Scenario generation for network optimization with uncertain demands

Abstract

The demand imposed on utility networks is a fundamentally uncertain quantity that varies in both space and time. Utilities have long dealt with this uncertainty by maintaining additional capacity. But, technological advances in measuring and especially transmitting water usage information allow further improvement in operational efficiency. Unlike the monthly or annual values obtained by automatic meter reading, smart water meters measure consumption at sub-daily intervals. This increased resolution, combined with measurements of bulk flows and network pressures, presents an opportunity to characterize the spatio-temporal distribution of demand. Once this representation of demand uncertainty is available, it can be used within optimization models to support decisions in network operations and planning. Several methods are available for dealing with uncertain parameters in optimization problems including sensitivity analysis, stochastic programming and robust optimization. This work deals with robust optimization, which characterizes uncertainty by scenarios or uncertainty sets. The robust solution to an optimization problem gives the best objective value in the worst case, which is feasible over the range of possible values. The choice of scenarios is a critical step in obtaining meaningful solutions with the right compromise between system performance (value of the objective function) and robustness to variations in uncertain parameters. This paper examines several techniques for generating scenarios of uncertain demands. Based on the spatial distribution of demands modeled as a multi-variable Gaussian with a mean vector and covariance matrix calculated at each time step, a technique is proposed to generate scenarios of demand which (a) cover a given probability level and (b) adhere to the spatial correlation of demands between nodes. The technique is compared with Monte Carlo sampling using the same co-variance matrix. Particular emphasis is given to the interpretation of scenarios and their effect on the solution to the operational problem of finding valve set points for a water network.