# Robust Meter Network for Water Distribution Pipe Burst Detection

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Detection Performance Measures

#### 2.2. Meter Network’s Robustness

- Step 1: For a given meter set with a total number of meters N, one of the meters (ith meter, i = 1, ..., N) is assumed to fail in a long time (e.g., for two days). No data measurement is received by the supervisory control and data acquisition (SCADA) system from the meter due to either meter malfunction or communication system failure. Thus, the ith meter’s DP is set to zero.
- Step 2: The network’s DP in the event of failure of the ith meter (DP
_{f,i}) is then calculated. - Step 3: The failed/impaired meter is then assumed to be back to its normal condition, and another meter is assumed to fail.
- Step 4: Steps 1–3 are repeated until the failure of all meters has been evaluated (i.e., DP
_{f,i}is calculated for i = 1, ..., N). - Step 5: CV of DP
_{f,i}(CVDP_{f}) is then calculated:$${\mathrm{CVDP}}_{f}=\frac{\mathsf{\sigma}\left({\mathrm{DP}}_{f,i}\right)}{avg\left({\mathrm{DP}}_{f,i}\right)},i=1,\dots ,\mathrm{N}$$_{f,i}) and σ(DP_{f,i}) are the average and standard deviation, respectively, of DP_{f,i}(i = 1, ..., N). - Step 6: Finally, MeterRob is calculated as:$$\mathrm{MeterRob}=1-{\mathrm{CVDP}}_{f}$$

_{f}with low σ(DP

_{f,i}) and high avg(DP

_{f,i}). Note that MeterRob has a value between 0 and 1 because CVDP

_{f}is less than 1. DP

_{f,i}is a ratio value between 0 and 1, so $\mathsf{\sigma}\left({\mathrm{DP}}_{f,i}\right)$ will always be smaller than avg(DP

_{f,i}).

#### 2.3. Multi-Objective Optimal Meter Placement (MOMP) Model

_{1}), minimize RF (F

_{2}), and maximize MeterRob (F

_{3}):

#### 2.4. Pipe Burst Detection Method: Western Electric Company (WEC) Rules

- Rule 1: Any single measurement is beyond the ±4σ CL.
- Rule 2: Two of three consecutive measurements are beyond the ±3σ WL.
- Rule 3: Four of five consecutive measurements are beyond the ±2σ WL.
- Rule 4: Eight consecutive measurements are beyond the ±1σ WL.

#### 2.5. Data Generation and Detection and False Alarm Matrix

_{n,m}is set to 1 if the mth meter detects the n burst event. The element value of 1 in F indicates a false alarm. To populate the two matrices, the WEC rules were applied to each meter’s data. A number of burst events are considered to have reliable DP and RF values. The matrices D and F are nburst × N and nnormal × N matrices, respectively. Note that N is nn for the pressure meter and np for a pipe flowmeter, where np is the total number of pipes.

_{l}= 1 (l = 1, 2, ..., N), and no meter is located there otherwise. Therefore, DP of a given meter set (meterset) is computed as follows:

## 3. Study Network

## 4. Results

#### 4.1. Pareto Optimal Solutions for Pipe Flow Meter Placement

#### 4.2. Pareto Optimal Solutions for Pressure Meter Placement

#### 4.3. Optimal Meter Placement Comparison

## 5. Summary and Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

Acronyms | |

WDS | Water Distribution System |

OMP | Optimal Meter Placement |

MOMP | Multi-objective Optimal Meter Placement |

DP | Detection Probability |

RF | Rate of False Alarms |

SPC | Statistical Process Control |

CV | Coefficient of Variation |

SCADA | Supervisory Control and Data Acquisition |

NSGA-II | Non-dominated Sorting Genetic Algorithm-II |

WEC | Western Electric Company Rules |

WL | Warning Limit |

CL | Control Limit |

DMA | District Metering Area |

Symbols | |

nmeter | Predefined total number of meters in the proposed MOMP model |

nburst | Total number of burst events |

nnormal | Total number of natural events |

N | Total number of meters |

nn | Total number of nodes |

np | Total number of pipes |

DP_{f,i} | Meter network’s DP in the event of failure of the ith meter |

CVDP_{f} | Coefficient of variation of DP_{f,i} |

avg(DP_{f,i}) | Average of DP_{f,i} |

σ(DP_{f,i}) | Standard deviation of DP_{f,i} |

MeterRob | Meter network’s robustness indicator and 1-CVDP_{f} |

F_{1} | The first objective function of the proposed MOMP model (DP) |

F_{2} | The second objective function of the proposed MOMP model (RF) |

F_{3} | The third objective of function the proposed MOMP model (MeterRob) |

μ | Time-varying mean value of the process variable (e.g., pipe flows and nodal pressures) |

σ | Time-varying standard deviation value of the process variable |

C | Burst discharge coefficient |

α | The burst flow power equation’s exponent |

D | Detection matrix (its subscript indicates the location of element in D) |

F | False alarm matrix (its subscript indicates the location of the element in F) |

M | Meter configuration matrix (its subscript indicates the location of the element in M) |

meterset | A given meter set |

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**Figure 2.**Projections of the Pareto optimal solutions of the pipe flow meters given nmeter = 3 in (

**a**) a three-dimensional plot and two-dimensional plots of (

**b**) DP and RF; (

**c**) DP and MeterRob; and (

**d**) RF and MeterRob. Non-dominated solutions with respect to the two objectives were identified in a post-optimization process and are marked with asterisks in Figure 2b–d.

**Figure 3.**Comparison of two Pareto optimal solutions for pipe flow meter placement with nmeter = 3 (circle points) and 5 (diamond points) in two-dimensional plots of (

**a**) DP and RF; (

**b**) DP and MeterRob; and (

**c**) RF and MeterRob. Non-dominated solutions with respect to the two objectives were identified in a post-optimization process and are marked with asterisk points for nmeter = 3 and cross marks for nmeter = 5.

**Figure 4.**Projections of the Pareto optimal solutions of pressure meters given nmeter = 3 in (

**a**) a three-dimensional plot and two-dimensional plots of (

**b**) DP and RF; (

**c**) DP and MeterRob; and (

**d**) RF and MeterRob. Non-dominated solutions with respect to the two objectives were identified in a post-optimization process and are marked with asterisk points in Figure 4b–d.

**Figure 5.**Comparison of two Pareto optimal solutions for pressure meter placement with nmeter = 3 (circle points) and 5 (diamond points) in two-dimensional plots of (

**a**) DP and RF; (

**b**) DP and MeterRob; and (

**c**) RF and MeterRob. Non-dominated solutions with respect to the two objectives were identified in a post-optimization process and are marked with asterisk points for nmeter = 3 and cross marks for nmeter = 5.

**Figure 6.**Meter locations of the three selected solutions given in Table 1.

**Figure 7.**Meter locations of the three selected solutions given in Table 2.

**Table 1.**DP, RF, and MeterRob values of the three selected solutions obtained for three pipe flow meter placements (nmeter = 3).

Solution Identifier | Objective Function Values | Meter Locations | ||||
---|---|---|---|---|---|---|

DP | RF | MeterRob | Meter 1 | Meter 2 | Meter 3 | |

Solution 1 | 0.9 | 0.091 | 0.846 | Pipe 1 | Pipe 16 | Pipe 33 |

Solution 2 | 0.9 | 0.111 | 0.908 | Pipe 1 | Pipe 18 | Pipe 5 |

Solution 3 | 0.9 | 0.256 | 0.988 | Pipe 1 | Pipe 2 | Pipe 79 |

**Table 2.**DP, RF, and MeterRob values of the three selected solutions for three pressure meter placements (nmeter = 3).

Solution Identifier | Objective Function Values | Meter Locations | ||||
---|---|---|---|---|---|---|

DP | RF | MeterRob | Meter 1 | Meter 2 | Meter 3 | |

Solution 1 | 0.875 | 0.044 | 0.9993 | Node 125 | Node 154 | Node 157 |

Solution 2 | 0.885 | 0.059 | 0.9989 | Node 97 | Node 135 | Node 155 |

Solution 3 | 0.894 | 0.071 | 0.9994 | Node 91 | Node 141 | Node 155 |

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**MDPI and ACS Style**

Jung, D.; Kim, J.H.
Robust Meter Network for Water Distribution Pipe Burst Detection. *Water* **2017**, *9*, 820.
https://doi.org/10.3390/w9110820

**AMA Style**

Jung D, Kim JH.
Robust Meter Network for Water Distribution Pipe Burst Detection. *Water*. 2017; 9(11):820.
https://doi.org/10.3390/w9110820

**Chicago/Turabian Style**

Jung, Donghwi, and Joong Hoon Kim.
2017. "Robust Meter Network for Water Distribution Pipe Burst Detection" *Water* 9, no. 11: 820.
https://doi.org/10.3390/w9110820