# Optimal Number of Pressure Sensors for Real-Time Monitoring of Distribution Networks by Using the Hypervolume Indicator

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. General Approach

#### 2.2. Definition of the Maximum Number of Sensors

^{24}, 2.24 × 10

^{25}and 1.31 × 10

^{26}possible combinations of sensors, respectively. Therefore, the maximum number of installed sensors, N

_{max}, is firstly defined, aiming at minimizing the computational burden by limiting the number of optimization problems to be solved.

#### 2.3. Establishment of the Discrete Set of Numbers of Sensors

_{max}, for which the optimization problem will be solved. The established set does not have to be a continuous sequence of natural numbers, for instance, it could be 1, 5, 10, 15,…, N

_{max}. The aim is to reduce the number of optimization problems to be solved whilst leaving enough results to characterize the total gain as a function of the number of sensors between 1 and N

_{max}. As such, different sets of the number of sensors can be considered, for instance, with an evenly distributed number of sensors between 1 and N

_{max}or with a higher density of observations in the lower number of sensors. The effect of different discrete sets of the numbers of sensors is assessed in this article.

#### 2.4. Optimization of the Pressure Sensor Locations

_{i,j}refers to the variation of the pressure in node j with respect to the variation of the pipe roughness coefficient of pipe i. Similarly, S2

_{i,j}refers to the variation of the pressure in node j with respect to the variation in the pipe burst size in node i. Further details regarding pressure sensitivity analysis can be found in the works by de Schaetzen et al. [20] and Lansey et al. [30].

_{1}is defined according to de Schaetzen et al. [20] by using a compromise programming formulation. It aims at maximizing the sensitivity to the pipe roughness coefficient that the sensors cover (f

_{A}) whilst ensuring the evenly spread geographical distribution of sensor locations by maximizing the entropy according to Shannon’s definition (f

_{B}). Functions f

_{A}and f

_{B}are defined as follows:

_{A}and f

_{B}are combined together into f

_{1}by taking the weighted sum of the two functions.

_{2}aims at directly maximizing the sensitivity of sensors to pipe burst events and can be computed as follows:

#### 2.5. Assessment of the Total Gain of Sensors

_{1}and f

_{2}since the methodology can be adopted to different optimization problem formulations with different dimensions). Furthermore, multiple optimal configurations might exist for a given number of sensors due to a trade-off between the objectives. This is represented in Figure 2b with a Pareto front of four optimal combinations of five sensors depicted with red dots. Note that the increase of objective function 3 will lead to the decrease of objective functions 1 and 2 as a trade-off between these contradictory objectives.

#### 2.6. Trade-Off Function Fitting

_{max}.

#### 2.7. Determination of the Optimal Number of Pressure Sensors

_{norm}. The number of sensors between 1 and N

_{max}should also be rescaled to [0,1], leading to the normalized number of sensors values, N

_{norm}. These values are represented in Figure 3 as black circles and squares, respectively, considering N

_{max}= 15 in this example. Finally, the differences Δ between HV

_{norm}and N

_{norm}are calculated, as exemplified with grey arrows and triangular markers in Figure 3. The optimal number of sensors, N

_{opt}, is selected as the number of sensors that presents the maximum difference.

_{max}= 7. The seven hypervolume values are presented both as blue or orange circles (either they are on the left or right side of the junction point). The corresponding fitted straight lines are represented with blue or red dashed lines, respectively. The number of sensors tested for the optimal number is in a blue diamond. The optimal number of sensors is the junction point that minimizes the weighted RMSE for the two linear parts of the hypervolume data.

_{opt}. Nonetheless, the Pareto front might not have yet been obtained (i.e., when N

_{opt}was not considered in the discrete set). In these cases, the optimization problem should be solved for the number of sensors equal to N

_{opt}.

## 3. Case Study

^{3}/h. The supplied area consists mainly of single-family houses with irrigated gardens, most of them with a swimming pool. The WDN also supplies a few golf courses. The hydraulic simulation model was developed in EPANET [48] and included four storage tanks, 4474 pipes and 4429 nodes, of which 2200 were consumers with an individual hourly demand pattern. Figure 5 presents the layout of the hydraulic network.

_{max}, was determined based on the intrinsic network characteristics, namely, on network length. A maximum of one sensor per kilometer of pipe network was adopted according to several Portuguese expert opinions, leading to N

_{max}= 70 sensors (i.e., one sensor per km).

_{max}whilst Set 3, Set 4 and Set 5 incorporated a higher density in the lower number of sensors. Furthermore, the number of observations highly varied between the sets: Set 2 and Set 3 contained only five numbers of sensors (leading to five optimization problems to be solved) whilst Set 5 contained 26 numbers of sensors (leading to a total of 26 optimization problems).

_{1}) and pipe burst events (f

_{2}).

_{m}= 0.05 and index parameter n

_{m}= 20; simulated binary crossover with probability p

_{c}= 0.95 and index parameter n

_{c}= 20.

_{1}and f

_{2}. For the sake of simplicity, the obtained Pareto fronts for 10, 20, 30 and 40 sensors are depicted in Figure 6. Each circle represents an optimal configuration of either 10, 20, 30 or 40 sensors.

_{i}, were considered to describe the hypervolume data as a function of the number of sensors, N:

_{i}to each set of hypervolume values by finding the optimal set of parameters. This was done by using MATLAB’s Curve Fitting Toolbox. A reasonable initial parameter guess was found by coarsely gridding the parameter space. This was carried out to cope method sensitivity to the initial parameters. Figure 7 presents in colored lines the estimated hypervolume values obtained by using each model F

_{i}. The original hypervolume values are represented as triangular black markers.

_{i}of each set of numbers of sensors in order to assist in deciding which function F

_{i}best describes the hypervolume data. The obtained results are presented in Table 3, with the best fitted function for each set of numbers of sensors presented in bold and shaded areas. Finally, the optimal number of sensors N

_{opt}(considered as the point of maximum curvature) was determined using an automatic detection technique. The two distinct techniques presented in Section 2.7 were used for each function F

_{i}of each set of numbers of sensors, with the obtained results presented in Table 3. Figure 8 presents the application of both Kneedle method and L-method techniques to the estimated hypervolume for the F

_{5}function of Set 1.

## 4. Discussion

_{1}(grey line) clearly fails to mathematically describe the trend of the hypervolume as a function of the number of sensors, whereas the remaining functions can approximately describe the hypervolume variation.

_{i}for each discrete set of numbers of sensors. The best fit corresponds to the smallest values of the RMSE. Complex models F

_{4}and F

_{5}(i.e., both with a larger number of parameters) presented the best fitting results (in bold) for most sets of numbers of sensors, except for Set 3, for which F

_{3}led to the lowest RMSE values.

_{i}to estimate the hypervolume value (note that results from F

_{1}were excluded from analysis). Technique 1 presented higher variability in the optimal number of sensors, ranging from 16 to 18. Technique 2 presented smaller variability, between 16 and 17. Therefore, and as long as selected functions F

_{i}(and respective parameters) can properly represent the hypervolume as a function of the number of sensors, a good estimation of N

_{opt}can be obtained (considering either the Kneedle method or the L-method).

_{i}, it was possible to verify that the optimal number was relatively stable for the different sets, and no differences greater than one were found. Thus, and as long as the selected discrete set of numbers of sensors can properly represent the hypervolume as a function of the number of sensors, a good estimation of N

_{opt}can be obtained. This allows for a significant reduction in the number of optimization problems to be solved as similar results are obtained when both five (in Set 2 and Set 3) or 26 (in Set 5) optimization problems are solved.

_{i}as the optimal number varied between 16 and 18 sensors. Furthermore, when the best function F

_{i}was considered for each set (results in bold in Table 3), the optimal number of sensors was equal to 16 or 17 (depending on the “knee/elbow” detection technique) regardless of the discrete set of numbers of sensors. Based on these results, the Kneedle method was recommended given the straightforward implementation and easier concept understanding.

_{1}), any discrete set of numbers of sensors and any “knee/elbow” detection technique.

## 5. Conclusions

- The hypervolume indicator can be used to characterize the total gain of installing pressure sensors whose locations are obtained by solving multi-objective optimization problems.
- A trade-off function can be derived, allowing the characterization of the total gain of installing sensors (i.e., hypervolume) as a function of the number of sensors. This trade-off function fitting process allows for a great reduction in the computational effort associated with the overall analysis, as similar results are obtained when solving both five and 26 multi-objective optimization problems.
- The obtained results are not affected by considering different trade-off functions or different sets of numbers of sensors as long as the selected function and the selected discrete set can properly represent the hypervolume as a function of the number of sensors.
- Both Techniques 1 and 2 lead to similar results. The use of Technique 1 (Kneedle method) is further recommended given the straightforward implementation and easier concept understanding.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Characterization of the total gain for a given number of sensors using the hypervolume indicator.

**Figure 3.**Determination of the optimal number of sensors based on the evolution of the hypervolume by using the Kneedle method.

**Figure 4.**Determination of the optimal number of sensors based on the evolution of the hypervolume using the second technique.

**Figure 7.**Estimated hypervolume values using different fitted functions F

_{i}and different discrete sets of numbers of sensors.

**Figure 8.**Optimal number of sensors for the F

_{5}function of Set 1 using (

**a**) the Kneedle method and (

**b**) the L-method.

Sets of Numbers of Sensors | Number of Observations | Objective Function Evaluations |
---|---|---|

$\begin{array}{c}Set1=\left[1,10,20,30,40,50,60,70\right]\end{array}$ | 8 | 400,000 |

$\begin{array}{c}Set2=\left[1,10,30,50,70\right]\end{array}$ | 5 | 250,000 |

$Set3=\left[1,10,20,30,70\right]$ | 5 | 250,000 |

$Set4=\left[1,5,10,15,20,25,30,70\right]$ | 8 | 400,000 |

$\begin{array}{c}Set5=\left[1,2,3,\dots ,24,25,70\right]\end{array}$ | 26 | 1,300,000 |

Fitting Function | a | b | c | d |
---|---|---|---|---|

${F}_{1}\left(N\right)=a{N}^{b}$ | 532,809 | 0.308 | - | - |

${F}_{2}\left(N\right)=\left(aN\right)/\left(b+N\right)$ | 2,007,063 | 6.465 | − | − |

${F}_{3}\left(N\right)=a+b{log}_{10}\left(N\right)-cN$ | 144,141 | 1,190,045 | 7835 | − |

${F}_{4}\left(N\right)=a{e}^{bN}+c{e}^{dN}$ | 1,665,878 | 0.001 | −1,733,051 | −0.129 |

${F}_{5}\left(N\right)=a{\left(N+b\right)}^{c}+d$ | −136,128,665 | 10.779 | −1.772 | 1,861,283 |

**Table 3.**RMSE and the optimal number of sensors N

_{opt}for the different fitted functions and discrete sets of sensors. The results for the best-fitted function of each discrete set are presented in bold and shaded area.

Number of Sensors’ Set | ${\mathit{F}}_{1}\left(\mathit{N}\right)$ | ${\mathit{F}}_{2}\left(\mathit{N}\right)$ | ${\mathit{F}}_{3}\left(\mathit{N}\right)$ | ${\mathit{F}}_{4}\left(\mathit{N}\right)$ | ${\mathit{F}}_{5}\left(\mathit{N}\right)$ | |
---|---|---|---|---|---|---|

$\begin{array}{c}Set1=\left[1,10,20,30,40,50,60,70\right]\end{array}$ | RMSE | 2.2 × 10^{5} | 6.0 × 10^{4} | 1.9 × 10^{4} | 2.0 × 10^{4} | 1.5 × 10^{4} |

N_{opt} (Kneedle method) | 20 | 17 | 16 | 18 | 17 | |

N_{opt} (L-method) | 20 | 16 | 16 | 17 | 16 | |

$\begin{array}{c}Set2=\left[1,10,30,50,70\right]\end{array}$ | RMSE | 2.7 × 10^{5} | 7.6 × 10^{4} | 2.2 × 10^{4} | 3.0 × 10^{4} | 2.6 × 10^{4} |

N_{opt} (Kneedle method) | 20 | 18 | 16 | 18 | 17 | |

N_{opt} (L-method) | 20 | 17 | 16 | 16 | 16 | |

$\begin{array}{c}Set3=\left[1,10,20,30,70\right]\end{array}$ | RMSE | 3.3 × 10^{5} | 8.9 × 10^{4} | 2.9 × 10^{4} | 1.0 × 10^{4} | 3.3 × 10^{4} |

N_{opt} (Kneedle method) | 19 | 17 | 16 | 17 | 17 | |

N_{opt} (L-method) | 20 | 16 | 16 | 16 | 16 | |

$\begin{array}{c}Set4=\left[1,5,10,15,20,25,30,70\right]\end{array}$ | RMSE | 2.8 × 10^{5} | 7.9 × 10^{4} | 6.1 × 10^{4} | 1.4 × 10^{4} | 2.2 × 10^{4} |

N_{opt} (Kneedle method) | 19 | 17 | 16 | 17 | 17 | |

N_{opt} (L-method) | 19 | 16 | 16 | 16 | 16 | |

$\begin{array}{c}Set5=\left[1,2,3,\dots ,24,25,70\right]\end{array}$ | RMSE | 4.1 × 10^{5} | 1.3 × 10^{5} | 6.4 × 10^{4} | 1.5 × 10^{4} | 2.4 × 10^{4} |

N_{opt} (Kneedle method) | 20 | 18 | 16 | 17 | 17 | |

N_{opt} (L-method) | 21 | 17 | 16 | 16 | 16 |

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Ferreira, B.; Carriço, N.; Covas, D.
Optimal Number of Pressure Sensors for Real-Time Monitoring of Distribution Networks by Using the Hypervolume Indicator. *Water* **2021**, *13*, 2235.
https://doi.org/10.3390/w13162235

**AMA Style**

Ferreira B, Carriço N, Covas D.
Optimal Number of Pressure Sensors for Real-Time Monitoring of Distribution Networks by Using the Hypervolume Indicator. *Water*. 2021; 13(16):2235.
https://doi.org/10.3390/w13162235

**Chicago/Turabian Style**

Ferreira, Bruno, Nelson Carriço, and Dídia Covas.
2021. "Optimal Number of Pressure Sensors for Real-Time Monitoring of Distribution Networks by Using the Hypervolume Indicator" *Water* 13, no. 16: 2235.
https://doi.org/10.3390/w13162235