Analysis of probabilistic constraints in the management of a hydroelectric power system
The paper deals with the problem of determining the amount of water to release from several reservoirs located in the same river basin, so as to satisfy the demand for electricity at minimal cost white respecting a set of constraints related to environment protection, flood control, wood floating and navigation. Would the demand for electricity and the river inflows be known for the entire period studied, that the problem could be written Minimize {Z (D - H (X)) / AX ≽ Ɵ, X ɛ ɼ} (1) Where Z is the cost function, D the demand for electrical energy, H(X) the hydroelectric production, AX ≽ Ɵ the set of constraints mentioned above, and r the set of admissible solutions. However, since the demand for electricity and the river inflows cannot be forecasted a long time in advance, and since they change significantly from one year to the next, the constraints AX ≽ Ɵ cannot always be respected. For instance, maintaining the level of reservoir i above Ɵi may be impossible in a period of very low flow. In order to find a solution to the problem, the constraints AX ≽ Ɵ were replaced by the probabilistic constraints Pr(AX ≽ Ɵ) ≼ B and the problem rewritten Minimize E {Z (D - H (X))} (2) s.t. Pr(AX < Ɵ) ≼ B (3) X ɛ ɼ (4) Where E denotes the mathematical expectation and Pr the probability. In the solution procedure, however, constraints (3) are not directly taken into account, but are replaced by additional terms in the objective function which penalize violations of the constraints. The parameters of these penalty terms are adjusted iteratively until constraints (3) are respected. More precisely, values are chosen for the parameters and the problem is solved by stochastic dynamic programming. Next, the operation of the reservoirs is simulated over hundreds of years to find out whether constraints (3) are respected with the parameters initially selected. If the constraints are too often violated or not enough, the values of the parameters are changed and the problem solved again. This approach has been applied to the St-Maurice River in Canada, where upper limits on the flow exist at different points along the river to prevent floods. The aim of the study was to find an operating policy that respects these upper limits 95 % of the time, and to determine what effects such policy has on the production of hydroelectric energy.