Continuous OptimizationPolynomial optimization for water networks: Global solutions for the valve setting problem
Introduction
As water systems age and resources to make repairs diminish, mathematical optimization techniques are increasingly applied to design and operational problems for water networks. Design problems include placement of pressure reducing valves to minimize pressure or leakage (Reis, Porto, & Chaudhry, 1997) and selection of pipes and pipe sizes for minimum cost rehabilitation (Bragalli, D’Ambrosio, Lee, Lodi, & Toth, 2012). Operational problems include finding set points for pressure reducing valves, and scheduling pumps to minimize electricity costs (Eck, Mevissen, 2015, Ghaddar, Naoum-Sawaya, Kishimoto, Taheri, Eck, 2015).
In the optimization of water networks the governing equations of energy and mass conservation act as constraints. Without considering any additional constraints, this set of equations has a unique solution provided frictional effects exist in the system (Berghout & Kuczera, 1997). Water network problems are formulated as non-linear and non-convex optimization problems. Hybrid simulation-optimization approaches have been developed to solve to water network problems by combining optimization with a simulator (Naoum-Sawaya, Ghaddar, Arandia, & Eck, 2015). Pure mathematical optimization approaches are also becoming more popular owing to the improvement of optimization solvers and interest in the problems. The survey of D’Ambrosio, Lodi, Wiese, and Bragalli (2015) presents various mathematical programing approaches to problems in the field of water network optimization. Most of the proposed methods find local solutions for these types of problems because of the non-linear and non-convex problem structure. However, global solutions or bounds remain of theoretical and practical interest.
A particular challenge to finding global solutions to optimization problems on water networks is modeling the energy lost due to friction along the pipes. In fluid dynamics it is often convenient to describe energy in terms of the fluid weight. This quantity is called the head and has units of length. Head lost due to friction with the pipe is given by the Darcy–Weisbach law: where l is the pipe length, ϕ is the pipe diameter, u is the average velocity, and g is the gravitational acceleration all in consistent units. The friction factor, f has different forms for the laminar, transitional, and turbulent flow regimes. In most practical systems the turbulent regime dominates and the friction factor is found from the Colebrook formula (White, 1999): Here ks is the height of roughness elements inside the pipe, is Reynold’s number and ν is the kinematic viscosity. Owing to the non-linear and implicit nature of Eq. (2) mathematical optimization approaches introduce some form of approximation, such as the Hazen–Williams formula (Association et al., 1989) The parameter C relates to the pipe roughness and is the volumetric flow rate.
In this paper, we approximate the head loss formulas of Darcy–Weisbach and Hazen–Williams using the quadratic expression developed recently in Eck and Mevissen (2015). The head loss equation is given as . For each pipe, values of a, b, and c are selected to minimize the relative error between Eq. (4) and Eq. (1) or (3) over the approximation range (the minimum and maximum flows). The head loss equation is given as: For each pipe, values of a and b are selected to minimize the relative error over the approximation range (the minimum and maximum flows). The form of Eq. (4) balances the limiting case of fully rough flow (Re → ∞), where a squared function applies, with improved accuracy obtained for more typical values of Re by including the linear term. For flow ranges encountered on water networks, the approximation error is typically less than 10%. Further details on the development and accuracy of the approximation are available in Eck and Mevissen (2015). As discussed subsequently, applying Eq. (4) results in non-linear and non-convex problems that may be tackled using polynomial optimization techniques.
We utilize the quadratic approximation to formulate the valve setting problem, the operational problem of choosing the set-point for pressure reducing valves, as a polynomial optimization problem. The physical motivation for studying valve control is to reduce leakage. Leakage may occur on distribution mains, service connections, or at the point of use. Although leakage localization and pipe replacement would be ideal, this is expensive and slow. Lowering system pressure by inserting control valves can reduce, though not eliminate, leakage before the pipes are replaced.
The valve setting problem has been studied extensively using mathematical programing starting with Sterling and Bargiela (1984), who used a Taylor-series approximation of the head loss curve to apply sequential linear programing. A sequential linear programing technique was also used by Germanopoulos and Jowitt (1989) and Jowitt and Xu (1990). In Vairavamoorthy and Lumbers (1998), the authors introduced sequential quadratic programing for the valve setting problem. A parallel computing technique using sequential quadratic programing was given by Alonso et al. (2000), where different time steps were assigned to each node. Metaheuristic approaches have also been applied to the valve setting problem. In Savic and Walters (1995), a genetic algorithm was used to find settings of isolating valves to minimize pressure heads. Additionally, Reis et al. (1997) finds optimal locations and settings for control valves. A multi-objective approach considering the number, location, and setting of valves was proposed by Nicolini and Zovatto (2009). A scatter-search algorithm was used by Liberatore and Sechi (2009). A cubic formulation of the valve setting problem was introduced in Eck and Mevissen (2013). The resulting formulation was solved using both a local approach, Ipopt, and a global method by using polynomial optimization. The approach was tested on the Pescara water network.
In this paper, we study global solutions for a formulation of the valve setting problem that uses quadratic pipe friction and is solved by using semidefinite relaxations. These global solutions are shown to be close to the ones obtained by a simulation software where the accuracy of the quadratic approximation was assessed by comparing the flows and pressures from the optimization model with the EPANET simulator. The main contributions of the paper are as follows:
- •
showing that global solutions may be found for two formulations of the valve setting problem by using quadratic pipe friction and semidefinite relaxations;
- •
formulating energy conservation as a quadratic rather than cubic constraints;
- •
exploiting the structure of the network to take advantage of sparsity of the polynomial program;
- •
demonstrating that the proposed method outperforms available solvers (e.g., Ipopt and Couenne) and the resulting solutions are close to the ones obtained by the EPANET simulator.
The remainder of this paper is organized as follows. The cubic formulation proposed in Eck and Mevissen (2013) and a novel quadratic formulation of the valve setting problem are presented in Section 2. The dense and the sparse semidefinite-based hierarchies are presented in Section 3. Computational results comparing the proposed approach with Ipopt, a local solver, and Couenne, a global solver, are presented for four benchmark networks in Section 4. Finally, a brief conclusion and future research directions are discussed in Section 5.
Section snippets
Mathematical formulation
In this section we examine the valve setting problem in water distribution networks to find settings for pressure reducing valves that minimize the total pressure on the network while also providing a minimum pressure. The physical motivation for minimizing pressure is to reduce pressure driven leakage and to decrease the frequency of pipe bursts.
The water network, comprised of a set of nodes N and a set of pipes E, is modeled as a graph with |N| vertices and |E| edges. Nodes are numbered
SDP hierarchy and exploiting sparsity
Both valve setting formulations described in the previous section are polynomial programs. In this section, we present the dense and the sparse SDP hierarchies that are utilized to solve these two formulations. Consider the general polynomial programing problem whose objective and constraints are multivariate polynomials: Let be the feasible set of [PP]. Then [PP] can be rephrased as The condition being
Computational results
This section presents numerical experiments conducted on four water networks using the formulations [VS-C] and [VS-Q] given in Section 2. The first network is a small size water network from our industrial partners, the second network is a benchmark network with 25 nodes, the third network is a medium size network (Pescara water network), and the fourth network is a large scale network (Exnet) taken from the literature. The valve placement locations for each network (i.e., the edges where the
Conclusions
In this paper, we apply polynomial optimization techniques to the valve setting problem to reduce leakage on aging systems. For the explored problem, an existing cubic and new quadratic formulation of the energy conservation constraints are examined. Global solutions were found for both formulations by exploiting sparsity and using the hierarchy of SDP relaxations. In our experiments, the quadratic formulation had better performance with an optimality gap of zero for three out of the four
Acknowledgment
We greatly thank the referees and the editor for their thorough and thoughtful comments that helped us improve the quality of the paper.
References (26)
- et al.
Mathematical programming techniques in water network optimization
European Journal of Operational Research
(2015) - et al.
A Lagrangian decomposition approach for the pump scheduling problem in water networks
European Journal of Operational Research
(2015) - et al.
Simulation optimization approaches for water pump scheduling and pipe replacement problems
European Journal of Operational Research
(2015) - et al.
Parallel computing in water network analysis and leakage minimization
Journal of Water Resources Planning and Management
(2000) Steel pipe: A guide for design and installation
(1989)- et al.
Network linear programming as pipe network hydraulic analysis tool
Journal of Hydraulic Engineering
(1997) - et al.
On the optimal design of water distribution networks: A practical MINLP approach
Optimization and Engineering
(2012) - Couenne (2011). (v. 0.4). https://projects.coin-or.org/Couenne (Accessed...
- et al.
Fast non-linear optimization for design problems on water networks
Proceedings of the 2013 world environmental and water resources congress
(2013) - et al.
Quadratic approximations for pipe friction
Journal of Hydro Informatics
(2015)
EXNET benchmark problem for multi-objective optimization of large water systems
Proceedings of the IFAC workshop onmodelling and control for participatory planning and managing water systems
SDPA project: solving large-scale semidefinite programs
Journal of Operations Research Society of Japan
Leakage reduction by excess pressure minimization in a water supply network
ICE Proceedings
Cited by (19)
Dynamically adaptive networks for integrating optimal pressure management and self-cleaning controls
2023, Annual Reviews in ControlOptimal control of water distribution networks without storage
2020, European Journal of Operational ResearchCitation Excerpt :Based on the optimized solutions, they derived and implemented the optimal control for the valves however, no pump control was considered. Ghaddar et al. (2017) consider the global optimization of valve settings for the minimization of pressure and apply semidefinite programming relaxations with a branch-and-bound scheme. However, they consider the solution for only one time step (i.e. for a single demand vector).
Role of specific energy in decomposition of time-invariant least-cost reservoir filling problem
2019, European Journal of Operational ResearchCitation Excerpt :Operational optimization of hydrodynamic systems has been, and continues to be, in the focus of extensive research. The area covers a broad range of applications ranging from drinking water processing and distribution (D’Ambrosio, Lodi, Wiese, & Bragalli, 2015; Ghaddar, Claeys, Mevissen, & Eck, 2017; Ghaddar, Naoum-Sawaya, Kishimoto, Taheri, & Eck, 2015; Naoum-Sawaya, Ghaddar, Arandia, & Eck, 2015) to wastewater treatment (Hou, Li, Xi, & Cen, 2015; Wei, He, & Kusiak, 2013), irrigation (Reca, Garca-Manzano, & Martnez, 2015) and energy production (Kusakana, 2016; Steffen & Weber, 2016). The field addresses the operational aspects of hydrodynamic systems where given the system topology, the task is to derive an operational policy for the active hydrodynamic components (pumps and valves) so that the related costs are minimal subject to operational constraints.
A Mixed-Integer Linear Programming Framework for Optimization of Water Network Operations Problems
2024, Water Resources Research