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Show faced by most other formulations. Further, computational efficiency is enhanced and problem size is reduced by directly solving for the needed storages and generator capacities, given desired yields and releases. The structure of the model formulated here, obviates the need to pose all mass balance equations as constraints in the overall optimization model. These are explicitly satisfied, during the system simulation, and are efficiently exploited by the sequent peak algorithm, and the heuristic to solve for the generator capacities. The structure of the model also allows an easy way to explicitly incorporate considerations for water rights or any other oddities of the reservoir system in the subprogram. As an example, it is possible to define conditional operating rules for the reservoir as part of the system simulation, and to directly perform any allocations of the releases. The model presented may thus be viewed as a hybrid optimization/ simulation model, and can be designed to have significant simulation capabilities. The nonlinear nature of the formulation also enables the consideration of loss/ benefit or other functional relationships that are more complex than those presented in this chapter. The hydro releases from each reservoir can also serve to represent regular downstream release from the reservoir. Minimum flow requirements can thus be modeled by specifying a lower bound on the hydro release. The requirement is satisfied at the monthly level through the monthly demand fractions specified for the annual hydro release. Further, situations where there are " off- stream" reservoirs or diversions from one reservoir to another are modeled by solving for the optimal fractions of streamflow to divert. The formulation also allows for the consideration of multiple yields ( M& I and irrigation to a number of demand areas) and provides for the inclusion of return flows from these supplies. Uses other than M& I and irrigation are readily accommodated by specifying them as either of these and specifying the appropriate unit benefits, monthly demand fractions and reliabilities. Costs of water treatment and supply do usually vary from one site to another. These are not explicitly embodied in the model. This however, is easily accomplished by including the requisite terms in the objective function, or by reducing the unit benefit from supply to each area by an appropriate amount. A final note concerns the global or local optimality properties of the solutions generated by the PSLP algorithm used to solve the optimization problem. Since we are maximizing, optimality is predicated on the concavity of the objective function, and convexity of the constraint set. Where these functions are evaluated through simulation, these properties can be established only qualitatively. Analysis of the results from the applications performed shows that the storage capacity is indeed generally a convex function of the decision variables ( yields) and the net revenue for hydropower is generally a concave function. Revenues from yields are linear. Revenues used for flood benefits were linear. The storage cost functions could however be convex or concave depending on the site characteristics. Consequently, limited efforts are suggested to explore different initial solutions to PSLP, to compare the local optima or saddle points evaluated by the algorithm, in an effort to seek global optima. The strategy used in the applications was to pick the best optimal solution upon constructing the following initial solutions for optimization - ( 1) all variables at their lower bound, ( 2) all variables at their upper bound, ( 3) all variables halfway between their bounds, and ( 4) some variables at their upper bound, others at their lower bound. The PSLP algorithm succeeded in constructing feasible optimal solutions in all cases. It was reassuring to find that the optimal solutions reported for the same problem from different initial solutions were very close. In some cases, there was indeed quite a bit of difference in the optimal solutions obtained from the 2 best initial solution vectors, and the worst initial solution vector. 120 |