Conjunctive And Multipurpose Operation Of Reservoirs Using Genetic Algorithms

Abstract

Optimal operation of reservoir systems is necessary for better utilizing the limited water resources and to justify the high capital investments associated with reservoir projects. However, finding optimal policies for real-life problems of reservoir systems operation (RSO) is a challenging task as the available analytical methods can not handle the arbitrary functions of the problem and almost all methods employed are numerical or iterative type that are computer dependent. Since the computer resources in terms of memory and CPU time are limited, a limit exists for the size of the problem, in terms of arithmetic and memory involved, that can be handled. This limit is approached quickly as the dimension and the nonlinearity of the problem increases.

In encountering the complex aspects of the problem all the traditionally employed methods have their own drawbacks. Linear programming (LP), though very efficient in dealing with linear functions, can not handle nonlinear functions which is the case mostly in real-life problems. Attempting to approximate nonlinear functions to linear ones results in the problem size growing enormously. Dynamic programming (DP), though suitable for most of the RSO problems, requires exponentially increasing computer resources as the dimension of the problem increases and at present many high dimensional real-life problems can not be solved using DP. Nonlinear programming (NLP) methods are not known to be efficient in RSO problems due to slow rate of convergence and inability to handle stochastic problems. Simulation methods can, practically, explore only a small portion of the search region. Many simplifications in formulations and adoption of approximate methods in literature still fall short in addressing the most critical aspects, namely multidimensionality, stochasticity, and additional complexity in conjunctive operation, of the problem. As the problem complexity increases and the possibility of arriving at the solution recedes, a near optimal solution with the best use of computational resources can be very valuable. In this context, genetic algorithms (GA) can be a promising technique which is believed to have an advantage in terms of efficient use of computer resources.

GA is a random search method which find, in general, near optimal solutions using evolutionary mechanism of natural selection and natural genetics. When a pool of feasible solutions, represented in a coded form, are given fitness according to a objective function and explored by genetic operators for obtaining new pools of solutions, then the ensuing trajectories of solutions come closer and closer to the optimal solution which has the greatest fitness associated with it. GA can be applied to arbitrary functions and is not excessively sensitive to the dimension of the problem. Though in general GA finds only the near optimal solutions trapping in local optima is not a serious problem due to global look and random search.

Since GA is not fully explored for RSO problems two such problems are selected here to study the usefulness and efficiency of GA in obtaining near optimal solutions. One problem is conjunctive operation of a system consisting of a surface reservoir and an aquifer, taken from the literature for which deterministic and stochastic models are solved. Another problem is real-time operation of a multipurpose reservoir, operated for irrigation (primary purpose) and hydropower production, which is in the form of a case study.

The conjunctive operation problem consists of determining optimal policy for a combined system of a surface reservoir and an aquifer. The surface reservoir releases water to an exclusive area for irrigation and to a recharge facility from which it reaches the aquifer in the following period. Another exclusive area is irrigated by water pumped from the aquifer. The objective is to maximize the total benefit from the two irrigated areas. The inflow to the surface reservoir is treated as constant in deterministic model and taken at 6 different classes in stochastic model. The hydrological interactions between aquifer and reservoir are described using a lumped parameter model in which the average aquifer water table is arrived at based on the quantity of water in the aquifer, and local drawdown in pumping well is neglected. In order to evaluate the GA solution both deterministic and stochastic models are solved using DP and stochastic DP (SDP) techniques respectively. In the deterministic model, steady state (SS) cyclic (repetitive) solution is identified in DP as well as in GA. It is shown that the benefit from GA solution converges to as near as 95% of the benefit from exact DP solution at a highly discounted CPU time.

In the stochastic model, the steady state solution obtained with SDP consists of converged first stage decisions, which took a 8-stage horizon, for any combination of components of the system state. The GA solution is obtained after simplifying the model to reduce the number of decision variables. Unlike SDP policy which gives decisions considering the state of the system in terms of storages, at reservoir, aquifer, and recharge facility, and previous inflow at the beginning of that period, GA gives decisions for each period of the horizon considering only the past inflow state of the period. In arriving at these decisions the effect of neglected state information is approximately reflected in the decisions by the process of refinement of the decisions, to conform to feasibility of storages in reservoir and aquifer, carried out in a simplified simulation process. Moreover, the validity of the solution is confirmed by simulating the operation with all possible inflow sequences for which the 8-stages benefit converged up to 90 % of the optimum. However, since 8 stages are required for convergence to SS, a 16-stage process is required for GA method in which the first 8 stages policy is valid. Results show that GA convergence to the optimum is satisfactory, justifying the approximations, with significant savings in CPU time.

For real-time operation of a multipurpose reservoir, a rule curve (RC) based monthly operation is formulated and applied on a real-life problem involving releases for irrigation as well as power production. The RC operation is based on the target storages that have to be maintained, at each season of the year, in the reservoir during normal hydrological conditions. Exceptions to target storages are allowed when the demands have to be met or for conserving water during the periods of high inflows. The reservoir in the case study supplies water to irrigation fields through two canals where a set of turbines each at the canal heads generate hydropower. A third set of turbines operate on the river bed with the water let out downstream from the dam. The problem consists of determining the the RC target storages that facilitate maximum power production while meeting the irrigation demands up to a given reliability level. The RC target storages are considered at three different levels, corresponding to dry, normal, and wet conditions, according to the system state in terms of actual (beginning of period) storage of the reservoir. That is, if the actual beginning storage of the reservoir is less than some coefficient, dry-coe, times the normal target storage the target for the end of the period storage is taken at the dry storage target (of the three sets of storages). Similarly the wet level is taken for the end of the period target if the actual beginning storage is greater than some coefficient, wet-coe, times the normal storage. For other conditions the target is the normal storage level.

The dry-coe and wet-coe parameters are obtained by trial and error analysis working on a small sequence of inflows. The three sets of targets are obtained from optimization over a 1000 year generated inflow sequence. With deterministic DP solutions, for small sequences of inflows, the optimization capability of GA-RC approach, in terms of objective function convergence, and generalization or robustness capability of GA-RC approach, for which the GA-RC benefit is obtained by simulating the reservoir operation using the previously obtained GA-RC solution, are evaluated. In both the cases GA-RC approach proves to be promising. Finally a 15 year real-time simulation of the reservoir is carried out using historical inflows and demands and the comparison with the historical operation shows significant improvement in benefit, i.e. power produced, without compromising irrigation demands throughout the simulation period.