# Comparison of Algorithms for the Optimal Location of Control Valves for Leakage Reduction in WDNs

^{1}

^{2}

^{*}

## Abstract

**:**

_{val}of SA, the search for the optimal combination of N

_{val}valves is carried out, while containing the optimal combination of N

_{val}− 1 valves found at the previous step. Therefore, only one new valve location is searched for at each step of SA, among all the remaining available locations. The latter algorithm consists of a multi-objective genetic algorithm (GA), in which valve locations are encoded inside individual genes. For the sake of consistency, the same embedded algorithm, based on iterated linear programming (LP), was used inside SA and GA, to search for the optimal valve settings at various time slots in the day. The results of applications to two WDNs show that SA and GA yield identical results for small values of N

_{val}. When this number grows, the limitations of SA, related to its reduced exploration of the research space, emerge. In fact, for higher values of N

_{val}, SA tends to produce less beneficial valve locations in terms of leakage abatement. However, the smaller computation time of SA may make this algorithm preferable in the case of large WDNs, for which the application of GA would be overly burdensome.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Fitness Evaluation of the Generic Location of Control Valves

_{L}. The former assessment comes straight away after the control valves have been installed in selected pipes of the WDN model. In the applications of this work, the total number of control valves is considered as a surrogate for the cost. The latter assessment requires determination of the daily pattern of optimal valve settings in the context of WDN extended period simulation (EPS).

_{p}pipes and n

_{n}nodes = n demanding nodes + n

_{0}sources. WDN operation can be represented through a succession of ${N}_{\Delta t}$ time slots, all featuring the same duration ∆t. At each time slot, in which source heads and nodal demands are assigned, pipe water discharges and nodal heads can be derived by solving the following equations, derived from previous works [17,18,21]:

**H**(n × 1) and

**Q**(n

_{p}× 1) are the unknown vectors of nodal heads at the demanding nodes and of pipe water discharges, respectively.

**H**(n

_{0}_{0}× 1) and

**d**(n × 1) are the assigned vectors of source heads and nodal demands, respectively. Matrixes

**A**and

_{12}**A**come from the incidence topological matrix

_{10}**A**, with size n

_{p}× n

_{n}. In the generic row of

**A**, associated with the generic network pipe, the generic element can take on the values 0, −1 or 1, whether the node corresponding to the matrix element is not at the end of the pipe, it is the initial node of the pipe, or it is the final node of the pipe, respectively. Starting from

**A**, matrix

**A**(n

_{12}_{p}× n) is obtained by considering the columns corresponding to the n network nodes with unknown head.

**A**(n × n

_{21}_{p}) is the transpose matrix of

**A**. Matrix

_{12}**A**(n

_{10}_{p}× n

_{0}) is obtained by considering the columns corresponding to the n

_{0}nodes with fixed head.

**A**(n

_{11}_{p}× n

_{p}) is a diagonal matrix, the elements of which identify the resistances of the n

_{p}network pipes through the following relationship:

_{i}is the diameter of the i-th pipe and where roughness coefficient k

_{i}, coefficient b

_{i}and exponents α, β and γ depend on the formula used to express pipe head losses. As an example, α = 2, β = 5.33 and γ = 2 in the Strickler formula. Matrix

**M**in Equation (1) is used to increase the resistance of the N

_{val}pipes fitted with the control valve. It is a diagonal matrix (n

_{p}× n

_{p}), in which the diagonal elements corresponding to the N

_{val}pipes fitted with control valve are equal to V

_{k}(k = 1, …, N

_{val}), whereas those corresponding to the n

_{p}-N

_{val}pipes without valve are equal to 1. Indeed, V

_{k}is the valve setting ranging from 0 to 1. For the generic pipe subject to a certain head loss, it represents the ratio of the pipe water discharge in the presence of the regulated valve to the water discharge in its absence. The extreme values of the range represent the fully closed and fully open valve, respectively.

**q**(n × 1) of leakage allocated to network demanding nodes can be calculated starting from the following Equation (3):

_{L}**Q**(n

_{L}_{p}× 1) of leakage outflows from WDN pipes can be assessed through the following relationship [6]:

_{i}and L

_{i}are the mean pressure head (ratio of pressure to specific weight of water) and the length, respectively. C

_{L,i}and n

_{leak}are empirical coefficients. The mean pressure head h

_{i}can be calculated as:

_{i,}

_{1}, H

_{i,}

_{2}and z

_{i,}

_{1}, z

_{i,}

_{2}are the heads and elevations, respectively, for the end nodes of the pipe.

**V**of valve settings V

_{k}(k = 1, …, N

_{v}) can be optimized to minimize the total leakage volume W

_{L,j}from the network, while meeting the minimum pressure head requirement h

_{des}at the demanding nodes:

_{L,j}at each time slot:

#### 2.2. Optimal Location through Sequential Addition of Valves (SA)

_{max}of valves installable in the network has to be fixed. The total number of steps in the algorithm is then N

_{max}+ 1. At Step 0 of this algorithm, the WDN has no control valves. At Step 1, the optimal location of 1 valve is searched for among the available locations in the WDN, i.e., all the n

_{p}pipes. To this end, the control valve is placed inside the WDN model at one potential location at a time. The algorithm described in Section 2.1 is applied, enabling assessment of the installation cost of the system (or rather the total number of control valves as a surrogate, banally coinciding with the step of SA) and assessment of W

_{L}. Then, the most beneficial valve, which yields the largest W

_{L}reduction compared to the no valve scenario, is detected. At Step 2, the optimal location of two control valves is searched for, while keeping the first optimal control valve obtained from Step 1 installed in the WDN. The second valve for the optimal combination of two valves is searched for among the available locations, i.e., the n

_{p}− 1 remaining pipes. The algorithm of Section 2.1 is applied considering n

_{p}− 1 combinations of two control valves and the most beneficial one is detected in terms of W

_{L}reduction compared to the 1 valve scenario. Other steps of SA can be carried out up to N

_{max}, always considering the following basic assumption: at the generic Step N

_{val}, the optimal combination of N

_{val}valves is searched for, while containing the optimal combination of N

_{val}− 1 valves detected at the previous step. Considering the number N

_{max}of control valves installable in the WDN, SA would require the following number ${C}_{v}^{*}$ of locations of control valves to be evaluated with the algorithm described in Section 2.1:

_{v}with no repetitions, which is given by:

_{L}values obtained through SA as a function of N

_{val}, with 0 ≤ N

_{val}≤ N

_{max}, a Pareto front of optimal solutions with 0 ≤ N

_{val}≤ N

_{max}is obtained.

#### 2.3. Optimal Location through Multi-Objective Genetic Algorithm (GA)

_{p}of potential locations of control valves. Each gene can take on two possible values, 0 and 1, which indicate the absence and presence of the control valve at the generic pipe, respectively. The number N

_{val}of valves for the generic solution is banally obtained by summing up the gene values.

_{L}. Both the objective functions need to be minimized inside the GA. Similar to the algorithm for SA, the results of the multi-objective optimization are Pareto fronts. Unlike SA, in which the Pareto fronts feature a maximum number of valves equal to N

_{max}depending on the number of SA steps, the GA Pareto fronts are made up of solutions with N

_{val}ranging from 0 to n

_{p}.

## 3. Application

#### 3.1. Case Studies

_{n}= 34 nodes (of which n = 32 with unknown head and n

_{0}= 2 source nodes with fixed head, i.e., Nodes 33 and 34) and of n

_{p}= 41 pipes. The features of the WDN demanding nodes in terms of ground elevation z, mean daily demand d, and desired pressure head h

_{des}for full demand satisfaction, are reported in Table 1. At the generic demanding node, the latter variable was set at the minimum value between 15 m and the daily lowest pressure head under uncontrolled conditions. This was done to avoid any pressure deficit increase at nodes that were pressure deficient ab initio. Table 2 reports the features of the pipes in terms of length L, diameter D and Strickler roughness coefficient k. The two source nodes, i.e., Nodes 33 and 34, have a ground elevation of 477.5 m a.s.l.

_{h}for nodal demands are reported in Table 3.

_{L}= 2.79 × 10

^{−8}m

^{0.82}/s and of the exponent n

_{leak}= 1.18 were assumed for all the pipes of the network. The values of C

_{L}and n

_{leak}lead to a daily leakage volume W

_{L}= 1242.6 m

^{3}, which is about 44% of the water volume leaving the source nodes, as evaluated in the real network. In the latter, the coefficient C

_{L}was reduced to 0.85 × 10

^{−8}m

^{0.82}/s, obtaining a daily leakage volume W

_{L}= 397.7 m

^{3}(about 20% of the water volume leaving the source nodes) without control valves.

_{n}= 46 nodes (of which n = 45 with unknown head and n

_{0}= 1 source node with assigned head, i.e., tank Node 46) and n

_{p}= 52 pipes. The features of the district demanding nodes and pipes are provided by Creaco and Pezzinga [16], who also reported the values of the leakage exponent n

_{leak}= 0.9 and of the leakage coefficient C

_{L}ranging from 0.2 to 1 m

^{1.1}/s in the various pipes. In this district, the leakage volume adds up to 118.7 m

^{3}, which is 6.1% of the water volume entering the district, in the no valve scenario.

#### 3.2. Results

_{max}= 10 control valves were searched for. This required 365 objective function evaluations.

_{L}as a function of N

_{val}obtained with SA and GA in the first leakage condition are reported in Figure 2. As for GA, two Pareto fronts are shown, the final one (after 50 generations) and that after seven generations. The latter is associated with 350 objective function evaluations (very close to SA). Though GA explored solutions with N

_{val}values up to n

_{p}(see Section 2.3), only the solutions with N

_{val}≤ 10 were reported in the graph for a consistent comparison with SA. This graph shows that the curve of SA and the final curve of GA have identical trend up to N

_{val}= 3. To the right of N

_{val}= 3, the curve obtained with GA dominates that of SA, in that it provides lower values of W

_{L}for each value of N

_{val}. Furthermore, the distance between the two curves increases as N

_{val}grew. This seems to suggest that, for low number of N

_{val}, SA can provide identical or close results to those of GA. Conversely, for high number of N

_{val}, the better performance of GA stands out, due to the wider exploration of the research space. However, this comes at an about seven times larger computational cost. As was expected, the curve of GA after seven generations features worse solutions than the final curve of GA to the right of N

_{val}= 3. The curve of GA after seven generations enables the results of GA to be compared with those of SA, given the same computational budget. Interestingly, neither curve prevails in absolute terms. In fact, identical solutions are observed up to N

_{val}= 3. GA after seven generations is slightly better for N

_{val}= 7, 9 and 10, while not offering any solutions for N

_{val}= 8. SA, instead, is better for N

_{val}= 5 and 6. Summing up the results in Figure 2, the better performance of GA stands out only with a higher computational budget.

_{L}values for the two algorithms. Whereas the valve positions suggested by SA and GA are the same up to N

_{val}= 4, the two algorithms behave differently starting from the optimal location of five valves. In fact, at Step 5, SA suggests adding valve in Pipe 33 to the optimal location of N

_{val}= 4 valves, in which the valve-fitted pipes are 3, 7, 14 and 27. Instead, for the sake of optimality, GA proposes valve elimination in Pipe 14 and valve insertion in Pipes 25 and 26, while moving from N

_{val}= 4 to N

_{val}= 5. The locations of SA and GA for N

_{val}= 5 valves are shown in Figure 3. As Table 4 shows, this yields a 4.36% benefit of GA compared to SA, in terms of W

_{L}. The percentage benefit of GA tends to grow up to almost 10% for N

_{val}= 10.

_{down}at the downstream node, which can be used as the time varying setting to be prescribed if the water utility chooses to perform the pressure regulation by means of pressure reducing valves (PRVs). In fact, PRVs has the downstream pressure head as setting. This variable is always equal to h

_{des}(that is 15 m) at the valve in Pipe 3 because the downstream node of Pipe 3, namely Node 32, is a terminal node of the network. It is always equal to h

_{des}also for the valve in Pipe 7, though the downstream node of this pipe, namely Node 3, is not terminal. This happens because the descending topography downstream of Pipe 7 facilitates the meeting of the pressure requirements. The case of the valve in Pipe 27, which has Node 9 as downstream node, is similar. However, this valve cannot reduce h

_{down}to h

_{des}at nighttime, even if it is almost fully closed (V ≈ 0).

_{L}as a function of N

_{val}obtained with SA and GA in the second leakage condition are reported in Figure 5. This figure leads to similar remarks as Figure 2, as far as the comparison of the two algorithms is concerned.

_{L}values and benefits of GA compared to SA. The comparison of Table 5 (low starting leakage) and Table 4 (high starting leakage rate) points out that the reduction in the starting leakage has the following minor effects:

_{L}provided by the two algorithms, as well as the benefits of GA. Similar to the first case study, these benefits stand out only for N

_{val}≥ 5. However, they are smaller (always below 4.32%), due to the different structure of the WDN. In fact, while the network of Santa Maria di Licodia is very interconnected, the network of the second case study is made up of quite independent branch structures fed by the main line, along which valves cannot be installed. This attenuates the non-linearities pertinent to optimal valve location.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**WDNs for the two case studies. Pipe numbers close to the pipes. Numbers of demanding nodes inside circles. (

**a**) First case study has source Nodes 33 and 34. (

**b**) Second case study has pump inflow at Node 1 and source Node 46. Main interconnection line is between the pump and source node.

**Figure 2.**First case study. Daily leakage volume W

_{L}as a function of N

_{val}for the two algorithms.

**Figure 3.**First case study. Optimal location of five valves for: (

**a**) SA; and (

**b**) GA. Valve locations indicated with thick lines.

**Figure 4.**First case study. Optimal location of three valves. For each valve, trend of: V (

**a**); and h

_{down}(

**b**) in the daily time slots.

**Figure 5.**First case study under modified leakage conditions. Daily leakage volume W

_{L}as a function of N

_{val}for the two algorithms.

**Figure 6.**Second case study. Daily leakage volume W

_{L}as a function of N

_{va}

_{l}for the two algorithms.

Node | z (m) | d (m^{3}/s) | h_{des} (m) | Node | z (m) | d (m^{3}/s) | h_{des} (m) |
---|---|---|---|---|---|---|---|

1 | 465 | 0 | 14.5 | 17 | 437.6 | 0.0007516 | 15 |

2 | 462 | 0 | 15 | 18 | 450 | 0.0006938 | 15 |

3 | 456 | 0.0001156 | 15 | 19 | 442 | 0.0011563 | 15 |

4 | 451.3 | 0.0001156 | 15 | 20 | 436.5 | 0.00075164 | 15 |

5 | 451 | 0.0002891 | 15 | 21 | 433.5 | 0.0008094 | 15 |

6 | 448.5 | 0.0002891 | 15 | 22 | 434 | 0.0008672 | 15 |

7 | 444 | 0.0006359 | 15 | 23 | 431.2 | 0.000925 | 15 |

8 | 446 | 0.0005781 | 15 | 24 | 436.8 | 0.0008672 | 15 |

9 | 445 | 0.0008672 | 15 | 25 | 435.8 | 0.0008672 | 15 |

10 | 442 | 0.0006938 | 15 | 26 | 438.6 | 0.0006938 | 15 |

11 | 438.6 | 0.0006359 | 15 | 27 | 440.3 | 0.0007516 | 15 |

12 | 437.5 | 0.0006938 | 15 | 28 | 430.1 | 0.0008672 | 15 |

13 | 448 | 0.0004047 | 15 | 29 | 465 | 0.0001734 | 11 |

14 | 435 | 0.0004625 | 15 | 30 | 445 | 0.0003469 | 15 |

15 | 441 | 0.0006938 | 15 | 31 | 454 | 0.0006938 | 15 |

16 | 394.8 | 0.0005781 | 15 | 32 | 435 | 0.0002313 | 15 |

Pipe | L (m) | D (mm) | k (m^{1/3}/s) | Pipe | L (m) | D (mm) | k (m^{1/3}/s) |
---|---|---|---|---|---|---|---|

1 | 352 | 250 | 75 | 22 | 110 | 125 | 65 |

2 | 314 | 175 | 65 | 23 | 214 | 150 | 65 |

3 | 1100 | 125 | 75 | 24 | 85 | 100 | 65 |

4 | 350 | 100 | 65 | 25 | 398 | 100 | 65 |

5 | 96 | 250 | 75 | 26 | 242 | 100 | 65 |

6 | 282 | 100 | 75 | 27 | 118 | 175 | 65 |

7 | 148 | 250 | 75 | 28 | 324 | 175 | 65 |

8 | 256 | 250 | 75 | 29 | 140 | 125 | 65 |

9 | 192 | 100 | 75 | 30 | 206 | 125 | 65 |

10 | 58 | 100 | 75 | 31 | 70 | 125 | 65 |

11 | 66 | 100 | 75 | 32 | 142 | 150 | 65 |

12 | 230 | 150 | 75 | 33 | 86 | 150 | 65 |

13 | 200 | 100 | 75 | 34 | 294 | 80 | 65 |

14 | 44 | 250 | 75 | 35 | 150 | 80 | 65 |

15 | 226 | 250 | 75 | 36 | 124 | 125 | 65 |

16 | 70 | 150 | 65 | 37 | 144 | 125 | 65 |

17 | 88 | 80 | 65 | 38 | 158 | 125 | 65 |

18 | 204 | 125 | 65 | 39 | 130 | 80 | 65 |

19 | 172 | 125 | 65 | 40 | 124 | 80 | 65 |

20 | 94 | 125 | 65 | 41 | 500 | 80 | 65 |

21 | 90 | 125 | 65 |

Time Slot (h) | Source 33–34 Head (m) | C_{h} (-) | Time Slot (h) | Source 33–34 Head (m) | C_{h} (-) |
---|---|---|---|---|---|

0–2 | 480.77 | 0.40 | 12–14 | 480.55 | 1.8 |

2–4 | 481.14 | 0.40 | 14–16 | 480.45 | 0.90 |

4–6 | 481.46 | 0.55 | 16–18 | 480.64 | 0.70 |

6–8 | 481.22 | 1.70 | 18–20 | 480.53 | 1.45 |

8–10 | 480.91 | 1.25 | 20–22 | 480.19 | 1.40 |

10–12 | 480.94 | 1.0 | 22–24 | 480.41 | 0.45 |

**Table 4.**First case study. Optimal valve locations and daily leakage volumes W

_{L}obtained with SA and GA. Benefits of GA in terms of W

_{L}reduction.

N_{val} | Valve Locations with SA | W_{L} with SA (m^{3}) | Valve Locations with GA | W_{L} with GA (m^{3}) | Benefits of GA (%) |
---|---|---|---|---|---|

0 | - | 1243 | - | 1243 | 0.00 |

1 | 27 | 1029 | 27 | 1029 | 0.00 |

2 | 27,7 | 885 | 7,27 | 885 | 0.00 |

3 | 27,7,3 | 805 | 3,7,27 | 805 | 0.00 |

4 | 27,7,3,14 | 751 | 3,7,14,27 | 751 | 0.00 |

5 | 27,7,3,14,33 | 725 | 3,7,25,26,27 | 693 | 4.36 |

6 | 27,7,3,14,33,4 | 708 | 3,7,8,25,26,27 | 672 | 5.02 |

7 | 27,7,3,14,33,4,2 | 692 | 3,7,8,23,25,26,27 | 647 | 6.38 |

8 | 27,7,3,14,33,4,2,41 | 680 | 3,4,7,8,23,25,26,27 | 630 | 7.38 |

9 | 27,7,3,14,33,4,2,41,6 | 670 | 3,4,7,8,24,25,26,27,33 | 614 | 8.36 |

10 | 27,7,3,14,33,4,2,41,6,30 | 659 | 2,3,4,7,8,24,25,26,27,33 | 598 | 9.29 |

**Table 5.**First case study under modified leakage conditions. Optimal valve locations and daily leakage volumes W

_{L}obtained with SA and GA. Benefits of GA in terms of W

_{L}reduction.

N_{val} | Valve Locations with SA | W_{L} with SA (m^{3}) | Valve Locations with GA | W_{L} with GA (m^{3}) | Benefits of GA (%) |
---|---|---|---|---|---|

0 | - | 398 | - | 398 | 0.00 |

1 | 27 | 338 | 27 | 338 | 0.00 |

2 | 27-7 | 282 | 7-27 | 282 | 0.00 |

3 | 27-7-3 | 257 | 3-7-27 | 257 | 0.00 |

4 | 27-7-3-24 | 235 | 3-7-24-27 | 235 | 0.00 |

5 | 27-7-3-24-8 | 228 | 3-7-25-26-27 | 217 | 4.97 |

6 | 27-7-3-24-8-23 | 221 | 3-8-10-25-26-27 | 206 | 6.73 |

7 | 27-7-3-24-8-23-20 | 215 | 3-8-10-23-25-26-27 | 201 | 6.33 |

8 | 27-7-3-24-8-23-20-2 | 209 | 3-4-8-10-23-25-26-27 | 196 | 6.47 |

9 | 27-7-3-24-8-23-20-2-4 | 204 | 3-4-8-10-20-23-25-26-27 | 191 | 6.54 |

10 | 27-7-3-24-8-23-20-2-4-41 | 200 | 3-4-8-10-20-23-24-25-26-27 | 187 | 6.68 |

**Table 6.**Second case study. Optimal valve locations and daily leakage volumes W

_{L}obtained with SA and GA. Benefits of GA in terms of W

_{L}reduction.

N_{val} | Valve Locations with SA | W_{L} with SA (m^{3}) | Valve Locations with GA | W_{L} with GA (m^{3}) | Benefits of GA (%) |
---|---|---|---|---|---|

0 | - | 119 | - | 119 | 0.00 |

1 | 49 | 114 | 49 | 114 | 0.00 |

2 | 49-45 | 113 | 15-45 | 112 | 1.14 |

3 | 49-45-15 | 107 | 15-25-45 | 107 | 0.00 |

4 | 49-45-15-5 | 103 | 5 -15-45-49 | 103 | 0.00 |

5 | 49-45-15-5-27 | 102 | 7-15-38-45-49 | 100 | 1.85 |

6 | 49-45-15-5-27-7 | 101 | 5 -7-15-25-38-45 | 96 | 4.32 |

7 | 49-45-15-5-27-7-38 | 95 | 5-7-25-32-38-39-45 | 95 | 0.09 |

8 | 49-45-15-5-27-7-38-52 | 94 | 5-7-15-32-38-39-45-49 | 92 | 2.61 |

9 | 49-45-15-5-27-7-38-52-33 | 94 | 5-7-15-25-27-32-38-39-45 | 90 | 3.92 |

10 | 49-45-15-5-27-7-38-52-33-30 | 93 | 5-7-15-27-32-33-38-39-45-49 | 89 | 4.05 |

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## Share and Cite

**MDPI and ACS Style**

Creaco, E.; Pezzinga, G.
Comparison of Algorithms for the Optimal Location of Control Valves for Leakage Reduction in WDNs. *Water* **2018**, *10*, 466.
https://doi.org/10.3390/w10040466

**AMA Style**

Creaco E, Pezzinga G.
Comparison of Algorithms for the Optimal Location of Control Valves for Leakage Reduction in WDNs. *Water*. 2018; 10(4):466.
https://doi.org/10.3390/w10040466

**Chicago/Turabian Style**

Creaco, Enrico, and Giuseppe Pezzinga.
2018. "Comparison of Algorithms for the Optimal Location of Control Valves for Leakage Reduction in WDNs" *Water* 10, no. 4: 466.
https://doi.org/10.3390/w10040466