Studying vulnerability in power grids due to extreme events is important. Particularly, the clusters present in the power grids can identify key components in the system as well as prepare systems for contingencies and failures. These clusters can be found in multiple ways dependent on the criteria such as equal size of clusters, minimum number of links between clusters, and maximum interactions between components inside the clusters etc. However, finding clusters dependent on multiple criteria is a novel way to define clusters in power grids. In this paper, the clusters in the IEEE 30 bus system are found such that the number of links between clusters are minimized and the power grid experiences maximum imbalance in power between two clusters. Partitioning in such a way can yield between-clusters links which are the key components in the system as their removal causes the most drastic power imbalance in the system.
Extreme events in power systems such as blackouts and failures have significant societal and economical impacts. Since, other infrastructures such as water, transportation, gas, communication etc. are reliant on power grid networks for electricity supply, failures in power grids can cause catastrophic consequences. The blackouts of the July 2, 1996 and August 10, 1996 in the Western United States cite{kosterev1999model}; the blackout of August 14, 2003 in the Midwest and Northeast United States and Ontario of Canada cite{force2004final}; the blackout of November 4, 2006 in the European power grid system all give examples of catastrophic failures that affected millions of consumers. There have been 14 major blackouts during the year from 2003 to 2015 as mentioned in cite{veloza2016analysis}. The November 4, 2006 blackout in the European power grid system affected 45 million people with a loss of 14.5 MW of load in just over 2 hours cite{veloza2016analysis}. Hence, analysis of the causes and scenarios leading upto these catastrophic failures are essential for smooth operation and stability of the power grid. As observed in the Northeast blackouts of 1965 and 2003, failure of one component may cause failure of other components and this process may proceed to cause a large portion of the power grid to fail. Hence, cascading failures one after the other may lead to blackouts in the power grid. Finding critical components in the power grid network is of utmost importance and will help power system planners in design and management of power grids. Power systems are complex structures made up of generators, transformers, buses, transmission lines etc. and while failures in any component is probable, cascading failures and blackouts are most probable in the transmission network. Other components such as buses may also undergo failures but generally have protection schemes such as redundancies and backups. Understanding and mitigating failures in power grids is challenging due to the large number of components and their complex interactions. Intensive research efforts have been focused on understanding the interactions in power grids such as power flow analysis, physics based studies, graph based analysis and so on. These enable predicting the propagation path of failures, identifying critical/vulnerable components in the power grid, predicting the size of failures, response to contingencies/deliberate attacks etc. However, anticipating such events in the power system is difficult due to the large number of possibilities in which the failures could occur. For example, in a small system of 5 components, there are 31 possibilities in which the failures could occur. As the system size increases, the possible ways in which the failures could occur increases as well. Hence, the failures have combinatorial possibilities. In this study, a heuristic graph based approach is used to identify the most extreme case of partitioning the power system into two clusters such that the disruption to the power flow is maximum i.e. one partition will have a large generation and will be overflowing with power whereas the second partition will have a small generation and will be deficient in power. The components that are {it textbf{between-clusters}} will then be identified as the most critical components in the system.Some studies such as cite{nakarmianalyzing, sanchez2014hierarchical} focus on partitioning the power grid into clusters/zones or communities such that interactions inside the clusters are maximum and the interactions between clusters are minimum. This partitioning process is dependent on the idea that managing smaller clusters is easier, flexible and can be done in a distributed manner. This network splitting process cite{sanchez2014hierarchical} is also used to control islanding and consequently blackouts. Further, the critical components in the system can be identified. However, clustering can be done in reverse i.e. cluster power grids based on power flow in the system such that, instead of maximal interactions inside clusters, there is minimum interactions inside clusters. The power imbalance achieved is the maximum possible between the two partitions and the component between the clusters are the key ones.
However, as mentioned, brute-force calculation to search the most consequential combinations of component failures for the contingencies that may lead to extreme events is not possible. In this study, a simplified and reduced objective technique for analyzing these combinations of extreme events is used. The scope of the combinations are limited and only searched to focus on events that cause extreme consequences which may arise from a small number of component outages. For this purpose, a graph based approach is used to find clusters that arise due to a small number of component outages but that result in a very large disruption in power flow between the clusters.
This method can be be used as a preliminary step after which intensive analysis using detailed power system models can be used to evaluate outages of the identified critical components that may result in an extreme event such as blackout.
In the next section, we provide the theory on spectral partitioning of graphs as the analysis in this study employs the spectral partitioning technique in power grids. In Section III, the problem and a heuristic solution is proposed, and in Section IV, analysis performed on a synthetic power grid is shown and discussed.
A power system can be represented as a graph: vertices (nodes) represent buses, and edges (links) represent transmission lines. Graph $mathcal{G}=(V,E)$, where $V$ is the set of the nodes in the graph (i.e., the set of buses in the power system) and $E$ is the set of transmission lines in the system. The power grid is assumed to have no loops and no multiple edges. Hence, for a simple graph, $V = {1, 2, cdots, n}$ where, $N$ is the number of vertices and, $E subset V times V$ where $(i, j) in E$ represents an edge (a transmission line) from node $i$ to node $j$. In spectral theory, the directions of the edges are not considered. Hence, the graphs are undirected and $(i, j) in E$ is true only if $(j, i) in E$. This topological representation of power grids does not consider the functional properties of the system such as power flow, admittances, connectivity etc. However, in this study, a sample power injection vector for an extreme event cite{lesieutre2006power} is used in the analysis with the undirected and unweighted representation of the system i.e. the functional property of the system is combined with the topological information of the system.
Next, the representation of the power system by an incidence matrix with a sample spectral partitioning is shown to emphasize on the use and properties of partitioning using spectral theory. The following incidence matrix $A$ represents a power grid with 4 nodes and 5 edges. Each row in the matrix represents an edge whereas each column represents a node in the power grid. This example is adopted from the work in cite{lesieutre2006power} to emphasize on the spectral technique.
An element $A(i, j)$ in the matrix $A$ is 1 if edge $i$ originates at node $j$, -1 if edge $i$ terminates at node $j$ and 0 if there is no edge between the nodes. This representation of the power grid is very useful and reveals information about partitions of the power grid that will be discussed next.
The Laplacian matrix is extensively used in graph theory. Further, it has a power engineering application that can be used for graph partitioning. There are two different kinds of Laplacian matrix cite{sanchez2014hierarchical} that can be associated with an undirected and unweighted simple graph $mathcal{G}=(V,E)$.
Both unnormalized and normalized graph laplacians represent non positive entries outside the diagonal. Sum of each column or row is zero. The normalized laplacian is independent of scale. There are many ways to find the laplacian matrix of a graph. One of them is by using $A^TA$. It has the following properties:
begin{enumerate}
item $L$ is real, square and symmetric with dimension equal to the number of nodes in the system.
item As seen in unnormalized and normalized laplacian definitions, the entries not in the diagonal are either 0 or -1.
item The diagonal element in each row is the negative of the sum of the entries not in the diagonal.
item It is not dependent on the orientation of the branches in the system.
item For connected graphs, it has one zero eigenvalue with eigenvector consisting of a vector of all ones or identical values.
item The remaining eigenvalues are positive. Hence, the matrix is positive semi-definite.
item The signs: positive or negative of the eigenvector of the second smallest eigenvalue is used for partitioning the graph. It is also called an indicator vector. The components corresponding to the positive signs belong to one group and the components corresponding to the negative signs belong to the other group. This second smallest eigenvector is called the Fiedler vector cite{pothen1990partitioning} and the second smallest eigenvalue is called the Fiedler eigenvalue.
end{enumerate}