C2. Model with Probabilistic Temperature Constraints The mixed integer model is deterministic in that it yields no measure of the reliability with which the stream standards will be met in the future since the transfer coefficient and mixed temperature rise calculations are usually based on the worst possible conditions, as determined by past stream flow records. The probabilistic formulation of the problem, however, specifically takes into account the stochastic nature of environmental conditions in developing the temperature standard constraints, and thus has the advantage of providing both an explicit statement of the risk of violating the temperature constraints and a means of avoiding the additional expense involved in preparing for statistically extreme environmental conditions. In the sample problem formulation, a critical factor in determining the stream temperature constraints is the temperature transfer coefficient which shows stream temperature response at a given point due to a unit temperature increase at the plant site. This factor is not constant over time, but rather exhibits variations because of the stochastic inputs which determine it, e.g., wind speed, air temperature and stream flow. ReVelle, Joeres and Kirby (7) have devised a method to attack constraints based on random stream flow as a straight linear programming problem. flow frequency histogram and determine probability distributions for stream flow for each month. Their technique is to examine a stream The temperature standards are expressed as: { Aik Ri [ & Yighj ] ≤ Ck, for all k, i j where Aik imum permissible excess temperature at stream point k. In the probabilistic formulation, the constraint is changed to require that the probability that the above constraint is met is greater than or equal to 1 where a is the fraction of the time that the stream standard may be exceeded. Such a probabilistic standard is not amenable to linear programming techniques. However, since Aik R1 and Ck are all func tions of the prevailing environmental conditions, which in the deterministic formulation are assumed to be at the worst possible level, it is possible to transform the probabilistic constraint into a linear equivalent by introducing Ak, R and C based on a stream flow with a specified low probability of occurrence a, i.e.: It is important to note that such a linear constraint containing an implicit probability consideration can be solved by the same computer model which has been developed, viz., the zero-one integer model. A subroutine could be added to the existing program to examine historical or synthetically generated stream flow for a given period of time, and for each month to choose a flow with a specified probability of occurrence a and build constraints using the corresponding values of Ak R and C. Each of the twelve (monthly) constraints so generated would then be examined, and the most binding one would be chosen to be used in solving the problem. This would assure that the stream temperature standards are met at least 100(1 - a)% of the time and would provide an idea of the cost of planning to meet the worst possible conditions, on which the deterministic constraints are based, as compared with conditions which occur less than 100% of the time. In addition, provision could be made in the program to perform regression analysis on stream flow and temperature so that the former uniquely determines the latter by a regression equation. This could also be done outside of the present program using an available computer code. A formulation for a model with probabilistic temperature constraints is presented in Table VIII. 59-068 - 71 pt. 3 ---13 -C 91 |