Optimal Operation Of Cascade Reservoirs Based On Linear Approximation And Storage Distribution Curve

Reservoir operation helps to make full use of water resources and maximize the comprehensive utilization benefit of water conservancy project,as well as contributes to a more reliable power system.Thus,the research on optimal operation of cascaded reservoirs is very important.The optimal scheduling of cascaded hydropower reservoirs is typically a non-convex and nonlinear programming problem,which is difficult to solve due to the complex coupled nonlinear relationships that more often induce non-convex feature,leading to local optimality.On the other hand,for the long-term stochastic operation of cascaded reservoirs considering runoff and other random factors,the number of combinations of discrete state variables increases exponentially with the increase of the number of reservoirs,causing the calculation too long or even unable to complete,known as the dimensional curse phenomenon.This paper focuses on the nonlinear factors and dimensional difficulty in operation problems of cascaded reservoirs,and proposes corresponding solutions.For the first issue,this paper presents two piecewise linearization methods,which transform nonlinear models into linear ones,which are then solved by using the linear programming(LP)or mixed integer linear programming(MILP).For the second issue,this paper designs a multi-reservoir storage distribution curve based on the hydraulic connection of cascaded reservoirs,and considers the cascade as an entirety during the recursive process of stochastic dynamic programming.The major contributions of this paper are summarized as follows.(1)Introduce a linearization method,denoted as PWL1 hereafter,which employs the special ordered sets(SOS)modeling tool that assigns SOS-constrained weights to the discrete points of a nonlinear function.This set of weights ensures that two adjacent discrete points are used to approximate the corresponding function interval.Implementing SOS constraints requires integer variables,so the PWL1 transforms the original model into an MILP one.A case study of short-term hydrothermal scheduling shows that the solution obtained by this optimization method is better than that of previous research works.(2)Present a linearization method denoted as PWL2,which formulates the hydropower output as a function of reservoir storage and release,which is then approximated by a series of planes.The PWL2 does not introduce integer variables and converts the original model into an LP one.Both PWL1 and PWL2 are used to solve a long-term scheduling of cascaded hydropower reservoirs.The comparison shows that the PWL1 is more accurate in approximation,and the PWL2 is more efficient in solution process.(3)Propose the multi-reservoir storage distribution curve(SDC),which represents the storage relationship between the whole cascade and the individual reservoirs.The SDC can be used to guide the long-term scheduling of cascaded reservoirs.The SDC parameters are estimated by fitting to deterministic long-term scheduling decisions simulated with historical runoffs,and the more accurate PWL1 is used to generate better scheduling decisions.(4)Based on the SDC and the stochastic dynamic programming(SDP),the SDCSDP method is proposed to solve the stochastic operation of cascaded reservoirs.The state variables in the recursive procedure of SDCSDP are the total storage of and total inflow into the cascade.To calculate the stage benefit of each combination,the total storage of cascade is allocated to individual reservoirs according to the SDC,and a one-stage benefit model is used to make appropriate adjustment and optimization.The PWL2 is used here to linearize hydropower output to improve the optimization efficiency.The case studies show that the SDCSDP is able to avoid the dimensional issue and solve stochastic operation problems of large-scale reservoir systems.