Approximation of Fuzzy Membership Functions in Water Distribution Network Analysis

Abstract

Design and analysis of water distribution networks (WDNs) is laden with uncertainty, both aleatory, i.e., natural randomness, such as variations in reservoir elevation heads, and epistemic, i.e., incomplete knowledge, imprecise data, and linguistic ambiguity such as that associated with the characterization of pipe resistance, nodal demands, and hydraulic responses. To accommodate aleatory uncertainty, stochastic analysis is applied to represent the input uncertainties and to estimate resulting uncertainty in nodal pressures and pipe flows. In the analysis of WDNs facing epistemic uncertainty, in particular, fuzzy set theory has widely been suggested as an alternative to stochastic analysis. This technique can identify the extreme values of unknown variables when uncertain input information ranges between prespecified extremes, and when the probability distribution of the information cannot be obtained. Current approaches for conducting fuzzy analysis of WDNs to support operations and design are computationally demanding, and thus limited in their applicability to large networks. Approximations of the gradients of equations that govern WDN analysis, with respect to nodal demands and pipe resistance, are identified herein and harnessed to accelerate fuzzy analysis of system hydraulics. The resulting WDN nodal pressures are inversely proportional to nodal demands, and depending on flow directions, proportional to pipe resistance. Results of fuzzy analyses for two realistically sized WDNs show that the proposed method performs with an acceptable level of accuracy and greatly reduces computational time, relative to existing fuzzy analysis approaches.