An entropy-based index for evaluation of reservoir performance

AN ENTROPY-BASED INDEX FOR EVALUATION OF RESERVOIR PERFORMANCE Antonio Boccafoschi1 and Bartolomeo Rejtano2 2Department of Cicil and Environmental Engineering, University of Catania, Catania (Italy 1boccafos@dica.unict.it, 2breitano@dica.unict.it Abstract Operation of water supply reservoirs during scarsity periods requires appropriate rules for the proactive reduction of water release, aiming to prevent very severe and unbalanced deficit concentrations. Operation rules need to be evaluated in order to ascertain that an improved time-spreading of release does not imply unacceptable increase of the overall deficit due to increased evaporation and overflow. Also, it is necessary to check unacceptable sacrifices on any user. Evaluation is a major step in the development of rules. Despite the multiobjective nature of the problem, the complexity and controversial nature of a multi-index approach and the need of skimming alternatives ask for a synthetic evaluation. A synthetic performance index Ip is introduced for a joint evaluation of the overall release and of the "appropriatedness" of the deficit distribution in time and space. It is defined as the product of a "base value" Rb expressing the overall release as ratio to overall demand, times a reduction coefficient Ce expressing the deficit concentration, in time and among users: Ip = Rb Ce The Shannon entropy S is borrowed from the communication theory as the base for the measure of the spreading goodness. According to Shannon, entropy measures how much uncertainty is in a discrete probability distribution Pi, with i=1,N. It is defined as follows: S = - Σ i =1,N (Pi ⋅ ln Pi) In order to transfer the entropy concept, the following specifications apply: - N is the number of “release cells”, equal to the sum of the number of time cells with non-zero demand, extended to all users; - Pi = Di / (Σ j=1,N Dj) is the probability of a droplet of deficit D to occur at cell i: Entropy is normalized with respect to its maximum, in order to obtain values in a 0-1 range. Its complement, measuring the non-uniformity of the distribution, is multiplied by (1-Rb) in order to weight it with the overall deficit. Its complement to unity is assumed finally to serve as the disuniformity coefficient Ce: Ce = 1 – (1-Rb) ∙ (1 – S/Smax) so that Ip = Rb ∙ [1-(1-Rb) ∙ (1-S/Smax)] The use of the proposed index is shown for the real case of the operation of Lentini Reservoir, in Sicily. Several operation rules for release allocation along time and among users are developed and compared by the Ip index, leading to a preference ranking. Finally, the index ranking is justified by reasonable arguments. Keywords Performance, entropy, reservoir, operation, reliability References [1] J Hashimoto et al., "Reliability, resilience and vulnerability criteria for water resource system performance evaluation", Water resource Research, Vol. 18 n.1, 1982. [2] Shannon C.E., "A mathematical theory of communication." The Bell system technical journal, vol. 27, July-October, 1948.