Scalable Pareto set generation for multiobjective co-design problems in water distribution networks: a continuous relaxation approach

Abstract

In this paper, we study the multiobjective codesign problem of optimal valve placement and operation in water distribution networks, addressing the minimization of average pressure and pressure variability indices. The presented formulation considers nodal pressures, pipe flows and valve locations as decision variables, where binary variables are used to model the placement of control valves. The resulting optimization problem is a multiobjective mixed integer nonlinear optimization problem. As conflicting objectives, average zone pressure and pressure variability can not be simultaneously optimized. Therefore, we present the concept of Pareto optima sets to investigate the trade-offs between the two conflicting objectives and evaluate the best compromise. We focus on the approximation of the Pareto front, the image of the Pareto optima set through the objective functions, using the weighted sum, normal boundary intersection and normalized normal constraint scalarization techniques. Each of the three methods relies on the solution of a series of single-objective optimization problems, which are mixed integer nonlinear programs (MINLPs) in our case. For the solution of each single-objective optimization problem, we implement a relaxation method that solves a sequence of nonlinear programs (NLPs) whose stationary points converge to a stationary point of the original MINLP. The relaxed NLPs have a sparse structure that come from the sparse water network graph constraints. In solving the large number of relaxed NLPs, sparsity is exploited by tailored techniques to improve the performance of the algorithms further and render the approaches scalable for large scale networks. The features of the proposed scalarization approaches are evaluated using a published benchmarking network model.

Penalty and relaxation methods for the optimal placement and operation of control valves in water supply networks

“…In particular, it is possible to prove that Problem (3) satisfies the above constraint qualification at every feasible point except a set of measure zero, once a small perturbation is applied to the optimization constraints; for the sake of brevity we omit the proof which can be found in the Appendix to [33].…”

A Novel Water Supply Network Sectorization Methodology Based on a Complete Economic Analysis, Including Uncertainties

“…This is achieved by the optimal location of both valves and CUs. As expected, the number of possible combinations CU/boundary valves grows combinatorially with the pipe extension and the network meshing configuration [2,3], which makes unfeasible any manual design and makes necessary the use of mathematical and computational techniques.…”

Decreasing the Discoloration Risk of Drinking Water Distribution Systems through Optimized Topological Changes and Optimal Flow Velocity Control

“…valve closures) to maximize self-cleaning capacity in (1) is a mixed-integer nonlinear programming (MINLP) problem. These class of optimization problems for a WDS are especially challenging because they combine large number of discrete decision variables with the difficulties of handling non-convex nonlinear hydraulic constraints (Pecci et al 2016). In addition, the problem in (1) has an objective function that is non-smooth at ±v min , resulting in unbounded gradients around ±v min .…”