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Show The limitation of the stretched thread method is that it addresses a simplistic reservoir optimization problem - single purpose release, inability to directly consider hydrologic abstractions ( e. g. evaporation), simple loss function, lack of economic orientation ( marginal values of variable target releases are likely to be lower than those for an equivalent ' safe draft1) and system conjunctive use perspective. The systems analytic formulations offer greater generality and flexibility with respect to these limitations. In the context of these arguments, it shall be demonstrated that the yield formulation is similar in spirit to the ' stretched thread method', and generalizations of the ' stretched thread method' shall be developed in the yield model context for reservoir operation. It shall be shown that reservoir regulation based on the yield formulation ( as discussed in Chapter 5) leads to optimal reservoir regulation that falls between the two limits of optimality described by the two physical objectives listed above. 6.1 The Stretched Thread Method - a review The review in this section, of the stretched thread method developed by Varlet ( 1923), is based directly on the presentation by Klemes ( 1979). Consider two mass curves Xe ( reservoir empty) and Xf ( reservoir full) of cumulative reservoir inflow ( Figure 6.1, Figure 2 of Klemes ( 1979)) that are displaced by the reservoir's active storage capacity Ss. The reservoir active storage S^ at any time t, is then the distance between Xe and the optimal cumulative outflow curve Y. The stretched thread method then considers the reservoir empty at the start of the simulation, and determines the optimal cumulative outflow as the shortest path through the corridor between Xe and Xf from t= 0 to t= T. The shortest path is represented as a set of piecewise continuous straight lines that are tangential to the corners of the region defined by Xe and Xf. Thus, only three conditions for regulation exist. For any subinterval of time At, the optimal release y0pt( At) is either ( 1) the mean inflow in that time period, ( 2) the sum of the mean inflow in the time period and a uniform depletion rate of storage over the time period, or ( 3) the difference between the mean inflow in the time period and a uniform replenishment rate for the reservoir. These conditions are stated as: ' E [ X I ; + A I I yODt< At) - \ 0TE[ x)\+ M + Ss/ At> 6 1 orE[ xf+ At - S/ At t s J These release rules are obvious from the observation that the corner points are defined strictly by St= 0, and by St= Ss. Thus, for optimal operation, the release is either the mean inflow in the time period, a constant release that leads to the eventual emptying of the reservoir, without a deficit, or a constant release that leads to the reservoir being filled with minimum spilling. The properties of the optimal release policy obvious from this operation are that ( 1) the optimal release depends on both past and future inflows, only to the extent to which these inflows define the location of the comer points of the Xe- Xf region, ( 2) the only storage values that influence the optimal release are the storage empty and storage full conditions and ( 3) as the reservoir size grows, the long term mean inflow provides a good approximation to the optimal release. 156 |