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Show proposed reservoir system is then considered for the flow record, with failure of water supply yields considered during the critical period. A proper pre- specification of yield reliabilities at each reservoir, can be achieved, at least approximately, by specifying the upper bounds of the demands considered on the reservoir as an aggregate demand, and then using the modified sequent peak procedure to determine cumulative required storage capacities, as a function of time. A rule for desired reliabilities ( e. g. 50% failure in 1 year, 75% of annual yield failure in two consecutive years and 100% of annual yield failure in 10 consecutive years) may then be applied for the selection of failure years ( and degrees of failure in each of those years), to cause a maximum reduction in the required storage capacity of the reservoir. This determination may be done through the use of a specially formulated optimization model, or through the use of an iterative heuristic. Of note here is the ' stretched thread' method advocated by Klemes ( 1979) and discussed in the next chapter. Another approach is also possible for the specification of yield reliabilities. Consider a rule of the form referenced above. Then one may assume that the year by year failure probabilities, within this 10 year period may be described by a probability distribution function ( e. g. the Beta distribution), such that the cumulative failure in 10 years corresponds to a 100% yield failure ( e. g if the annual yield is 100, over 10 consecutive years a shortfall of 100 units occurs). Then, the parameters of the probability distribution function may be introduced as decision variables in the optimization model, for each yield area. Constraints may then be specified to ensure that these parameters are selected such that the rule for desired reliabilities is satisfied ( e. g. probability content of any 1 year corresponds to less than 50 % failure, the probability content of any two consecutive years is less than 75% failure, etc.). The index of the first year of the 10 year failure period at each site may also be introduced as a decision variable. Exploratory attempts with this strategy were pursued using the Beta distribution as a vehicle. While, the results obtained were promising in terms of the general trends indicated, computational complexity was increased and computational efficiency decreased. For the examples pursued, the optimal reservoir storage capacities were not significantly different from those obtained with pre-specification of the yield reliabilities. While this could be due to the fact that the area used in the applications is largely hydrologically homogeneous, the added complexity of this exercise for a screening model was unjustified. Another problem observed was that the use of a monotonic probability distribution as a failure model, does not allow for much increased flexibility in operation, and can indeed indicate a need for failures of yield in some critical years that are not necessary. Alternately, if a preliminary identification of the joint critical period at the reservoir sites is made a priori, decision variables that represent the degree of yield failure at each site, in each of the critical years can be introduced. Constraints that specify the contractual terms for yield reliability for each demand area can then also be specified. This allows us to attain a fairly high degree of generality without an excessive increase in the mathematical size of the optimization problem. An example of how this formulation could be developed is presented by considering two reservoirs in parallel, meeting a common irrigation demand. Assume a single ' firm' yield from each reservoir. The contractual obligation requires no more than 50% yield failure in any year, no more than 75% of annual yield failure in 2 consecutive years, and no more than 100% of annual yield over a 10 year period. Let us assume that the joint critical period at the two reservoir sites is 14 years. Then define as decision variables qfvs ( y= l„. 14, s= 1.. 2) that represent the probability of failure of the yield Is from reservoir s, for year y. Each of these variables is bounded 122 |