Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 330 https://internationalpubls.com Optimizing Phasor Measurement Units Placement in Power Networks Using Betweenness Centrality Saravanan M 1 , Sujatha R 2, Sundareswaran R 3, Venkat Narayanan G 4 * 1PG and Research Department of Mathematics, Mannar Thirumalai Naciker College, Madurai, Tamil Nadu, India, email: msaran81@gmail.com 2School of Science and Humanities(Mathematics),Shiv Nadar University, ,Chennai, Tamilnadu, India, email: sujathar@snuchennai.edu.in 3Department of Mathematics, SSN College of Engineering, Chennai, Tamilnadu, India, email: sundareswaranr@ssn.edu.in *4 Department of Mathematics, St. Joseph’s College of Engineering, Chennai, Tamilnadu, India, email: gvenkatnarayanan@gmail.com Article History: Received: 15-09-2024 Revised: 23-10-2024 Accepted: 03-11-2024 Abstract: Introduction: Centrality concepts play a crucial role in determining the significance of a vertex based on its necessity. A network always prefers the shortest path, also known as the geodesic path, for information transmission between nodes. Betweenness centrality, which reduces geodesic distance, makes it easier to strategically position phasor measurement units (PMUs) in a power network. We present a research paper that describes an algorithm specifically designed for PMU placement in power networks. The algorithm's performance is assessed by running it on the widely used IEEE 14 and IEEE 30 buses, allowing for comparison with results obtained using other methodologies. Keywords: Power network, Power domination, centrality, betweenness, centrality 1. Introduction The importance of graph theory in power network optimization is that it allows the effective control of the complicated systems that result in transmitting electric power from producers to consumers. The networks can be described as graphs, with nodes representing different entities such as generators and substations and edges representing connections such as transmission lines. A few key areas have been outlined here: optimal power flow, fault detection and restoration, distribution network optimization, and smart grid communication and control. Graph theory plays an important role in helping engineers deal with these issues. In electric power systems, phasor measurement units (PMUs) are specialized tools for monitoring and analysing the dynamic behaviours of a system in real time. They measure the voltage and current phasors at specific points on the grid, providing synchronized time-stamped information essential for efficient operation, control and protection. Key components and functions of a Phasor Measurement Unit include Phasor Measurement, Wide Area Monitoring, Time Stamping, Power System Stability, and Control. Haynes et al. [8] first introduced the dominance problem in electrical networks. Kirchhoff's law and Ohm's law are both used to derive this algorithm. Within a graph, dominance refers to a subset S of a vertex set V such that every vertex not in S is adjacent to at least one vertex in S. The same can be mailto:msaran81@gmail.com mailto:81@gmail.com mailto:sujathar@snuchennai.edu.in mailto:sundareswaranr@ssn.edu.in mailto:aranr@ssn.edu.in mailto:gvenkatnarayanan@gmail.com mailto:anan@gmail.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 331 https://internationalpubls.com defined for power domination in power networks. A PMU-observed vertex is one whose voltage and phase angle have been measured by PMU. The paper uses concepts of centrality to determine the best positions for placing PMUs on graphs representing power networks. Centrality, as a metric in graph theory, is crucial in measuring the importance or influence of nodes within a graph. This metric provides invaluable information that helps us understand real world systems such as social networks, biological networks, and transportation networks. Centrality concepts used in the analysis of graphs include Eigen vector centrality, Closeness centrality, Degree centrality, Betweenness centrality, and Katz centrality. Networks often employ the shortest path for information transmission through nodes, with high centrality regulating data communication among other nodes. The inefficiency of degree centrality that uses the node’s degrees as a basis for placing PMUs on the power network graphs was shown by Baldwin et al. [3]. They recommended using concepts related to spanning trees as an alternative method instead. In contrast to that notion of betweenness centrality is computed based on geodesic paths and identifies such nodes that serve as bridges or intermediaries between different parts of a graph. Highly centralized betweenness facilitates interactions and communication across diverse groups or clusters. The position of a PMU at a power node makes it interact with all other vertices in the surrounding area. The optimal placement of PMUs within the power network can be effectively determined through minimizing the geodesic distance, which is based on the betweenness centrality concept. Vertices are said to be neighbours when their geodesic distance is one. Gago et.al [7] provides a mathematical basis, interconnections and restrictions of betweenness centrality in their paper. It is possible to optimize many scenarios using this method, such as dynamic water distribution networks [12], vast sparse networks [13], extensive social networks [9], and identification of protein complexes [1]. In 2019, the discourse on cost optimization focused on the use of dominance centrality over betweenness centrality for PMU placement [5]. In this paper, a methodology is introduced that represents an electric network as a power network graph. To tackle the challenges of PMU placement in the power network, the paper utilizes betweenness centrality. The study is summarized in Section 1, while Section 2 provides an explanation of graph theory and the technical terms used in the paper. Section 3 presents an algorithm for PMU placement within a power network. The algorithm’s performance is evaluated on standard IEEE 6, IEEE 14, and IEEE 30 buses in Section 4. Section 5 compares the outcomes of the proposed method with those of other techniques. Finally, Section 6 concludes the study and discusses its broader implications. 2. Graph theory basic definitions In a graph G=(V,E), V is a non-empty set of vertices and E is a set of edges contained in V X V. The path consists of n vertices. A cycle is defined as having the same starting and ending vertices. The graph’s order corresponds to the number of vertices, whereas its size represents the number of edges. A graph is said to be connected when every pair of vertices in it has a path connecting them. If there is no such path, the graph is considered disconnected. The minimum length of a path between two vertices is referred to as the distance. The geodesic path denotes the shortest path between any two nodes, and its length is termed the geodesic number. All the terms mentioned are defined according to [4]. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 332 https://internationalpubls.com 2.1. Power domination integrity The graph that corresponds to a power network is defined as a power network. The power domination of a power network is defined as a subset S of the vertex set V such that every vertex and every edge in that power network is observed by S. The minimum cardinality of a power dominating set is known as the power dominance number γP. In [8], the rules to find the observed vertices are provided as follows: 1. Any vertex that is incident to an observed edge is observed. 2. Any edge joining two observed vertices is observed. 3. If a vertex is incident to a total of k >1 edges and if k –1 of these edges are observed, then all k of these edges is observed. 2.2 Centralities In social networks, the centrality concepts are employed to assess the importance of a vertex. The degree centrality, introduced by Freeman [6], relies on the degree of a vertex in a graph. Closeness centrality [6] of a vertex u is determined by the reciprocal of the sum of the shortest paths from x to all other vertices. Furthermore, Freeman [6] defined betweenness centrality as the amount of information that passes through a vertex.. ∑    𝑙≠𝑚≠𝑛 𝐺𝑙𝑛𝑚 𝐺𝑙𝑚 where 𝐺𝑙𝑛𝑚 denotes the number of geodesic paths from vertex 𝑖 to vertex 𝑚 via vertex 𝑛 and 𝐺𝑙𝑗 denotes the number of geodesic paths from the vertex 𝑙 to vertex 𝑚. 3. Betweeness centrality in PMU placing An observed vertex is a node that can be measured by a PMU. An observed network is one where all nodes are observed. A power dominating set P is a collection of PMU placed nodes where each node is either in P or observes at least one vertex in P. The power domination number of a power graph is the minimum number of power dominating sets required. To place a PMU in a power network the following algorithm is used: 1. Find the betweenness centrality of each nodes and arrange it in descending order by centrality. 2. Place the PMU in the node which has highest centrality. 3. Apply rule 1, 2 and 3 and check all the vertices are observed. 4. If all the vertices are observed, then the set of PMUs placed nodes are required power dominating set. Otherwise, repeat step 2. 4. Implementation of proposed method in IEEE buses In this section, we implement the above algorithms in IEEE buses. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 333 https://internationalpubls.com 4.1. IEEE 6 bus The line diagram of IEEE bus 6 is given in Fig. 1 and The power network diagram of IEEE 6 bus is given in the Fig 2. Figure 1: Line Diagram of IEEE 6 bus system Figure 2: IEEE 6 bus - Power network graph The table 1 shows the betweenness centrality (BC) and degree of all vertices in the IEEE 6 power network graph. The vertices v4 and v6 exhibit the highest betweenness centrality among the given vertices. Therefore, it is recommended to position the first PMU at either node v4 or v6. Since all the vertices are already observed, placing one PMU at either v4 or v6 would suffice. Table 1: Calculation of edge integrity of a fuzzy graph Vertex BetweenessCentrality Placement v1 0 - v2 1 - v3 1.5 - v4 2.5 1 v5 1.5 - v6 2.5 - 4.2 IEEE 14 bus Given in Fig.3 is the line diagram for the IEEE 14 bus system and the power network graph of IEEE 14 bus is given in Fig.4. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 334 https://internationalpubls.com . Figure 3: Line diagram of IEEE 14 bus Figure 4: IEEE 14 bus - Power Network Graph Table 2: IEEE 14 bus BC values Vertex Betweeness Centrality Placement Vertex Betweeness Centrality Placement v1 0 - v8 0 - v2 5.833 - v9 21 2 v3 0 - v10 4.667 - v4 24.5 - v11 3.667 - v5 21 1 v12 0 - v6 20 3 v13 5.667 - v7 12 - v14 6.667 - According to the data provided in table 2, vertex v4 has the highest betweenness centrality. Therefore, the first PMU should be placed in vertex v4. This PMU will observe the vertices v2, v3, v5, v7 and v9, as all of these vertices are connected to vertex v4. However, since there are still some vertices that are not observed, the next PMUs should be placed in the vertices with the next highest betweeness centrality values. Hence, the next PMUs should be placed in vertices v5 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 335 https://internationalpubls.com and v9. However, it is important to note that these two vertices are already observed by vertex v4 and the distance from the newly placed PMU to these vertices is not v2. Therefore, the PMU at vertex v4 should be removed. At this point, the PMUs should only be placed in vertices v5 and v9. However, vertices v12 and v13 are still not observed. Therefore, the next placement should be in vertices v6, which has the next highest betweenness centrality value. As a result, the PMU placed in vertex v5 will observe the vertices v1, v2, v4, and v5. The PMU placed in vertex v9 will observe the vertices v7, v9, v10, and v14. Lastly, the PMU placed in vertex v6 will observe the vertices v6, v11, v12, and v13. By following rules 2 and 3, the remaining vertices v3 and v8 are observed. Consequently, all the vertices in the power network graph are observed. By placing 3 PMUs in vertices v5, v6, and v9, the network is fully observed. 4.3 IEEE 30 bus The line diagram of IEEE 30 is given in Fig. 5 . Figure 5: Line diagram of IEEE 30 bus Figure 6: IEEE 30 bus - Power network graph Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 336 https://internationalpubls.com Table 3: IEEE 30 bus B C values Vertex Betweeness Centrality Placement Vertex Betweeness Centrality Placement v1 1 - v16 10.417 - v2 40.5 - v17 15.917 - v3 4 - v18 11.417 - v4 89.75 3 v19 11.417 - v5 1 - v20 26.25 - v6 176.583 1 v21 0 - v7 8.5 - v22 34.917 - v8 0 - v23 31.25 - v9 28 - v24 56.417 6 v10 115.667 2 v25 48.833 - v11 0 - v26 0 - v12 87.5 4 v27 76.833 5 v13 0 - v28 72.833 - v14 0 - v29 0 - v15 54 - v30 0 - According to the information provided in table 3, we will begin by placing the first PMU in vertex v6, which will observe the vertices v2, v4, v6, v7, v8, v9, v10, and v28. Additionally, based on rule 1 and 2, vertex v11 will also be observed. Next, we will place the second and third PMUs in vertices v10 and v4, respectively, in order to observe the vertices v3, v4, v10, v12, v17, v20, v21, and v22. Following rule 1 and 2, vertices v1 and v5 will also be observed. For the fourth and fifth PMUs, we will place them in vertex v12 and v27 to measure the values in vertices v12, v13, v14, v15, v16, v25, v27, v28, v29 and v30. At this stage, vertex v26 will also be observed based on rule 1 and 2. Since all the vertices adjacent to vertex v28 have already been observed, placing any additional PMU in this location will not be effective. Moving on to the next maximum betweenness value, we will place a PMU in vertex v24 and another in vertex v5 to observe the vertices v18, v23, and v24. Following rule 1 and 2, vertex v19 will also be observed. Therefore, with a total of 7 PMUs, we will be able to observe all the vertices in the power network. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 337 https://internationalpubls.com 5. Results and Discussions To tackle the issue of high costs, it is essential to reduce the number of PMUs and implement measures originating from the vertices. There are several well-known techniques for calculating the number of PMUs and determining their suitable placements. By comparing the outcomes of the proposed method with other popular methods like power domination integrity [11], topology transformation [10] and Immunity genetic algorithm [2] the results can be organized in a tabulated manner. Table 4: Number of PMUs in various methods Method IEEE 14 IEEE 30 Betweenness Centrality method 3 7 Power domination method [11] 3 7 Topology transformation method [10] 3 7 Immunity Genetic algorithm [2] 3 7 6. Conclusion Saravanan et al. [11] proposed a methodology involving two distinct steps to determine the minimum number of PMUs and their respective placements. However, a more efficient approach can be achieved by utilizing Betweenness centrality. By calculating the betweenness centrality for all nodes and arranging the PMU placement from highest to lowest, this task can be effectively completed. The results obtained through this technique are consistent with those obtained through other methods, as indicated in Table 4. 7. Acknowledgment The authors expresses gratitude to the editor and anonymous referees for reviewing this manuscript and providing helpful suggestions and comments References [1] J. Ahn, D. H. Lee, Y. Yoon, Y. Yeu, and S. Park, Improved method for protein complex detection using bottleneck proteins. In BMC Medical Informatics and Decision Making, volume 13, pages 1–9. Springer, 2013. [2] F. Aminifar, C. Lucas, A. Khodaei, and M. Fotuhi-Firuzabad, Optimal placement of phasor measurement units using immunity genetic algorithm. IEEE Transactions on power delivery, 24 (3):1014–1020, 2009. [3] T. Baldwin, L. Mili, M. 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