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Show a strictly concave function. The type of cost function specified by equation ( 5.11) is strictly monotone increasing. Under these conditions the simulation procedure described above leads to a globally optimal solution for Gs. The heuristic adopted to solve for the optimal hydropower plant capacity performed better in terms of computation time, than a few gradient- newton schemes that were tested. An alternate numerical scheme could however be used for the purpose. 5.3.3 Exogenous Evaluation of Flood Control Benefits No new procedures for estimating flood control benefits given a desired flood storage were devised. For the Lower Bear River applications, the flood storage was provided as a measure for the control of the level of the Great Salt Lake. In this case, retention of a certain volume of water upstream, leads to a direct reduction in the level of the Great Salt Lake, and translates into a direct benefit, without the need for flood routing exercises. The reader is referred to Loucks et al ( 1981, pp. 238- 247) for a brief review of relevant procedures. 5.3.4 Optimization Model for a Multi- purpose Reservoir Procedures for estimating storage capacities, costs and revenues in terms of specified values of the decision variables of the optimization model were delineated in the preceding discussion. The optimization model formulated has a maximum of (( M+ I)* R+ 3) decision variables irrespective of the number of years of record over which reservoir operation is desired. Thus for a single M& I yield, a single irrigation yield, a firm release for hydropower, and provision of flood and dead storage, only 5 decision variables result. The number of decision variables in this case would be 5 independent of the length of the reservoir operation period. Computer storage requirements ( to store inflows, releases, storage values etc.) would indeed grow as the length of the operation period is increased. However this growth is only linear, and does not impact the computational efficiency of the algorithm significantly. The convergence of the MSP procedure is governed largely by the critical period in the record. Addition of a number of years of record thus does not have a marked effect on the time taken by the algorithm to converge. As shall be seen from the following presentation of the constraints and objective function of the resulting optimization model, the penalty for the condensation achieved in problem size is a nonlinear problem formulation that is implicit in the decision variables. Constraints: ( 1) TSC - The total storage capacity Ts of reservoir s should lie between some specified upper ( TSjinax) and lower ( TSjrnm) bounds, that recognize physical limitations on size and existing storage respectively. The total storage capacity Ts is defined in terms of the decision variables Mrsm, Ir si, Ds and Fs using the sequent peak procedure ( see equation 5.7). T . £ T < T ( 5.13) s, min s s, max v ' It should be noted that the active ( and hence the total) storage capacity is also implicitly dependent on the yield reliabilities specified for each yield. The yield model, which is the original basis of this formulation, considers pre- specification of the yield 110 |