M.Sc Thesis
M.Sc Student Schwartz Rafael Inclusion of Uncertainty in Water Distribution Dystems Optimal Design and Operation Department of Civil and Environmental Engineering Professor Avi Ostfeld

Abstract

Least cost design and operation of water distribution systems (WDS) is a well-known problem in the literature. The formulation of the problem started by deterministic modeling and later by more sophisticated stochastic models that incorporate uncertainties related to the problem’s parameters. The stochastic nature of WDS excludes the credibility of a model based entirely on known data while in order to obtain an accurate representation of the problem uncertainty, consideration of multiple possible outcomes is required. Due to this, the stochastic algorithms result in intractable problems which grow exponentially with each time step and outcome taken into consideration.

This work advances two stochastic algorithms to deal with the full formulation of the least cost design and operation of WDS. These algorithms reduce the complexity by formulating a worst-case oriented deterministic formulation and by adding constraints which result in a linear growth to the problem. Both algorithms linearize the problem formulation which allows to solve for large network with minimal computational effort.

An algorithm which simplifies the exponential growth of the stochastic programming problem is the robust counterpart. The non-probabilistic robust counterpart approach utilizes the correlation between stochastic parameters to construct and account for the uncertainty in the problem. The uncertainty is expressed through a user-defined ellipsoidal uncertainty set which accounts for possible realizations of uncertainty according to the correlation between the stochastic parameters. By this, it eliminates the necessity to make assumptions as to the probability function assigned to the uncertain data. The correlation characteristics (e.g. positive or negative) along with the required reliability level determine the feasible range for the group of stochastic parameters defined by the problem. Each uncertain constraint is replaced by a deterministic equivalent which accounts for these two parameters.

Housh et. al (2013) presented the limited multi-stage stochastic programming (LMSP) which reduces the problem complexity without losing any of the data defined by the scenarios. This is accomplished by merging groups of decision variables into clusters. The clusters allow reduction of the number of decision variables as a function of the number of clusters. The clustering method will raise the network cost due to the additional constraints imposed on the problem (equating decision variables). A tradeoff is considered between the computational complexity and the optimality of the objective value to the number of clusters considered.