Due to the economic crisis and water scarcity happening in recent years, many researchers have focused on the water distribution network optimization problem. On this specified subject, all recent research has applied stochastic meta-heuristic algorithms to solve these sets of problems. In this study, the application of a novel deterministic physically inspired heuristic algorithm for minimizing the cost of pipe-sizing in the water distribution system (WDS) is investigated. In fact, the algorithm used in this research is the modified central force optimization algorithm to solve the water distribution network problem called CFOnet. The approach is applied to optimize the design of the Kadu and Khorramshahr networks. For this purpose, CFOnet method is programmed in MATLAB and interfaced with the hydraulic simulation model, EPANET. The obtained solutions in this study are compared with those stochastic methods in the WDS optimization literature. This comparison shows that CFOnet is more efficient in obtaining lower cost than other optimization methods for solving the two mentioned WDSs, while it enjoys the merits of a deterministic optimization method.

INTRODUCTION

A significant portion of the water supply system cost is relevant to the water distribution network (Swamee & Sharma 2008). Thus, finding the optimal cost for implementation of these networks is economically helpful in water supply system design. In the past few decades, researchers have focused on utilizing meta-heuristic algorithms, such as genetic algorithm (GA), simulated annealing (SA) and particle swarm optimization (PSO), to evaluate water network designs, and have shown that these algorithms are successful in finding the most economic cost of water networks. But, due to their stochastic nature, there is no guarantee that the global optimum is found. Also, the obtained solution in each run is not always the same; therefore, several runs are necessary to ensure that the solutions are identified as good quality. Moreover, the required computation time to reach near-optimal configuration is a great limiting problem in using these algorithms in real water distribution systems (WDSs). These limitations have led researchers to combine these algorithms with the deterministic math-based approaches. Cisty (2010) and Haghighi et al. (2011) proposed GA-LP (combination of GA and linear programming) and GA-ILP (combination of GA and integer-linear programming) models, respectively, to solve the WDS problem. They have shown that their proposed models are better from the computational efficiency point of view compared with completely stochastic algorithms.

Recently, a new approach to the optimization has been introduced, called central force optimization (CFO). This method is a new meta-heuristic algorithm for multidimensional optimization which performs based on the rules of gravity (Formato 2007). Since this physical law, in which masses move towards the gravitational field, is deterministic, CFO's equations are inherently deterministic. Therefore, in CFO, multiple runs for the sake of finding its performance are not required, and every CFO run with the same setup will result the same (Formato 2007). This algorithm has been successfully applied to a variety of problems, among them: antenna optimization (Formato 2010a); drinking water distribution networks (Haghighi & Ramos 2012); and improving the global search ability of Standard CFO (Ding et al. 2012). This research has shown high efficiency of this optimization method, and also in some cases, the researchers have enhanced CFO performance through creating some modifications in the method.

Since the problem of WDS has never been solved by any completely deterministic meta-heuristic method, utilization of the CFO has been investigated in this research. Since this method is still under development, in the present research, a modified CFO algorithm is introduced to optimize the design cost of water distribution networks (CFOnet). For this purpose, CFOnet is linked to the hydraulic simulation model, EPANET 2.0 (Rossman 2000), in MATLAB. The obtained optimum design costs for two WDSs are compared with the solutions published using other approaches.

METHODS

The basic hydraulic equations involved in EPANET are the mass and energy conservation (Rossman 2000). For each junction, the conservation of mass can be expressed as: 
formula
1
where and are the inflows and outflows of the node; and is the external demand.
For each closed loop, the conservation of energy can be expressed as: 
formula
2

where is head loss in the pipe i and is the total number of loops in the system.

The head loss in the pipe which is located between the and junctions is equal to the difference between nodal head at both ends: 
formula
3
Also, a form of Hazen–Williams (HW) equation is used herein to estimate the head loss, which was proposed by Savic & Walters (1997), described as: 
formula
4
where, is the pipe flow (m3/s), is the Hazen–Williams roughness, Di is the diameter of the pipe (m), and Li is the length of the pipe (m).

is a numerical value which depends on the units used in the equation and acts as a dimensionless conversion factor. Herein the ω in EPANET 2.0 is equal to 10.667 (Eusuff & Lansey 2003).

Optimization model formulation

In this research, the mathematical model for the optimal design cost of WDS is formulated as (Savic & Walters 1997): 
formula
5
where is the total cost of the network; is the cost of the pipe i with the diameter and the length; and is the total number of pipes in the network.
Also the design and hydraulic constraints are given as: 
formula
6
 
formula
7
 
formula
8
where and are the minimum and maximum existing commercial diameters, respectively; is the flow velocity in pipe i; and are allowable minimum and maximum velocities in the pipes, respectively; is the pressure in node; and are allowable minimum and maximum pressure points in nodes, respectively; and is the total number of nodes in the system.
To consider the above constraints, an additive penalty function approach is used, and the penalty cost for an infeasible solution is calculated based on the distance away from the feasible solution area. So, the overall objective function is formulated as: 
formula
9
where and and v(Pj) are penalty functions for violation of velocity and pressure constraints, respectively, and is the violation factor. Herein Ct is computed with calibrated and penalty functions are considered as below: 
formula
10
 
formula
11
 
formula
12
 
formula
13

CFO algorithm

CFO makes a set of probes to fly through the space over a set of time steps. A decision space is defined by, where is the decision variable and is the number of the problem's dimensions. and are the minimum and maximum decision variables. Each probe moves under the influence of an accelerated force created by the gravitational attraction of masses in decision space. The total acceleration experienced by probe produced by each of the other probes on it at step () is given as: 
formula
14
where is probe's fitness at time step. Index j is the iteration number, which is valued in the range from one to (the maximum number of iterations). Index p is the probe number, which is valued in the range from one to (the total numbers of probes). Each of the other probes also has associated with it the fitness . Likewise, , and G are the CFO constants, which usually are equal to 2 (Formato 2007). Also, is the Unit Step function, which determines whether probe k applied the attraction force on probe p or not. U in maximizing the objective function is defined by the below equation, according to which probe k could attract probe p if it has a greater mass than that of probe : 
formula
15
And is the distance between the position of probes p and, which is given by: 
formula
16
where and are the positions of probe k and p in the dimension , respectively. The acceleration causes probe p to move from position at step to position at step j according to the below equation: 
formula
17
Figure 1 demonstrates the gravitational metaphor for a three-dimensional space with four probes in a maximizing problem in which the probes' fitness is shown by the black filled circles; the size of the circles is proportional to the mass of probes (their fitness value). Movement of each probe is restricted to the bounded feasible region. However, when some of the probes fly out, a retrieving mechanism such as the below scheme is used to act on these errant probes and reposition them in the feasible region: 
formula
18
 
formula
19
Figure 1

Typical 3-D CFO decision space in a maximizing problem (Formato 2007).

Figure 1

Typical 3-D CFO decision space in a maximizing problem (Formato 2007).

where is the repositioning factor and More detailed information on CFO can be seen in (Formato 2010a, 2010b).

CFOnet algorithm and hydraulic model implementation

Herein CFO code is developed in MATLAB and linked with EPANET Toolkit. The CFO parameters are assumed to be as the above-mentioned default amounts, because the change in parameters has no significant influence in solving the WDS problem. To define initial probes, a user-defined probe distribution is used in this study, in which probes are defined by a deterministic method. In this regard, the commercial diameters matrix is replicated to the number of pipes quantity in each considered WDS, and 42 probes are generated via displacement diameters in this permutation. Also, the initial acceleration vectors are set to zero. The CFO operation flowchart to solve the WDS problem is shown in Figure 2.
Figure 2

Steps in CFO method for WDS design.

Figure 2

Steps in CFO method for WDS design.

To evaluate the CFO method for solving the water network problem, two WDSs are selected. Kadu and Khorramshahr WDSs, which include a range of different decision space topologies and make a suitable evaluation of CFO method performance for solving the WDS problem, are assessed. The CFO method is implemented to optimize the design cost of these WDSs.

Given the poor performance of CFO to optimize these networks, it seems reasonable to apply a few modifications to this method. The modified CFO algorithm is called CFOnet, and the differences from the original CFO are explained herein.

The performance of the CFOnet algorithm is evaluated in the two above-mentioned networks. In CFOnet, the constant parameters such as , , G, and the initial acceleration vectors are set the same as in CFO. Also, to define the initial probes, the above-mentioned user-defined probe distribution method is applied. After the first iteration of CFOnet, the new diameters, via a subscript coded in Visual Basic program, were replaced in the input file of EPANET to simulate the considered network. The constraints of the conservation of mass and energy are satisfied by EPANET (Liong & Atiquazzaman 2004). To check hydraulic constraints, another subscript code developed in Visual Basic program is used. This subscript reads the output file of EPANET and specifies hydraulic characteristics of the network and computes penalty values. So the objective function is evaluated for each probe and these fitness values are compared with each other and the least value is determined. This least cost replaces the best ever obtained cost. Thereafter, new acceleration is computed by CFOnet for each probe. Since the main purpose is to minimize the objective function in this research, in Equation (14) is rectified as the below formula: 
formula
20
Since the computed acceleration values are big in the WDS problem and cause all probes to be thrown out of the decision space, a normalization operator is applied to decelerate the probes. The maximum and minimum bounds for this normalization operator ( and ) depend on the existing commercial diameters in the specified WDS. Zero acceleration values would not be normalized. At the beginning of running the algorithm, values of and are chosen as large amounts to increase the convergence rate to the region of the global solution. These values are set for a specified number of iterations and thereafter are decreased to the lower amounts. The time in which and are reduced () depends on the behavior of the algorithm in the specified WDS. In fact is the time in which the algorithm traps in a local optimum solution. So should be calibrated for each WDS. To determine initial , firstly, commercial diameters are arranged in ascending order and then twice the amount of the maximum difference between the two consecutive diameters is considered as . Also, is given by: 
formula
21
In addition, after iteration, and are reduced to a quarter of their initial value. Therefore, according to the normalized acceleration values, probe positions have changed by Equation (17) and then new diameters are obtained. Due to the discreteness of the WDS problem, it is decided to eliminate the parameter in Equations (18) and (19), and the errant probes are repositioned by the following simple definitions: 
formula
22
 
formula
23
The input file for EPANET is updated using new diameters. This process is continued until the number of iterations reaches , which in this research is set to 10,000. As for prevention of probe trapping phenomena in the local optimum solutions, a mutation operator is defined in CFOnet. In this case, during running the algorithm, when the best cost remains constant after a specified number of iterations, the deterministic mutation operator acts on the probes. For this purpose, firstly, the existing commercial diameters matrix replicates to the number of pipes quantity in the considered network, which makes a larger matrix called the sized diameters matrix. The mutated probes are moved to the new positions, which are produced by applying Swap, Insertion and Reversion operators on the sized diameters matrix. The mutated rate is considered as 15% herein. The CFOnet operation flowchart is shown in Figure 3. Also, the main differences between CFOnet and conventional CFO are demonstrated in Table 1 for more clarification.
Table 1

The different procedures in CFOnet and CFO methods

Operator   
methodRepairing the errant probesNormalizing the accelerationsMutation of the probes
 CFO Returning the errant probes considering (Equations (18) and (19)) Unavailable Unavailable 
 CFOnet Returning the errant probes without using (Equations (22) and (23)) Normalizing the accelerations regarding the existing commercial diameters Replacing the probes with new probes which are defined by user deterministically 
Operator   
methodRepairing the errant probesNormalizing the accelerationsMutation of the probes
 CFO Returning the errant probes considering (Equations (18) and (19)) Unavailable Unavailable 
 CFOnet Returning the errant probes without using (Equations (22) and (23)) Normalizing the accelerations regarding the existing commercial diameters Replacing the probes with new probes which are defined by user deterministically 
Figure 3

Steps in CFOnet method for WDS design.

Figure 3

Steps in CFOnet method for WDS design.

Kadu network

Kadu network is a looped WDS initially presented by Kadu et al. (2008), which comprises 26 nodes, 34 pipes and two reservoirs. The heads of reservoirs 1 and 2 are 100 and 95 m, respectively. The additional information on this network can be found in Kadu et al. (2008). The HW coefficient is considered to be 130 for all pipes. Table 2 represents 14 commercially available diameters, which makes 1434 possibilities to solve this network. The layout of the network is shown in Figure 4.
Table 2

Pipe cost data in Kadu network (rupees (Rs))

Diameter (inches)Cost (Rs/m)Diameter (inches)Cost (Rs/m)Diameter (inches)Cost (Rs/m)
150 1,115 400 4,255 750 11,874 
200 1,600 450 5,172 800 13,261 
250 2,154 500 6,092 900 16,151 
300 2,780 600 8,189 1,000 19,395 
350 3,475 700 10,670   
Diameter (inches)Cost (Rs/m)Diameter (inches)Cost (Rs/m)Diameter (inches)Cost (Rs/m)
150 1,115 400 4,255 750 11,874 
200 1,600 450 5,172 800 13,261 
250 2,154 500 6,092 900 16,151 
300 2,780 600 8,189 1,000 19,395 
350 3,475 700 10,670   
Figure 4

Schematic of Kadu network.

Figure 4

Schematic of Kadu network.

Khorramshahr network

Khorramshahr network was first presented by Samani & Zanganeh (2010). This network consists of 58 pipes, 39 nodes, a single fixed head reservoir and two booster pumps arranged in 17 loops. In this research, pipe-sizing of this network is optimized and the reservoir head and pumps specification are set as optimum conditions, which have been obtained by Samani & Zanganeh (2010). Nine discrete pipe sizes for the network, represented in Table 3, cause the algorithm to have the solution space consisting of 958 possibilities. The amount is considered to be 120 for all the pipes. Also, the minimum hydraulic head requirement is set to 30 m for all of the nodes. Additional information on this network can be found in Samani & Zanganeh (2010). With regard to the results of that research, the minimum and maximum allowable flow velocities in pipes are considered to be 0.1 m/s and 2.04 m/s, respectively. The layout of the network is shown in Figure 5.
Table 3

Commercial diameters (mm) in Khorramshahr network

DiameterCost (units/m)DiameterCost (units/m)DiameterCost (units/m)
25 205 124 450 222 590 
75 411 142 470 248 610 
111 430 160 490 279 700 
DiameterCost (units/m)DiameterCost (units/m)DiameterCost (units/m)
25 205 124 450 222 590 
75 411 142 470 248 610 
111 430 160 490 279 700 
Figure 5

Layout of Khorramshahr network.

Figure 5

Layout of Khorramshahr network.

RESULTS AND DISCUSSION

The convergence behavior of the CFO and CFOnet methods to optimize pipe-sizing in Kadu network is shown in Figure 6 (regarding the huge amount of the initial 11 iterations cost values, it is decided not to show them in this figure). The best solution reduced from 358 × 106 (Rs) to 291 × 106 (Rs) after running the first 11 iterations of CFOnet. In this network the calibrated for CFOnet method is 1,000 iterations. The comparison of the CFOnet results with those obtained using other optimization techniques in the literature with the same is shown in Table 4.
Table 4

Comparison of the solutions of Kadu network

PipeDiameter (mm)
Kadu et al. (2008) Haghighi et al. (2011) Mohammadi-Aghdam et al. (2015) CFOnet
1,000 1,000 900 900 
900 900 900 900 
400 400 500 350 
350 350 250 300 
150 150 150 150 
250 250 200 300 
800 800 900 800 
150 150 150 150 
400 400 600 600 
10 500 500 700 600 
11 1,000 1,000 900 900 
12 700 700 700 700 
13 800 800 500 500 
14 400 400 450 500 
15 150 150 150 150 
16 500 500 450 500 
17 350 350 300 350 
18 350 350 450 400 
19 150 150 500 500 
20 200 150 150 150 
21 700 700 600 600 
22 150 150 150 150 
23 400 450 150 150 
24 400 400 400 450 
25 700 700 500 500 
26 250 250 150 200 
27 250 250 350 350 
28 200 200 350 250 
29 300 300 150 250 
30 300 300 300 250 
31 200 200 200 150 
32 150 150 150 150 
33 250 200 200 150 
34 150 150 150 150 
Method used GA GA-ILP PSO CFOnet 
Cost (Rs) 131,678,935 131,312,815 130,666,043 126,535,915 
Eval. 36,000 4,440 22,000 259,476 
PipeDiameter (mm)
Kadu et al. (2008) Haghighi et al. (2011) Mohammadi-Aghdam et al. (2015) CFOnet
1,000 1,000 900 900 
900 900 900 900 
400 400 500 350 
350 350 250 300 
150 150 150 150 
250 250 200 300 
800 800 900 800 
150 150 150 150 
400 400 600 600 
10 500 500 700 600 
11 1,000 1,000 900 900 
12 700 700 700 700 
13 800 800 500 500 
14 400 400 450 500 
15 150 150 150 150 
16 500 500 450 500 
17 350 350 300 350 
18 350 350 450 400 
19 150 150 500 500 
20 200 150 150 150 
21 700 700 600 600 
22 150 150 150 150 
23 400 450 150 150 
24 400 400 400 450 
25 700 700 500 500 
26 250 250 150 200 
27 250 250 350 350 
28 200 200 350 250 
29 300 300 150 250 
30 300 300 300 250 
31 200 200 200 150 
32 150 150 150 150 
33 250 200 200 150 
34 150 150 150 150 
Method used GA GA-ILP PSO CFOnet 
Cost (Rs) 131,678,935 131,312,815 130,666,043 126,535,915 
Eval. 36,000 4,440 22,000 259,476 
Figure 6

Convergence behavior of CFO and CFOnet for Kadu network.

Figure 6

Convergence behavior of CFO and CFOnet for Kadu network.

The CFO method finds the optimum cost of 282,474,720 (Rs) after 25,452 function evaluations. This solution has a fairly big difference from the best results in the literature, so it is decided to eliminate it from Table 4. With regard to the CFO result, it shows a relatively poor performance to solve this problem. This is due to CFO's high convergence speed, so that all probes quickly move to the central gravitational field and lose their accelerations. Therefore, CFO unavoidably is unable to decrease the cost anymore.

As shown in Figure 6, the convergence behavior of deterministic proposed CFOnet is improved as compared with the CFO, so that the best cost decreases to 126,535,915 (Rs) after 259,476 function evaluations. This obtained solution is better than the widely obtained answers as shown in Table 4. By simulating Kadu network with the obtained optimum diameters using EPANET, it can be observed that all the pressure values meet the requirements in all nodes of the network shown in Table 5. It should be noted that since the computers utilized in the mentioned researches were not identical in terms of processing speed, the run times are not considered herein.

Table 5

Nodal pressures (m) for Kadu network

NodePressureNodePressureNodePressureNodePressure
Reservoir 1 89.17 15 88.29 22 80.46 
Reservoir 2 91.13 16 82.06 23 82.71 
98.29 10 88.19 17 90.24 24 83.24 
95.06 11 86.36 18 86.33 25 80.05 
87.83 12 85.12 19 85.36 26 80.00 
86.18 13 84.65 20 82.95   
87.78 14 94.15 21 83.18   
NodePressureNodePressureNodePressureNodePressure
Reservoir 1 89.17 15 88.29 22 80.46 
Reservoir 2 91.13 16 82.06 23 82.71 
98.29 10 88.19 17 90.24 24 83.24 
95.06 11 86.36 18 86.33 25 80.05 
87.83 12 85.12 19 85.36 26 80.00 
86.18 13 84.65 20 82.95   
87.78 14 94.15 21 83.18   

The result of implementation of the CFOnet method to solve Khorramshahr network is shown in Figure 7 (regarding the huge amount of the cost values of the initial 75 iterations, it is decided not to show them in this figure). By running CFOnet, within the first 75 iterations, the best solution reduced to 10.8 × 106 (thousand Iran rials) from 109 × 106 (thousand Iran rials). In this network for CFOnet is calibrated as 300 iterations. The lowest cost found by CFO is 22,974,703 (thousand Iran rials) for this network. This solution is obtained after 25,410 function evaluations, and CFO is unable to improve it anymore. So with regard to the CFO result, it is decided to eliminate the CFO curve in Figure 7. By applying the CFOnet model, after 41,454 evaluations, the best obtained cost became 1,474,807 (thousand Iran rials). Samani & Zanganeh (2010) obtained the best cost as 1,535,371 (thousand Iran rials) after 14 iterations for this network. It can be concluded that CFOnet produces a significant improvement in the best cost as compared with the one obtained by LP technique as shown in Table 6. Also, the obtained optimal diameters for this network are listed in Table 7. All the hydraulic characteristics are satisfactory in the optimum design of Khorramshahr network, as reported in Tables 8 and 9.
Table 6

Solutions for pipe cost of Khorramshahr network

AuthorsSamani & Zanganeh (2010) Present work
Method used LP CFOnet 
Cost (thousand Iran rials) 1,535,371 1,474,807 
AuthorsSamani & Zanganeh (2010) Present work
Method used LP CFOnet 
Cost (thousand Iran rials) 1,535,371 1,474,807 
Table 7

Optimal diameters (mm) of Khorramshahr network

PipeSamani & Zanganeh (2010) Present workPipeSamani & Zanganeh (2010) CFOnet
222 222 30 111 25 
160 222 31 25 111 
142 124 32 25 25 
124 160 33 25 75 
111 111 34 75 75 
25 25 35 142 111 
142 111 36 75 160 
75 160 37 75 160 
75 124 38 75 25 
10 75 25 39 25 111 
11 111 75 40 75 25 
12 75 25 41 75 75 
13 75 25 42 111 75 
14 25 111 43 111 160 
15 25 160 44 25 25 
16 75 25 45 25 25 
17 25 25 46 75 25 
18 142 75 47 75 25 
19 25 75 48 111 111 
20 75 160 49 25 25 
21 75 25 50 25 160 
22 75 75 51 75 111 
23 111 160 52 75 111 
24 75 25 53 111 124 
25 75 25 54 25 25 
26 25 75 55 75 75 
27 25 25 56 75 75 
28 124 25 57 25 25 
29 75 160 58 75 75 
PipeSamani & Zanganeh (2010) Present workPipeSamani & Zanganeh (2010) CFOnet
222 222 30 111 25 
160 222 31 25 111 
142 124 32 25 25 
124 160 33 25 75 
111 111 34 75 75 
25 25 35 142 111 
142 111 36 75 160 
75 160 37 75 160 
75 124 38 75 25 
10 75 25 39 25 111 
11 111 75 40 75 25 
12 75 25 41 75 75 
13 75 25 42 111 75 
14 25 111 43 111 160 
15 25 160 44 25 25 
16 75 25 45 25 25 
17 25 25 46 75 25 
18 142 75 47 75 25 
19 25 75 48 111 111 
20 75 160 49 25 25 
21 75 25 50 25 160 
22 75 75 51 75 111 
23 111 160 52 75 111 
24 75 25 53 111 124 
25 75 25 54 25 25 
26 25 75 55 75 75 
27 25 25 56 75 75 
28 124 25 57 25 25 
29 75 160 58 75 75 
Table 8

Nodal pressures (m) in Khorramshahr network

NodePressureNodePressureNodePressure
Reservoir 14 33.58 27 39.70 
1j 50.34 15 49.23 28 35.10 
50.27 16 33.30 29 32.45 
47.54 17 40.66 30 31.23 
46.55 18 42.29 31 39.19 
46.45 19 42.83 32 30.52 
45.93 20 32.19 33 35.78 
36.62 21 41.04 34 37.43 
50.14 22 45.00 35 36.73 
33.32 23 42.15 36 34.78 
10 46.46 24 41.61 37 36.32 
11 46.35 25 41.27 38 35.18 
12 46.26 26 40.42 39 33.79 
13 42.90 26j 39.54   
NodePressureNodePressureNodePressure
Reservoir 14 33.58 27 39.70 
1j 50.34 15 49.23 28 35.10 
50.27 16 33.30 29 32.45 
47.54 17 40.66 30 31.23 
46.55 18 42.29 31 39.19 
46.45 19 42.83 32 30.52 
45.93 20 32.19 33 35.78 
36.62 21 41.04 34 37.43 
50.14 22 45.00 35 36.73 
33.32 23 42.15 36 34.78 
10 46.46 24 41.61 37 36.32 
11 46.35 25 41.27 38 35.18 
12 46.26 26 40.42 39 33.79 
13 42.90 26j 39.54   
Table 9

Velocities (m/s) in Khorramshahr network

PipeVelocityPipeVelocityPipeVelocity
1.65 21 1.32 41 0.97 
1.50 22 2.01 42 0.54 
1.67 23 0.36 43 1.00 
0.58 24 1.01 44 1.04 
1.09 25 0.72 45 1.27 
2.04 26 0.87 46 0.84 
0.40 27 2.04 47 0.55 
1.79 28 1.57 48 1.83 
0.62 29 0.57 49 1.65 
10 0.21 30 0.56 50 0.1 
11 1.94 31 1.13 51 0.31 
12 1.16 32 0.73 52 0.51 
13 1.90 33 0.86 53 0.51 
14 0.47 34 0.68 54 0.37 
15 0.52 35 1.00 55 0.31 
16 1.16 36 0.81 56 0.86 
17 2.04 37 0.30 57 0.42 
18 0.51 38 0.18 58 0.41 
19 0.11 39 1.29   
20 1.42 40 0.69   
PipeVelocityPipeVelocityPipeVelocity
1.65 21 1.32 41 0.97 
1.50 22 2.01 42 0.54 
1.67 23 0.36 43 1.00 
0.58 24 1.01 44 1.04 
1.09 25 0.72 45 1.27 
2.04 26 0.87 46 0.84 
0.40 27 2.04 47 0.55 
1.79 28 1.57 48 1.83 
0.62 29 0.57 49 1.65 
10 0.21 30 0.56 50 0.1 
11 1.94 31 1.13 51 0.31 
12 1.16 32 0.73 52 0.51 
13 1.90 33 0.86 53 0.51 
14 0.47 34 0.68 54 0.37 
15 0.52 35 1.00 55 0.31 
16 1.16 36 0.81 56 0.86 
17 2.04 37 0.30 57 0.42 
18 0.51 38 0.18 58 0.41 
19 0.11 39 1.29   
20 1.42 40 0.69   
Figure 7

Convergence behavior of CFOnet for Khorramshahr network.

Figure 7

Convergence behavior of CFOnet for Khorramshahr network.

Generally, the obtained results using the proposed CFOnet method as compared with the conventional CFO have a better quality in the two investigated networks. CFOnet shows the solution about 55.2% better than CFO in Kadu network. Moreover, this improvement in Khorramshahr network was about 93.6%. In both networks CFOnet shows a considerable performance rather than the CFO. The main cause of this improvement is due to applying the deterministic mutation operator in CFOnet, which replaces the trapped probes in the local optimum point with new previously defined probes.

It can be observed from this research that CFOnet obtains the better quality solutions in two specified WDSs as compared with the best solutions found before. With regard to Table 4, the optimum cost obtained by CFOnet for Kadu network compared with those obtained by GA, GA-ILP and PSO methods shows an approximate improvement of 3.9%, 3.64% and 3.16%, respectively. Also, the optimum cost obtained by CFOnet for Khorramshahr network as compared with the LP method result (Table 6) shows an approximate improvement of 3.94%.

It should be noted that the number of objective function evaluations in the previous research cannot be compared directly with this study, due to the fact that their calculations are stochastic based and the results are obtained after several runs. By contrast, every CFOnet run with the same parameters returns entirely the same solutions (Formato 2007) and this method is performed only once. In addition, in CFOnet there is no need to calibrate several parameters (only is set), so it saves time significantly.

CONCLUSIONS

The present research assessed the optimization of WDSs. Application of some of the stochastic algorithms to solve this problem was shown in the literature previously. In this research, as a remedy for the deficiencies of utilizing stochastic optimization algorithms, a proposed deterministic approach (CFOnet) was used to find the optimal design cost of WDSs. The major differences between CFOnet and the conventional CFO method were in injecting new probes within the optimization process via a deterministic mutation operator and also in normalizing the acceleration values of probes in a defined range. Since probes in the proposed CFOnet method shared their own global and individual information, this method could be accounted as an intelligent swarm algorithm. Also, the mutation operator enhances increasing the global exploration ability and convergence velocity of this approach. Evaluating CFOnet to optimize the design of Kadu and Khorramshahr networks showed that this model gives the desired results in this kind of problem. For these two networks, the least lowest cost found by CFOnet was better than the best previous investigation. So the results of this comparison indicated that the CFOnet method was more efficient in solving the WDS optimization problem than other methods, while enjoying the merits of a deterministic optimization method. This study is a first effort to optimize WDSs utilizing a deterministic evolutionary algorithm; hence, only the pipe-sizing optimization of two looped networks with a single demand loading was presented. Deterministic optimization of the complex looped networks with multi-objective functions is highly recommended for the subject of future research. Also, the application of the CFOnet to select the optimum rehabilitation alternatives of WDSs during the operational period can be considered for future studies. The CFOnet code could be available upon request of researchers for their better understanding.

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