system dependng on ther relatve poston n the cascade. The results for the water dscharge are shown n fg. 7.

13 Water dscharge of reservor 3 (hm 3 ) Water dscharge of reservor 2 (hm 3 ) Water dscharge of reservor 1 (hm 3 ) Tme (h) fgure 7. Water dscharge of plants. The sold lne denotes NLNP results whle the dashed lne denotes LNP results. As we can see n fg. 7, the LNP results show that the water dscharge change from the mnmum value, zero, uckly to the upper value thus gnorng the head change effect, whle the NLNP results show that, consderng the head change effect, the hydro generaton n the two upstream reservors s postponed n order to uckly reach hgh reservor storage levels and, conseuently, ncrease hydro generaton effcency. To compare LNP wth NLNP, the man results are summarzed n table 3. Table 3 Comparson of LNP wth NLNP Results Reservor Method Average volume 3 ( hm ) Average dscharge ( hm 3 / h) Total proft 3 ($ 10 ) CPU (s) LNP NLNP

14 The average water dscharge s as expected the same wth both optmzaton methods but the average volume n the two upstream reservors s superor wth the proposed NLNP approach. Thus, wth NLNP we have a larger total proft for the hydroelectrc power utlty, almost 6% representng $, wth neglgble addtonal CPU tme reured. 5. Concluson As the tradtonal monopolstc scenery for the electrc energy makes way to a compettve energy market, an mproved plannng s crucal for an electrc utlty to face compettveness. We propose and compare optmzaton methods based on dynamc, lnear and non-lnear network programmng, to solve the short-term hydro schedulng problem wth head-senstve cascaded reservors. Numercal smulaton results show that non-lnear network programmng s justfed for representng a better soluton. The comparson gves a neglgble extra computatonal effort n a realstc cascaded hydro system where the head depends on the water volume stored. References [1] V.M.F. Mendes, L.A.F.M. Ferrera, & S.J.P.S. Marano, Short-term hydro schedule wth head-dependent approach by a nonlnear model, Proc. 8th Portuguese-Spansh Congress on Electrcal Engneerng, Algarve, Portugal, 2003, [2] J.P.S. Catalão, S.J.P.S. Marano, V.M.F. Mendes, & L.A.F.M. Ferrera, Short-term hydro schedulng: A comparson of lnear wth non-lnear network mathematcal programmng, Proc. 3rd IASTED Internatonal Conference on Power and Energy Systems, Marbella, Span, 2003, [3] J.P.S. Catalão, Short-term operatonal plannng of hydroelectrc power systems (n Portuguese), master thess, Insttuto Superor Tecnco, Lsbon, Portugal, 2003.

15 [4] L.A.F.M. Ferrera, T. Andersson, C.F. Imparato, T.E. Mller, C.K. Pang, A. Svoboda, & A.F. Vojdan, Short-term resource schedulng n mult-area hydro-thermal power systems, Electrc Power and Energy Systems, 11(3), 1989, [5] X. Guan, E. N, R. L, & P.B. Luh, An optmsaton-based algorthm for schedulng hydrothermal power systems wth cascaded reservors and dscrete hydro constrants, IEEE Transactons on Power Systems, 12(4), 1997, [6] C. Lyra, & L.R.M. Ferrera, A multobjectve approach to the short-term schedulng of a hydroelectrc power system, IEEE Transactons on Power Systems, 10(4), 1995, [7] S.M. Amado, & C.C. Rbero, Short-term generaton schedulng of hydraulc mult-reservor mult-area nterconnected systems, IEEE Transactons on Power Systems, 2(3), 1987, [8] A.J. Wood, & B.F. Wollenberg, Power generaton, operaton and control (NY: John Wley & Sons, 1996). [9] A.I. Cohen, & V.R. Sherkat, Optmzaton-based methods for operatons schedulng, Proceedngs of the IEEE, 75(12), 1987, [10] C. L, E. Hsu, A.J. Svoboda, C. Tseng, & R.B. Johnson, Hydro unt commtment n hydro-thermal optmzaton, IEEE Transactons on Power Systems, 12(2), 1997, [11] M.R. Pekutowsk, T. Ltwnowcz, & R.J. Frowd, Optmal short-term schedulng of a large-scale cascaded hydro system, IEEE Transactons on Power Systems, 9(2), 1994, [12] A.J. Conejo, J.M. Arroyo, J. Contreras, & F.A. Vllamor, Self-schedulng of a hydro producer n a poolbased electrcty market, IEEE Transactons on Power Systems, 17(4), 2002, [13] G.W. Chang, M. Aganagc, J.G. Waght, J. Medna, T. Burton, S. Reeves, & M. Chrstofords, Experences wth mxed nteger lnear programmng based approaches on short-term hydro schedulng, IEEE Transactons on Power Systems, 16(4), 2001, [14] J. García-González, E. Parrlla, J. Baruín, J. Alonso, A. Sáz-Chcharro, & A. González, Under-relaxed teratve procedure for feasble short-term schedulng of a hydro chan, IEEE PowerTech, Bologna, Italy, [15] O. Nlsson, & D. Sjelvgren, Mxed-nteger programmng appled to short-term plannng of a hydro-thermal system, IEEE Transactons on Power Systems, 11(1), 1996,

16 [16] X. Guan, A. Svoboda, & C. L, Schedulng hydro power systems wth restrcted operatng zones and dscharge rampng constrants, IEEE Transactons on Power Systems, 14(1), 1999, [17] E. N, X. Guan, & R. L, Schedulng hydrothermal power systems wth cascaded and head-dependent reservors, IEEE Transactons on Power Systems, 14(3), 1999, Bographes J.P.S. Catalão receved the electromechancal engneerng degree from the Unversty of Bera Interor (UBI), Covlha, Portugal n 1998 and the M.Sc. degree n electrcal and computer engneerng from the Insttuto Superor Tecnco (IST), Lsbon, Portugal n Snce 1999 he has been wth UBI, Department of Electromechancal Engneerng, where currently he s a Research Assstant and a Ph.D. student n electrcal engneerng. S.J.P.S. Marano receved the electrcal and computer engneerng and the M.Sc. degrees from the Insttuto Superor Tecnco (IST), Lsbon, Portugal n 1990 and 1994, respectvely, and the Ph.D. degree from the Unversty of Bera Interor (UBI), Covlha, Portugal n He s currently an Assstant Professor at UBI and Head of the Department of Electromechancal Engneerng. V.M.F. Mendes receved the electrcal power system engneerng degree and the M.Sc. and Ph.D. degrees n electrcal and computer engneerng from the Insttuto Superor Tecnco (IST), Lsbon, Portugal n 1977, 1987 and 1994, respectvely. Snce 1997 he has been wth Insttuto Superor de Engenhara de Lsboa (ISEL), where currently he s a Professor Coordnator n charge of the Economc and Management Secton at the Department of Electrcal Engneerng and Automaton. L.A.F.M. Ferrera receved the electrcal engneerng degree from the Insttuto Superor Tecnco (IST), Lsbon, Portugal n 1977 and the MSEE and Ph.D. degrees from the Georga Insttute of Technology, Atlanta n 1983 and 1986, respectvely. From 1986 to 1989 he was wth the Pacfc Gas and Electrc Company, San Francsco, where he was a major developer of the Hydro-Thermal Optmzaton program. Snce 1989 he has been wth IST, Department of Electrcal Engneerng and Computers, where currently he s an Assocate Professor.