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Show ^ min = pt, minVp 1** 2 - ViyP t = trti+ 1, i= 2,3 ( 6.8) <* max " Vp tm* 4* S where ct_ t j ^ is a lower bound on the reliability of the yield yp considered. Note that the full yield yp is provided only in a time period defined by the years t4 and t5. A smaller yield CL> t m^ p is provided through the critical period defined through the years t\ and t2- Thus in this formulation total rather than incremental yields are considered in a manner analogous to the stretched thread method. The formal inclusion of the above operation rule in the yield model ( Loucks et al, 1981 or Chapter 5) necessitates the specification of a variable yp for the yield for each purpose, a set of n variables 0Cpt j, corresponding to na pieces for the optimal yield from the stretched thread method, and ( na+ l) variables t[ denoting the beginning and ending years for each period of constant yield for each purpose. Constraints to ensure that t{+\ is greater than t[ would also be needed. Such an extension is considerably easier with the simulation based yield model described in Chapter 5, than with the Linear programming formulation of Loucks et al ( 1981). Yields for various purposes, their reliabilities, their economic values and their upper and lower limits may directly be considered in this context. The primary difficulty in using this model is still the need to a priori specify the number of segments na of the optimal yield policy for each purpose. It may be possible to develop reasonable estimates of na through a prior application of the stretched thread method on a simplified situation. An alternate course of action is however appealing. Consider a relaxation of the requirement that the optimal release be piecewise linear. Note that the marginal economic value of the yield for a given purpose may be quite minimal below a certain level of reliability. This implicitly defines an upper limit on the value of yp and a lower limit on the duration t4 to t5 as defined above. Recall that the critical consideration in reservoir regulation is the inflow volumes during the working cycles of the reservoir ( periods between reservoir full and reservoir empty). Further note that the modified sequent peak formulation records reservoir storages precisely through a recognition of accumulated deficits at the specified demand level, relative to the reservoir capacity and the accumulated inflow. Then, preliminary applications of the sequent peak algorithm with the data set may be used to loosely define a range of years in the record where failures of the yield yp are permissible. The reliability ctpt of the yield yp in each of the years t, thus identified may then be introduced as a decision variable in the model of Chapter 5. The resulting model can then be used to determine the optimal release pattern for each purpose through a determination of yp and Opt values. For the multi- reservoir case, the discussion in section 5.3.8 relative to pre- specifying yield reliabilities is relevant, and constraints of the form indicated by equations 5.32 and 5.33 need to be employed. Joint rule curves by reservoir and purpose may then be developed by analyzing the results from the application of the modification described above of the Chapter 5 yield model. The derivation of these rule curves follows a strategy similar to that outlined by Loucks et opt. l opt. i opt, 4 161 |