APPENDIX C

FORMULATIONS FOR PROPOSED FUTURE MODELS

C1.

Mixed Integer Model

In the mixed integer formulation, presented in Table VII, the variables which indicate the facilities to be built, X, remain zero-one

In other

integers as before, whereas the variables now represent the amount of ij power to be transmitted from each plant alternative to each load center, and are no longer zero-one integers but continuous variables. words, instead of simply indicating whether or not any power is transmitted from plant alternative i to load center j, Yíj in the mixed integer formulation represents exactly how much power is assigned to each plantload center link in the system. The other mathematical symbols are defined in the same way as in Section 2.3 and Appendix A. A solution of the mixed integer formulation using the implicit enumeration technique would supply answers to much larger problems than those solved by the zero-one integer model previously described. Work on this type of computer algorithm is the next developmental step in this field.

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. Their technique is to examine a stream flow frequency histogram and determine probability distributions for stream flow for each month. The temperature standards are expressed in the deterministic formulation as:

where

Aik is the temperature transfer coefficient, R, is the mixed stream temperature rise per MW of plant production, and Ck is the maximum permissible excess temperature at stream point k. In the probabilistic formulation, the constraint is changed to require that the proba

bility that the above constraint is met is greater than or equal to

1

a, 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, R; and Ck are all functions 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