Water loss is a phenomenon frequently observed within water distribution systems (WDSs), that is considerably worsened by an excessive pressure throughout the network. As an alternative option to pipe replacement, the use of pumps working as turbines, throttle control valves (TCVs), or pressure reduction valves (PRVs) can be used to reduce leakage. For a preassigned number of these devices, their positions and settings can be chosen to minimize the water losses in the network or to minimise the costs associated with the leakage. On the other hand, for a preassigned reduction in leakage, the number, the position and the setting of valves could be optimized in order to minimize their installation and maintenance costs. Based on these observations, a procedure for the optimal choice of the number, position and setting of PRVs is devised. The procedure is aimed at reducing the whole cost associated with water loss in urban WDSs, due to the background leakage from joints, and the purchase, installation and maintaining of the PRVs themselves. The effectiveness of the procedure, which is based on the physical modelling of leakage from pipe joints as well as on the use of a genetic algorithm, is proven using a small but realistic example.

INTRODUCTION

Water distribution systems (WDSs) are affected by several issues relating to: quantity (Salgado et al. 1988; Wagner et al. 1988; Germanopoulos & Jowitt 1989; Puust et al. 2010) and quality (Pirozzi et al. 2002; Cozzolino et al. 2005a, 2005b, 2006, 2011). This paper is devoted to the resolution of a quantity problem such as water losses in WDSs. The work describes a novel procedure, which aims at the reduction of water losses through joints by means of valves. Indeed, for pipelines connected by joints, a considerable proportion of water loss occurs because of incorrect assembly of joints or the fatigue and ageing of the material used to ensure a watertight seal (Covelli et al. 2015).

The leakage discharges depend on the local pressure and can be modelled by means of appropriate pressure-discharge equations. This observation suggests that the correct management of undetectable leaks could be achieved by minimising the nodal pressures throughout the WDS (Thornton 2003; Ulanicki et al. 2008; Hunaidi 2010). This purpose can be accomplished by means of pressure reduction valves (PRVs). In particular, PRVs should be placed and set to approximate an ideal condition in which the nodal pressure heads are very close to the minimal values strictly required to satisfy the local water demand.

Obviously, the optimal positioning and setting of the valves are strongly connected tasks. In the literature, many works have been devoted to the optimisation of valve setting with fixed positions, considering stationary water demands (Germanopoulos & Jowitt 1989) or time variable discharges required at the nodes (namely, extended period simulation (EPS), Germanopoulos 1995; Vairavamoorthy & Lumbers 1998; Tabesh & Hoomehr 2007) with or without the possibility of achieving the regulation of valves by means of a specific real time control (Reis et al. 1997; Araujo et al. 2006; Campisano et al. 2010; Ali, 2015; Creaco & Pezzinga 2015a, 2015b). In particular, Creaco & Pezzinga (2015a, 2015b) have proven that the closure of some isolation valves in the network can improve effectiveness of control valve regulation. Sometimes, without the closure of suitably identified isolation valves, the water can bypass the control valve and this affects negatively service pressure regulation, and therefore, this aspect should be accounted for in the computations.

In order to minimise simultaneously the number of valves and the volume of water lost during a daily cycle, Nicolini & Zovatto (2009) used a multiobjective genetic algorithm (GA) (NSGA-II), considering setting values variable in time. However, their approach does not appear suitable to identify the number of valves to be used. As an alternative, it is possible to consider an approach based on the minimisation of the total costs connected with the water leakage and the pressure management. In fact, we observe that by increasing the number of valves there is a reduction of the volume of water dispersed within the WDS and the costs associated with these losses, but there is also an increase in the costs of purchasing, installation and maintenance of valves. As a result, it is possible to identify the number and the setting of valves that will allow a reduction of the whole cost related to both the volume of water dispersed and the valves.

Starting from these observations, a novel approach is proposed in this work aimed at identifying the number, the position and the settings of fixed set-point PRVs for the reduction of the daily and yearly water losses from joints, and correspondingly costs due to the loss of the profits that could be earned from the sale of the water lost through background leakage. This work does not account for the possibilities of using PRVs capable of changing the reduced pressure values in time (e.g. over a day and night), nor for controlling in real time PRVs based on remote pressure readings. Nevertheless, as the majority of situations are those in which the PRVs setting is fixed, the proposed approach is still useful in the technical field.

In particular, the approach is based on the coupling of an original GA which is known for its robustness for handling the optimization of WDSs (Ali 2015), and an hydraulic EPS model for the analysis of WDS behaviours in the presence of valves. The GA is first used to optimise the positioning and settings of given numbers of fixed set-point PRVs, by minimizing the value assumed by a properly defined fitness function (FF) without affecting the hydraulic reliability of the network. Then, coupled with a cost-benefit analysis, the model is also used to determine the optimal number of valves to be used. In particular, depending on the pressures available at the joints of each pipe in the WDS, the adopted FF takes into account both the costs due to the water volume lost from the joints and holes, and the costs for purchasing/installing/managing a given number of fixed set-point PRVs.

Outline of the proposed optimisation procedure

The optimisation procedure proposed in this work is based on the possibility that background water losses from WDSs can be consistently reduced by decreasing the head pressures along the links of the network to values higher but not far from those strictly needed to allow the water distribution node by node (Walski et al. 2006). In particular, this objective can be achieved by deploying inside the WDS a fixed number of fixed set-point PRVs with the appropriate valve settings. Going into more detail, once the number of PRVs has been fixed, their optimal locations and settings are evaluated by means of a GA. Then, for each fixed number of PRVs, once their optimal positions and settings have been evaluated, the optimal number of valves to be used is evaluated by a cost-benefit analysis.

In particular, within the GA, the pressure heads at nodes are evaluated by a hydraulic pressure-driven WDS modeller (Cozzolino et al. 2005a, 2005b) based on the gradient method (Todini & Pilati 1987; Todini 2003; Elhay et al. 2014) able to consider both PRVs and TCVs (Salgado et al. 1988), where the formula by Wagner et al. (1988) is adopted, given by: 
formula
1
where Qj,act = actual discharge supplied at node j; Qj,serv = users' demand at node j; Hj,serv = service head at node j (i.e. the minimum head needed to completely satisfy the users' demand at node j); Hj,min = elevation of the lowest emitter existing at node j; Hj = calculated head at node j.

Then, in order to test the proposed procedure and to carry out the above reported operation, the water demand coefficient relative to the node j and to the mth time step in which the day can be subdivided (Gargano & Pianese 2000; Cozzolino et al. 2005a, 2005b) was considered variable during the day, following a given pattern, while the water levels in the reservoirs were considered to be known and constant in time. To ensure the delivery of the desired amount of water, minimum and daily constant values of the nodal heads capable of ensuring the satisfaction of the local water demands were considered.

From a theoretical point of view, the location of each valve is defined by two parameters, namely, the network link and the valve's position along the link. Of course, not all the positions are feasible, for either economical or technical reasons; hence, a limited set of admissible positions (existing manholes, for example) must be chosen a priori, link by link. Aiming at demonstrating the proposed procedure, in this work the admissible positions of the valves have been fixed everywhere in the middle of the pipes, even though different choices are possible. In fact, if a PRV is placed in the most upstream section of a pipe, the effects of pressure reduction affect the whole link, and then the leakages in the same pipe may be reduced. However, if the pipe where the PRV is introduced is very long and the elevation pattern is very steep, it may need more PRVs in one or more pipes downstream from where the PRV was placed. Further, for simplicity, each pipe in the WDS is considered eligible for PRV positioning.

The FF considered in the optimization procedure

For a given geometry of the network and demand pattern, a GA is able to explore the space of the candidate solutions (each characterised, for a given number of valves, by their positioning and setting), attempting to minimise the properly chosen objective function. This function conceptually consists of two parts: the first is related to the costs to be incurred, and the second consists of penalties which are introduced when one or more constraints are not satisfied.

In the present case, the costs considered in the FF are both the costs directly related to the leakage and the costs of the valves. To evaluate the leakage costs, an EPS is carried out for each candidate solution, characterised by a given demand pattern, and the whole water volume dispersed from the WDS during the considered simulation period is evaluated. For demonstration purposes, in the present case, the simulation period is assumed to be one day long, and the users' demand pattern is representative of a typical daily demand pattern during the year. In turn, the whole yearly amount of water saved by the WDS is evaluated by multiplying the daily water volume lost, given by Covelli et al. (2015), by the number of days in a year (365).

The FF to be minimised (expressed in €) is defined by the sum: 
formula
2
where 
formula
3
is the real cost (evaluated at the end of a period of time NEWT years long) and 
formula
4
is the penalty related to the penalties introduced when one or more pressure constraints are not fully satisfied during the EPS. The symbols in Equation (4) have the following meaning: Hj,serv is the value of the minimum pressure head needed at node j to satisfy completely the water demand of the local users, referred to the pipe axis; is the piezometric head at node j during the mth time step of the EPS [metres above mean sea level (m amsl)]; Nnodes is the number of delivery nodes of the WDS; and p [€/(m year)] is the specific penalty coefficient, set equal to 1020 €/(m year), introduced to penalize the operating conditions under which the pressure constraint is not satisfied.
In Equation (3), the symbols have the following meanings: 
formula
5
are the costs associated with the loss of earnings as a result of the water lost from the pipes; NEWT is the expected working time of the valves before their substitution [years]; Wlost is the expected volume of water lost over one year [m3/year]; is the initial cost for a single cubic metre of water lost [€/m3]; is the annual growth rate of the price of the water volume that could be delivered to users if no longer dispersed. 
formula
6
are the costs related to the purchase, installation and maintenance of the selected set of PRVs, where:
  • NPRV is the number of PRVs considered in the simulation, and ν=1,2, …,NPRV;

  • FCPRV_Maint. are the final costs sustained for the maintenance of both the PRV and manhole, evaluated at the end of the expected PRV working time by using the formula: 
    formula
    7
    where ICPRV_Maint. and are the costs of sustaining the first year of maintenance and the annual growth rate of the maintenance costs, respectively, for both the PRV and manhole;
  • FCPRV_Inst. are the final costs for PRV purchasing, for the construction of the related manhole and for installing the PRV in the manhole, evaluated at the end of the expected PRV working time by using the relationship: 
    formula
    8
    where is the annual growth rate of the money spent for the repayment of a loan received from any third party for purchasing the PRV, constructing the associated manhole, and installing the PRV in the manhole, and ICPRV_Inst. is given by: 
    formula
    9
    where ICPRV_Purch. is the cost initially sustained for both PRV purchasing and installation in the specifically constructed manhole, and ICManhole is the cost initially sustained for the construction of the manhole in which the PRV will be installed.

Obviously, the values of ICPRV_Purch. and ICManhole and, then, of ICPRV_Inst. and FCPRV_Inst. depend on the size and type of valve to be used, and then on the diameter of the pipe where the valve has to be installed, as well as ICPRV_Maint. and then FCPRV_Maint.. As a consequence, the costs depend on the diameter of pipes where the valves have to be installed and the number of valves to be used.

It is to be expected that, with the increase of the number of PRVs, on the one hand the volume of water lost into the soil tends to be reduced and the costs associated with these losses decrease; on the other, the costs described by are increased. As a result, the optimal number of valves to be positioned within the network can be chosen solely on the basis of a cost-benefit analysis.

The GA used to characterise the optimal positions and settings of assigned numbers of PRVs

In this paper, given a specified number of valves, NPRV, a traditional GA was used to optimise the valve positioning and settings. The GA, originally proposed and used by some authors to carry out the optimised design/rehabilitation of urban and rural drainage networks (Cimorelli et al. 2013, 2014, 2016; Palumbo et al. 2014; Cozzolino et al. 2015), is based on a binary Gray encoding for the properties of the single individuals. An iterative procedure was developed, consisting, in cascade, of a selection procedure based on an exponential rank selection, a uniform crossover with probability of crossover pc, a bitwise mutation process with probability of mutation pm and a partial elitism technique (with up to two elements eventually preserved at each step of the procedure). These parameters were determined by a preliminary analysis made for this particular case study. Generally speaking, the GA parameters must be determined through sensitivity analysis for each case study. Indeed, the GA exploration should be consistent with the complexity of the search space which depends on the number of candidate solutions and also on the complexity of the WDS in terms of topology and need for multiple PRV installation.

In particular, in the case study considered in this paper, pc and pm were set equal to 1 and 0.01, respectively. The population size was Ni = 100 individuals, and the maximum number of GA iterations was Nit = 400. Consequently, 40,000 FF evaluations were carried out for each case examined (see Figure 1).
Figure 1

Logic flux of the instructions to carry out for each solution of the GA.

Figure 1

Logic flux of the instructions to carry out for each solution of the GA.

An example of application of the proposed procedure

Brief description of the case study used to illustrate the proposed procedure

The proposed procedure was applied to a small realistic WDS (named Pi.Co.Sa.), designed to serve a population of 31,500 inhabitants. In Figure 2, the layout of the considered WDS is depicted, composed of NL = 32 cast iron mains and Nnodes = 24 nodes.
Figure 2

The layout of the Pi.Co.Sa. network used to test the proposed procedure for optimisation.

Figure 2

The layout of the Pi.Co.Sa. network used to test the proposed procedure for optimisation.

The network is supplied by two tanks: for the sake of simplicity, the water surface elevations in the tanks were considered to be constant and equal to 213.15 m amsl and 195.00 m amsl, respectively.

The ground elevation zgr ranges between 130.93 and 214.81 m amsl (see Table 1), whereas the height of the buildings ranges between 4.0 and 18.0 m. In particular, the ground elevations at the nodes decrease from north-west to south-east. Thus, the values of the ideal pressure heads in the water column hj,min were set close to 24 m for every node, except for nodes 2, 3, and 9, for which the hj,min values were fixed close to 14 m, and nodes 1 and 8, where the two reservoirs are located. Because of the ground elevations and pipe diameters, the pressure heads are, without control valves, higher than that strictly needed to guarantee the delivery of water at each node for the overall south-east area.

Table 1

Ground elevations zgr, daily averaged discharges Qa delivered at the nodes, ideal pressure heads and pipeline centroid depths δj (inclusive of that initially lost along the pipes not using PRVs)

Nodezgr [m amsl]Qa [l/s]hj,min [m]δj [m]Nodezgr [m amsl]Qa [l/s]hj,min [m]δj [m]
214.81 – – 1.66 13 166.23 10.4034 23.60 1.40 
193.94 1.9964 13.34 1.66 14 157.52 2.5731 23.82 1.18 
189.23 2.5731 13.80 1.20 15 155.24 1.9964 23.80 1.20 
178.91 2.5731 23.95 1.05 16 154.09 1.9964 23.80 1.20 
180.21 10.4034 23.45 1.55 17 149.02 2.5731 23.84 1.16 
176.86 10.4034 23.65 1.35 18 147.08 2.5731 23.84 1.16 
168.93 2.5731 23.95 1.05 19 149.31 2.5731 23.84 1.16 
196.04 – – 1.04 20 147.65 2.5731 23.67 1.33 
170.76 1.9964 13.96 1.04 21 140.51 1.9964 23.95 1.05 
10 164.60 2.5731 23.82 1.18 22 141.65 1.9964 23.95 1.05 
11 167.38 10.4034 23.67 1.33 23 138.81 1.9964 23.95 1.05 
12 168.25 10.4034 23.82 1.18 24 130.93 1.9964 23.95 1.05 
Nodezgr [m amsl]Qa [l/s]hj,min [m]δj [m]Nodezgr [m amsl]Qa [l/s]hj,min [m]δj [m]
214.81 – – 1.66 13 166.23 10.4034 23.60 1.40 
193.94 1.9964 13.34 1.66 14 157.52 2.5731 23.82 1.18 
189.23 2.5731 13.80 1.20 15 155.24 1.9964 23.80 1.20 
178.91 2.5731 23.95 1.05 16 154.09 1.9964 23.80 1.20 
180.21 10.4034 23.45 1.55 17 149.02 2.5731 23.84 1.16 
176.86 10.4034 23.65 1.35 18 147.08 2.5731 23.84 1.16 
168.93 2.5731 23.95 1.05 19 149.31 2.5731 23.84 1.16 
196.04 – – 1.04 20 147.65 2.5731 23.67 1.33 
170.76 1.9964 13.96 1.04 21 140.51 1.9964 23.95 1.05 
10 164.60 2.5731 23.82 1.18 22 141.65 1.9964 23.95 1.05 
11 167.38 10.4034 23.67 1.33 23 138.81 1.9964 23.95 1.05 
12 168.25 10.4034 23.82 1.18 24 130.93 1.9964 23.95 1.05 

The WDS delivers a daily average users demand equal to 91.15 l/s, corresponding to 70% of water distributed each day, because the leakage discharge is supposed to be about 30% of the total daily average discharge delivered equal to 130.21 l/s.

In Table 1 the ground elevations zgr, the daily averaged discharges Qa delivered at the nodes (inclusive of that initially lost along the pipes not using PRVs), the ideal pressure heads and pipeline centroid depths δj are given for each node of the network.

In Table 2, the geometric characteristics of the pipes (i.e. the length L and the internal diameter Di) are summarised.

Table 2

The geometric characteristics of the pipes of the WDS used to test the procedure proposed

LinkInitial nodeFinal nodeL [m]Dint. [mm]LinkInitial nodeFinal nodeL [m]Dint. [mm]
500 706.4 17 13 14 700 151.4 
700 203.2 18 14 15 350 151.4 
450 203.2 19 15 16 350 203.2 
350 504.0 20 10 17 700 80.0 
350 99.8 21 11 18 700 99.8 
350 80.0 22 12 19 700 80.0 
450 99.8 23 13 20 700 254.4 
10 700 80.0 24 13 21 990 80.0 
11 381 254.4 25 14 21 700 80.0 
10 13 350 402.8 26 17 22 450 99.8 
11 13 350 305.6 27 19 23 450 99.8 
12 500 80.0 28 21 24 450 99.8 
13 10 500 80.0 29 20 21 700 80.0 
14 10 11 200 151.4 30 17 18 200 125.6 
15 11 12 200 151.4 31 18 19 200 99.8 
16 12 13 300 151.4 32 19 20 200 125.6 
LinkInitial nodeFinal nodeL [m]Dint. [mm]LinkInitial nodeFinal nodeL [m]Dint. [mm]
500 706.4 17 13 14 700 151.4 
700 203.2 18 14 15 350 151.4 
450 203.2 19 15 16 350 203.2 
350 504.0 20 10 17 700 80.0 
350 99.8 21 11 18 700 99.8 
350 80.0 22 12 19 700 80.0 
450 99.8 23 13 20 700 254.4 
10 700 80.0 24 13 21 990 80.0 
11 381 254.4 25 14 21 700 80.0 
10 13 350 402.8 26 17 22 450 99.8 
11 13 350 305.6 27 19 23 450 99.8 
12 500 80.0 28 21 24 450 99.8 
13 10 500 80.0 29 20 21 700 80.0 
14 10 11 200 151.4 30 17 18 200 125.6 
15 11 12 200 151.4 31 18 19 200 99.8 
16 12 13 300 151.4 32 19 20 200 125.6 

To evaluate the head losses along the links, the Hazen-Williams' roughness formula was used. In particular, the value of the dimensionless coefficient CH−W of that formula was set wherever used as equal to 120.

To test the GA, the optimisation of the positions and settings of fixed numbers of PRVs was accomplished by considering both the water demand at the delivering nodes and the discharges lost along the links throughout the day as variable. The total flow rates supplied to the network do not change because of the installation and operation of the PRVs. However, the presence of the PRVs change the patterns of both the discharges flowing within the pipes of the WDS and nodal pressure heads and, then, the volume of water lost at the joints. In Table 3, the daily demand coefficients (DCm)j are shown, where m and j are the generic time interval during the day and the generic node of the WDS, respectively. For the sake of simplicity, the demand coefficients (DCm)j were considered constant from one node to another and equal to the demand coefficients (DCm)net assumed for the whole network.

Table 3

The daily water demand coefficients (DCm)net

m123456789101112
(DCm)net 0.45 0.42 0.40 0.35 0.38 0.67 1.26 2.52 2.10 1.49 1.33 1.34 
13 14 15 16 17 18 19 20 21 22 23 24 
(DCm)net 1.52 1.68 1.35 0.81 0.79 0.77 1.14 0.98 0.76 0.58 0.46 0.45 
m123456789101112
(DCm)net 0.45 0.42 0.40 0.35 0.38 0.67 1.26 2.52 2.10 1.49 1.33 1.34 
13 14 15 16 17 18 19 20 21 22 23 24 
(DCm)net 1.52 1.68 1.35 0.81 0.79 0.77 1.14 0.98 0.76 0.58 0.46 0.45 

The leaking discharges at the pipe joints were calculated by means of the approach proposed by Covelli et al. (2015), where the calibrated values ξ = 0.56 were considered. Note that these values were chosen by a trial and error procedure, such that the total leakage volume amounted to approximately 30% of the daily volume in the network (estimated background water losses). Furthermore, the value of the head pressure at each joint of each link of the WDS was calculated very simply by considering the hypothesis that the ground elevations and pipeline depths were linearly variable between the extreme nodes of the pipes and the pipe joints where modelled as emitters.

The costs ICPRV_Purch. were given by surveyed providers. They are reported in the second column of Table 4 as a function of the nominal pipe size (NPS).

Table 4

The costs related to the provision, installation and maintenance of PRVs

NPSICPRV_Purch.ICManholeICPRV_Inst.FCPRV_Inst.ICPRV_Maint.FCPRV_Maint.FTCPRVITRPRV
[mm][€][€][€][€][€/year][€][€][€]
60 867.00 3,118.00 3,985.00 5,059.89 86.70 2,448.69 7,508.58 265.85 
80 890.00 3,160.00 4,050.00 5,142.43 89.00 2,513.64 7,656.07 271.08 
100 969.00 3,199.60 4,168.60 5,293.02 96.90 2,736.77 8,029.79 284.31 
125 1,415.00 3,251.20 4,666.20 5,924.84 141.50 3,996.41 9,921.25 351.28 
150 1,817.00 3,302.80 5,119.80 6,500.79 181.70 5,131.79 11,632.58 411.87 
200 2,828.00 3,406.40 6,234.40 7,916.03 282.80 7,987.18 15,903.21 563.08 
250 4,074.00 3,508.80 7,582.80 9,628.14 407.40 11,500.63 21,128.77 748.10 
300 5,960.00 3,611.20 9,571.20 12,152.88 596.00 16,832.95 28,985.83 1,026.29 
350 7,424.00 3,705.20 11,129.20 14,131.13 742.40 20,967.75 35,098.88 1,242.74 
400 10,436.00 3,805.60 14,241.60 18,083.05 1,043.60 29,474.60 47,557.66 1,683.86 
450 12,411.00 3,905.60 16,316.60 20,717.75 1,241.10 35,052.63 55,770.39 1,974.65 
500 14,613.00 4,008.00 18,621.00 23,643.73 1,461.30 41,271.79 64,915.52 2,298.45 
600 18,888.00 4,210.40 23,098.40 29,328.84 1,888.80 53,345.76 82,674.60 2,927.24 
700 25,343.00 4,412.80 29,755.80 37,781.97 2,534.30 71,576.74 109,358.71 3,872.04 
NPSICPRV_Purch.ICManholeICPRV_Inst.FCPRV_Inst.ICPRV_Maint.FCPRV_Maint.FTCPRVITRPRV
[mm][€][€][€][€][€/year][€][€][€]
60 867.00 3,118.00 3,985.00 5,059.89 86.70 2,448.69 7,508.58 265.85 
80 890.00 3,160.00 4,050.00 5,142.43 89.00 2,513.64 7,656.07 271.08 
100 969.00 3,199.60 4,168.60 5,293.02 96.90 2,736.77 8,029.79 284.31 
125 1,415.00 3,251.20 4,666.20 5,924.84 141.50 3,996.41 9,921.25 351.28 
150 1,817.00 3,302.80 5,119.80 6,500.79 181.70 5,131.79 11,632.58 411.87 
200 2,828.00 3,406.40 6,234.40 7,916.03 282.80 7,987.18 15,903.21 563.08 
250 4,074.00 3,508.80 7,582.80 9,628.14 407.40 11,500.63 21,128.77 748.10 
300 5,960.00 3,611.20 9,571.20 12,152.88 596.00 16,832.95 28,985.83 1,026.29 
350 7,424.00 3,705.20 11,129.20 14,131.13 742.40 20,967.75 35,098.88 1,242.74 
400 10,436.00 3,805.60 14,241.60 18,083.05 1,043.60 29,474.60 47,557.66 1,683.86 
450 12,411.00 3,905.60 16,316.60 20,717.75 1,241.10 35,052.63 55,770.39 1,974.65 
500 14,613.00 4,008.00 18,621.00 23,643.73 1,461.30 41,271.79 64,915.52 2,298.45 
600 18,888.00 4,210.40 23,098.40 29,328.84 1,888.80 53,345.76 82,674.60 2,927.24 
700 25,343.00 4,412.80 29,755.80 37,781.97 2,534.30 71,576.74 109,358.71 3,872.04 

For the sake of simplicity, the initial costs ICManhole were evaluated by using the relationship ICManhole = (3,000 + 2 NPS), where ICManhole is expressed in Euros and NPS is expressed in millimetres. They are reported in the third column of Table 4. In turn, the costs ICPRV_Maint., depending on the NPS value of the pipe, were evaluated by using the relationship ICPRV_Maint. = 0.1 ICPRV_Purch.. They are reported in the sixth column of Table 4.

Table 4 also shows: (i) the (initial) whole costs ICPRV_Inst. (fourth column); (ii) the (final) whole costs FCPRV_Inst. (fifth column); (iii) the (final) whole costs FCPRV_Maint. (seventh column); (iv) the (final) total costs FTCPRV (eighth column), given by: 
formula
10
and (v) the evaluations of the first year whole cost to sustain the repayment of a loan eventually received from any third party for purchasing the PRV, constructing the associated manhole, installing the PRV in the manhole, and maintaining the PRV (ITRPRV) (ninth column). This value was evaluated by means of the formula: 
formula
11
where represents the interest rate applied for repayment of the loan.

All of the calculations were carried out considering NEWT = 25 years and interest rates rPRV_Purch·_&_Inst. = = = 0.01.

By inspection of the values reported in the final columns of Table 4, it is possible to observe that, in the case examined here, the total costs , though depending on the NPS value, are small in any case. Similarly, the costs resulting from positioning and operating a set of NPRV valves are very small as well. Moreover, for illustrative purposes, in this work the value of was assumed to be 0.818 €/m3, which is representative of the water costs accounted for by the end users present in the Campania area (southern Italy).

DISCUSSION OF THE RESULTS

The optimal positioning and setting of fixed numbers of valves were found for an increasing number of PRVs, first with reference to a case in which all of the costs associated with the valves could be considered null, and then for cases in which it was necessary to consider these costs. The results obtained are shown in Table 5 and Figure 3.
Table 5

The optimal positioning and setting of the assigned numbers of PRVs, obtained without considering the costs of the valves

NPRVVlost [m3/year]FF [€]Positioning [number of link]Setting [m amsl]
1,231,878.60 28,460,017.38 – – 
1,049,174.54 20,073,786.09 23 
1,010,678.77 18,321,959.24 5; 1 36.9; 6.1 
959,448.65 15,974,668.94 18; 17; 1 25.3; 35.7; 6 
938,367.31 15,045,879.41 10; 18; 1; 27 35.3; 25.3; 6; 24.1 
923,858.93 14,391,410.34 17; 1; 27; 10; 9 35.8; 6.1; 24.1; 35.3; 30.8 
NPRVVlost [m3/year]FF [€]Positioning [number of link]Setting [m amsl]
1,231,878.60 28,460,017.38 – – 
1,049,174.54 20,073,786.09 23 
1,010,678.77 18,321,959.24 5; 1 36.9; 6.1 
959,448.65 15,974,668.94 18; 17; 1 25.3; 35.7; 6 
938,367.31 15,045,879.41 10; 18; 1; 27 35.3; 25.3; 6; 24.1 
923,858.93 14,391,410.34 17; 1; 27; 10; 9 35.8; 6.1; 24.1; 35.3; 30.8 
Figure 3

The trends of the FF and the yearly volume of water lost from the WDS with the number of deployed PRVs, without consideration of the valve costs.

Figure 3

The trends of the FF and the yearly volume of water lost from the WDS with the number of deployed PRVs, without consideration of the valve costs.

The trends for the FF and the minimum yearly volume of water lost from the WDS seem to be very consistent with what could be hypothesised. When the number of valves NPRV is increased, not only does the yearly volume lost decrease but, due to the progressive redundancy of the valves introduced in the network (which is not very large), the rate of reduction decreases as well. Furthermore, because in this case the total costs are proportional to the yearly water volume lost, the same results can also be observed for the FF.

The trends observed suggest that the GA adopted in this work should also be able to minimize the FF if the costs of the valves are taken into account.

Then, the proposed procedure was tested with reference to the same WDS but also considering, in the evaluation of the FF, the values CPRV for the assigned number of PRVs (see Table 6). Because the goal to achieve was minimally changed by the introduction of these costs the results of the optimisation procedure do not change (see Tables 5 and 6). This result is an additional demonstration that the procedure developed in this work seems to choose optimally the positions and settings of the assigned number of valves in a WDS, independently of whether the valve costs are taken into account or not.

Table 6

The optimal positioning and setting of the assigned number of PRVs by also taking into account their costs

NPRVVlost [m3/year]Leakage costs [€]PRV costs [€]FF [€]Positioning [number of link]Setting [m a.m.s.l.]
1,231,878.60 28,460,017.38 28,460,017.38 – – 
1,049,174.54 24,239,016.54 55,770.39063 24,294,786.93 23 
1,010,678.77 23,349,651.01 82,674.60156 23,432,325.61 5; 1 36.9; 6.1 
959,448.65 22,166,084.61 102,517.1016 22,268,601.71 18; 17; 1 25.3; 35.7; 6 
938,367.31 21,679,043.61 147,809.5781 21,826,853.18 10; 18; 1; 27 35.3; 25.3; 6; 24.1 
923,858.93 21,343,857.47 163,712.7813 21,507,570.25 17; 1; 27; 10; 9 35.8; 6.1; 24.1; 35.3; 30.8 
NPRVVlost [m3/year]Leakage costs [€]PRV costs [€]FF [€]Positioning [number of link]Setting [m a.m.s.l.]
1,231,878.60 28,460,017.38 28,460,017.38 – – 
1,049,174.54 24,239,016.54 55,770.39063 24,294,786.93 23 
1,010,678.77 23,349,651.01 82,674.60156 23,432,325.61 5; 1 36.9; 6.1 
959,448.65 22,166,084.61 102,517.1016 22,268,601.71 18; 17; 1 25.3; 35.7; 6 
938,367.31 21,679,043.61 147,809.5781 21,826,853.18 10; 18; 1; 27 35.3; 25.3; 6; 24.1 
923,858.93 21,343,857.47 163,712.7813 21,507,570.25 17; 1; 27; 10; 9 35.8; 6.1; 24.1; 35.3; 30.8 

Despite the PRVs cost contribution to the FF, because of the economic savings resulting from the decreased water losses, for increasing numbers of PRVs, the FF values decrease.

For increasing NPRV values, the trend at which the FF varies shows a tendency to reach the maximum of the net benefit (defined as the difference between the economic value of the volume of water that is not lost and the cost of the valves necessary to achieve the water savings).

This tendency is confirmed by the trends shown in Figure 4, in which a cost-benefit analysis is carried out to compare the direct costs needed for WDS rehabilitation by the introduction of the given set of PRVs with the direct benefits of leakage reduction for a 25-year period.
Figure 4

The cost–benefit curve.

Figure 4

The cost–benefit curve.

This analysis, although it does not consider the environmental aspects, such as the greenhouse gas emissions (Venkatesh 2012), but only the economic saving, seems suitable for evaluating the optimal number of valves to use. In addition, the approach appears suitable for solving the problem of optimally choosing the number, positions and settings of PRVs to introduce in a WDS for reductions in water losses.

For example, Figure 4 shows that the rate at which the net benefits grow with the investment costs for PRVs purchasing and operation tend to decrease when the investment costs increase. Indeed, when the number of PRVs is equal to 4–5, the growth rate of the net retractable benefits from the investment made for the valves tends toward zero.

Even though the approach proposed in this work is generally applicable to any WDS, the results obtained for the case study examined are not general, and additional work must take place, for different cases, to evaluate the number of PRVs strictly needed to achieve the maximum net benefit.

CONCLUSIONS

In many cases of practical interest, the reduction of leakage in WDSs can be obtained by reducing the pressure through the system: this, in turn, leads to the reduction of the costs associated with the loss of water. Special devices, such as the PRVs, can be used to regulate the pressure, but their use introduces additional costs due to the implementation and operation of the valves. In this paper, a procedure for the optimal positioning and setting of increasing sets of valves was considered, based on the minimisation of the total costs associated with the water losses and valve implementation. The procedure was demonstrated using a realistic small network, showing its applicability to real-world cases.

It is well known that the reduction of water leakage leads to energy savings in pumped systems, and the pressure management actions performed lead to cost savings, which arise from the reduced rate of pipe breakage. The approach proposed in the present work, even though these sources of financial savings are not explicitly taken into account, is completely general. Thus, to minimise simultaneously the water loss and the energy and pipe rehabilitation costs, the economic analysis framework considered in this paper can be easily enriched by using a multi-objective function to consider these additional beneficial effects.

ACKNOWLEDGEMENTS

The present work was developed with financial contributions from the Campania Region, L.R. n.5/2002 – year 2008 – within the project ‘Methods for the evaluation of security of pressurized water supply and distribution systems with regard to the contamination, even intentional, of water to be distributed, and the optimized dimensioning of the same', prot. 2014.0293987 dated 29.04.2014 – CUP: E66D08000060002.

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