Abstract
This study aims at the development of an optimization model based on a model calibration, using artificial immune systems (AIS) for quantifying and locating water loss in water distribution networks (WDNs) without using observed pressure data as used by previous studies in the related literature. The modified Clonal Selection Algorithm (modified Clonalg), a class of AIS, was used as a heuristic optimization technique in the model. EPANET 2, a widely known WDN simulator, was used in conjunction with the model. The model was applied to four-loop and six-loop virtual WDNs under steady-state conditions in order to test its performance in water loss detection in both pipes and nodes. Also, sensitivity analysis of the modified Clonalg was performed according to mutation coefficient to test its search capability in this optimization problem. The results showed that the model appeared to be promising in terms of water loss detection in WDNs.
INTRODUCTION
Water is of vital importance for all living creatures because it is a life source. This also increases the economic value of water. In this regard, the water loss detection comes to prominence to prevent water loss in water distribution networks (WDNs). Water loss in the WDNs occurs due to unauthorized water consumptions, meter inaccuracies, and leakages or bursts in pipes and nodes. Misiunas (2003) and De Silva et al. (2011) devised the most commonly used leakage detection techniques in WDNs, which are Static Mass Balance (Mounce et al. 2003), State Estimation (Andersen & Powell 2000), Transient Analysis (Covas et al. 2004; Savic et al. 2005), Acoustic Methods (Tafuri 2000). Within these techniques, various sensors, meters, monitoring systems and measurements (flow rate, pressure and temperature sensors detecting quasi-static signals, acoustic sensors detecting sound waves/noises, electromagnetic sensors using Faraday's law of induction, infrared thermography, transmitters emitting radio frequencies, manometers, ultrasonic flow meters utilizing propagation time of the ultrasonic signals, SCADA, tracing substances, etc.) are used and installed into WDNs. Therefore, the water loss detection is an expensive and difficult task. In order to facilitate the task, the optimization model was developed in this study based on a model calibration, using modified Clonal Selection Algorithm (modified Clonalg), a class of artificial immune systems (AIS). There are many studies such as Vitkovsky et al. (2000), Tabesh et al. (2009), Prasad (2010), Nasirian et al. (2013), Kang & Lansey (2014), Ribeiro et al. (2015), and Sanz et al. (2015) regarding water loss detection in WDNs based on a model calibration in the related literature. In these studies, many observed flow rates and pressure measurements in the field were used to detect water loss at some WDN points. The success of water loss detection of WDNs depends on the number of field measurements (flow rate and pressure). More field measurements facilitate the detection of greater water loss. But this increases the cost of the task due to lack of equipment. Thus, herein, the most important aim is to detect the maximum amount of water loss by using the minimum required field measurements. Within this aim, in this study, the model does not need to observe pressures in the field, and it uses observed flow rates in the initial points of pipes (not measured throughout pipes) and observed nodal demands in WDN nodes for detecting water loss including leakages in all pipes and unauthorized water consumptions in all nodes. Although there are many unknown leakages in WDN pipes which cannot be calculated by the continuity equation with the observed data, the model can detect locations and amounts of water loss. This demonstrates that the model minimizes the number of required field measurements used in a model calibration (pressures are not used) for detecting water loss in all pipes and WDN nodes. In order to test a search capability of the modified Clonalg in this optimization problem, the sensitivity analysis was carried out according to mutation coefficient. The model was applied to four-loop and six-loop virtual WDNs under steady-state conditions in order to test its performance in water loss detection. The results showed that the model can detect both locations and amounts of water loss in all pipes and of WDN nodes.
MATERIALS AND METHODS
Model formulation
Four-loop WDN scenario
This network consists of 10 actual nodes including nodes 1–9, and a reservoir, 13 pipes with four loops, and is fed by the gravity from a reservoir with a 75 m fixed head. Pipe roughness coefficient (Cp) was 100 in all pipes. Nodes 10–21 are pseudo nodes representing leakages in pipes. Also, unauthorized water consumption was added to base demands in nodes 1–9 as water loss. In this WDN, total amount of water loss is 85 l/s (Lmax) (total inflow of 450 l/s – total base demands of 365 l/s). Node and pipe data, operating data, and the layouts of four-loop WDN are shown in Tables 1 and 2, Figures 2 and 3, respectively. As seen in Figure 3, leakages in pipes 1, 2, 5, and 8 (pseudo nodes 12, 10, 11, and 17, respectively) can be calculated by Equation (3) with the observed data (e.g., pseudo node 12 = 198.33 l/s − (44 l/s + 81.78 l/s + 67.55 l/s)) while leakage in pipes 3, 4, 6, 7, 9, 10, 11 and 12 (pseudo nodes 13, 15, 14, 16, 20, 18, 21 and 19, respectively) cannot be calculated because the flows are not measured throughout pipes.
Node and pipe data of four-loop WDN
Node . | Elevation (m) . | Base demand (l/s) . | Pipe . | Length (m) . | Diameter (mm) . | Cp . |
---|---|---|---|---|---|---|
Reservoir | 75 | – | 1 | 500 | 400 | 100 |
1 | 0 | 50 | 2 | 500 | 400 | 100 |
2 | 0 | 40 | 3 | 500 | 300 | 100 |
3 | 0 | 40 | 4 | 500 | 300 | 100 |
4 | 0 | 50 | 5 | 500 | 300 | 100 |
5 | 0 | 40 | 6 | 500 | 200 | 100 |
6 | 0 | 30 | 7 | 500 | 200 | 100 |
7 | 0 | 40 | 8 | 500 | 300 | 100 |
8 | 0 | 35 | 9 | 500 | 200 | 100 |
9 | 0 | 40 | 10 | 500 | 200 | 100 |
11 | 500 | 200 | 100 | |||
12 | 500 | 200 | 100 | |||
13 | 1,000 | 600 | 100 |
Node . | Elevation (m) . | Base demand (l/s) . | Pipe . | Length (m) . | Diameter (mm) . | Cp . |
---|---|---|---|---|---|---|
Reservoir | 75 | – | 1 | 500 | 400 | 100 |
1 | 0 | 50 | 2 | 500 | 400 | 100 |
2 | 0 | 40 | 3 | 500 | 300 | 100 |
3 | 0 | 40 | 4 | 500 | 300 | 100 |
4 | 0 | 50 | 5 | 500 | 300 | 100 |
5 | 0 | 40 | 6 | 500 | 200 | 100 |
6 | 0 | 30 | 7 | 500 | 200 | 100 |
7 | 0 | 40 | 8 | 500 | 300 | 100 |
8 | 0 | 35 | 9 | 500 | 200 | 100 |
9 | 0 | 40 | 10 | 500 | 200 | 100 |
11 | 500 | 200 | 100 | |||
12 | 500 | 200 | 100 | |||
13 | 1,000 | 600 | 100 |
Operating data of four-loop WDN
Node . | Observed demand (l/s) . | Water loss in node (l/s) . | Pipe . | *Observed flow rate (l/s) . | Leakage in pipe (l/s) . |
---|---|---|---|---|---|
Reservoir | – | – | 1 | 198.33 | 5 |
1 | 56 | 6 | 2 | 195.67 | 5 |
2 | 44 | 4 | 3 | 68.7 | 3 |
3 | 43 | 3 | 4 | 67.55 | 4 |
4 | 57 | 7 | 5 | 78.98 | 3 |
5 | 45 | 5 | 6 | 30.98 | 0 |
6 | 36 | 6 | 7 | 35.75 | 4 |
7 | 46 | 6 | 8 | 81.78 | 5 |
8 | 38 | 3 | 9 | 30.78 | 0 |
9 | 43 | 3 | 10 | 36.49 | 4 |
11 | 25.27 | 4 | |||
12 | 26.73 | 5 | |||
13 | 450 | – |
Node . | Observed demand (l/s) . | Water loss in node (l/s) . | Pipe . | *Observed flow rate (l/s) . | Leakage in pipe (l/s) . |
---|---|---|---|---|---|
Reservoir | – | – | 1 | 198.33 | 5 |
1 | 56 | 6 | 2 | 195.67 | 5 |
2 | 44 | 4 | 3 | 68.7 | 3 |
3 | 43 | 3 | 4 | 67.55 | 4 |
4 | 57 | 7 | 5 | 78.98 | 3 |
5 | 45 | 5 | 6 | 30.98 | 0 |
6 | 36 | 6 | 7 | 35.75 | 4 |
7 | 46 | 6 | 8 | 81.78 | 5 |
8 | 38 | 3 | 9 | 30.78 | 0 |
9 | 43 | 3 | 10 | 36.49 | 4 |
11 | 25.27 | 4 | |||
12 | 26.73 | 5 | |||
13 | 450 | – |
*All observed flow rates were assumed to be measured only at initial points of pipes in WDNs.
Although there are no leakages in pipes 6 and 9, they were assumed as pseudo nodes (nodes 14 and 20) due to their being leak candidates.
Six-loop WDN scenario
This network consists of 13 actual nodes including nodes 1–12, and a reservoir, 18 pipes with six loops, and is fed by the gravity from a reservoir with a 75 m fixed head. Pipe roughness coefficient (Cp) was 100 in all pipes. Nodes 13–29 are pseudo nodes representing leakages in pipes. Also, unauthorized water consumption was added to base demands in nodes 1–12 as water loss. In this WDN, total amount of water loss is 91 l/s (Lmax) (total inflow of 686 l/s – total base demands of 595 l/s). Node and pipe data, operating data, and the layouts of six-loop WDN are shown in Tables 3 and 4, Figures 4 and 5, respectively. As it is seen in Figure 5, leakages in pipes 1, 2, 5, 8, and 13 (pseudo nodes 15, 13, 14, 20, and 25, respectively) can be calculated by Equation (3) with the observed data (e.g., pseudo node 15 = 348.51 l/s − (47 l/s + 190.7 l/s + 107.8 l/s)) while leakages in pipes 3, 4, 6, 7, 9, 10, 11, 12, 14, 15, 16, and 17 (pseudo nodes 16, 18, 17, 19, 23, 21, 24, 22, 28, 26, 29, and 27, respectively) cannot be calculated because the flows are not measured throughout pipes.
Node and pipe data of six-loop WDN
Node . | Elevation (m) . | Base demand (l/s) . | Pipe . | Length (m) . | Diameter (mm) . | Cp . |
---|---|---|---|---|---|---|
Reservoir | 75 | – | 1 | 500 | 500 | 100 |
1 | 0 | 50 | 2 | 500 | 500 | 100 |
2 | 0 | 45 | 3 | 500 | 300 | 100 |
3 | 0 | 40 | 4 | 500 | 300 | 100 |
4 | 0 | 35 | 5 | 500 | 300 | 100 |
5 | 0 | 50 | 6 | 500 | 200 | 100 |
6 | 0 | 60 | 7 | 500 | 300 | 100 |
7 | 0 | 55 | 8 | 500 | 400 | 100 |
8 | 0 | 40 | 9 | 500 | 250 | 100 |
9 | 0 | 55 | 10 | 500 | 200 | 100 |
10 | 0 | 60 | 11 | 500 | 150 | 100 |
11 | 0 | 55 | 12 | 500 | 250 | 100 |
12 | 0 | 50 | 13 | 500 | 200 | 100 |
14 | 500 | 150 | 100 | |||
15 | 500 | 200 | 100 | |||
16 | 500 | 150 | 100 | |||
17 | 500 | 200 | 100 | |||
18 | 1,000 | 750 | 100 |
Node . | Elevation (m) . | Base demand (l/s) . | Pipe . | Length (m) . | Diameter (mm) . | Cp . |
---|---|---|---|---|---|---|
Reservoir | 75 | – | 1 | 500 | 500 | 100 |
1 | 0 | 50 | 2 | 500 | 500 | 100 |
2 | 0 | 45 | 3 | 500 | 300 | 100 |
3 | 0 | 40 | 4 | 500 | 300 | 100 |
4 | 0 | 35 | 5 | 500 | 300 | 100 |
5 | 0 | 50 | 6 | 500 | 200 | 100 |
6 | 0 | 60 | 7 | 500 | 300 | 100 |
7 | 0 | 55 | 8 | 500 | 400 | 100 |
8 | 0 | 40 | 9 | 500 | 250 | 100 |
9 | 0 | 55 | 10 | 500 | 200 | 100 |
10 | 0 | 60 | 11 | 500 | 150 | 100 |
11 | 0 | 55 | 12 | 500 | 250 | 100 |
12 | 0 | 50 | 13 | 500 | 200 | 100 |
14 | 500 | 150 | 100 | |||
15 | 500 | 200 | 100 | |||
16 | 500 | 150 | 100 | |||
17 | 500 | 200 | 100 | |||
18 | 1,000 | 750 | 100 |
Operating data of six-loop WDN
Node . | Observed demand (l/s) . | Water loss in node (l/s) . | Pipe . | *Observed flow rate (l/s) . | Leakage in pipe (l/s) . |
---|---|---|---|---|---|
Reservoir | – | – | 1 | 348.51 | 3 |
1 | 55 | 5 | 2 | 282.49 | 0 |
2 | 47 | 2 | 3 | 120.21 | 3 |
3 | 42 | 2 | 4 | 107.8 | 4 |
4 | 36 | 1 | 5 | 106.29 | 4 |
5 | 56 | 6 | 6 | 44.29 | 3 |
6 | 65 | 5 | 7 | 111.58 | 2 |
7 | 61 | 6 | 8 | 190.7 | 2 |
8 | 44 | 4 | 9 | 90.23 | 5 |
9 | 58 | 3 | 10 | 44.43 | 0 |
10 | 63 | 3 | 11 | 20.46 | 2 |
11 | 58 | 3 | 12 | 87.87 | 5 |
12 | 54 | 4 | 13 | 56.47 | 0 |
14 | 20.47 | 5 | |||
15 | 48.2 | 1 | |||
16 | 18.67 | 5 | |||
17 | 43.33 | 3 | |||
18 | 686 | – |
Node . | Observed demand (l/s) . | Water loss in node (l/s) . | Pipe . | *Observed flow rate (l/s) . | Leakage in pipe (l/s) . |
---|---|---|---|---|---|
Reservoir | – | – | 1 | 348.51 | 3 |
1 | 55 | 5 | 2 | 282.49 | 0 |
2 | 47 | 2 | 3 | 120.21 | 3 |
3 | 42 | 2 | 4 | 107.8 | 4 |
4 | 36 | 1 | 5 | 106.29 | 4 |
5 | 56 | 6 | 6 | 44.29 | 3 |
6 | 65 | 5 | 7 | 111.58 | 2 |
7 | 61 | 6 | 8 | 190.7 | 2 |
8 | 44 | 4 | 9 | 90.23 | 5 |
9 | 58 | 3 | 10 | 44.43 | 0 |
10 | 63 | 3 | 11 | 20.46 | 2 |
11 | 58 | 3 | 12 | 87.87 | 5 |
12 | 54 | 4 | 13 | 56.47 | 0 |
14 | 20.47 | 5 | |||
15 | 48.2 | 1 | |||
16 | 18.67 | 5 | |||
17 | 43.33 | 3 | |||
18 | 686 | – |
*All observed flow rates were assumed to be measured only at initial points of pipes in the WDN.
Although there are no leakages in pipes 2, 10 and 13, they were assumed as pseudo nodes (nodes 13, 21 and 25) due to their being leak candidates.
RESULTS AND DISCUSSION
The model minimizes the objective function (see Equation (2)) for detecting water loss (unauthorized water consumption in actual nodes and leakages in pipes) in WDNs, and the average of the best (minimum) objective function values after 20 runs is zero (see Table 5). As seen in Tables 6 and 7, leakages and water demands predicted by the model, and actual leakages and observed water demands are almost the same. Also, within hydraulic calculations, pressures in all actual nodes were obtained appropriately, and any negative pressure did not occur in nodes. In order to analyze the sensitivity of the model, ρ (decay coefficient), one of the most important parameters of the modified Clonalg, was selected because of playing a critical role in search capability in optimization problems. α (mutation rate) changes values of genes of all antibodies (AD and L) during the mutation process, and it is very sensitive to ρ. Therefore, parameter ρ affects the model calibration directly because AD and L are decision variables of the model calibration in this study. For 8, 9, and 10 values of ρ, the model was applied to six-loop WDN. The results demonstrated that the modified Clonalg used in the model shows a steady and stable performance for water loss detection in the WDN (see Table 7).
Parameters and performances of the modified Clonalg used for the optimization
WDN . | NAb . | β . | ρ . | Probability rate . | Iteration number . | Min. f (l/s) . | Max. f (l/s) . | *Mean f (l/s) . | *Mean run time (min) . |
---|---|---|---|---|---|---|---|---|---|
Four-loop | 30 | 1 | 8 | 0.1 | 20,000 | 1.02 × 10−4 | 2.14 × 10−4 | 1.52 × 10−4 ± 3.3 × 10−5 | 186.7 ± 4.1 |
Six-loop | 30 | 1 | 8 | 0.1 | 20,000 | 5.2 × 10−4 | 20.3 × 10−4 | 13.9 × 10−4 ± 4.2 × 10−4 | 256.4 ± 6.9 |
30 | 1 | 9 | 0.1 | 20,000 | 2.36 × 10−4 | 20.7 × 10−4 | 8.99 × 10−4 ± 5.1 × 10−4 | 253.5 ± 0.4 | |
30 | 1 | 10 | 0.1 | 20,000 | 3.29 × 10−4 | 19.5 × 10−4 | 8.64 × 10−4 ± 5.1 × 10−4 | 258.9 ± 13.7 |
WDN . | NAb . | β . | ρ . | Probability rate . | Iteration number . | Min. f (l/s) . | Max. f (l/s) . | *Mean f (l/s) . | *Mean run time (min) . |
---|---|---|---|---|---|---|---|---|---|
Four-loop | 30 | 1 | 8 | 0.1 | 20,000 | 1.02 × 10−4 | 2.14 × 10−4 | 1.52 × 10−4 ± 3.3 × 10−5 | 186.7 ± 4.1 |
Six-loop | 30 | 1 | 8 | 0.1 | 20,000 | 5.2 × 10−4 | 20.3 × 10−4 | 13.9 × 10−4 ± 4.2 × 10−4 | 256.4 ± 6.9 |
30 | 1 | 9 | 0.1 | 20,000 | 2.36 × 10−4 | 20.7 × 10−4 | 8.99 × 10−4 ± 5.1 × 10−4 | 253.5 ± 0.4 | |
30 | 1 | 10 | 0.1 | 20,000 | 3.29 × 10−4 | 19.5 × 10−4 | 8.64 × 10−4 ± 5.1 × 10−4 | 258.9 ± 13.7 |
NAb: Number of population Ab. β: Multiplying coefficient for the cloning. ρ: Decay coefficient.
*Average of 20 runs.
Comparison of mean predicted and actual water loss (leakages and unauthorized consumption) in nodes and pipes of four-loop WDN
Node . | *Mean predicted demand (l/s) . | Observed demand (l/s) . | *Mean pressure (m) . | Pipe . | *Mean predicted leakage (l/s) . | Leakage (l/s) . |
---|---|---|---|---|---|---|
1 | 56.000 ± 0.004 | 56 | 69.21 ± 0.000 | 1 | 5.001 ± 0.028 | 5 |
2 | 44.000 ± 0.008 | 44 | 64.75 ± 0.001 | 2 | 4.999 ± 0.031 | 5 |
3 | 43.001 ± 0.007 | 43 | 64.86 ± 0.001 | 3 | 3.015 ± 0.065 | 3 |
4 | 56.998 ± 0.004 | 57 | 62.36 ± 0.000 | 4 | 4.007 ± 0.056 | 4 |
5 | 45.005 ± 0.006 | 45 | 61.60 ± 0.001 | 5 | 2.974 ± 0.020 | 3 |
6 | 35.995 ± 0.007 | 36 | 57.31 ± 0.003 | 6 | 0.043 ± 0.023 | 0 |
7 | 46.003 ± 0.006 | 46 | 61.35 ± 0.001 | 7 | 3.975 ± 0.023 | 4 |
8 | 37.996 ± 0.007 | 38 | 57.10 ± 0.003 | 8 | 4.978 ± 0.011 | 5 |
9 | 43.000 ± 0.004 | 43 | 54.56 ± 0.001 | 9 | 0.047 ± 0.017 | 0 |
10 | 3.971 ± 0.020 | 4 | ||||
11 | 3.994 ± 0.072 | 4 | ||||
12 | 4.996 ± 0.067 | 5 |
Node . | *Mean predicted demand (l/s) . | Observed demand (l/s) . | *Mean pressure (m) . | Pipe . | *Mean predicted leakage (l/s) . | Leakage (l/s) . |
---|---|---|---|---|---|---|
1 | 56.000 ± 0.004 | 56 | 69.21 ± 0.000 | 1 | 5.001 ± 0.028 | 5 |
2 | 44.000 ± 0.008 | 44 | 64.75 ± 0.001 | 2 | 4.999 ± 0.031 | 5 |
3 | 43.001 ± 0.007 | 43 | 64.86 ± 0.001 | 3 | 3.015 ± 0.065 | 3 |
4 | 56.998 ± 0.004 | 57 | 62.36 ± 0.000 | 4 | 4.007 ± 0.056 | 4 |
5 | 45.005 ± 0.006 | 45 | 61.60 ± 0.001 | 5 | 2.974 ± 0.020 | 3 |
6 | 35.995 ± 0.007 | 36 | 57.31 ± 0.003 | 6 | 0.043 ± 0.023 | 0 |
7 | 46.003 ± 0.006 | 46 | 61.35 ± 0.001 | 7 | 3.975 ± 0.023 | 4 |
8 | 37.996 ± 0.007 | 38 | 57.10 ± 0.003 | 8 | 4.978 ± 0.011 | 5 |
9 | 43.000 ± 0.004 | 43 | 54.56 ± 0.001 | 9 | 0.047 ± 0.017 | 0 |
10 | 3.971 ± 0.020 | 4 | ||||
11 | 3.994 ± 0.072 | 4 | ||||
12 | 4.996 ± 0.067 | 5 |
*Average of 20 runs.
Comparison of mean predicted and actual water loss (leakages and unauthorized consumption) in nodes and pipes of six-loop WDN
Node . | *Mean predicted demand (l/s) . | Observed demand (l/s) . | *Mean pressure (m) . | Pipe . | *Mean predicted leakage (l/s) . | Leakage (l/s) . | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
ρ = 8 . | ρ = 9 . | ρ = 10 . | ρ = 8 . | ρ = 9 . | ρ = 10 . | ρ = 8 . | ρ = 9 . | ρ = 10 . | ||||
1 | 54.995 ± 0.01 | 54.999 ± 0.00 | 54.998 ± 0.00 | 55 | 70.74 ± 0.00 | 70.74 ± 0.00 | 70.74 ± 0.00 | 1 | 2.989 ± 0.06 | 2.978 ± 0.03 | 2.986 ± 0.04 | 3 |
2 | 47.001 ± 0.02 | 47.003 ± 0.01 | 47.001 ± 0.01 | 47 | 66.39 ± 0.00 | 66.39 ± 0.00 | 66.39 ± 0.00 | 2 | 0.056 ± 0.03 | 0.040 ± 0.02 | 0.037 ± 0.02 | 0 |
3 | 41.991 ± 0.02 | 41.993 ± 0.01 | 41.994 ± 0.01 | 42 | 62.18 ± 0.00 | 62.18 ± 0.00 | 62.18 ± 0.00 | 3 | 2.934 ± 0.05 | 2.949 ± 0.03 | 2.952 ± 0.04 | 3 |
4 | 35.954 ± 0.02 | 35.958 ± 0.02 | 35.957 ± 0.02 | 36 | 49.15 ± 0.02 | 49.15 ± 0.02 | 49.15 ± 0.01 | 4 | 3.981 ± 0.06 | 4.001 ± 0.03 | 4.000 ± 0.03 | 4 |
5 | 55.988 ± 0.01 | 55.991 ± 0.01 | 55.992 ± 0.00 | 56 | 67.77 ± 0.00 | 67.77 ± 0.00 | 67.77 ± 0.00 | 5 | 3.984 ± 0.08 | 3.998 ± 0.05 | 3.987 ± 0.05 | 4 |
6 | 65.012 ± 0.01 | 65.006 ± 0.01 | 65.005 ± 0.01 | 65 | 60.59 ± 0.00 | 60.59 ± 0.00 | 60.59 ± 0.00 | 6 | 2.977 ± 0.14 | 2.980 ± 0.09 | 2.996 ± 0.09 | 3 |
7 | 61.006 ± 0.01 | 61.005 ± 0.01 | 61.005 ± 0.00 | 61 | 52.21 ± 0.01 | 52.21 ± 0.00 | 52.21 ± 0.00 | 7 | 2.047 ± 0.13 | 2.030 ± 0.09 | 2.021 ± 0.10 | 2 |
8 | 43.988 ± 0.02 | 43.986 ± 0.01 | 43.983 ± 0.01 | 44 | 42.68 ± 0.02 | 42.68 ± 0.02 | 42.68 ± 0.01 | 8 | 2.037 ± 0.09 | 2.025 ± 0.05 | 2.025 ± 0.06 | 2 |
9 | 58.003 ± 0.02 | 57.999 ± 0.01 | 58.002 ± 0.01 | 58 | 62.12 ± 0.00 | 62.12 ± 0.00 | 62.12 ± 0.00 | 9 | 4.882 ± 0.07 | 4.920 ± 0.06 | 4.917 ± 0.06 | 5 |
10 | 62.996 ± 0.01 | 62.998 ± 0.01 | 62.998 ± 0.01 | 63 | 54.30 ± 0.00 | 54.30 ± 0.00 | 54.30 ± 0.00 | 10 | 0.058 ± 0.04 | 0.029 ± 0.03 | 0.028 ± 0.03 | 0 |
11 | 58.006 ± 0.01 | 58.006 ± 0.01 | 58.004 ± 0.01 | 58 | 44.82 ± 0.01 | 44.82 ± 0.01 | 44.82 ± 0.00 | 11 | 2.076 ± 0.11 | 2.053 ± 0.06 | 2.059 ± 0.07 | 2 |
12 | 53.997 ± 0.01 | 53.997 ± 0.01 | 53.997 ± 0.01 | 54 | 37.33 ± 0.01 | 37.33 ± 0.01 | 37.33 ± 0.00 | 12 | 4.924 ± 0.08 | 4.938 ± 0.06 | 4.936 ± 0.06 | 5 |
13 | 0.143 ± 0.06 | 0.128 ± 0.06 | 0.129 ± 0.06 | 0 | ||||||||
14 | 4.733 ± 0.11 | 4.781 ± 0.14 | 4.766 ± 0.10 | 5 | ||||||||
15 | 1.255 ± 0.13 | 1.220 ± 0.16 | 1.241 ± 0.11 | 1 | ||||||||
16 | 4.865 ± 0.17 | 4.864 ± 0.13 | 4.856 ± 0.10 | 5 | ||||||||
17 | 3.123 ± 0.17 | 3.127 ± 0.13 | 3.131 ± 0.10 | 3 |
Node . | *Mean predicted demand (l/s) . | Observed demand (l/s) . | *Mean pressure (m) . | Pipe . | *Mean predicted leakage (l/s) . | Leakage (l/s) . | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
ρ = 8 . | ρ = 9 . | ρ = 10 . | ρ = 8 . | ρ = 9 . | ρ = 10 . | ρ = 8 . | ρ = 9 . | ρ = 10 . | ||||
1 | 54.995 ± 0.01 | 54.999 ± 0.00 | 54.998 ± 0.00 | 55 | 70.74 ± 0.00 | 70.74 ± 0.00 | 70.74 ± 0.00 | 1 | 2.989 ± 0.06 | 2.978 ± 0.03 | 2.986 ± 0.04 | 3 |
2 | 47.001 ± 0.02 | 47.003 ± 0.01 | 47.001 ± 0.01 | 47 | 66.39 ± 0.00 | 66.39 ± 0.00 | 66.39 ± 0.00 | 2 | 0.056 ± 0.03 | 0.040 ± 0.02 | 0.037 ± 0.02 | 0 |
3 | 41.991 ± 0.02 | 41.993 ± 0.01 | 41.994 ± 0.01 | 42 | 62.18 ± 0.00 | 62.18 ± 0.00 | 62.18 ± 0.00 | 3 | 2.934 ± 0.05 | 2.949 ± 0.03 | 2.952 ± 0.04 | 3 |
4 | 35.954 ± 0.02 | 35.958 ± 0.02 | 35.957 ± 0.02 | 36 | 49.15 ± 0.02 | 49.15 ± 0.02 | 49.15 ± 0.01 | 4 | 3.981 ± 0.06 | 4.001 ± 0.03 | 4.000 ± 0.03 | 4 |
5 | 55.988 ± 0.01 | 55.991 ± 0.01 | 55.992 ± 0.00 | 56 | 67.77 ± 0.00 | 67.77 ± 0.00 | 67.77 ± 0.00 | 5 | 3.984 ± 0.08 | 3.998 ± 0.05 | 3.987 ± 0.05 | 4 |
6 | 65.012 ± 0.01 | 65.006 ± 0.01 | 65.005 ± 0.01 | 65 | 60.59 ± 0.00 | 60.59 ± 0.00 | 60.59 ± 0.00 | 6 | 2.977 ± 0.14 | 2.980 ± 0.09 | 2.996 ± 0.09 | 3 |
7 | 61.006 ± 0.01 | 61.005 ± 0.01 | 61.005 ± 0.00 | 61 | 52.21 ± 0.01 | 52.21 ± 0.00 | 52.21 ± 0.00 | 7 | 2.047 ± 0.13 | 2.030 ± 0.09 | 2.021 ± 0.10 | 2 |
8 | 43.988 ± 0.02 | 43.986 ± 0.01 | 43.983 ± 0.01 | 44 | 42.68 ± 0.02 | 42.68 ± 0.02 | 42.68 ± 0.01 | 8 | 2.037 ± 0.09 | 2.025 ± 0.05 | 2.025 ± 0.06 | 2 |
9 | 58.003 ± 0.02 | 57.999 ± 0.01 | 58.002 ± 0.01 | 58 | 62.12 ± 0.00 | 62.12 ± 0.00 | 62.12 ± 0.00 | 9 | 4.882 ± 0.07 | 4.920 ± 0.06 | 4.917 ± 0.06 | 5 |
10 | 62.996 ± 0.01 | 62.998 ± 0.01 | 62.998 ± 0.01 | 63 | 54.30 ± 0.00 | 54.30 ± 0.00 | 54.30 ± 0.00 | 10 | 0.058 ± 0.04 | 0.029 ± 0.03 | 0.028 ± 0.03 | 0 |
11 | 58.006 ± 0.01 | 58.006 ± 0.01 | 58.004 ± 0.01 | 58 | 44.82 ± 0.01 | 44.82 ± 0.01 | 44.82 ± 0.00 | 11 | 2.076 ± 0.11 | 2.053 ± 0.06 | 2.059 ± 0.07 | 2 |
12 | 53.997 ± 0.01 | 53.997 ± 0.01 | 53.997 ± 0.01 | 54 | 37.33 ± 0.01 | 37.33 ± 0.01 | 37.33 ± 0.00 | 12 | 4.924 ± 0.08 | 4.938 ± 0.06 | 4.936 ± 0.06 | 5 |
13 | 0.143 ± 0.06 | 0.128 ± 0.06 | 0.129 ± 0.06 | 0 | ||||||||
14 | 4.733 ± 0.11 | 4.781 ± 0.14 | 4.766 ± 0.10 | 5 | ||||||||
15 | 1.255 ± 0.13 | 1.220 ± 0.16 | 1.241 ± 0.11 | 1 | ||||||||
16 | 4.865 ± 0.17 | 4.864 ± 0.13 | 4.856 ± 0.10 | 5 | ||||||||
17 | 3.123 ± 0.17 | 3.127 ± 0.13 | 3.131 ± 0.10 | 3 |
*Average of 20 runs.
The model could detect both locations and amounts of water loss in all nodes and pipes of the WDNs without using observed pressure data (see Equation (1)). Also, although there are many unknown leakages in WDNs, which cannot be calculated by Equation (3) with the observed data, they were detected successfully. These demonstrate that the model minimizes the number of required field measurements used in a model calibration (pressures are not used) for detecting water loss in all pipes and nodes of WDNs. Moreover, the sensitivity analysis of the modified Clonalg was carried out according to ρ (decay coefficient) to test its search capability in the optimization problems. The results showed that the model appeared to be significantly successful and feasible for water loss detection in WDNs. In future studies, the performance of this model needs to be explored in different WDNs.