On the Optimal Control of a Cascade of Hydro-Electric Power Stations

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1 On the Optmal Control of a Cascade of Hydro-Electrc Power Statons M.C.M. Guedes a, A.F. Rbero a, G.V. Smrnov b and S. Vlela c a Department of Mathematcs, School of Scences, Unversty of Porto, Portugal; b Centre of Mathematcs, Unversty of Porto, Department of Mathematcs and Applcatons, School of Scences, Unversty of Mnho, Portugal; c REN - Servços, S.A., Portugal In ths paper we present a model for a cascade of hydro-electrc power statons where some of the statons have reversble turbnes. The objectve of our work s to optmze the proft of power produton. The problem s consdered n the framework of dscrete-tme optmal control and s solved usng numercal methods. The smulaton uses real data. Keywords: Energy Polcy and Plannng, Natural Resources, Optmal Control 1. Introducton 2. Problem statement 3. Computaconal experments and results 4. Conclusons 5. Acknowledgements 6. References Index 1. Introducton Water s becomng a scarce resource and ts use has attaned, n more advanced countres, a certan degree of sophstcaton. Ths has mpact on how the water s used to produce electrc energy. The management of multreservor systems has attracted the attenton of many researchers Labade, 2004; Ladurantaye et al, 2009, e.g.). It s especally mportant f there s also a possblty of reusng the downstream water n a stuaton of drought. Ths may be mplemented n modern reversble hydroelectrc power statons, assocated wth reservors along a rver basn wth a cascade structure, where t s possble both to dscharge water from upstream to produce electrc power and to pump from downstream to refll an upstream reservor. Here we present a model for a cascade of hydro-electrc power statons where some of the statons have reversble turbnes. There are restrctons on the water level n the reservors and the objectve s to optmze the proft of power producton. The problem s consdered n the framework of a dscrete-tme optmal control and s solved usng numercal methods. The smulaton uses real data. The paper s organzed as follows: the model s presented and the problem s stated Correspondng author. Emal: 1

2 2 n secton 2; n secton 3 the computatonal experments are descrbed and obtaned results are presented; secton 4 contans some conclusons. 2. Problem statement Consder a cascade of hydro-electrc power statons lke the one shown n Fgure 1. The dynamcs of water volumes, V t), n the reservors = 1, 2,..., I, at tme t, s Fgure 1. Example of a cascade of hydro-electrc power statons. descrbed by the followng dscrete-tme control system V t) = V t 1) + a q t) s t) + t = 1, 2,..., T, = 1, 2,..., I, V 0) = V n = 1, 2,..., I, m M q m t) + n N s n t), 1) where V n = V1 n,..., V I n ) s the vector of ntal stored water volumes n the reservors = 1,..., I, M represents the set of reservor ndces from whch the water flow comes to reservor, from pumpng or turbnng, N s the set of reservor ndces contrbutng to the spllway to reservor, qt) = q 1 t),..., q I t)), and st) = s 1 t),..., s I t)), t = 1, 2,..., T, are the controls, representng the turbned/pumped volumes of water and spllways for each reservor at tme t. The controls and the water volumes satsfy the followng constrants: ζ h t) h 0 ) q 0P Z mn q t) q 0T h t)/h 0 ) 1 2, 2) Z t) Z max, 3) V n a V T ), 4)

3 3 where Z t) = Z 0 V t) + α V 0 and ) β 1 h t) = Z t) max {Z j t), ξ }. Here j stands for the number of the downstream reservor recevng water from reservor, h t) are the dfferences between water levels see Fgure 2), V 0, = 1, 2,..., I, are the mnmal water volumes; Z t), = 1, 2,..., I, are the water levels n the reservors; Z 0, Zmn, and Z max stand for the mposed nomnal, mnmal and maxmal water levels meters above sea level) respectvely; h 0, = 1, 2,..., I, are nomnal heads, and ξ, = 1, 2,..., I, are talwater levels; q 0T, = 1, 2,..., I, and q 0P, = 1, 2,..., I, are the nomnal turbned and pumped water volumes; a, = 1, 2,..., I, are the ncommng flows; fnally α, β, ζ, = 1, 2,..., I, are postve constants. Consder the followng dscrete-tme optmal control problem wth mxed constrants. The functonal, representng the proft, has the form P q, s, V n ) = T 0 I ) prcet) r t) dt. =1 The head losses n reservor at nstant t, h t), are gven by h t) = h 0T ) q t) 2 q 0T. The functons r t), = 1, 2,..., I, are gven by r t) = { 9.8 q t) h t) h T t)) µ T 1 φ ), f q t) 0, 9.8 q t) h t) + h P t)) 1/µ P 1 φ ), f q t) < 0. The functons r t) connect the amounts of turbned water and the values of the gross head. The dynamcs s descrbed by dscrete-tme optmal control system 1) wth constrats 2)-4). The optmal values V n, = 1, 2..., I, gve the mean volumes of water that are necessary to keep n the reservors when the ncommng flows are a, = 1, 2,..., I. For example, n the case of the two reservor system shown n Fgure 3, the respectve optmzaton problem has the form:

4 4 Fgure 2. Two cascade reservors. P q, s, V 0 ) = Fgure 3. Two cascade reservors. T 0 2 ) prcet) r t) dt max, =1 V 1 t) = V 1 t 1) + a 1 q 1 t) s 1 t), V 2 t) = V 2 t 1) + a 2 q 2 t) s 2 t) + q 1 t) + s 1 t), V 0) = V n = 1, 2, Z t) = Z 0 V t) β + α V 0 1), = 1, 2 h 1 t) = Z 1 t) max {Z 2 t), ξ 1 }, h 2 t) = Z 2 t) ξ 2, ζ 1 h1 t) h q 2 t) q 0T 2 Z mn ) q 0P 1 q 1 t) q1 0T h2 t)/h 0 2) 1 2, Z t) Z max, = 1, 2 V n a V T ), = 1, 2. h1 t)/h 0 1) 1 2,

5 5 where t = 1, 2,..., T. A typcal one day prce functon, prcet), s shown n Fgure 3. Fgure 4. One day real market prces of electrcty. It should be noted that the hgh varablty of prces certanly has a great nfluence on the economcally effcent use of water n the reservors to produce energy. The restrctons are determned not only by economcal reasons of producng electrcty, but also by ecologcal reasons and other uses of the reservor water by the nearby populaton. It s known that there s a hgher use of electrcty at 13h and 21h whch s related wth domestc consumpton and daly cycles, and we can see that the prce always ncrease at those tmes. One can then expect that ths fact has nfluence n the water management. In the next secton we study the above two reservor system as well as the more nvolved four reservor system shown n Fgure 5. The problem of the proft optmzaton ncludes two man ssues: one s how to control the turbned/pumped water flows and the other one s how to project a cascade of hydro-electrc power statons. In partcular we study the effectveness of ntroducng a reversble lnk L between reservors 2 and 4. Fgure 5. Four cascade reservors.

6 6 3. Computaconal experments and results Computatonal experments wth both models were fulflled wth real data of the water levels and flows, as well as the market prces of electrcty. The tme perod consdered was one day, 24 hours, because of the great varablty of ntra-day electrcty prces. Several type of days were tred, such as dry, mldly wet and wet days, as well as dfferent days of the week. Only a sample of these results s presented. The optmzaton problems were solved usng a penalty functon method. The problems had to be solved numercally because ther complexty does not allow for an analytcal soluton to be found. In the case of two reservors for a very dry day the results are shown n Fgure 6. Fgure 6. Example for two reservors: proft and power staton controls. The calculatons were done wth the market prces of electrcty shown n Fgure 4. It should be notced that the hydroelectrc power statons assocated wth the two reservors only produce electrcty when the prce s hgh enough to justfy that producton. The system chooses to produce energy manly at meals tme. As t was a very dry day, the system had a small amount of water to manage. Because of ths, power staton 1 beng reversble pumped when the prce was lower, allowng the reuse of the water from reservor 2. Pumpng requred a certan cost but ths ncreased the amount of avalable water n reservor 1 allowng to dscharge more, even out of peak hours, augmentng the proft. The varaton of the water flows assocated wth power statons 1 and 2 can be seen n Fgure 6. Before 8 o clock the system only pumps and costs money, but from 10 o clock onwards, the system produces energy and recovers gvng a proft. It should be noted that the Proft n Fgure 6, s net proft. The optmal trajectory of the volume of water n the reservors can be seen n Fgure 7. For the more complex cascade of four reservors Fgure 5), agan wth the same day market prces for the electrcty, the obtaned results are presented n Fgure 8 Lnk L s ncluded). It can be notced a smlar behavour as n the prevous case: electrcty s produced when hgh prces justfy the producton. Now, reservors 3

7 7 Fgure 7. The optmal trajectory. and 4 are reversble and because of that water s pumped at dawn as n the prevous case. Fgure 8. Four cascade reservors: power staton controls. We also consder an ntutve water management scheme, that s, all the water s used to produce electrcty when ts prce reaches the hghest value and pumpng s the opton when the prce s low enough, allowng later to use a bgger volume of water for energy producton. The results wth ths naf polcy are presented n Fgure 9. From Fgure 10 we can see that the control algorthm used provdes an ntellgent water management wth a fnal optmal proft much better than the smple one. For a 24 hour perod the proft obtaned usng the optmal polcy, was e and the proft wth the naf polcy was e. The above consderatons show that the use of optmal control methods can be mportant to manage water n the best way. Now let us llustrate how ths model can be used to plan a cascade of hydroelectrc power statons. For example, we study the utlty of lnk L n the cascade

8 8 Fgure 9. A naf control polcy wth four cascade reservors. Fgure 10. Comparson wth a smple polcy. of four reservos as shown n Fgure 5. The same optmal control problems were solved wth and wthout lnk L see Fgure 5). The results are summarzed n the followng table: Wet Average Dry Cascade Inflow m 3 ) Proft ke) Wth lnk L Turbned Flow m 3 ) Pumped Flow m 3 ) Proft ke) Wthout lnk L Turbned Flow m 3 ) Pumped Flow m 3 )

9 Optmzaton 9 The proft n the case wth the lnk L, wth two reversble power statons, has better values than n the case where the lnk L s out, even f there s no lack of water. For a dry day, the proft obtaned wth lnk L has approxmately doubled the one wthout lnk L. Snce the water to be managed by the system s very lttle, the ncluson of a reversble reservor s essental to ts reuse. For a wet day, the dsposable water s enough. Snce the level of water n each reservor s nearer the maxmum admssble level, t s more dffcult to manage the water and the stuaton becomes less flexble. Anyway, the lnk s advantageous because the system contnues to reuse the water of the reservor 2 havng always a bgger proft. We can conclude that the ncluson of a reversble reservor s advantageous, and t shall be as more advantageous as less water the system has, that s, as far away s the volume of water from ts upper lmt. 4. Conclusons A cascade of hydroelectrc power statons was consdered wth a possblty of turbnng and pumpng n some of the power statons. Ths was translated nto a dscrete-tme optmal control problem whch was solved numercally. The data used n our numercal experments were real. The developed approach can be used to plan and to manage cascade power statons. 5. Acknowledgements Ths problem was suggested by Redes Energétcas Naconas REN). We want to thank Dr. Mara Natála Tavares and Eng. Mara Helena Azevedo for proposng the basc model that has been studed for some tme at REN. Referêncas [1] T. W. Archbald,C.S. Buchanan, L. C. Thomas and K. I. M. McKnnon, Controllng mult-reservor systems, European Journal of Operatonal Research,Elsever, Vol. 1293)March 2001), pp [2] A. Korobenkov, A. Kovacec, M. McGunness, M. Pascoal, A. Perera and S. Vlela, Optmzng the proft from a complex cascade of hydroelectrc statons wth recrculatng water, accepted for publcaton Aprl 2010) n MICS Journal [3] Internatonal Atomc Energy Agency, Valoragua - A model for the optmal operatng strategy of mxed hydrothermal generatng systems - Users manual for the manframe computer verson1992) [4] J. W. Labade, Optmal Operaton of multreservor systems: state-of-the-art revew, Journal of Water Resources Plannng and Management, Vol. 1302), 2004, pp [5] D. De Ladurantaye, M. Gendreau, Potvn and Jean-Yves, Optmzng profts from hydroelectrcty producton, Computers and Operatons Research, Vol. 362)2009), pp [6] P.A.J.Lautala, On the modellng and solvng of an optmal control problem of hydro-electrc power plant systems,internatonal Journal of Systems Scence, Vol. 105)May 1979), pp [7] N.R.Mzyed, J.C.Lofts and D.G.Fontane, Operaton of large multreservor systems usng optmalcontrol theory, Journal of Water Resources Plannng and Management, Vol. 1184)July/August 1992), pp [8] H.X. Phu, On the optmal-control of a hydroelectrc power-plant, Systems and Control Letters, Vol. 83)January 1987), pp [9] H.X. Phu, Optmal-control of a hydroelectrc power-plant wth unregulated spllng water, Systems and Control Letters, Vol. 102)1988), pp [10] K. Rechert, Optmal-control of a hydroelectrc power-plants - Problems, Concepts, Solutons, Brown Bover Revew, Vol. 647)1977), pp

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