# Identification of Critical Nodes in Water Distribution Networks

Thapelo Mosetlhe, Yskandar Hamam, D Shengzhi, Eric Monacelli

► To cite this version:

Thapelo Mosetlhe, Yskandar Hamam, D Shengzhi, Eric Monacelli. Identification of Critical Nodes in Water Distribution Networks. 2nd International Symposium on Water, Ecology and Environment, Oct 2021, Beijing, China. pp.012004, 10.1088/1755-1315/987/1/012004. [hal-04778982](https://hal.science/hal-04778982v1)

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PAPER • OPEN ACCESS

# Identification of Critical Nodes in Water Distribution Networks

To cite this article: Thapelo C. Moselhe et al 2022 IOP Conf. Ser.: Earth Environ. Sci. 987 012004

View the [article online](#) for updates and enhancements.

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# Identification of Critical Nodes in Water Distribution Networks

Thapelo C. Mosetlhe¹,²*, Yskandar Hamam²,³, Shengzhi D² and Eric Monacelli⁴

¹Department of Electrical and Mining Engineering, University of South Africa,
Florida, South Africa

²Department of Electrical Engineering, Tshwane University of Technology, Pretoria,
South Africa

³École Supérieure d'Ingénieurs en Électrotechnique et Électronique, Noisy-Le-Grand,
France

⁴Laboratoire d'Ingénierie des Systémes de Versailles, Université Paris-Saclay, Vélizy,
France

*Email: mosetlhethapelo@yahoo.com

**Abstract.** In this work, the problem of the identification of critical nodes in water distribution networks (WDNs) is investigated. A scheme is then put forward to identify critical nodes in WDNs. Mathematical formulations for the scheme put forward are provided in this work. Results show that the application of the scheme yields consistent outputs when applied to the various WDNs test cases. Furthermore, critical nodes could be ranked in order of their criticality based on the sensitivity index.

## 1. Introduction

Water distribution networks (WDNs) management relies much on the availability of information (measurements). The sizes of WDNs brings about the challenge in terms of taking measurements on all the nodes and/or pipes. In addition, the cost associated with the deployment of these infrastructures is also a concern. Therefore, there is a need to identify critical points to ensure that a limited number of measurements are taken.

Diao et al. [1] proposed the modelling of the water distribution system as an undirected and directed graph in other to identify critical components. Their results showed that 77 % of critical links are pipes connected to the actuator nodes. The importance of the identification of critical nodes is brought out in [2]. Measurements on critical remote nodes are used to ensure that the pressure of the system is maintained within limits. Page et al. [3] proposes a parameter-less P-controller with the input being measured from a critical node. The measurement from the critical enables the control of the pressure in the WDNs. An adaptive reference control scheme was used with partial pressure information from the critical nodes; however, the scheme was able to meet the pressure requirement [4]. In [5], a clustering-based approach was put forward to identify critical nodes and edges. The scheme put forward was able to identify the top five critical nodes and edges. In [6], critical links are identified as pipes with leakage flow above a certain threshold.

The works that have been done clearly show that critical nodes and edges must be identified in WDNs. This is to ensure that adequate steps can be taken timeously in the case that there is a need. Furthermore, this also alleviates the costs associated with the deployment of measuring instruments. As a result, this

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Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
work proposes a novel greedy approach to identify critical nodes. The scheme is based on running iterative hydraulic simulations while varying water demand.

The rest of the paper is organised as follows: Section 2 presents the modelling and the formulation of the greedy algorithm. Results and their discussion are presented in Section 3 and some conclusions are drawn in Section 4.

## 2. Modelling

### 2.1. Water Distribution Model

Generally, the solution (hydraulic simulations) of the water distribution networks (WDNs) can be obtained by an iterative solution of the system of Equation in (1).

$$ 
\begin{aligned}
C_l Q + L &= 0 \\
A Q - C_s^T h_s - C_l^T h_l &= 0
\end{aligned}
\qquad (1) $$

In Equation (1), Q and L are the flows through the pipes and the quantity of water demand at the withdrawal nodes [7, 8]. The pressure head at the supply nodes is represented by $h_s$ while $h_l$ denotes the pressure head at the withdrawal nodes. A is the diagonal matrix of the pressure losses along the pipes. $C_s$ and $C_l$ represents the components of the decomposed incidence matrix. The incidence matrix is defined as:

$$ C_{ij} = \begin{cases} +1, & \text{if flow in branch } j \text{ leaves node } i \\ -1, & \text{if flow in branch } j \text{ enters node } i \\ 0, & \text{if branch } j \text{ is not incident to node } i \end{cases} \qquad (2) $$

### 2.2. Identification of Critical Nodes

Due to budgetary and physical constraints, there is a need to estimate the system pressure from limited nodal measurements. In WDNs, critical nodes are used as indicators for state of operation of the network. These are sensitive and are susceptible to the occurrence of over and under pressure as the nodal withdrawal varies. It is well known that the variation of the demand on the network influence the pressure in the WDNs. The sensitivity of a node can be determined using Equation (3).

$$ S = \frac{\partial h_i}{\partial L_i} \qquad (3) $$

and

$$ \frac{\partial h_i}{\partial L_i} \approx \frac{h_{ref} - h_i}{\Delta L} \qquad (4) $$

where,

$$ \Delta L = L_{base} - L_{current} \qquad (5) $$

$L_{base}$ is the base demand for the network and $L_{current}$ is the current demand/variation of the demand. A vector can now be defined with the ranking of the nodes with the highest variation as

$$ S = [S_1 \ S_2 \ \dots \ S_n] \qquad (6) $$

where $S_1$ is the node with the highest variation and $S_n$ has the lowest sensitivity index. A threshold of the variation could be defined as $\theta$. Therefore, nodes that have the variation that is less than the threshold could be eliminated from Equation (6) taking into account that the number of critical nodes must be limited to a fixed number z and the resultant be denoted by $\bar{S}$. For a demand pattern over a time
period (m periods), the critical node index vector could be defined. A matrix with dimension m × z comprising critical nodes could be defined as

$$CNI = \begin{bmatrix} S_{1,1} & S_{1,2} & \dots & S_{1,z} \\ S_{2,1} & S_{2,2} & \dots & S_{2,z} \\ \vdots & \vdots & \vdots & \vdots \\ S_{m,1} & S_{m,2} & \dots & S_{m,z} \end{bmatrix} \quad (7)$$

The final ranking can be defined as a vector with a node having the highest variation and appearing in most rows as the most critical (first element in the vector). The reverse will be the least critical node and last in the vector. The solution to the identification of critical nodes can be determined using the greedy algorithm and its structure is shown in Algorithm 1.

<table><caption>**Algorithm 1.** Greedy Algorithm for Identification of Critical Nodes.</caption><tr><td>Input</td><td>:</td><td>Network parameters</td></tr><tr><td>Input</td><td>:</td><td>Define base demand over period</td></tr><tr><td>Results</td><td>:</td><td>Critical Nodes</td></tr><tr><td>Initialisation</td><td></td><td></td></tr><tr><td>CNI</td><td>←</td><td>0</td></tr><tr><td>Convergence</td><td>←</td><td>False</td></tr><tr><td>while</td><td>!True</td><td>do<br/>Run hydraulic simulation<br/>S<sub>i</sub> = [S<sub>1</sub>, S<sub>2</sub>, ..., S<sub>n</sub>]<br/>do if Length of S<sub>i</sub> > then<br/>z<br/>S<sub>i</sub> ← S<sub>i</sub> (row=1; column=1; z)<br/>else<br/>S<sub>i</sub> ← S<sub>i</sub><br/>CNI (row = I, column =all) ← S<sub>i</sub><br/>if end of period then<br/>Convergence ← True</td></tr><tr><td>Reshape</td><td>CNI</td><td>to be vector CNI<sub>1</sub>, CNI<sub>2</sub>, ..., CNI<sub>n</sub>, node with highest variation</td></tr><tr><td>Return</td><td>CNI<sub>1</sub></td><td></td></tr></table>

### 3. Results and Discussion

To evaluate the appropriateness of the proposed scheme, two case studies are used. The base demand is adapted from [6] and it is shown in Figure 1. It can be easily seen that the water demand has got two peaks. In general, the two peaks correspond with the morning and the evening rush hours.
<figure><img src="image_8.png" /><figcaption>**Figure 1.** Residential demand pattern [9].</figcaption></figure>

The first case study depicted in Figure 2 comprises 70 nodes of which three (1, 69, 70) are the supply nodes and 108 pipes. Parameters of the nodes and pipes can be obtained in [10, 11, 12]. The second consists of 300 nodes and 447 pipes [13]. Of the 300 nodes, 3 are configured as fixed-head nodes (supply nodes) as shown in Figure 3.

<figure><img src="image_9.png" /><figcaption>**Figure 2.** Case Study 1: Water Distribution Network.</figcaption></figure>

The results of the implementation of Algorithm 1 as applied to case study 1 are depicted in Figures 3 and 4. The number of critical nodes is restricted to 7. It can be seen that for both the maximum and minimum demands, the same sets of nodes (59, 64, 61, 48, 56, 21 and 50) are critical. Furthermore, it is evident that node 59 is the most critical node throughout the testing set.
<figure><img src="image_10.png" /><figcaption>**Figure 3.** Nodal head variation at maximum demand.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Figure 4.** Nodal head variation at minimum demand.</figcaption></figure>

The number of critical nodes for the case study in [8,9,13] are restricted to 10 nodes. The implementation of Algorithm 1 for the demand pattern shown in Figure 1. Evidently, the set of nodes consisting of 273, 257, 259, 275, 274, 222, 221, 220, 219 and 213 for maximum demand is critical whereas 257, 273, 259, 274, 275, 222, 221, 220, 219 and 213 is critical at minimum demands as shown in Figures 5 and 6. The results obtained for this case study show that the proposed scheme identified the same sets of nodes for the maximum and minimum, however, the order of the sensitivity of the nodes varied. This shows that the proposed greedy algorithm is unbiased.

<figure><img src="image_12.png" /><figcaption>**Figure 5.** Nodal head variation at maximum demand.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>**Figure 6.** Nodal head variation at maximum demand.</figcaption></figure>

**4. Conclusion**

In this paper, the solution to the identification of the critical node in a water distribution network is proposed. A greedy algorithm scheme is put forward to identify critical nodes in WDNs. Two case studies are used to investigate the effectiveness of the proposed scheme. It was found that the same sets of nodes were identified as critical for numerical experiment 1. Although the sensitivity index was found to be different, the order of their criticality does not change. In numerical experiment 2, the same sets of nodes are identified as critical, however, the order of their criticality differs for maximum and minimum demand scenarios.
**5. References**

[1] Kegong Diao, Raziyeh Farmani, Guangtao Fu, and David Butler. 2014 Vulnerability assessment of water distribution systems using directed and undirected graph theory. In *11th International Conference on Hydroinformatics (HIC 2014)*, pp 1-8, City University of New York (CUNY)

[2] Daniele Laucelli, Luigi Berardi, Rita Ugarelli, Antonietta Simone, and Orazio Giustolisi 2016 Supporting real-time pressure control in oppegard municipality with wdnetxl. *Procedia Engineering*, **154**, 71-9

[3] Philip R Page, Adnan M Abu-Mahfouz, and Sonwabiso Yoyo 2016. Real-time adjustment of pressure to demand in water distribution systems: Parameter-less p-controller algorithm. *Procedia Engineering*, **154** 391-7

[4] Carsten Skovmose Kallesoe, Tom N_rgaard Jensen, and Rafa l Wisniewski 2015 Adaptive reference control for pressure management in water networks. In *2015 European Control Conference (ECC)*, pp 3268-73. IEEE

[5] Vaibhav Gaur, OP Yadav, Gunjan Soni, and APS Rathore 2020. Identification of critical nodes and edges in a network based on clustering. *Procedia Manufacturing*, **51** 1298-1304

[6] Adedeji, Kazeem B., Yskandar Hamam, Bolanle T. Abe, and Adnan M. Abu-Mahfouz 2017 Leakage detection and estimation algorithm for loss reduction in water piping networks. *Water* **9**(10) 773

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[8] Adedeji, Kazeem B., Yskandar Hamam, and Adnan M. Abu-Mahfouz 2019 Impact of pressure-driven demand on background leakage estimation in water supply networks *Water* **11**(8) 1600

[9] Lawrence K Letting, Yskandar Hamam, and Adnan M Abu-Mahfouz 2017 Estimation of water demand in water distribution systems using particle swarm optimization. *Water* **9**(8) 593

[10] Thapelo C Mosetlhe, Yskandar Hamam, Shengzhi Du, Eric Monacelli, and Adedayo A Yusuff 2020 Towards model-free pressure control in water distribution networks. *Water* **12**(10) 2697

[11] Adedoja, Oluwaseye S., Yskandar Hamam, Baset Khalaf, and Emmanuel R. Sadiku 2020 Development of an Algorithm for the Estimation of Contamination Sources in a Water Distribution Network *IEEE Access* **8** 200412-9

[12] Kazeem Adedeji 2018 Development of a leakage detection and localization technique for real-time applications in water distribution networks', DTech thesis, Tshwane University of Technology, Pretoria, 2018 [Online]. Available: http://tutvital.tut.ac.za:8080/vital/access/manager/Repository/tut:4620?query=kazeem&queryType=vitalDismax. [Accessed: 03-Mar-2021]

[13] Adedoja, Oluwaseye S., Yskandar Hamam, Baset Khalaf, and Rotimi Sadiku 2019 Development of a contaminant distribution model for water supply systems *Water* **11**(7) 1510