Optimal operations of a reservoir with a deterministic future
The problem for operations of reservoirs is to choose on a day to day basis the value of the release at the dam location. The choice of the value of that discharge is conditioned by a criterion of satisfaction of one or several objectives. These objectives are defined in one or several points in the system on the river, or the rivers, downstream from the point, or the points, of release. Typical objectives may be to maximize electric production, or to minimize damage due to flooding downstream from the dams or due to shortages of water in the rivers at diversion points for municipal water supply or other uses, etc. adapted to the concerns of the managers, and relatively intuitive. The approach described in this article pursues the reasoning of Massé (1946) but generalizes it and therefore makes it more applicable. At first we look at the case of a single reservoir, located directly on the stream for the production of electric energy. In this case the target-point (the point where an objective function is to be evaluated) coincides with the point of release. This was precisely the problem studied by Massé (1946) in his classical two volumes on "Reserves and the Regulation of the Future". We pursue his reasoning but we use a more appropriate mathematical procedure which will allow us to obtain more general results. The same results are derived using two different approaches. The first one is more intuitive and uses the concept of marginal value to secure the necessary condition of optimality to be satisfied by the releases. The second procedure is more mathematical and uses, basically, the method of Calculus of Variations, generalized to the case where there are inequality constraints that must be satisfied. In the case of a single reservoir one shows that the optimality condition provides the rigorous proof of the graphical method of Varlet (1923). The results of Massé are generalized to the case where the objective function is evaluated downstream from the point of release and the management strategy must account for the phenomenon of propagation of discharges in the streams. Again in this case the results are obtained in two ways, (1) by the economic reasoning on the marginal values and (2) with the Constrained Calculus of Variations. Massé had found that the optimal policy for the releases was the one that maintained the marginal benefit constant in time. That applied for the case of a single reservoir and where the target-point coincides with the point of release. If B{x(t),t} is the instantaneous benefit obtained from making the release at the dam at a rate x(t) at time t, then the optimality condition is mathematically: b{x(t),t}=L=constant with time where b{x(t),t} is the marginal benefit, i.e. the partial derivative of B{x(t),t} with respect to the argument x. L is a constant, which in the mathematical formulation of the problem is the Lagrange multiplier associated with the mass balance constraint to be satisfied over the selected horizon of operations. In other words the cumulative volume of releases over the time horizon must be equal to the cumulative volume of inflows plus the drop in reservoir storage between the initial and final times. Economically the marginal benefit is the incremental benefit realized by making an extra release of one unit of water, given that the rate of release was x(t). Typically the marginal benefit decreases as the rate of release increases and that is often referred to as the "law of decreasing returns". For the case of electric production the marginal benefit will depend on the amount of releases made through the turbines but also on the season of year or day of week or hour of day. The price of electricity is higher in winter than it is in summer. It is higher during peak hours during the week than it is on weekends, etc. If on the other hand the marginal benefit is only a function of the release, and not a function of time, then the constancy of the marginal benefit with time is equivalent to the constancy of the release with time. Optimality becomes synonymous with regulation, i.e. releasing at a constant rate. It is only under these conditions that the graphical method of Varlet is applicable. In the graphical domain of cumulative volume of releases versus time, the optimal "trajectory" is a straight line where such a strategy is feasible i.e. does not make the reservoir more than full nor less than empty. When the objective is evaluated at a point downstream from the point of release and the marginal benefit (or cost) has a seasonal character, neither the graphical procedure of Varlet nor the mathematical result of Massé apply. For this more general case the derived optimality condition states that it is no longer an instantaneous marginal benefit that must remain constant in time. What must remain constant in time is a time integrated and weighted value of the marginal benefit (or damage) between the time the release is made and a later time. That later time is the release time plus the memory of the propagation system. The memory time is the time that must lapse before an upstream release is no longer felt at the target point downstream. The longer the distance between the release point and the target-point the longer is the memory of the propagation system. At the downstream point the damage depends on the discharge at that point, which is of course related to the release rate but also to the lateral inflows in between from tributaries and on the amount of attenuation that happens between the point of release and the target-point downstream. The integrand at dummy integration time t' is the marginal damage at that time multiplied by the instantaneous unit hydrograph at that time. Mathematically the integrand is: f{q(t'),t'}*k(t'-t) where f is marginal damage, q(t') is discharge at target point and k(.) is instantaneous unit hydrograph of propagation between release and target points. This integrand is to be integrated between time t of the release and time t + M, where M is the memory of the system. It is that integral that we have called the "Integrated Marginal Future" (or IMF for short) value that must remain constant in time. That optimality condition applies as long as the trajectory remains in the feasible domain bounded by the constraints of the problem, the "interior domain". When on a bound, the IMF value does not remain constant but must vary monotonically in a given direction, i.e. increases or decreases with time, depending on the constraint on which the solution rests.