CO l" ri,, 11I © I L\( : (:olll l'''i l'r .-\ pplictl iOIl . ... 10 Pr()n',. .... (:OIlI r()1. \' ic.:IlIl ;1. . -\ ria. I OPTIMAL CONTROL ALGORITHM FOR HYDROPOWER PLANTS CHAIN SHORT-TERM OPERATION A. F. Sakr and H. T. Dorrah Abstract. This paper presents a new optimization algorithm for solving the scheduling problem of hydroelectric power plant chain . Hydropower plants within the same ri ver basin tend to be highly interdependant in the short-term. These interdependancies are accommodated within the mode l incorporating some hydraulic properties in the flow model of the river between the plants . A continuous dynamic model is developed that , inheren- tly, considers the time lags nature of the problem . The hydropower plants operation problem is t hen formulated us ing the optimal control t heory in order to maximize the benefit of a large power system comprising hydrostorage and thermal power plants us ing marginal price approach. It is shown that the optimization problem is reduced to the Prntryagin ' s Maximum Principle which can be efficiently solved by a suggested methodo - logy. Fina ll y, it is demonstrated , with an application to the Egyptian power system having five hydrostorage plants , that the proposed technique is pragmatical for real case studies , developing optimal operation policy matchable with any specified elec - tric load demand and providing a considerable improvement in the short -t erm energy ge - neration. Keywords. Digital computer applications ; Hydraulic systems.; Maximum principle; Modeling; Opt i mal con t rol; Power generation; Water resources. INTRODUCTION The problem of optimal hydropower production can be class ified int o three categories . These are (Vide Fig . l): i) Long-term operat i on schedule: This is the problem of maxim izin g the hydroenergy over an extended period of time and obta i ning an operat ing schedule for intervals of time much longer than the time lags between the storages in series. The problem with such a long duration is cha ra cter ize d by its stochastic nature arising from the undeterministic inflow from the river (Turgeon (1980». ii) Medium and short -te rm schedule: This problem is seek ing for the optimal solution over a time duration that is not too small as in the dispatch problem to assume system stat i onarity and not too long to ignore ti me lags between sto - rages. A dynamic model for the system is essen - tial to cons ide r the effect of the time lag in the de pendence of each storage on others . Most of the work in this d ir ect ion is based on employing a d is- crete model of the form (1 ) where Xi(k) is the water content of reservoir i at start of inter v al k, Ui (k) denotes the re le ase from reservoir i dur - ing period k, is the losses from th e reservoir i dur - ing period k, and li designates the time lag between reser- vo irs i-l and i. The above model together with the system constr - aints enables the use of the dynamic programing or one of its successive approximate methods (Bellman and Dreyfus (1962), Murray and Yakowitz (1979». Hi. " iii) Load dispatch: Load dispatch problem is concerned with the econo - mi cal operation of power systems compr ising con - vential hydrostorage and thermal power plants . This is carried out for the purpose of load d ispat ch at a g i ven loading condition during a small interval of time to the extend that the dynamic behav i or of the system can be ignored . The formulation of the problem is static in nature (Ghoneim and Colleagues (1981». Power dispatch problems are simple mathe - matical exercise and today many power sys tem s are operating under economic dispatch. This paper considers the short - term hydropower plant operating strategy problem which is highly affected by the dynamic behavior of the hydro re - servoir chain system. The short - term strategy is considered as one level of the infinite horizon op - timal strategy elucidated in Fig.l . The intera c- tion with other levels is the final and initial system conditions as well as the fixed amount of water during the time interval obtained in the higher levels of optimization. From th is view point , the algorithm is an open - loop control s cheme . The goal of this control scheme is to obtain t he most economical util iz at i on of the who le power generating system including hydro and thermal units. Thermal power generating plants are taken i nto con - sideration by means of marginal price curve , as if the thermal stations are purchasing hydropower with di fferent prices at d iff eren t l oading cond iti ons to encourage hydropower production dur ing peak per - i ods. This leads thermal stations to operate uni- formly , wh i ch certainly reduces their electrical energy uni t price. System Description The system under consideration, Fig. 2, consists of n cascaded reserv oi rs wit h power plants, locate d on one rive r basin . For this chain, let us intro- duce t he following notations : Hi The head upstream the i th reservoir measured