Hi the topic of my project is cluster synchronization in complex power grid. I need just the simulation part. if you can get more suitable paper for the simulation, then you can do from that paper as well. the main theme of the project is to making the clusters of the power grids. then synchronising those clusters. Please i need the detail information
untitled IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 29, NO. 5, SEPTEMBER 2014 2229 Hierarchical Spectral Clustering of Power Grids Rubén J. Sánchez-García, Max Fennelly, Seán Norris, Student Member, IEEE, Nick Wright, Graham Niblo, Jacek Brodzki, and Janusz W. Bialek, Fellow, IEEE Abstract—A power transmission system can be represented by a network with nodes and links representing buses and electrical transmission lines, respectively. Each line can be given a weight, representing some electrical property of the line, such as line admittance or average power flow at a given time. We use a hierarchical spectral clustering methodology to reveal the internal connectivity structure of such a network. Spectral clustering uses the eigenvalues and eigenvectors of a matrix associated to the network, it is computationally very efficient, and it works for any choice of weights. When using line admittances, it reveals the static internal connectivity structure of the underlying network, while using power flows highlights islands with minimal power flow disruption, and thus it naturally relates to controlled islanding. Our methodology goes beyond the standard -means algorithm by instead representing the complete network substructure as a dendrogram. We provide a thorough theoretical justification of the use of spectral clustering in power systems, and we include the results of our methodology for several test systems of small, medium and large size, including a model of the Great Britain transmission network. Index Terms—Clustering, power system analysis computing. I. INTRODUCTION T HE idea of decomposing (partitioning, tearing, splitting,clustering) a power network into smaller parts goes back to the concept of diakoptics introduced by G. Kron in the 1950s [1], and it was followed by many others, e.g., [2]–[4]. The initial motivation was the limited memory and computation speed of early computers. Later on, the advent of parallel computing re- sulted in a very significant research effort aimed at splitting the power system model into smaller parts to be solved in parallel. Decomposing large interconnected networks into loosely- connected zones that could be more easily managed often uses the concept of electrical cohesiveness as expressed by electrical distance between network nodes [5] to define network zones. Generally, network splitting can enable a flexible, distributed and adaptable power system control that utilizes the concept of smart grids [6]. Manuscript received June 21, 2013; revised November 29, 2013; accepted February 09, 2014. Date of publication March 17, 2014; date of current version August 15, 2014. This work was supported by the EPSRC grants EP/G059101/1 and EP/G060169/1. Paper no. TPWRS-00806-2013. R. J. Sánchez-García, M. Fennelly, N. Wright, G. Niblo, and J. Brodzki are with the School of Mathematics, University of Southampton, Southampton SO17 1BJ, U.K. (e-mail:
[email protected]). S. Norris and J. W. Bialek are with the School of Engineering and Com- puting Sciences, Durham University, Durham DH1 2PE, U.K. (e-mail: Janusz.
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TPWRS.2014.2306756 Network splitting is also used in controlled islanding that aims to prevent the spreading of large-area blackouts. The most popular methodologies proposed include keeping together slow-coherent generators to avoid the loss of transient stability [7] and graph-search based approaches, e.g., [8], [9]. Recently a novel methodology of spectral clustering has been proposed for controlled islanding that is based on recent advances in graph theory [10], [11]. The main contribution of this paper is to adapt the spectral clustering methodology [12], [13] to the context of power trans- mission networks, and explain how this technique can be used to provide a real-time, versatile, analytical or visualization tool for power transmission systems. Our methodology reveals the internal structure of a given network with respect to any choice of electrical parameter that can be associated to a transmission line, such as line admittance, power flow or other. It may also take a preferred number of clusters as input, or this number can be obtained directly from the data. The output is a hierarchical clustering of the network according to the connection strengths given by the chosenweighting. Thismay be thought of as a func- tional decomposition of the system into smaller subsystems of highly connected buses. Our methodology goes beyond bisec- tion or recursive bisection techniques by providing an all-in-one decomposition of the network into any number of clusters. In- deed, we keep all the underlying hierarchical substructure, in- stead of just a partition of the network, by using a hierarchical clustering algorithm as our final step. This way we are able to represent simultaneously the different levels in the clustering, and reveal the scale-dependence of the resulting data, which represents the functional hierarchy of the network. This paper should be thought as a proof-of-concept of the underlying spec- tral clustering technique in the power engineering context, and we have taken special care in explaining its mathematical basis. We illustrate the potential of our methodology in a small test network in full detail, and in other larger test networks in some detail. The paper is the result of a collaboration between power en- gineers and mathematicians, as we strongly believe that there is a wealth of knowledge in graph and operator theory that is not widely known in the power engineering community and which could be usefully employed to solve practical power en- gineering problems. This paper is an attempt in that direction. II. PRELIMINARIES A. Graphs Representing a Power Transmission Grid 1) Terminology and Notation: A power network can be naturally represented as a graph: vertices (nodes) represent buses, and edges (links) represent electrical connections. We 0885-8950 © 2014 IEEE. Translations and content mining are permitted for academic research only. Personal use is also permitted, but republication/ redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. 2230 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 29, NO. 5, SEPTEMBER 2014 write for a graph with vertex set and edge set . In what follows we will only consider simple graphs, where no loops and no multiple edges are allowed. This assumption does not restrict the generality of our consideration as multiple edges can be replaced by equivalent single edges. Since the graph is finite and simple, we can write , where is the number of vertices (nodes), and , where represents an edge (a trans- mission line or a transformer) from vertex to vertex . Since spectral clustering ignores edge directions, we will assume that all graphs are undirected: if and only if . 2) Edge Weights: The topological structure of the graph does not capture the functional information about the power grid. To include this information, we use edge weights. An edge weight is a function such that 1) for all ; 2) if ; 3) . We use the notation if , and for the so-called weighted vertex degree. The purely topological structure of the graph is returned when we select the weight function for all , recovering the classical adjacency matrix in this case. Note that the edge weights must be nonnegative and symmetric to have a Laplacian (see Section II-B). 3) Weights in Power Grids: To study the functional struc- ture, in a very restricted sense, of a power grid we may use the following edge weight functions: • Topology: for all ; this measures pure connectivity of the network. • Admittance: , where and are the line resistance and reactance respectively; this measures the strength of connections (electrical distance). In this paper we will use the DC network model in which these weights are . • Average power flow: , where is the (real) power flow from to (if the network is lossless this is simply the power flow ); this weight measures the importance of a line in a given oper- ating condition—a small flow means that the line is more likely to be removed when clustering. Note that the first two weights are static, i.e., they are constant for a given power system, while the power flow is dynamic, as it changes according to actual operating conditions. Edge weights can be interpreted as a penalty for cutting the corresponding line when clustering, but also as a measure of the connection strength, as strongly connected vertices are more likely to be clustered together. Thus the admittance based clus- tering will reveal the internal structure (electrical distance) of the network while the power flow based clustering will reveal islands which, when separated, disrupt as least as possible the power flow in the network, and thus can be especially useful for preventive islanding purposes [11]. B. Graph Laplacians The Laplacian matrix is widely used in graph theory and it has a clear power engineering interpretation. There are twomain kinds of Laplacian matrix that can be associated with an undi- rected weighted simple graph . 1) Unnormalized Laplacian: The Laplacian of is the matrix defined as if ; if and ; otherwise. (1) It is a real symmetric matrix with non-positive entries outside the diagonal, and the sum of each column (or row) is zero. If the weights are equal to inverse line reactances (using the DC network model), then the unnormalized Laplacian is simply equal to the well-known nodal admittance matrix (neglecting shunt susceptances). There is no established power engineering interpretation of the Laplacian when the weights are equal to power flows. 2) Normalized Laplacian: The normalized Laplacian is the matrix , where is the diagonal matrix with nonzero entries . That is, if ; if and ; otherwise. (2) The normalized Laplacian is scale-independent, and it is more advantageous for clustering purposes (see Section IV-A). C. Eigenvalues of the Laplacian The eigenvalues of the (normalized or unnormalized) Laplacian matrix satisfy the following key properties [14]: 1) all the eigenvalues are nonnegative real numbers; 2) 0 is an eigenvalue with multiplicity equal to the number of connected components (islands) in the graph. We write the eigenvalues of as , and the eigenvalues of as . From property (2) above, we know that , or , if and only if is connected. From now on we will assume that our graph is connected. The eigenvalues of are scale-dependent and have no a priori upper bound (multiplying all weights by a scalar multiplies all the eigenvalues by the same scalar). However, the eigenvalues of satisfy the inequality for all . III. SPECTRAL CLUSTERING A. Introduction and Terminology By clusteringwemean the identification of groups of vertices in a graph (called clusters) such that the vertices in a cluster are highly connected among themselves (taking edge weights into consideration) but weakly connected to vertices in other clusters. By spectral clustering we mean any clustering proce- dure that uses the Laplacian eigenvalues and eigenvectors [12]. A theoretical justification for spectral clustering is deferred to Section