| The southern hilly region is an important part of China in society and economy.It is strategically significant to ensure the water supply in southern hilly areas for the sustainable and high-quality development of China.The precipitation in this region is abundant but unevenly distributed in time and space.A reservoir with a replenishment pumping station is the most characteristic and representative water supply system in this region.However,the existing optimization methods cannot adapt to the characteristics of this type of system.On the one hand,the operation rules in the previous models are not suitable for this type of system.On the other hand,classic mathematical programming algorithms will cause "curse of dimension" while the meta-heuristic algorithms may have difficulty in judging logic conditions in the constraint of reservoir operation rules.Therefore,it is of practical value and academic significance to optimize the operation of the multi-reservoirs-and-multi-pumping-stations system in the southern hilly region of China.The proposed methods are as follow.(1)For a reservoir,an optimization model is developed in order to minimize the annual sum of squared water shortage.The amount of water supply and water spill of each period are two types of decision variables in the model,which are subject to the annual available water in the reservoir and the operation rule of reservoir.For a pumping station,an optimization model is developed in order to minimize the annual sum of squared water shortage as well.The only decision variable in this model is the pumping volume of each period,which is restricted by the annual water pumping right and the pumping capacity in each period.There is only one type of state variable in each model above,so the one-dimension dynamic programming(DP)is adopted to solve the models.For example,the reservoir model was applied to the Daquan Reservoir in Nanjing,China.After optimization,although the frequency of water shortage increased,any severe drought in a period could be avoided.The maximum water shortage of each period was reduced by 77.8%in a normal year and was reduced by 83.9%in a dry year,respectively.(2)A system of one reservoir and two pumping stations consists of a reservoir,a pumping station replenishing water to the reservoir from a river and a pumping station lifting water to other recipients from the reservoir.An optimization model is developed for this type of system in order to minimize the annual sum of squared water shortage.There are three types of decision variables in the model including the amount of water supply and water spill of the reservoir and the pumping volume of each pumping station during each period.These decision variables are subject to several constraints including the available water of the system(reservoir inflows and water pumping rights),the requirement of water transfer and the operation rule of reservoir.There are two types of state variables in the model,so the dynamic programming successive approximation(DPSA)is adopted to solve it.The method was applied to the Hewang Reservoir and its two pumping stations in Nanjing,China.After optimization,the water storage,diversion and transfer functions of the system could be maximized and the utilization rate of water resources could be improved.Specifically,the amount of annual water spill could be reduced,so the deficit of irrigation could be reduced by 5.9%and the deficit of water transfer could be reduced by 2.0%,respectively.(3)A system of parallel reservoirs that share the joint water rights from a water diversion line is a commonly used type of water supply system in the southern hilly region of China.An optimization model is developed for this type of system in order to minimize the annual sum of squared water shortage.The model is restricted by the total water diversion right of the system,the available water in each reservoir,the operation rule of reservoirs,etc.A decomposition and dynamic programming aggregation method(DDPA)for this type of parallel system is proposed.The algorithm can transform the L+1 dimensional dynamic programming problem into L calculations of one-dimensional dynamic programming for the subsystem models and one calculation of one-dimensional dynamic programming for the aggregation model.The proposed model and algorithm were applied in a system of four parallel reservoirs in Nanjing,China.It was shown that the method can optimize the allocation of the limited water diversion right for the four reservoirs in order to reduce the water spill and water shortage of the whole system.Specifically,the annual water shortage of the system could be reduced by 76.5%and the maximum water shortage of each period could be reduced by 98.9%in a dry year.(4)A system of in-series reservoirs with water replenishment pumping stations is another commonly used type of water supply system in the southern hilly region of China.This type of system can lift water from a river and distribute water among the cascade reservoirs through a water transfer line.An optimization model is developed for this type of system in order to minimize the annual sum of squared water shortage.The model is restricted by the available water of system,operation rules of reservoirs and water replenishment rights of pumping stations,etc.A decomposition and dynamic programming aggregation method(DDPA)for this type of cascade system is proposed.Different from the parallel system,the algorithm can transform the L+1 dimensional dynamic programming into a nested procedure that L calculations of one-dimensional dynamic programming for the subsystem models are embedded into the calculation of one-dimensional dynamic programming for the aggregation model.The method was applied to a system of dual reservoirs and dual pumping stations in Nanjing,China.It was shown that the method could realize the redistribution of water resources in the system and make up for possible shortages with surpluses among the reservoirs.Specifically,the annual water spill of the Shanhu Reservoir was reduced in a dry year and the saving water could be used to reduce the amount of water replenishment from the river by 2.5%and eliminate the water shortage in the Niqiao Reservoir.Furthermore,the DDPA method was compared with PSO and GA.From the perspective of optimality,DDPA can obtain the global optimal solution while the results of PSO and GA were stochastic and related to the parameters selection.From the perspective of efficiency,the computing time of DDPA was only 14.3%and 15.6%of that of PSO and GA,respectively.In conclusion,a series of optimization models is developed in this paper for the operation of multi-reservoirs-and-multi-pumping-stations systems in the southern hilly region of China.The operation rule of reservoirs is integrated into these models for adapting to the local water resources characteristics and relieving the contradiction of seasonal water shortage and water abandonment.A series of DP algorithms is proposed to solve these models;these algorithms can handle the logic conditions in the constraint of reservoir operation rules appropriately and overcome the "curse of dimension" in the multi-dimensional models.Therefore,this research can be applicable to the optimal water allocation of reservoirs in the humid regions and may provide a new reference for solving similar multi-dimensional optimization problems. |
