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Show 5.3.7 Solution of Multi- reservoir Optimization Model The nonlinear optimization model formulated was solved using the PSLP algorithm of Zhang et al ( 1985). Table 5.2 illustrates the structure of the nonlinear part of the Jacobian matrix of the problem. All the variables appear nonlinearly in the objective function. However, unlike the single reservoir problem, not all decision variables appear in every constraint. First, the M& I and irrigation yield variables appear linearly in the total demand constraints for each type for the relevant demand area. Each of the decision variables appear in the same manner in each type of nonlinear constraint ( TSC, FAS, DSC, GSC). The set of decision variables for an upstream reservoir appear in each of the nonlinear constraints for the reservoir and all reservoirs downstream of the reservoir. Thus, for reservoirs in parallel, the decision variables for the reservoir, appear only in the nonlinear constraints for the reservoir. The only linkage in the problem is then through the M& I and irrigation demand constraints. For reservoirs in series, the decision variables for the most downstream reservoir appear only in the nonlinear constraints for that reservoir, while the decision variables for the upstream reservoir appear in the nonlinear constraints for both the upstream and the downstream reservoir. Trade- offs in supplying a given demand through yields from different reservoirs, as well as the impact of developing a certain yield or instituting a flow regulation ( e. g. through H § ) at an upstream reservoir, on the storage capacities and economics of the downstream reservoir are thus accounted for. While the algorithm does conjunctive operation of reservoirs in a limited sense, the decisions on storage capacity and evaluations of reservoir system economics are done in a conjunctive manner. Since, the determination of storage capacities is done through a simulation of the reservoir system, it should be possible to devise and specify more sophisticated conjunctive operation rules for the reservoir system to follow. 5.3.8 Discussion of the Formulation A general model formulation is presented for investigating optimal reservoir and hydropower plant sizes on a river basin, M& I and irrigation supply scenarios and quantifying recreation and flood control benefits. The model is flexible in that it can be applied to 1 reservoir or demand or to a number of candidate sites. All the purposes outlined in the original scope of work have been explicitly modeled. A very concise representation of the reservoir screening problem is achieved without loss of generality. For one Bear River basin application ( 432 months, 7 reservoirs, 9 hydro releases, 4 M& I demand areas, 3 Irrigation demand areas, 9 M& I releases, 10 Irrigation releases), the formulation leads to only 43 variables and 29 constraints. By comparison the linear programming based yield model of Loucks et al ( 1981) would lead to greater than 16500 decision variables and 9800 constraints for a comparable problem. The actual number of variables and constraints in the linear programming formulation would be even greater than indicated here, once piecewise linearization to accommodate the nonlinear functions ( e. g. hydropower, storage- area ) is implemented. Loucks et al ( 1981) indicate that the yield models presented by them are larger than the chance- constrained models, but are much smaller than the stochastic design models. The problem of excessive computational resources faced by most other formulations has thus been circumvented, by focusing on annual yields for all purposes, with pre- specified reliability and monthly demand fractions, and by decomposing the problem into two parts. The level of analysis is thus monthly, while the optimization needs to only consider the aggregate annual values as decision variables. From a computational standpoint, as additional reservoirs or additional years of operation are considered, the growth in optimization problem's size is almost linear. This is in contrast to the quadratic growth and intractability with larger networks 119 |