- Research
- Open Access
A network science-based k-means++ clustering method for power systems network equivalence
- Dhruv Sharma^{1}Email author,
- Krishnaiya Thulasiraman^{2},
- Di Wu^{3} and
- John N. Jiang^{1}
- Received: 8 March 2018
- Accepted: 3 April 2019
- Published: 22 April 2019
Abstract
Network equivalence is a technique useful for many areas including power systems. In many power system analyses, generation shift factor (GSF)-based bus clustering methods have been widely used to reduce the complexity of the equivalencing problem. GSF captures power flow on a line when power is injected at a node using bus to bus electrical distance. A more appropriate measure is the one which captures what may be called the electrical line distance with respect to a bus termed as relative bus to line distance. With increase in power transactions across different regions, the use of relative bus to line distance becomes appropriate for many analyses. Inspired by the recent trends in network science on the study of network dynamics based on the topological characteristics of a network, in this paper, we present a bus clustering method based on average electrical distance (AED). AED is independent of changes in location of slack bus and is based on the concept of electrical distance introduced in the context of molecular chemistry and pursued later for applications in social and complex networks. AED represents the AED from a bus to buses of the transmission line of interest. We first propose an AED-based method to group the buses into clusters for power systems network equivalence using k-means clustering algorithm integrated with silhouette analysis. One limitation of this method is that despite its speed, sometimes it may yield clusters of inferior quality compared to the optimal solution. To overcome this limitation, we next present our improved clustering method which incorporates a seeding technique that initializes centroids probabilistically. We also incorporate a technique in our method to find the number of clusters, k, to be given as input to our clustering algorithm. The resulting algorithm called AED-based k-means++ clustering method yields a clustering that is O(logk) competitive. Our network equivalence technique is next described. Finally, the efficacy of our new equivalencing technique is demonstrated by evaluating its performance on the IEEE 300-bus system and comparing that to the performance of our AED-based method (Sharma et al. in Power network equivalents: a network science-based k-means clustering method integrated with silhouette analysis. In: Complex networks and their application VI, Lyon, France. p. 78–89, 2017) and the existing GSF-based method.
Keywords
- Average electrical distance
- Power systems network equivalence
- Generation shift factor
- k-means++ algorithm
Introduction
The past few decades have witnessed a new movement of interest and research in the study of complex networks, i.e., networks whose structure is irregular, complex and dynamically evolving in time such as power grids, communication networks, biological networks, social networks, etc. Investigating dynamics in such complex networks requires an understanding of the interaction between network topology and specific domain constraints. For example, the study of power grids requires basic circuit laws, relating voltages and currents, to be incorporated along with the network topology. In this paper, we explore this interaction in the context of deriving simplified equivalent networks as representations of large power grids. In this paper, the terms power systems, power system network, and power network are used interchangeably. Power network equivalencing has been studied in the literature with different objectives. Our objective is to study the equivalents appropriate for electrical market analysis.
Electrical market analysis involving power exchange is becoming more and more complex due to the size and degree of interconnections in modern power systems due to economic, political, and environmental reasons [1]. With such inherent complexities and information deficiency, it is difficult for participants to make operational and market decisions at buses that are sensitive to the condition of transmission lines. For these reasons, it is necessary to develop an enhanced method to support decisions, particularly, those sensitive to major and critical transmission lines. The analysis can be computationally challenging, especially, when a full AC implementation approach is used [2]. Compared to the full AC analysis, a simplified analysis of the network can be done by means of full network DC power flow model [3]. Although the full AC analysis would be the most accurate approach, the DC approach allows network operators to make informed dispatch decisions thereby saving time and effort required. The enormous size of power system networks makes full network DC analysis computationally taxing. To reduce the computational burden and to simplify the analysis of electricity markets, several network equivalence models have been used [2, 4–6].
Various approaches for network equivalencing have been presented in [7–12]. These approaches follow the traditional method of eliminating less important elements from the system on the basis of geographic and electric parameters. Such elimination results in partitioning of the network into three clusters of buses: a cluster of internal buses, a cluster of external buses, and a cluster of boundary buses that divide the external buses from the internal buses. Due to their trivial impact on the internal system, remote generators and transmission lines connected to the boundary buses may be eliminated with minor impact on decisions. However, irrespective of the bus demarcation, market analysis of large power system networks requires retaining the desired buyer/seller pairs corresponding to operating zones in order to understand the impact of power flows on transmission lines from the buyer/seller pairs at various buses. Hence, in this study, we do not eliminate the external buses but rather focus on the line flows between various operating areas. These line flows are called tie-line flows.
In this paper, the metric developed for the study has properties similar to resistance distance which has been widely used since the early days of electrical circuit theory. There are several books on the basics of circuit analysis that deal with resistance distance and topological formula for resistance distance, for example [13]. However, recently this concept gained increased importance in view of its applications in areas outside electrical circuit theory [14–20].
In recent studies, network equivalencing has been done using generation shift factor (GSF)-based methods [5, 6, 21, 22]. In these methods, buses with similar impact on the interconnecting tie-line flows evaluated using GSFs are grouped together. In order to improve the efficiency and accuracy of bus clustering, [6] use is made of k-means algorithm based on GSF to cluster the buses. However, GSFs are sensitive to the change in the location of slack generator. Network operators supervising different regions of the interconnection might not be aware of the slack bus change, and hence there could be discrepancies in decision making, which can provoke an impact on regional power transactions. Therefore, our main objective in this paper is to develop a new clustering method as well as new network equivalent that overcomes the limitation of the GSF-based methods.
Our work in this paper includes our previous work [23] and its enhancements. The rest of this paper is organized as follows. “Preliminaries” section gives an introduction to all the basic concepts that are of interest in the development of techniques discussed in subsequent sections. We introduce, in “Average electrical distance” section, the concept of average electrical distance (AED) and discuss a relationship between AED and GSF. Also, the relevance of our work in the context of social network analysis is given in “Implications and relevance for social network analysis” section. “k-means algorithm” section gives an introduction to the k-means algorithm for clustering, followed by “AED-based k-means bus clustering method” section presenting our proposed AED-based k-means clustering method, which, however, has certain limitations. In “AED-based k-means++ bus clustering method” section, we present our improved AED-based k-means++ clustering method, to overcome the said limitations. The new method uses a seeding technique [24] for initialization of centroids. We call this augmented algorithm as AED-based k-means++ algorithm. We also incorporate in this method silhouette analysis to determine the number k of clusters to be given as input to the clustering method. The resulting AED-based k-means++ method yields clustering which is O(logk) competitive. “Power network equivalencing based on AED-based k-means++ clustering method” section presents how we use clustering to develop a power network equivalent suitable for market analysis. To demonstrate the efficacy of our approach for clustering and equivalencing, the rest of “Power network equivalencing based on AED-based k-means++ clustering method” section is concerned with experimental and comparative evaluations of our network equivalence method applied on the 300-bus system. We conclude this paper with “Conclusion” section, highlighting the main findings of this work, their implications, and our continuing attempt to find an efficient network equivalencing technique for power systems and its implications in the context of social network analysis.
Preliminaries
In this section, we introduce the basic concepts of Laplacian matrix of a graph, electrical distance, and generation shift factor.
Laplacian matrix of a graph
Consider a graph \(G=(V,E)\) with vertex set \(V=\{0,1,\ldots ,n\}\). Edge \(e\in E\) connecting vertices i and j is denoted by (i, j). We assume there are no loops on any vertices and there are no parallel edges connecting the vertices. In this paper, the terms, vertices and nodes, as well as links and edges, will be used interchangeably. Let an edge (i, j) be assigned a weight \(w_{ij}\), a positive real number. If there is no edge connecting i and j then \(w_{ij}=0\). Two vertices i and j are adjacent if there is an edge (i, j). A vertex j is incident on vertex i if there is an edge connecting i and j. The degree of a vertex i denoted by deg(i) is the sum of the weights of the edges incident on i.
In the eigenvalue approach, the pseudo-inverse \(Y^+\) of Y is used. The properties of G are studied in terms of the elements of \(Y^+\). This approach is quite popular among mathematicians [25, 26]. In this paper, we follow the determinant approach which is popular in the electrical engineering community. In this approach, we first remove a row and the corresponding column from the Laplacian matrix, Y. Let us assume that the vertex labelled o called the datum node or slack node is removed. The resulting matrix denoted by Y(o) is called a reduced Laplacian matrix. The reduced Laplacian matrix Y(o) of Y in Fig. 1b is shown in Fig. 1c. It can be shown that the matrix Y(o) is non-singular and it has several other properties, for example, [13].
In all discussions in this paper, we will use \(Y=[y_{ij}]\) to denote the reduced Laplacian of the network. We wish to draw attention to Cetinay et al. [27] where the authors investigate the impact of topology on power flow using spectral graph theory.
Electrical distance
Generation shift factor
Given a power network with two or more interconnected areas, in power market analysis, a simpler equivalent network is needed which preserves the flows across the lines (edges) connecting different areas. In this context, the concept of GSF [5] was introduced and used in determining power network equivalents.
Average electrical distance
In the rest of the paper, we discuss our improved clustering method based on AED and compare the results with the existing GSF-based clustering method.
Calculation of AED
Algorithm given below to determine AED for each bus in the network with respect to the tie-lines of interest includes the following main steps:
- 1.
Creation of bus impedance matrix According to the system data, bus admittance matrix is first developed. Then, bus impedance matrix, shown in (16), is created by calculating the inverse of bus admittance matrix.
- 2.
Calculation of Thevenin impedance Using the elements of bus impedance matrix obtained in Step 1, the Thevenin impedances for each pair of buses are calculated according to (17).
- 3.
Calculation of AED After calculating the Thevenin impedances, AEDs between buses and tie-lines of interest are calculated using (18). The results of AEDs are used to create a matrix as shown in Fig. 6. In this matrix, each row corresponds to a tie-line and each column corresponds to a bus in the system. Thus, an element in the matrix corresponds to AED from a bus to a tie-line in the system.
Implications and relevance for social network analysis
In the graphical representation of a social network, link weights are all unity. Different types of metrics/measures are defined to determine certain properties of links or nodes. For example, a measure called PageRank is used to rank the nodes in terms of their importance [28]. As another example, the betweenness measure of a link is used to determine the importance of a link. A link e is considered more important than another link \(e'\), if the fraction of the total number of messages that flow through e is greater than that for \(e'\). In view of the connection between random walk and current flow in a resistance network [18], this fraction is in fact equal to the current through the link when a unit current is injected at a node. Taking advantage of this connection in [15], a measure similar to GSF is used to determine the betweenness measures of links. A detailed discussion of many of these measures is given in [29].
The concept of role discovery in networks was first studied in sociology [30, 31]. In this context, roles considered are social roles. Thus, role discovery has become an important topic in social network analysis. Recently, role discovery has been studied in other settings such as online social networks, technological networks, biological networks, web graph, etc.
In [32], a comprehensive review of literature on role discovery in network has been given. This paper discusses the problem of identifying clusters in a network such that all nodes in each cluster are equivalent in some sense. Two types of equivalence are considered: graph-based equivalence and feature-based equivalence. Several challenges that arise in the application of role discovery in non-static network such as dynamic and streaming graphs are also discussed in [32].
Our work in this paper is about clustering in a power network and its application in deriving a simplified approximate equivalent network that preserves flows along certain lines. The GSF and AED are measures that are defined for each node with respect to certain lines. If we set the line weights to unity, then these measures in the context of social networks capture the fraction of total messages that flow through a link when messages arrive (or injected) at a node. Therefore, the work presented in this paper is relevant to social network studies. For example, a problem of interest is to determine clusters such that the total flow carried by inter-cluster links is optimized. Once such clusters are identified, we can determine simplified approximate equivalent network as explained in “Power network equivalencing based on AED-based k-means++ clustering method” section that can be used to predict the flows across the clusters. Further discussion of these ideas is given in “Conclusion” section.
k-means algorithm
k-means algorithm is one of the most popular clustering techniques in unsupervised learning tasks. Given a set of nodes or buses, this algorithm has been efficiently used to partition a network into k clusters [33]. This is based on the optimal placement of centroid for the respective cluster in a network [34].
The process becomes iterative in order for the clusters to reach a local minimum which is dependent on the initial selection of the reference buses. The k-means algorithm keeps on adjusting the centroids after each partition making it more dynamic to the changes. The k-means algorithm is explained in Algorithm 2.
- 1.
Selecting initial cluster centroids Cluster formation is initialized by selecting k centroids, i.e., \(\mu _1,\mu _2,\ldots ,\mu _k\) in the network. These centroids act as initial reference points for the buses to be assigned to an appropriate cluster.
- 2.
Grouping buses into clusters For the selected k centroids, based on the Euclidean distance, a bus is assigned to a cluster which has its centroid closest to the bus.
- 3.Recalculating centroid positions After all the buses are assigned to respective clusters, the new centroid of each cluster is recalculated as shown in (22).$$\begin{aligned} \mu _j=\frac{1}{n_j}\sum _{i=1}^{n_j}x_{ij} \end{aligned}$$(22)
- 4.
Evaluating objective function in (21) After all buses are grouped into the clusters in Step 2, the potential function in (21) is evaluated.
- 5.
Iterations of algorithm Steps 2–4 are repeated until the centroid of each cluster ceases to change its position with further iterations.
AED-based k-means bus clustering method
In this section, we discuss our AED-based improved clustering method that uses k-means algorithm for power system network equivalence. This method uses a simple iterative technique known as Lloyd’s algorithm for finding a locally minimal solution [34]. Further, it utilizes AED as a measure of distance between the buses in the system. Integration of AED makes k-means algorithm more relevant to the power system network study. This is because AED gives a measure of distance of a bus with respect to a tie-line. This algorithm proves to be sufficiently accurate for the independent analysis done by various utilities on their networks.
AED-based k-means algorithm
AED-based k-means++ bus clustering method
In this section, we introduce AED-based improved k-means++ algorithm which can be used for clustering of large power system networks.
k-means++ algorithm
We would like to point out that k-means++ algorithm, due to its initialization process, produces starting centroids uniformly distributed for different iterations compared to k-means algorithm starting centroids. This is illustrated in [36].
AED-based k-means++ algorithm
Different from commonly used k-means++ algorithm, the improved AED-based algorithm groups the buses in a power system into various clusters based on the closeness between buses and the centroid of each cluster in terms of AEDs. The objective of the improved AED-based k-means++ algorithm is also to minimize the same potential function, \(\phi\), as shown in (23).
To achieve the objective described in (23) with the initial seeding as shown in (24), the improved AED-based algorithm, shown in Algorithm 3, includes the following main steps:
- 1.
Cluster initialization Cluster formation is initialized by selecting one centroid \(\mu _1\), chosen uniformly at random in the network. This centroid acts as initial reference point for the buses to be assigned to an appropriate cluster. Those buses that are closer to \(\mu _1\) are later assigned to the same cluster.
- 2.Determining cluster centroids Given the centroids, \(\mu _1,\ldots ,\mu _{j-1}\), a new centroid, \(\mu _j\), is chosen and each bus, i, is selected with probabilitywhere \(D(i)=\text {min}|d(uv,i)-d(uv,\mu _r)|\), where \(r=1,\ldots ,j-1\).$$\begin{aligned} \frac{D(i)^2}{\sum _{i=1}^{n}{D(i)^2}}, \end{aligned}$$
- 3.
Step 2 is repeated until we get all the centroids.
- 4.
k-means algorithm Proceed as with the standard k-means algorithm (Algorithm 2) with AED used as distance measure as discussed in “AED-based k-means bus clustering method” section.
Silhouette value analysis
An important problem in the application of k-means algorithm is to determine appropriate value of k. This problem has been extensively studied in the mathematical statistics literature [37, 38]. The authors in [39] identified certain best performing methods. In [37], authors determined through extensive simulation studies that all the best performers do quite well in selecting the appropriate number of clusters to be selected. In our work, we use the silhouette value analysis proposed in [40] that is also among the best performers.
Silhouette value analysis is a graphical partitioning technique [41] allowing an appreciation of the relative quality of clusters. In our study, the silhouette value analysis is used to enhance the quality of clusters identified by the improved AED-based k-means++ clustering algorithm. The main steps of the silhouette value analysis, which are incorporated into the improved AED-based k-means algorithm are explained below:
Algorithm 4: Silhouette value analysis algorithm
- 1.Evaluating closeness between buses in a cluster In this step, the average closeness between the buses in cluster j is evaluated by calculating AED measure, \(ac_{ij}\), with respect to the tie-lines in the network. This is shown in (25).$$\begin{aligned} ac_{ij}=\frac{1}{n}\sum _{\begin{array}{c} r=1 \\ r\ne i \end{array}}^{n_j}|d(uv,i)-d(uv,r)| \end{aligned}$$(25)
- 2.Evaluating closeness between each bus and clusters In this step, the minimum of average closeness of a bus i with respect to each cluster \(m\ne j\) is evaluated which is given by \(eb_{im}\).where \(n_m\) is the number of buses in cluster m.$$\begin{aligned} eb_{im}=\frac{1}{n_m}\sum _{r=1}^{n_m}|d(uv,i)-d(uv,r)|,\quad m=1,\ldots ,k-1;\quad m\ne j \end{aligned}$$(26)
- 3.Calculating average silhouette coefficient of all the clusters The silhouette coefficient of a bus indicates whether its placement is in an appropriate cluster. Silhouette coefficient \((s_{im})\) of bus i can be calculated asThe average of silhouette coefficients, \(s_{im}\), of buses is evaluated as \(s_i\). The average silhouette coefficient, \(s_i\), of all the buses in the whole network is evaluated to give a perspective of average closeness of all the buses to their neighbouring clusters. The coefficient is in a range \([-\,1, +\,1]\), where \(+\,1\) indicates that buses are far away from their closest neighbouring clusters; while \(-1\) indicates that the buses are closer to their neighbouring clusters.$$\begin{aligned} s_{im}=\frac{eb_{im}-ac_{ij}}{\text {max}(ac_{ij},eb_{im})},\quad m=1,\ldots ,k-1;\quad m\ne j \end{aligned}$$(27)
- 4.
Selecting value of k based on silhouette value analysis Steps 1, 2 and 3 are repeated for different values of k and the one with average silhouette coefficient closest to \(+\,1\) is selected.
Flowchart
Power network equivalencing based on AED-based k-means++ clustering method
Power network equivalents based on aggregation of buses in a cluster
Case studies using tie-lines
In this section, we first demonstrate in “39-bus system (Algorithm 3 and method of “AED-based k-means bus clustering method)” section the efficacy of the proposed AED-based improved k-means++ clustering method by comparing it with our previous clustering method [23] (“AED-based k-means bus clustering method” section) on the IEEE 39-bus system. Then in “300-bus system” section, we use IEEE 300 bus system to show the superiority of the proposed method compared to the widely used GSF-based clustering method [5, 6, 21, 22].
39-bus system (Algorithm 3 and method of “AED-based k-means bus clustering method” section)
Comparison of clusters identified by two different AED-based clustering methods
Tie-line 25–26 and Tie-line 17–18 | |||
---|---|---|---|
Clusters identified by AED-based k-means algorithm (“AED-based k-means bus clustering method” section) | Clusters identified by proposed AED-based improved k-means++ algorithm (Algorithm 3) | ||
Clusters | Buses | Clusters | Buses |
SA 11 | 26, 28, 29, 38 | SA 11 | 26, 28, 29, 38 |
SA 12 | 27 | SA 12 | 27 |
SA 13 | 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 32, 33, 34, 35, 36 | SA 13 | 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 32, 33, 34, 35, 36 |
SA 21 | 25, 37 | SA 21 | 25, 37 |
SA 22 | 1, 2, 30, 39 | SA 22 | 1, 2, 30 |
SA 23 | 18 | SA 23 | 39 |
SA 24 | 3, 4, 5, 6, 7, 8, 9, 31 | SA 24 | 3, 18 |
SA 25 | 4, 5, 6, 7, 8, 9, 31 |
In our study, the two tie-line scenario is used to compare the proposed method with the method of “AED-based k-means bus clustering method” section. In this scenario, any two tie-lines out of four are utilized to connect the two areas in the system. Under this scenario, the clusters are identified using the two methods which are based on average AEDs of each bus with respect to the two tie-lines connected. The two clustering methods follow different algorithms, and hence, the clusters obtained using the two methods are different as observed in Table 1. This may affect the accuracy of tie-line flows in equivalent networks compared to the tie-line flows in the original network. The comparison of accuracy of tie-line flows in the equivalent network is demonstrated using tie-line flow analysis. Further, we also analyse the quality of clusters in the equivalent networks based on the similarity of buses in each cluster.
Tie-line flow analysis (Algorithm 3 and method of “AED-based k-means bus clustering method” section) The tie-line flow analysis is based on the comparison of accuracy of net tie-line flows in the equivalent networks with those in the original network. The equivalent networks are created using Algorithm 3 and method of “AED-based k-means bus clustering method” section. Different cases with combinations of two tie-lines are studied. In this study, the tie-line flows in the original network are calculated using GSFs, and those in the equivalent networks are calculated using average GSFs for each cluster.
Comparison of net tie-line power flows in the original network and AED-based equivalent networks for 39 bus system
Case no. | Tie-line combination | Original network | AED-based equivalent networks | |||
---|---|---|---|---|---|---|
Flow (MW) | AED-based k-means algorithm (“AED-based k-means bus clustering method” section) | AED-based k-means++ algorithm (Algorithm 3) | ||||
Flow (MW) | % deviation | Flow (MW) | % deviation | |||
Case 1 | TL25–26/TL17–18 | 41.30 | 41.50 | 0.48 | 41.34 | 0.10 |
Case 2 | TL25–26/TL4–14 | 35.78 | 35.81 | 0.09 | 35.75 | 0.09 |
Case 3 | TL25–26/TL6–11 | 41.50 | 40.50 | 2.40 | 40.49 | 2.43 |
Case 4 | TL17–18/TL4–14 | 10.67 | 19.73 | 84.90 | 18.85 | 76.66 |
Case 5 | TL17–18/TL6–11 | 23.58 | 40.52 | 71.82 | 32.76 | 38.93 |
Case 6 | TL4–14/TL6–11 | 32.81 | 32.67 | 0.43 | 32.77 | 0.12 |
It can be observed from Fig. 10 that for each set of two tie-lines, the similarity of buses in clusters created using proposed AED-based k-means++ algorithm is clearly more compared to the buses in clusters created using previous AED-based method. Based on similarity of buses in the clusters, the created equivalent network has net tie-line flows very similar to the original network which can be observed from Table 2. Thus, modifying the clustering method by replacing our method of “AED-based k-means bus clustering method” section with the proposed AED-based improved k-means++ algorithm provides much accurate results for network clustering schemes.
300-bus system
To demonstrate the efficacy of our Algorithm 3, we compare it with the widely used GSF-based clustering method. We use different two tie-line scenarios in the 300-bus system and the two methods, based on average AEDs and average GSFs respectively, are used to identify clusters to obtain equivalent networks using k-means++ algorithm. Further, to validate the use of k-means++ algorithm, we compare Algorithm 3 with the AED-based method of “AED-based k-means bus clustering method” section that uses k-means algorithm. In this study, we use different two tie-line scenarios in the 300-bus system for clustering. These methods yield different clusters for the same network and in order to compare the two clustering methods, we analyse the accuracy of tie-line flows in the equivalent networks. We also analyse the quality of clusters in these equivalent networks in terms of similarity of buses in each cluster.
Comparison of net tie-line power flows in the original network and those in GSF and AED-based equivalent networks obtained using Algorithm 3 for 300 bus system
Case no. | Tie-line combination | Original network | GSF-based equivalent network | AED-based equivalent network (Algorithm 3) | ||
---|---|---|---|---|---|---|
Flow (MW) | Flow (MW) | % deviation | Flow (MW) | % deviation | ||
Case 1 | TL19–87/TL8–14 | 448.09 | 375.31 | 16.24 | 392.85 | 12.33 |
Case 2 | TL19–87/TL4–16 | 1051.79 | 1019.24 | 3.09 | 1032.31 | 1.85 |
Case 3 | TL19–87/TL62–144 | 55.34 | 156.84 | 183.41 | 57.93 | 4.68 |
Case 4 | TL8–14/TL62–144 | 450.86 | 391.79 | 13.10 | 487.01 | 8.02 |
Case 5 | TL4–16/TL62–144 | 994.46 | 490.70 | 50.66 | 973.19 | 2.14 |
Comparison of net tie-line power flows in the original network and the AED-based equivalent networks obtained using the Algorithm 3 and method of “AED-based k-means bus clustering method” section
Case no. | Tie-line combination | Original network | AED-based equivalent networks | |||
---|---|---|---|---|---|---|
Flow (MW) | AED-based k-means algorithm (“AED-based k-means bus clustering method” section) | AED-based k-means++ algorithm (Algorithm 3) | ||||
Flow (MW) | % deviation | Flow (MW) | % deviation | |||
Case 1 | TL19–87/TL8–14 | 448.09 | 292.39 | 28.08 | 392.85 | 12.33 |
Case 2 | TL19–87/TL4–16 | 1051.79 | 978.73 | 6.95 | 1032.31 | 1.85 |
Case 3 | TL19–87/TL62–144 | 55.34 | 89.87 | 62.39 | 57.93 | 4.68 |
Case 4 | TL8–14/TL62–144 | 450.86 | 324.26 | 28.08 | 487.01 | 8.02 |
Case 5 | TL4–16/TL62–144 | 994.46 | 922.42 | 7.24 | 973.19 | 2.14 |
It can be observed from Fig. 12 that for each set of two tie-lines, buses in clusters created based on AEDs are more similar than those in clusters created based on GSFs since the average of standard deviations of AEDs in clusters for each set of two tie-lines is smaller than the average of standard deviations of corresponding GSFs. Based on the similarity and closeness of buses in the clusters, the created equivalent network has net tie-line flows very similar to that of the original network. It can be observed from Table 3, that the net tie-line flows in the equivalent network of 300-bus system created by bus clustering method using AED-based improved k-means++ algorithm are more accurate than those in the equivalent network created by GSF-based bus clustering method. Similar results comparing AED-based k-means algorithm [23] and the proposed k-means++ algorithm are shown in Table 4.
Conclusion
In this paper, we have presented an AED-based improved bus clustering method for network equivalence of large interconnected power systems. The method utilizes AED-based improved k-means++ algorithm for grouping similar buses together to form clusters on the basis of their respective AEDs. The new algorithm is obtained by augmenting the AED-based k-means algorithm to probabilistically initialize the centroids of clusters thereby, as in [26], improving the accuracy of the algorithm. The use of silhouette analysis along with improved k-means++ algorithm has resulted in further maximizing the accuracy of the clusters. The proposed method has been compared with our previous method [23] on the IEEE 39-bus system. It has been shown that when compared to the full network, the net tie-line flows in the equivalent networks created using the proposed method are more accurate than those in the equivalent networks created using our previous method. Also, the proposed method yields a better cluster quality which shows that the buses in clusters formed using the proposed method are more closely connected than those in the clusters formed using our previous AED-based method.
Moreover, the proposed method has been compared with the widely used GSF-based clustering method [6] on the IEEE 300-bus system. It has been shown that the net tie-line flows in the network with different combinations of tie-lines are more accurate for the equivalent network obtained using the proposed AED-based improved k-means++ clustering method than the one obtained using GSF-based clustering method as well as the one obtained with the AED-based k-means algorithm. Further, the results of the cluster quality analysis show that the buses in the clusters obtained using proposed method are more closely connected. Thus, the reduced network obtained using the proposed method gives a better representation of the original network compared to the widely used GSF-based clustering method.
In “Implications and relevance for social network analysis” section, we discussed the relevance and implications of our work in the context of social network analysis. We conclude this paper by pointing to an application to what is called the community detection problem in social networks. A community in a social network is a collection of closely related nodes with respect to a closeness measure. A detailed discussion of the community detection problem is given previously [29]. Electrical distance is a measure of closeness of two nodes when the links are assigned weights that capture the characteristics of interest. The Kirchhoff index [45–47] of a cluster is the sum of electrical distances between all pairs of nodes in the cluster. The smaller the Kirchhoff index of a cluster, the closer are the nodes in the cluster. On the other hand, the sum of the AEDs of all the nodes in a cluster is a measure of the total flow across inter-cluster links. Let us call this as inter-cluster Kirchhoff index. Then the smaller the inter-cluster Kirchhoff index of two clusters, the less closely connected are the nodes in the two clusters. A problem of interest is designing a clustering algorithm that determines clusters such that the Kirchhoff index of each cluster and inter-cluster Kirchhoff index of each pair of clusters are within pre-specified limits. The clusters so determined will provide a solution to the community detection problem in social networks. This is a fairly complex problem involving the solution of a bi-criteria optimization problem, Kirchhoff indices of clusters and inter-cluster indices of pairs of clusters, optimization problem. We are currently studying design of approximation heuristics of this community detection problem. This problem is also related to the problem of partitioning power networks with the aim of containing the impact of cascades due to failures in the network.
Declarations
Authors’ contributions
DS, DW and JNJ conceived of the presented idea of average electrical distance (AED) for network equivalencing. DS and JNJ developed the theory and DS performed the computations related to AED-based network equivalencing method and its comparison with the existing generation shift factor (GSF)-based method. KT, DW and JNJ verified the results and encouraged DS to extend the study by integrating k-means++ algorithm and apply it on a bigger network. JNJ and KT supervised the findings of this work. All authors discussed the results and contributed to the final manuscript. All authors read and approved the final manuscript.
Acknowlegements
Not applicable.
Competing interests
The authors declare that they have no competing interests.
Availability of data and materials
The datasets supporting the conclusions of this article are included within the article and its additional files. The IEEE 39-bus system analysed in this study is commonly known as “The 10-machine New-England Power System”. This system’s parameters are specified by Athay et al. [42] and are published in a book titled “Energy Function Analysis for Power System Stability” [43]. The IEEE 300-bus system test case was developed by the IEEE Test Systems Task Force under the direction of Mike Adibi in 1993. The dataset for this system is available at [44]. Simulation codes related to the study will be available upon request.
Funding
Not applicable.
Publisher’s Note
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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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