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Show between 0 and 0.5 ( the largest permissible failure in any year). We need in addition constraints of the form °- 75S'. - X foV * W! « 2 ° y= 1 - 13 ( 532) 2 2 s- l s « l to enforce the contractual requirement for each of 2 consecutive years, and 2 2 10 LvII s= l 8= 1 y= l to enforce the contractual requirement for each 10 year period. I Z 1U S1.- XX^ y. 1.*° y- 1- 4 < 5- 33> The new reliability decision variables qfyS appear nonlinearly in every constraint of a reservoir downstream of s ( including s). The demand constraints also become nonlinear. Where the Jacobian is evaluated through numerical derivatives of the functions in question, this apparently imposes an increased computational burden. However, noting that the change in the sequent peak simulation, occurs only after each year y, for each variable qfyS, the derivatives may be calculated efficiently by numerical means, through a perturbation of the value of qfYS during the application of the sequent peak algorithm. 5.4 Lower Bear River Applications Several applications of the model were performed using data for the Lower Bear River basin. The results of three of these applications are reported here. The first example is an application to 2 reservoirs in series - ONEIDA ( 1) and AMALGA ( 2). The second example has 2 reservoirs in parallel - MILLCREEK ( 1) and AVON ( 2). The third example considers 7 sites. Figures 5.1 through 5.3 illustrate the reservoir configurations for these examples. Table 5.3 summarizes major economic and physical parameter values for the applications. Only one level of reliability r, was considered for each of the M& I and Irrigation yields. Figures 5.4 through 5.7 illustrate the average annual flows at some of these sites. Yield failures were considered for two critical periods in the 1943- 1978 streamflow record, for each of the demand areas, for each purpose. A 5% failure of the M& I yield was considered in each of years 7,8,9,25,26 and 27 of the record. No return flows into the stream system from M& I demands were considered. A 3.125% failure of each irrigation yield was considered in each of years 3 through 6,9 through 12,22 through 24, and 27 through 31. A 25% failure of the Irrigation yield was considered in years 7 and 25, and a 50% failure in years 8 and 26. The failure patterns and years were determined using the contractual requirements indicated by the Utah Division of Water Resources, and using the sequent peak procedure with yields set at their upper bounds. An irrigation use efficiency and return flow factors over a maximum lag of three months were estimated from the historical record for each irrigation demand area and receiving stream section. Efficiency of irrigation water use ranged from 62% to 80%. Return flows from irrigation were typically distributed as 50%, 30% and 20% of the potential return flow, over the three months of lag of return flow. Variable monthly demand fractions were estimated from the historical use for each demand and each purpose and were accordingly specified. The unit M& I benefits were taken to be $ 75/ Acre- foot, the unit irrigation benefits to be $ 50/ Acre- 123 |