The operating rules that we described allocated the contribution of each of the two reserv...

The operating rules that we described allocated the contribution of each of the two reservoirs; to current needs to historical flow records. Still other ways exist to allocate the proportional contribution of the two reservoirs toward the water requirement of the city.

One way (Rule III) is to allocate the relative contribution of each reservoir in proportion to the fraction of total system storage that exists in each reservoir at the end of the previous period. That is, the fraction of the water requirement from each reservoir matches the fraction of the total storage in that reservoir, and so on. This allocation followed for a particular reservoir unless the previous storage stage plus current inflow in the reservoir is smaller than the portion of the requirement that the reservoir is supposed to fill. In this case both storage and current inflow are completely contributed toward the requirement. This rule, which is keyed to current reservoir conditions, is likely to perform even better than the previously discussed rules (I and II). Rule III was also discussed in the text

Still another way (Rule IV) to allocate the relative contributions between reservoirs is even more complicated but could improve performance still further. Let us now consider the fraction of total water in the system projected to be in a particular reservoir at the end of the current period if no release is made from that or any other reservoir. That is, the fraction of the water requirement furnished from each reservoir is to match the fraction of projected system storage in that reservoir at the end of the current month if no releases are made. The projected storage in a particular reservoir at the end of the month is calculated as the sum of the previous storage (end of last month) plus the projected inflow without any release toward the water requirement. The projected system storage is the sum of these projected reservoir storages over all reservoirs. This allocation rule would be followed unless insufficient water is available from a particular reservoir to meet its designated contribution—in which case only available water (storage plus current inflow) would be contributed. This rule is tuned even more tightly than previous rules to current conditions

(a) Using the notation in the chapter, write the six simulation equations for Rule III. Define αt and (βt both verbally and with mathematics.

(b) Writing the simulation equations for Rule IV requires more machinery. First) it will be necessary to establish the projected inflow for each month in the long record. Hydrologists have observed that flow in a. given basin, are serially correlated month to month. For instance, the flow in a particular month seems to be well predicted by the immediately preceding flow. Often, the best predictive equation for the flow in a particular month is based only on the flow, in the single preceding month. An equation of the form

Where

can be used to establish the projected flow in the current month.

(1) This predictive equation needs to be developed for every pair of months of the year, beginning with the January—December pair and ending with the December—November pair. By reference to a particular pair or months (and hence to any pair of months), explain how you would use the long record available to you on each stream to establish the predictive equation for each pair of months.

(2) Having established 12 predictive equations, you can now proceed to develop the simulation equations. For purposes or these equations we will distinguish two different flows in month t for each reservoir. We illustrate for reservoir 1

I1t = the recorded or "filled-in" flow in month t reservoir 1 these flows constitute the historical record used in the simulation; and

 = the predicted as opposed to actual flow in month t into reservoir 1 as established by the equation that predicts flow in month t from flow in month t – L

With this notation, you can write αt and βt as well as the six simulation equations