Water Distribution Networks (WDN) are the main component of industrial and urban water distribution systems and are currently formed by pipes, nodes, and loops. In the present paper a deterministic Mathematical Programming approach is proposed, aiming to minimize the cost of looped WDN, considering known pipe lengths and a discrete set of available commercial diameters. The optimization model constraints are mass balances in nodes, energy balances in loops and hydraulic equations, in such a way that no additional software is needed to find the appropriated pressure drops and water velocities. Generalized Disjunctive Programming is used to reformulate the discrete optimization problem to a Mixed Integer Non-Linear Programming (MINLP) problem. GAMS (General Algebraic Modeling System) environment is used to solve This is a previous version of the article published in
Water distribution networks (WDNs) are an important part of water distribution systems and are responsible for water transportation from the reservoirs to the demand nodes at adequate pressure and velocity. In the present paper, the synthesis of WDN is treated as an optimization problem with a mixed integer nonlinear programming formulation. The objective function to be minimized is the total network cost, considering installation and energy costs, with unknown flow directions, which is the novelty in the paper. Disjunctive programming and linearization techniques are used in the model formulation to avoid nonlinear and nonconvex problems. Two case studies are used to test the model's applicability. Results show that operational costs can represent a significant part of the total cost in sustainable networks. In the first case study, the total cost was better than the literature results (US$ 2,272,538.85 vs. US$ 2,272,387.49) and the operational costs represent ¼ of the total WDN costs. In the second case study, the operation cost corresponds to almost 2/3 of the total WDN cost. These results show the importance of considering operational costs in the WDN design. Also, the consideration of unknown flow directions can lead to better results for the network topology.
Water Distribution Networks (WDN) are important systems for industrial processes and urban centers. WDN can be formed by reservoirs, pipes, nodes, loops, and pumps and its complete design can be formulated as an optimization problem. The majority of published papers in the open literature use meta-heuristics for problem solution, as well as hydraulic simulators to calculate pressures and velocities. In the present study, a Mixed Integer Non-Linear Programming (MINLP) model was developed to the synthesis of WDN considering the minimization of the WDN total cost, given by the sum of installation and operational costs, which is the novelty in the paper. All the hydraulic calculations were included in the model (mass and energy balances and velocity and pressure upper and lower bounds), avoiding the use of additional software. Reformulation techniques are applied to the model considering the use of logarithms and disjunctive programming. Two case studies extracted from real WDN were used to test the model and global optimization techniques were employed to achieve the results. The results obtained show that the operational costs play an important role in the WDN system design.
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The design of water distribution networks (WDN) can be formulated as an optimization problem. The objective function, normally, is the network cost, given by the installation cost, which depends on the pipe diameters and by the operation cost, given by the pumping costs associated to the network, which depends on the hydraulic pumps necessary in the system. The water demand can be variable in the network nodes and this variability can be modeled by a finite set of scenarios generated by a normal distribution. In the present paper a disjunctive Mixed Integer Nonlinear Programming (MINLP) formulation optimization problem is proposed to model the design of WDN under uncertainties in the nodes demand. Flow directions are considered unknown and a deterministic approach is used to solve the problem in three steps. Firstly, the problem is solved considering only a nominal value to each uncertain parameter. In the second step, the problem is solved for all the scenarios, being the scenario independent variables fixed to the solution achieved in the first step, which is a deterministic solution. Finally, all the scenarios are solved without fixing any variable value, in a stochastic approach. Two case studies were used to test the model applicability and global optimization techniques were used to solve the problem. Results show that the stochastic solution can lead to optimal solutions for robust and flexible WDN, able to work under distinct conditions, considering the nodes demand uncertainties.
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