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

Abstract

In this paper, we investigate the application of penalty and relaxation methods to the problem of optimal placement and operation of control valves in water supply networks, where the minimization of average zone pressure is the objective. The optimization framework considers both the location and settings of control valves as decision variables. Hydraulic conservation laws are enforced as nonlinear constraints and binary variables are used to model the placement of control valves, resulting in a mixed-integer nonlinear program. We review and discuss theoretical and algorithmic properties of two solution approaches. These include penalty and relaxation methods that solve a sequence of nonlinear programs whose stationary points converge to a stationary point of the original mixed-integer program. We implement and evaluate the algorithms using a benchmarking water supply network. In addition, the performance of different update strategies for the penalty and relaxation parameters are investigated under multiple initial conditions. Practical recommendations on the numerical implementation are provided.

Outer approximation methods for the solution of co-design optimisation problems in water distribution networks

“…Moreover, only n l n p variables are involved in nonlinear functions, while the remaining n l (n n + n p ) + 2n p unknowns appear linearly in all the constraints. This is the main difference between the problem formulation for optimal valve placement proposed in the present work and the one considered in Eck and Mevissen (2012); Dai and Li (2014); Pecci et al (2016a). In fact, energy conservation across pipes and valves is modelled by nonlinear constraints (1a) and linear constraints (1c), (1d).…”

“…Using case studies, we show that the new method has much smaller computational complexity compared to other approaches used in literature for the solution of optimal valve placement and operation in water distribution networks (Eck and Mevissen, 2012;Dai and Li, 2014;Pecci et al, 2016a). Furthermore, the application of OA to the new problem formulation for the co-design control optimisation in WDNs has enabled the convergence to a (local) solution in large scale networks.…”

“…In fact, energy conservation across pipes and valves is modelled by nonlinear constraints (1a) and linear constraints (1c), (1d). In comparison, in Eck and Mevissen (2012); Dai and Li (2014); Pecci et al (2016a), this is expressed by two nonlinear inequalities, one of them being a polynomial expression of degree 3 involving both flow and hydraulic head variables. In conclusion, the introduction of the vectors η t together with linear constraints (1c) and (1d) results in a simplification of the problem formulation.…”

“…Mass and energy conservation laws are enforced as optimisation constraints at each node and pipe of the network, respectively. The bidirectional nature of flow across pipes complicates the mathematical formulation of energy conservation laws, resulting in nonconvex constraints Eck and Mevissen (2012); Dai and Li (2014); Pecci et al (2016a). An additional physical constraint from the control valves enforces the pressure differential across the control valves to be in the same direction as the flow through the valve.…”

“…An additional physical constraint from the control valves enforces the pressure differential across the control valves to be in the same direction as the flow through the valve. This constraint has so far been modelled by introducing highly nonlinear nonconvex equations and two links with unidirectional flows to model each physical pipe -see Eck and Mevissen (2012); Dai and Li (2014); Pecci et al (2016a). This problem formulation includes high-order nonconvex constraints and it is difficult to solve.…”