Improved WDN Design by Coupling Optimal Pipe Sizing and Isolation Valve Placement ^†

Abstract

This paper presents a novel methodology for the coupled optimization of pipe sizing and isolation valve placement in water distribution networks (WDNs). By employing a bi-objective genetic algorithm, the methodology searches for the best solutions in the trade-off between minimizing the average demand shortfalls caused by segment isolations and minimizing the total installation costs of pipes and valves. The optimization also incorporates a constraint that ensures the telescopic distribution property of pipe diameters, guaranteeing that the pipe diameters narrow down from source(s) to external areas. The outcomes are compared with the traditional design approach, which entails, in separate steps, the least-cost optimization for pipe sizing and the placement of isolation valves at all (N-rule) or all but one (N-1 rule) pipes connected to the generic junction.

1. Introduction

The reliability of water distribution networks (WDNs), specifically against component failures, hinges mainly on two factors: the layout of isolation valves and the hydraulic capacity of the pipes [1]. Isolation valves play a critical role in limiting the extent of the network segment isolated from the source(s) in the event of a failure [2]. Concurrently, the hydraulic capacity of the pipes, determined by their diameter, significantly affects the performance of the parts of the WDN that remain connected to the source(s) during such isolation events. While the optimal placement of isolation valves for a preassigned pipe design [3] and the optimal pipe sizing with a preassigned valve distribution (typically one valve close to either pipe end) [4] have been investigated, the interdependence of the two designs is mainly overlooked in the scientific literature.

This research gap has inspired the present paper, which proposes a novel method for the coupled design of pipes and isolation valves by adopting a bi-objective optimization to minimize the average demand shortfall caused by segment isolation and the total cost of pipes and valves. The novel method is applied to a real WDN. The results are compared to a traditional decoupled design approach, which first applies a minimum-cost WDN pipe design and then valve placement based on N and N-1 rules.

2. Materials and Methods

The real WDN of a town in Northern Italy [3] was adopted as a case study, to which the novel coupled method and the traditional decoupled approach were applied, as described in the following subsections.

2.1. Coupled Optimization of Pipes and Isolation Valves

The multi-objective genetic algorithm from MATLAB was used to find the optimal pipe diameters and isolation valve placement simultaneously in the trade-off between minimizing the total cost (C_tot) and the average demand shortfalls (D_avg). The two objective functions are expressed as follows:

$$f_1=C_{tot}=C_p+C_v$$

(1)

$${f_2=D}_{avg}={\displaystyle \frac{{\sum }_{i=1}^{N_{seg}}{D}_i}{N_{seg}}}$$

(2)

where C_p and C_v are the total cost of pipes and valves, respectively; N_seg is the number of segments with at least one pipe; and D_i is the demand shortfall associated with the i-th segment.

The number of individuals’ genes in the genetic algorithm was equal to three times the number of pipes. The first third of the genes corresponds to pipe diameters, and the rest concerns the valves close to each pipe’s start and end nodes. The genes regarding pipe diameters could take integer values from 1 to 9, representing the nine possible pipe diameters according to

Table 1. List of diameters and their corresponding cost for unit length of pipes and valves.

ID	1	2	3	4	5	6	7	8	9
Diameter (mm)	50	100	150	200	300	400	500	600	800
Unit Cost of Pipes (EUR/m)	17.45	27.9	44.35	66.8	129.7	216.6	327.5	462.4	804.2
Unit Cost of Valves (EUR)	153	685	1644	3061	7349	13,680	22,152	32,843	61,138

. The decision variables for isolation valves could take values of 0 and 1, indicating the presence or absence of each valve, respectively.

Furthermore, solutions that did not provide the minimum required pressure (i.e., 15 m for this case study) in all nodes were penalized in both objective functions. In addition, a novel constraint was implemented within the genetic algorithm to guarantee the telescopic pipe diameter distribution, in which at each node under peak demand conditions, the hydraulically downstream pipes have equal or smaller diameters than the hydraulically upstream pipes.

2.2. Decoupled Optimization of Pipes and Isolation Valves

In the decoupled approach, the optimal pipe diameters were found by employing a single objective genetic algorithm to minimize the cost of pipes. The same penalization for the objective functions and the constraint of telescopic diameter distribution, described in the previous subsection, were also implemented in this optimization.

After obtaining the pipe diameter design with minimum cost, the N and N-1 rule valve placements [5] were applied to the WDN layout. While for the N-rule, the isolation valve layout is unique, for the N-1 rule, there are numerous different potential layouts. Therefore, another single objective optimization was used to find the N-1 rule isolation valve placement that results in the lowest average demand shortfall. The objective function to minimize was identical to that expressed in Equation (2). Notably, the N-1 rule for valve placement was considered a linear constraint in the optimization, expressed as follows:

$$A_v^T\times X^T=\left[\begin{array}{c}{K}_1-1\\ \begin{array}{c}{K}_2-1\\ ⋮\end{array}\\ K_N-1\end{array}\right]$$

(3)

where X is the individual generated by the genetic algorithm, A_v is the submatrix of incidence matrix A associated with all potential valve pipes, and K_i (namely, node degree) denotes the number of pipes connected to the i-th node of the WDN.

In this genetic algorithm, the number of individuals’ genes was equal to two times the number of pipes, which represent the valves in the proximity of each pipe’s start and end nodes.

3. Results

The Pareto front of solutions obtained from the coupled method and solutions from the decoupled method are shown in Figure 1a. The front shows a sharp decrease at the beginning, and then the marginal benefit of increasing the cost starts to diminish. As can be observed, the solutions from the coupled method dominate the ones from the decoupled method with the N and optimal N-1 rules for valve placement. Moreover, 1000 random solutions with different N-1 rule valve placements were evaluated to show the range of their performance, all of which were outperformed by the solutions from the coupled method and the optimal N-1 rule. Furthermore, as Figure 1b shows, in optimizations, as the total cost increases, the costs of pipe and valves increase at a similar rate. Yet, the share of valve cost from the total cost rises from about 3% to roughly 30% from the solutions with the lowest to highest total cost. Moreover, the increase in the total cost is initially more dedicated to isolation valves, and therefore, the number of valves increases rapidly. Then, after this sharp increase in the number of valves, the increase in the total cost is mainly due to increasing pipe diameters.

Table 2. Comparison of the solutions selected from coupled optimization and the decoupled method with N and optimal N-1 rule isolation valve layout.

Solution	Ctot (EUR)	Davg (L/s)	Cp (EUR)	Cv (EUR)	Nv
N-rule	612,837	4.78	455,245	157,593	186
Optimal N-1 rule	547,951	4.93	455,245	92,706	119
Coupled method	545,748	2.04	468,817	77,131	123

compares a solution derived from the coupled method and the two solutions obtained from the decoupled approaches. The selected solution from the coupled method features a similar total cost as the optimal N-1 rule solution and a much lower cost than the N-rule solution. Yet, the coupled method solution achieves a significantly lower average demand shortfall, less than half of that obtained with the decoupled solutions. Additionally, this solution exhibits higher costs for pipe diameters and, instead, significantly lower costs associated with the isolation valves, compared to the decoupled solutions, even with a slightly higher number of valves (N_v = 123) than the optimal N-1 rule decoupled solution (N_v = 119).

4. Discussion

The novel method proposed in this study for the coupled design of pipe diameters and the isolation valve layout showed better performance than the traditional decoupled approach. Specifically, in the coupled method, the enhancement in WDN reliability is possible through a simultaneous cost-effective combination of increasing pipe diameters and improving valve placements. Thus, the coupled method allows for exploring solutions with different ratios of pipe and valve costs that result in a higher WDN reliability with the same total cost as the WDN designs obtained from the decoupled approach. Furthermore, between the solutions obtained from the decoupled approach, the optimal N-1 rule showed comparable performance to the N-rule solution in terms of the average demand shortfall while featuring a significantly lower cost. However, finding the optimal N-1 rule solution requires an additional optimization step. Further investigation and application in a different case study, a more detailed description of the methodology, and deeper discussions of the results can be found in the journal paper of [6].