Decision support system for the optimal design of district metered areas

adjD

set of the supplied DMAs in adjacency to the isolated DMA

boundD

set of boundary nodes of the isolated DMA

C_B

construction cost

C_B,max

budget limit for construction cost

C_FM

unit cost of flow-meter

C_L

cost due to water leakages

C_SV

unit cost of shut-off valve

C_WL

unit cost of water losses

Ĥ_i

minimum required head at i-th node

H_i

total head at i-th node in the partitioned network

H_i,nd

total head at i-th node in the starting configuration

I_RD

Resilience Deviation Index

I_RD,max

maximum value of I_RD over time

MaxCon

maximum number of customer connections per DMA

MinCon

minimum number of customer connections per DMA

NDMA

number of DMAs

NDMA_max

maximum number of DMAs

N_FM

number of flow-meters

ngen

number of generations

NHV

normalized hyper volume

N_leak

number of emitter nodes in the network

N_n

number of demand nodes

N_SV

number of shut-off valves

N_tot

number of network nodes

P_i,t

pressure at i-th node at time step t

P_min

minimum pressure target for proper customer service

popsize

size of the population

Q_d

water demand of d-th DMA

Q_d,max

maximum water demand for DMA

Q_d,min

minimum water demand for DMA

Q_i

total water demand at i-th node

Q_ij

demand served by link i-j

Q_tot

total water demand of customers

r

annual discounting rate

sp_δ

length of the shortest-path a network water source and the entry node of δ

SupD

set of DMAs supplied by at least one network water source

T

number of time steps in the hydraulic simulation

TotCon

total number of customer connections

tw_ij

topological distance associated with link i-j

unSupD

set of DMAs not supplied by any network water source

w_B

weight for the construction cost

w_L

weight for the cost due to water leakages

α

emitter coefficient

β

pressure exponent

δ

generic DMA in unSupD

Δt

duration of time step

Δz_ij

absolute difference between ground elevations of node i and j

η_m

distribution index for mutation

η_x

distribution index for crossover

θ

random value between 0 and 1

λ_d

acceptance probability

φ

blending factor

Among the best practices for the improvement of efficiency in water distribution network (WDN) management, the establishment of ‘supply clusters’ or district metered areas (DMAs) (Cheong 1993) is particularly suggested. This technique, also known as ‘network segmentation’ or ‘sectorization’, involves partitioning the whole system into hydraulically discrete areas through the closure of network pipes (e.g., by using shut-off valves).

The main benefit related to sectorization is an increased control on the network operation that is achieved through the continuous monitoring of the water flows entering each DMA. This also results in an improved capacity to assess the current level and the evolution of the water leakage in every area of the network. Therefore, the partitioning of the WDN into DMAs is crucial for identifying the most vulnerable areas in order to schedule leak detection activities (Morrison et al. 2007).

Furthermore, in recent years the integration of sectorization with other pressure management techniques has gained increasing attention in the scientific and professional literature. In particular, combination with the use of pressure reducing valves (PRVs) has been investigated, as it can boost the efficiency of the pressure regulation provided by these devices (Creaco & Pezzinga 2014). For example, the delimitation of network areas characterized by similar hydraulic requirements is extremely useful for setting the PRVs while ensuring the minimum pressure levels. However, as highlighted in some studies, sectorization itself can be considered as a pressure management technique, because the closure of pipes can produce significant pressure drops, and therefore, mitigation of the leakages (De Paola et al. 2014). Other interesting benefits are offered in terms of protection of the water quality against incidents or malicious attacks (Di Nardo et al. 2014a).

On the other hand, there are some drawbacks that must be carefully taken into account in the establishment of DMAs, especially regarding the hydraulic performance of the partitioned network. The opening of network loops due to the closure of pipes may result in a significant reduction of system reliability. This situation can lead to the violation of the delivery of the minimum level of service when unexpected increases or changes in the spatial distribution of the water demand occur. Particular attention should also be paid to the water quality, water age in this case, which could be negatively affected by the reduced velocities and the more constrained paths that are allowed to the network flows.

Together with the intrinsic complexity of the WDNs, these issues make sectorization very difficult to implement in real cases. For this reason, many researchers have investigated this problem with different goals and optimization techniques. A number of examples have focused mainly on graph-based approaches in which the network topology was primarily taken into account (Tzatchkov et al. 2006). In other examples, one or more additional criteria were considered, such as the minimization of the ‘edge cut’ (i.e., the number of pipes to close) and the WDN hydraulics (Sempewo et al. 2009; Di Nardo & Di Natale 2011; Alvisi & Franchini 2013).

Gomes et al. (2012a, b) introduced the use of simulated annealing (SA) and user-defined criteria for setting the boundaries of DMAs. Furthermore, they introduced the minimum requirements in terms of water velocity and pressure through pipe reinforcements and replacements. However, the developed methodology was not completely automated, and a significant input by the operator was required in the optimization.

A complex methodology based on multi-agent clustering was developed by Herrera et al. (2012), which considered both the network features (node elevations and water demands) and economic issues (i.e., the edge cut). Nevertheless, it was only applicable to cases in which the number of DMAs was lower than the number of source nodes.

Diao et al. (2013) focused on the detection of ‘community structures’, namely groups of nodes characterized by a high density of links between them. The developed method was based on the evaluation of the ‘modularity index’, the use of which has also been investigated in other works (Scibetta et al. 2013; Giustolisi & Ridolfi 2014). In particular, Giustolisi & Ridolfi (2014) contemplated tailoring of the modularity index in order to overcome resolution problems in the application to water networks.

However, in these approaches, the system topology was mainly considered while less emphasis was given to the hydraulic performance of the partitioned network and to the other advantages offered by the sectorization. More technical principles were adopted in the studies by Di Nardo et al. (2014a) and Ferrari et al. (2014). Finally, the protection of water quality as design criteria was particularly addressed in Grayman et al. (2009) and Di Nardo et al. (2014b). In fact, the establishment of DMAs also allows the water utility to isolate the areas eventually affected by contamination due to incidents or malicious attacks. In addition, it facilitates the collection of samples for the necessary physical-chemical analysis and it reduces the complexity of the mixing of different water sources (Herrera Fernandez 2011).

Despite the number of contributions, there are still some authors who identify shortcomings in approaches developed so far, as highlighted by Hajebi et al. (2014), who criticized them for one or more of the following: (i) the applicability to small networks only; (ii) the neglected consideration of the ground elevations among the design criteria; (iii) the lack of objective functions related to the hydraulics of the network. The same authors introduced a multi-objective approach based on the use of NSGA-II (Deb et al. 2002). Nevertheless, the large number of considered objectives significantly complicated the selection of the best solution.

To this aim, De Paola et al. (2014) introduced a comprehensive approach for the automatic and multi-objective sectorization of WDNs. An innovative perspective was proposed, namely, the use of sectorization as a pressure management technique for mitigating water leakages.

In this paper, a more detailed description of the optimization algorithm is provided and discussed through an application to a simple case from the literature. A calibration procedure that was followed for setting the parameters of the meta-heuristic optimizer is presented as well. Finally, an in-depth analysis of the results is carried out through the technical evaluation of the obtained solutions.

A summary of the proposed approach is presented here, as well as some novel developments and details of the optimization algorithm previously introduced by De Paola et al. (2014). Three categories of design criteria are considered in this methodology, which takes into account the topology, the operational cost and the hydraulics of the WDN. However, the mathematical formulation of the optimization problem is focused on the two main aspects involved in the design of the network segmentation. In fact, a total cost function (TCF) is compared to the change in the hydraulic behaviour of the WDN in terms of reduction of system resilience.

According to the provided definition, the minimization of the TCF is likely to produce lower water pressures. Therefore, this objective is clearly in conflict with the preservation of the hydraulic reliability of the WDN. The latter is usually evaluated in terms of ‘network resilience’ (Todini 2000; Prasad et al. 2003; Prasad & Park 2004; Farmani et al. 2005).

The NSGA-II is initialized with a randomly generated population of solutions, each one consisting of the following decision variables:

• the number of DMAs (NDMA) in the current solution (integer, ranged between 1 and NDMA_max);

• the node indices for a number of DMA centroids equal to NDMA (integer, ranged between 1 and the total number of network nodes, N_tot);

• a blending factor (φ) for the evaluation of a ‘topological distance’ between nodes (real, ranged between 0 and 1).

Starting from these variables, the corresponding designs of the DMAs are created according to the procedure described later in the text. Before, it is important to note some advantages provided by this feature of the proposed approach. First of all, the number of DMAs is often considered as an input parameter, while here it is obtained as a result of the optimization. Therefore, less knowledge about the problem is required and the analysis is not biased by preliminary assumptions. In addition, the characterization of every solution requires a limited number of decision variables, that is also independent of the size of the network. In fact, the number of decision variables is always equal to NDMA + 2. This is a finite and relatively low quantity, and it applies both to small and large networks. Consequently, the methodology can be successfully applied to problems of any size without significant differences in the computational effort, except for the share required by the hydraulic simulations.

Once all the nodes are assigned to the DMAs, a possible sectorization of the network is designed. This is introduced in the hydraulic model by closing the pipes connecting different DMAs. However, in some cases, this operation may lead to the isolation of the areas of the network which may be disconnected from any of the water sources. For example, this occurs when the number of network water sources is lower than the desired number of DMAs. In such cases, the supply must be provided by re-opening the most appropriate connection between different DMAs (i.e., cascade supply).

First, two groups of DMAs are identified, namely those that are supplied (SupD) and those that are not (unSupD). Then, the following steps are applied to all the δ in unSupD. (i) The presence of adjacent supplied DMAs is checked. If this is verified, these DMAs are included in the set adjD, while their boundary nodes having connections with those of δ are assigned to the set boundD. Otherwise, δ is skipped and queued. (ii) The FWA is used again for evaluating the shortest-paths from the water sources in the DMAs of adjD to the nodes in boundD. The weights on the graph edges are assumed equal to zero for pumps and valves, while for the initially open pipes the hydraulic resistance is adopted. The presence of check-valves is also taken into account, and the corresponding weight is set to infinity (like that of closed pipes) in the direction opposite to that allowed by the valve. (iii) The connection belonging to the shortest-path with minimum length is opened. (iv) The entry node of δ (the boundary node of δ belonging to the shortest-path) is labeled as ‘source’ node. (v) The length of the shortest-path (sp_δ) is stored. (vi) δ is moved from unSupD to SupD.

The described sequence is repeated until unSupD is emptied. Furthermore, when calculating the shortest-paths from the source nodes (i.e., the entry nodes) of the indirectly supplied DMAs, the length of the shortest-paths must be increased by the quantities sp_δ.

This process makes it possible to run the hydraulic simulation on the sectorized network, which is required to evaluate the objective functions and the constraints. The NSGA-II operates the selection of the best solutions in order to improve the objectives iteration by iteration until the desired number of generations is reached. In the end, an optimal set is obtained (Pareto front, PF) that is an approximation of the true Pareto front (true-PF) for the problem. Together with the evaluation of technical indicators about the solutions (e.g., number and diameters of closed pipes, reduction of water leakages, etc.), the PF provides the wanted decision support.

The water demands assigned to the nodes and the levels of the reservoirs vary according to a daily time pattern. Pressure-dependent leakages are introduced as well through the assignment of leakage coefficients to the pipes. Nevertheless, in order to run the hydraulic simulation with EPANETpdd, an equivalence is established for modelling the water leakages through emitter nodes. The same emitter exponent equal to 1.18 is assumed, while different coefficients are assigned depending on the length and the number of the pipes connected to the nodes. As a result, a 20% leakage is estimated under uncontrolled conditions.

Starting from the total demand and reasonable values of the per-capita consumption (200 litres/inhabitant/day) and number of inhabitants per customer connection, an estimate of TotCon is provided (approximately 1,900 connections). Other relevant quantities adopted in this analysis and the unit costs are reported in Tables 1 and 2. In particular, equal emphasis is given to the required investment and the cost of the water leakages in the TCF (w_B = w_L = 0.5). The unit costs were retrieved through a market survey among Italian manufacturers, and they were estimated on the basis of the closest commercial values. However, it must be highlighted that the reported values are only intended to provide a reasonable combination of parameters for the evaluation of the objectives, and that they do not involve any loss of generality.

In the present work, the parameters of the NSGA-II are not assigned a priori, but they are estimated through a calibration analysis. To this aim, 90 different test cases (see Table 3) are considered, which consisted of different combinations of the parameters at varying levels. For each of them, several iterations were carried out in order to improve the optimization. In particular, ‘short’ tests (30, 50 or 100 generations) were repeated five times, while longer tests were iterated only three times. Then, the total number of tests was 432, with a total of 3,429,000 evaluations of the objective functions and of the constraints.

The calibration is carried out with the aim of finding the combination of parameters that provides the best approximation of the true-PF. The procedure described in Wang et al. (2014) is adopted for this purpose. This methodology is based on the ranking of the algorithms according to the normalized hyper volume (NHV) (Deb 2011), that is a robust measure of the performance of the multi-objective optimization algorithm. The results show that the best combinations of parameters (NHV = 0.9641) is that consisting of a population size of 50 and 100 generations, with the distribution indices for crossover and mutation being, respectively, 1 and 20. The performance of this algorithm is significantly better than that of the second best (NHV = 0.7799), in which the largest values of popsize and ngen are assumed (respectively, 100 and 1,000). This outcome is apparently in contrast to the general assumption according to which the performance of the optimization algorithms improve with the number of evaluations. However, in this case, the detected anomaly can be ascribed to the reduced size of the considered network. In fact, in this small example, it is difficult to maintain diversity in the provided solutions, and this increases the probability of converging to local minima. However, this aspect is properly taken into account by running the optimization model several times.

The obtained values of the TCF range between 72,611 and 73,838, while I_RD,max goes from 0 (no sectorization) to 0.107. This means that the reduction of the system resilience induced by the partitioning is 10.7% in the worst case, in which a minimum water pressure of 30.1 m is determined at the most critical node.

Most solutions consist of just two DMAs (Figure 4(a)), and in many cases the required investment (Figure 4(b)) is somewhat lower than the assumed budget limit. The technical feasibility of the sectorization designs is also emphasized in Figure 4(d), which shows that the maximum number of pipes to close is seven. Moreover, these pipes are predominantly those characterized by the smallest diameters, as specified in best practices.

Figure 4(c) displays the effect of the partitioning of the network on the reduction of the water leakages. Potential reduction levels are estimated to be between 0.30 and 1.81%, and they are found to be highly correlated to the values of the TCF. However, the obtained correlation might depend on several features (e.g., the weights in the TCF, the unit costs, the network topology), whose effects should be evaluated case by case.

An innovative model for the optimal sectorization of water distribution networks is discussed in this work. The presented methodology follows a multi-objective approach, and it takes into account several design criteria related to the topology, the economics and the hydraulics of the network. In particular, the possible use of sectorization as a pressure management technique for the mitigation of water leakages is addressed.

However, in order to improve the understanding of the optimization process, only two objectives having a clear meaning are considered in the optimization problem. A set of constraints related to the minimum service requirements, financial issues and best practices is introduced as well.

The combination of an evolutionary algorithm with graph analysis tools allows a simple implementation of the model via software. Furthermore, the particular selection of the decision variables, as well as the intermediate procedures included in the algorithm, are intended to provide the applicability to a broad range of cases, regardless of the size of the analysed networks.

The intervention of the analyst is only required in the assignment of the input parameters, most of which are related to essential features of the network operation. Furthermore, the application of the model provides decision aiding in the optimal design of DMAs, by providing feasible solutions representing the best trade-offs between the two adopted objectives: the total operational cost against the reduction of the hydraulic reliability of the network.

The effectiveness of the methodology is demonstrated on an example from the literature. The developed application is focused on both the calibration of the optimization algorithm and the technical description of the outcomes. The obtained results demonstrate the potential of the proposed model in providing feasible solutions to the problems, which are also characterized by the independence on the number of water sources in the network, reduced impact on the network resilience and low cost compared to the assumed budget. The possibility of achieving a reduction in the water leakages through the simple closure of network pipes is emphasized as well.