Testing linear solvers for global gradient algorithm

Steady-state Water Distribution Network models compute pipe flows and nodal heads for assumed nodal demands, pipe hydraulic resistances, etc. The nonlinear mathematical problem is based on energy and mass conservation laws which is solved by using global linearization techniques, such as global gradient algorithm (GGA). The matrix of coefficients of the linear system inside GGA belongs to the class of sparse, symmetric and positive definite. Therefore a fast solver for the linear system is important in order to achieve the computational efficiency, especially when multiple runs are required. This work aims at testing three main strategies for the solution of linear systems inside GGA. The tests are performed on eight real networks by sampling nodal demands, considering the pressure-driven and demand-driven modelling to evaluate the robustness of solvers. The results show that there exists a robust specialized direct method which is superior to all the other alternatives. Furthermore, it is found that the number of times the linear system is solved inside the GGA does not depend on the specific solver, if a small regularization to the linear problem is applied, and that pressure-driven modelling requires a greater number which depends on the size and topology of the network and not only on the level of pressure deficiency

Pipe Network Modelling for Analysis of Flow in Porous Media

In this paper, a new matrix framework is developed for the simulation of flow and pressure in porous media. In this framework, the pressure gradient formulation in porous media is used instead of the energy equation in pipe network modeling and an artificial pipe network has been employed to find the pressure head profile in porous media. Two explicit and implicit formulations are advanced for linear and nonlinear flow analysis which the latter is an application of the Newton-Raphson algorithm. Both formulations consist of a solution of a linear system of equations and updating the flow vector. While the explicit method needs one iteration, the implicit method requires at least 20 iterations to converge with acceptable accuracy. For testing these formulations, four different types of network configurations tested. Results show that application of the implicit method in pipe network modeling with more pipes has higher accuracy than the explicit formula.The accepted manuscript in pdf format is listed with the files at the bottom of this page. The presentation of the authors' names and (or) special characters in the title of the manuscript may differ slightly between what is listed on this page and what is listed in the pdf file of the accepted manuscript; that in the pdf file of the accepted manuscript is what was submitted by the author

Particle Swarm Optimization for Hydraulic Analysis of Water Distribution Systems

The analysis of flow in water-distribution networks with several pumps by the Content Model may be turned into a non-convex optimization uncertain problem with multiple solutions. Newton-based methods such as GGA are not able to capture a global optimum in these situations. On the other hand, evolutionary methods designed to use the population of individuals may find a global solution even for such an uncertain problem. In the present paper, the Content Model is minimized using the particle-swarm optimization (PSO) technique. This is a population-based iterative evolutionary algorithm, applied for non-linear and non-convex optimization problems. The penalty-function method is used to convert the constrained problem into an unconstrained one. Both the PSO and GGA algorithms are applied to analyse two sample examples. It is revealed that while GGA demonstrates better performance in convex problems, PSO is more successful in non-convex networks. By increasing the penalty-function coefficient the accuracy of the solution may be improved considerably

Hydraulic Analysis of Water Distribution Network Using Shuffled Complex Evolution

Hydraulic analysis of water distribution networks is an important problem in civil engineering. A widely used approach in steady-state analysis of water distribution networks is the global gradient algorithm (GGA). However, when the GGA is applied to solve these networks, zero flows cause a computation failure. On the other hand, there are different mathematical formulations for hydraulic analysis under pressure-driven demand and leakage simulation. This paper introduces an optimization model for the hydraulic analysis of water distribution networks using a metaheuristic method called shuffled complex evolution (SCE) algorithm. In this method, applying if-then rules in the optimization model is a simple way in handling pressure-driven demand and leakage simulation, and there is no need for an initial solution vector which must be chosen carefully in many other procedures if numerical convergence is to be achieved. The overall results indicate that the proposed method has the capability of handling various pipe networks problems without changing in model or mathematical formulation. Application of SCE in optimization model can lead to accurate solutions in pipes with zero flows. Finally, it can be concluded that the proposed method is a suitable alternative optimizer challenging other methods especially in terms of accuracy

Testing evolutionary algorithms for optimization of water distribution networks

Water distribution networks (WDNs) are one of the most important elements of urban infrastructure and require large investment for construction. Design of WDNs is classified as a large combinatorial discrete nonlinear optimization problem. The main concerns associated with the optimization of such networks are the nonlinearity of the discharge-head loss relationships for pipes and the discrete nature of pipe sizes. Due to these issues, this problem is widely considered to be a benchmark problem for testing and evaluating the performance of nonlinear and heuristic optimization algorithms. This paper compares different techniques, all based on evolutionary algorithms (EAs), which yield optimal solutions for least-cost design of WDNs. All of these algorithms search for the global optimum starting from populations of solutions, rather than from a single solution, as in Newton-based search methods. They use different operators to improve the performance of many solutions over repeated iterations. Ten EAs, four of them for the first time, are applied to the design of three networks and their performance in terms of the least cost, under different stopping criteria, are evaluated. Statistical information for 20 executions of the ten algorithms is summarized, and Friedman tests are conducted. Results show that, for the two-loop benchmark network, the particle swarm optimization gravitational search and biology and bioinformatics global optimization algorithms efficiently converge to the global optimum, but perform poorly for large networks. In contrast, given a sufficient number of function evaluations, the covariance matrix adaptation evolution strategy and soccer league competition algorithm consistently converge to the global optimum, for large networks.The accepted manuscript in pdf format is listed with the files at the bottom of this page. The presentation of the authors' names and (or) special characters in the title of the manuscript may differ slightly between what is listed on this page and what is listed in the pdf file of the accepted manuscript; that in the pdf file of the accepted manuscript is what was submitted by the author