Comparative Studies of Clustering Techniques for Real-Time ... - arXiv

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Comparative Studies of Clustering Techniques for Real-Time Dynamic Model Reduction Emilie Hogan† , Eduardo Cotilla-Sanchez‡ , Mahantesh Halappanavar† , Zhenyu Huang† , Guang Lin∗ , Shuai Lu† , Shaobu Wang†

arXiv:1501.00943v1 [physics.soc-ph] 5 Jan 2015

†

Pacific Northwest National Laboratory; [email protected] ‡ Oregon State University; [email protected] ∗ Purdue University; [email protected]

Abstract—Dynamic model reduction in power systems is necessary for improving computational efficiency. Traditional model reduction using linearized models or offline analysis would not be adequate to capture power system dynamic behaviors, especially the new mix of intermittent generation and intelligent consumption makes the power system more dynamic and non-linear. Realtime dynamic model reduction emerges as an important need. This paper explores the use of clustering techniques to analyze real-time phasor measurements to determine generator groups and representative generators for dynamic model reduction. Two clustering techniques – graph clustering and evolutionary clustering – are studied in this paper. Various implementations of these techniques are compared and also compared with a previously developed Singular Value Decomposition (SVD)-based dynamic model reduction approach. Various methods exhibit different levels of accuracy when comparing the reduced model simulation against the original model. But some of them are consistently accurate. From this comparative perspective, this paper provides a good reference point for practical implementations.

I. I NTRODUCTION Power engineers rely on simulation to accomplish operation and planning tasks. Dynamic simulation tools are extensively used to simulate the dynamic behavior of power systems when it is subject to disturbances such as faults, sudden loss of transmission paths, and loss of generation or load. Good models and accurate parameters are essential for credible computer simulation. In some cases, inaccurate computer models and simulations would result in optimistic decisions, which put the electric infrastructure in jeopardy. The extreme consequences of such optimistic decisions are massive outages such as the August 1996 western US system breakup [1]. In many other cases, inaccurate models and simulations lead power engineers to make pessimistic decisions, which result in reduced asset utilization. Given the complexity and large footprint of the power system, a power company needs to reduce the model to its own service territory. The goal of such model reduction is to reasonably represent the external system with a simplified smaller model so that the analysis can be performed more efficiently [2], [3]. This is especially of particular interest for real-time power system operation, such as on-line dynamic security assessment [4]–[6]. Traditionally, model reduction is performed in the steadystate context, largely ignoring dynamics in the external system,

or performed off-line where scenarios may be different from the real-time conditions. This has served the power system reasonably well when the power system behaviors are more predictable. However, this practice is no longer adequate because the power system is becoming more and more dynamic and non-linear due to the new mix of intermittent generation and intelligent consumption. Model reduction has to evolve to handle such non-linear and dynamic behaviors. Dynamic model reduction has been studied extensively in the past. “Coherency” is the most common concept adopted in identifying groups of dynamic devices such as generators for model reduction purposes. Coherency can be determined by using a linearized model around an operating point [7], [8] or by analyzing offline simulated dynamics [9]. Either way it is not able to capture real-time operating conditions, which renders the reduced model useless for real-time analysis. Recent developments and deployment of phasor technologies present an opportunity to perform dynamic model reduction in real time, because high-speed phasor measurement captures the majority of power system dynamics that are of high interest for power system operation and planning purposes. Ref [10] proposed a Singular Value Decomposition (SVD)-based method that can be used for real-time dynamic model reduction. Such a method also preserves a certain level of non-linearity in the reduced model. Following this line of research, this research team continues the development of real-time dynamic model reduction techniques using clustering methods and compares the performance with the SVD-based approach. The general idea is to use real-time phasor measurements to cluster generators based on similar behaviors and then choose a representative generator from each cluster. The final reduced model would contain only the representative generators. We investigate two clustering methods – graph clustering [11] and evolutionary clustering [12], [13]. Variants of implementations of these two methods are tested alongside the SVD method and the well-known k-means algorithm on fault scenarios within the IEEE 50-machine system [14]. A workflow diagram showing how each method works with the given data types is shown in Figure 1. The results show various levels of accuracy for these methods, but many possess the accuracy adequate for power system operation and planning purposes. From this comparative perspective, this paper provides a good reference point for practical implementations.

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PMUs, also called synchrophasors, collect data synchronously across the power grid. These units are deployed to many systems throughout the grid and are time synchronized so that measurements taken at different locations can be correlated together. The data that we care about, collected from the PMUs, is rotor angle data. These data are collected at the millisecond resolution, between 30 and 120 samples per second, and gives a picture of the oscillation of the rotor angle at each generator. For each D generator a timeE series of rotor (1) (2) (m) (j) ~ angles is collected, δi = δi , δi , . . . , δi , where δi is the rotor angle of generator i at the j th time step. The other type of data that we care about is the Ybus matrix for the system. This is also known as the nodal admittance matrix. It is an n×n matrix (where n is the number of buses in the system). The i, j entry in the Ybus matrix is the equivalent admittance between bus i to bus j, and zero if there is no direct connection. A. Our Test System In this paper we use a small model system for method validation. Specifically, we use the IEEE 145-bus, 50-machine system [14] found in Figure 2. In this system there are 16 internal area machines and 34 external machines. We choose generator 37 at Bus 130 in the internal area to be our reference machine. We model all generators using the classic model for machine dynamics with a second order swing equation. For our tests we created five simulated PMU data sets using five different three-phase, short circuit faults within the system. The faults last for 60 ms and then the line is tripped to clear the fault. The oscillations of the rotor angles of the external system generators are recorded and analyzed for model reduction using various methods described in the next section. We additionally consider pre-fault Ybus data to get a connectivity picture of the system without any disturbances. Fig. 1: Workflow showing how the three methods use two different data sources to go perform model reduction.

The remainder of the paper is organized as follows: Section II describes the system and data used in implementing and comparing the proposed model reduction methods; Section III provides an overview of the two clustering techniques; Section IV presents the comparative study approach and results; and Section V concludes the paper with suggested future research. II. T EST S YSTEM AND DATA There are many types of data that can be collected from power grid systems. In our work we explore the use of two of these types of data. Primarily we use data collected by Phasor Measurement Units (PMUs), but in one of the tested algorithms we make use of Ybus information. The Ybus contains a different type of information than the PMU data. It is a full description of network connectivity and impedances. While the PMU data is more of an “online” data stream the Ybus is known a priori making it more of an “offline” source.

Fig. 2: The IEEE 50 generator system.

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III. OVERVIEW OF C LUSTERING T ECHNIQUES In this section we describe the varied methods we use for model reduction and characteristic generator identification. Here we will give basic details and refer to more detailed papers for a full description [10], [15], [16]. A. Graph Methods The graph clustering methods we describe here – recursive spectral bipartitioning, and spectral clustering – are not new. However, we do make modifications to the bipartitioning method to make it more robust. Additionally, these methods have not previously been used with power grid data to perform model reduction. Before we explain graph methods for clustering or partitioning we must discuss how to create a graph from either of our two power grid data sets. In both cases, the vertices of the graph are generators and/or buses and edges indicate a physical connection or an amount of similarity. It is straightforward to create a graph from the Ybus data since it is already a symmetric matrix: each row/column corresponds to a generator or bus and then there is an edge between i and j if Yi,j is nonzero. The weight of the edge is the quantity Yi,j . To create a graph from PMU data is not so straightforward. For each generator we consider its time series data vector, ~δi ∈ Rm , where m = number of time steps recorded. We then calculate a distance matrix D = (dij )ni,j=1 . The entries in D are given by dij = dist(~δi , ~δj ) for a given distance or pseudo-distance function on Rm . Once we have this matrix we create an - or k-nearest neighbor graph. We either connect each generator to all other generators which are distance less than  away, based on the entries in D, or we connect to its k closest generators. Details of the types of distances we use and the graph construction are presented in [16]. Both types of graph clustering that we test use spectral (eigenvalue) properties of the weighted Laplacian matrix associated with the graph. The weighted Laplacian, L = (Lij )ni,j=1 , is defined as follows:  P i=j k6=i wik Lij = −wij i 6= j The entries on the diagonal, Lii , are given by the sum of all edge weights on edges incident to vertex i. Off-diagonal entries, Lij , are the negative weight on edge eij . If an edge is absent then we treat it as an edge of weight zero. We define the weight of an edge to be a similarity between the endpoint vertices, based on the distance (high distance means low similarity, and low distance means high similarity). 1) Recursive Spectral Bipartitioning: The most basic type of spectral graph clustering or partitioning is recursive spectral bipartitioning [17]. This algorithm utilizes the eigenvector for the second smallest eigenvalue of the weighted Laplacian matrix. Clearly, since each row of the Laplacian matrix sums to zero, there is a zero eigenvalue. It is not difficult to show that L is positive semi-definite, so zero is in fact the smallest eigenvalue. The second smallest eigenvalue is called the algebraic connectivity of the graph, and its associated

eigenvector, commonly called the Fiedler vector after Miroslav Fiedler who first defined the theory of algebraic connectivity and its relation to graph partitioning [18], [19], has properties which define a partition of the vertices of the graph into two groups. The Fiedler vector contains positive and negative values. By partitioning the associated vertices into two sets, one in which the value in the Fiedler vector is negative and one in which it is positive, we obtain a graph partition which minimizes the sum of the edge weights between the two partitions [19]. Traditionally the vertices of the graph are partitioned into those which have positive vs negative values as just discussed. This can lead to unbalanced partitions as there is no guarantee that half of the vertices will have positive values while the other half have negative values. Another possibility is to partition based on the median of the Fiedler vector. We have experimented with both of these methods. One further degree of freedom in this method is how to continue the partitioning. Using the Fiedler vector to partition the vertices of the graph into to disjoint sets is only the first step. In traditional recursive spectral bipartitioning each of these two sets is then partitioned in half using the Laplacian of the subgraph induced by each set of vertices. If this process of repeatedly splitting each set is continued for N steps, this will yield 2N clusters. In our work we have taken a more targeted approach to splitting up the clusters. Instead of arbitrarily splitting each cluster into two at each step, we search among all current clusters for one that is least “tight”, for a suitable definition of tight, and just split that one. We define tight to be related to either the sum of all pairwise distances in that cluster, the average of all pairwise distances, or simply the size of the cluster. As with how to split the Fiedler vector (zero or midpoint) we also experimented with these three possible splitting schemes. All of our results are summarized in Section IV. In the remainder of this paper we may refer to recursive spectral bipartitioning as “Fiedler partitioning” for short, because of the prominence of the Fiedler vector in its implementation. 2) Spectral Clustering: For general spectral clustering we use more than just the one eigenvector for the second smallest eigenvalue. Instead, we look at the first k eigenvectors. Each row of an eigenvector corresponds to a vertex in the graph from which the weighted Laplacian was formed. So, if we form an n × k matrix where the columns are the first k eigenvectors, then each row can be thought of as a new vector representation of each of the vertices. In spectral clustering we take these new vector representations and use k-means to cluster them, and thereby clustering the vertices themselves. The pipeline illustration in Figure 3 shows this sequence of steps. For a more in depth discussion of spectral clustering, including other variants which we do not investigate here, see [20]. B. Evolutionary Clustering The evolutionary clustering approach for power grid data aims to improve upon other traditional clustering methods by incorporating objectives that capture particularly relevant trade-offs from the power systems perspective [21]. We have previously done a comparison of graph clustering with evolutionary clustering [16], however the comparison was not

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j −wij

i

Graph

Laplacian n × n matrix

x1 x2 x11 x21 x12 x22

xk xk1 xk2

x1n x2n

xkn

k-means on rows

Clustering of vertices

First k eigenvectors {xi }ni=1

Fig. 3: An illustration of the steps in spectral clustering

done on model reduction, only clustering. Additionally, the comparison was not based on power system responses to the clustering. In Section IV we describe our more rigorous comparison scheme for this paper. The experimental setup for evolutionary clustering is similar to the other methods that we comparatively discuss in this paper. Here we interpret distance as the electrical cohesiveness metric proposed in [22]. The electrical distance between objects a and b is derived from the portion of the power flow Jacobian that measures the incremental change in voltage phase angle difference between nodes a and b (∆θa − ∆θb ) for an incremental active power (P ) transaction between nodes a and b. This power flow Jacobian is itself derived from a combination of the Ybus , and generation and load information. We solve the power flow problem to obtain the power flow Jacobian. In this particular instance we set up a ’nominal’ power flow Jacobian by assuming that power injections are small increments, so the Jacobian is basically inherited from the Ybus structure. The initial population that the algorithm evolves is constructed from a combination of random solutions and a set of solutions that are pre-calculated from a classic k-Means clustering algorithm [23]. The set of solutions are improved generation after generation with standard evolutionary processes such as mutations and recombination. The algorithm selects good clusters according to the following fitness function: f = ECI · BCCI · CCI · CSI

(1)

where the indices ECI and BCCI capture the extent of electrical cohesiveness within clusters and between clusters, respectively. CCI gives a goal number of clusters and CSI maintains balanced cluster sizes. These individual indexes evaluate from zero to one depending upon the extreme conditions for each goal. For example, ECI scores zero for a clustering solution where all buses are clustered together and one for a clustering solution where each bus belongs to a different cluster (atomistic solution). This iterative method uses the same exogenous input (k) that determines the optimal number of clusters that will receive maximum score by the CCI index. Our ideas here could be viewed as a initial step to integrate compactness and separation in the context of power systems. This concept is later formalized and generalized in [24].

C. Methods Previously Used in Model Reduction 1) K-Means: Since the PMU data is vector data, which can be thought of as points in Rn for some n, we will also compare against k-means clustering. K-means is a standard clustering technique for data in Rn [25] and is widely used in many applications [26], [27]. The general idea is a recursive algorithm which, in each step, computes centroids of each of the clusters and then reassigns points to the cluster whose centroid it is closest to. This needs an initialization of cluster centroids which is often done randomly. The algorithm then runs as follows: assign each point to the cluster whose randomly chosen centroid is closest, recompute centroids, reassign points to clusters, recompute centroids, etc. This is repeated for some predetermined number of steps, or until clusters don’t change and the algorithm has converged. One problem with k-means is that there is no guarantee that it will terminate in a globally optimal clustering. k-means is essentially a gradient descent method, and these are well known to have the possibility of getting stuck in a local minimum. Since the initialization is done randomly there can be multiple clusterings from the same input data. 2) SVD Method: Singular Value Decomposition (SVD) is a method of matrix factorization [28]. Taking an initial n × m matrix, M , the SVD method factorizes M into a product of three matrices, M = U ΣV ∗ , where U (n × n) and V (m × m) are unitary matrices and Σ is an n × m diagonal matrix. Clustering rows of an n × m matrix using an SVD method has been shown to have promise [29]. Our comparison of model reduction techniques looks particularly at a method designed for the power grid application which makes use of SVD which is first utilized in [10]. This method can be summarized as follows. a) Step 1: Use SVD to extract the first few principal components. Suppose ~δ1 , ~δ2 , . . . , ~δi , . . . , ~δn are the n vectors corresponding to rotor angle readings of generators within the system to be reduced. The term ~δi is a normalized (subtract mean, divide by standard deviation) m-dimensional row vector representing the dynamics of generator i following a dis(1) (2) (m) turbance. Its elements are time series: δi , δi , . . . , δi . Define: δ = [~δ1 ; ~δ2 ; . . . ; ~δn ] where each ~δi is considered as a row vector. Here, δ is an n × m matrix. Suppose x = [~x1 ; ~x2 ; . . . ; ~xi ; . . . ; ~xr ] is the first r principal components from SVD where ~xi is an m-dimensional row vector.

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b) Step 2: Determine characteristic generators. Analyze the similarity between ~δi and ~xj ; those ~δi with the highest similarity to x are selected to form a subset of δ, which is called a set of characteristic generators. c) Step 3: Use characteristic generators to describe the full model. Responses of non-characteristic generators are approximated by linear combinations of characteristic generators so that dynamic equations of non-characteristic generators are canceled and a reduced model is obtained.

Index number 1 2 3 4

Code fiedler kmeans evolution e f e k su si a

Meaning Recursive spectral bipartitioning k-means clustering Evolutionary clustering Euclidean distance FFT pseudo-distance -nearest neighbor graph k-nearest neighbor graph Split Fiedler clusters on sum of distances Split Fiedler clusters on size of cluster Split Fiedler clusters on average distance

TABLE I: Explaining the clustering method codes in the comparison figures (Figures 4-13).

D. Choosing Cluster Representatives In all but the SVD methods above the algorithms output a set of clusters. However, in the context of model reduction what we need is a set of characteristic generators. We achieve this by choosing one representative generator from each cluster. Essentially we choose the centroid of each cluster based on the PMU data. |Ci | Given a cluster Ci = {δij }j=1 we find the average time series * P|Ci | (k) +m j=1 δij δCi = . |Ci | k=1

This is the true centroid, however it’s highly unlikely to be the time series for a generator in the given cluster. So, we then find which generator’s time series is closest to this centroid based on a given distance function. For both graph clustering methods we use the distance function that was used to create the graph. The evolutionary clustering algorithm differs from the rest since it clusters based on pre-fault Ybus data rather than postfault PMU data. The idea is that this yields a clustering indicating the generators that should swing together rather than the generators that do swing together. However, when choosing representative generators we are using the post-fault PMU data. We are effectively answering the question: “Given that these |Ci | generators should behave similarly, which one is most representative of the group given the current actual behavior?” In this way we can do the clustering step, which may be expensive, ahead of time, and then choose representatives based on the current behavior. IV.

C OMPARATIVE S TUDY A PPROACH AND R ESULTS

A. Method of comparison There are 16 machines in the internal system, which are {G28, G29} ∪ {G37, G38, G39, . . . , G50}. In order to judge the accuracy of the reduced model we simulate both the full system of 34 external generators and the reduced system of representative generators. Responses of the 16 internal generators are recorded during both simulations. Let δif (t) be the rotor angle response of the ith machine to the full order model simulation (original model) at time t, and δir (t) be the same for the reduced order model. We select G37 as the reference machine in this case. Then, define the following

two metrics shown in (2) and (3) to measure the mismatch of response curves of the original system and the reduced system. Z t2 1 r f (2) Ja (i) = δi (t) − δi (t) dt t2 − t1 t1 s Z t2 h i2 1 Js (i) = (3) δir (t) − δif (t) dt t2 − t1 t1 The metric Ja (i) in (2) is the L1 norm between δir and δif , whereas the metric Js (i) in (3) is the L2 norm between these time dependent functions. In the next section we apply these measures to compare the methods we described in Section III applied in five different fault scenarios in the 34 generator system. B. Results of comparison For each of our tested model reduction methods we calculated both Ja (i) and Js (i) for all 16 internal system generators. Then, we take the average of the Ja (i) and the Js (i) over all of the internal machines. We did this for each of the 5 fault scenarios in the IEEE50 system. Here we will summarize this comparison. In each of the fault scenarios we found that there was a different winner, however the top ten methods in each are consistently dominated by the Fiedler method. Figures 4 – 8 show the average Ja values for the top 10 clustering methods (blue diamonds). In these figures and tables we refer to recursive spectral bipartitioning as Fiedler methods. Table I explains the method codes found in these figures. The error bars represent one standard deviation above and below the average. Notice that in a majority of scenarios the standard deviation on the Fiedler methods is small. A small standard deviation means that the Ja (i) values are all close to the average. This indicates that the model reduction produced by the Fiedler method more consistently approximates the full system. We see similar results in Figures 9 – 13 which show the average Js values. Here, averages are indicated by red diamonds. This is an admittedly small test system and a limited number of scenarios. However, in all of these cases the only method which is consistently accurate is a variant of the Fiedler method. We plan to pursue further scenarios in larger systems in future work.

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Fig. 4: Top 10 clustering methods for scenario F1, average Ja values and standard deviation.

Fig. 7: Top 10 clustering methods for scenario F4, average Ja values and standard deviation.

Fig. 5: Top 10 clustering methods for scenario F2, average Ja values and standard deviation.

Fig. 8: Top 10 clustering methods for scenario F5, average Ja values and standard deviation.

Fig. 6: Top 10 clustering methods for scenario F3, average Ja values and standard deviation.

Fig. 9: Top 10 clustering methods for scenario F1, average Js values and standard deviation.

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Fig. 10: Top 10 clustering methods for scenario F2, average Js values and standard deviation.

Fig. 13: Top 10 clustering methods for scenario F5, average Js values and standard deviation.

V.

C ONCLUSION

In this paper we provide a small survey of model reduction techniques along with a comparison of these methods against the IEEE 50 generator test system. We compare two graph methods – recursive spectral bipartitioning and spectral clustering – as well as an SVD method, evolutionary clustering, and the standard k-means algorithm. Our comparison shows that in these scenarios the recursive method outperforms the other methods. In addition, the accuracy of the recursive method is typically more consistent across the whole test system. In future work we plan on exploring other graph clustering methods, both standard and novel. We will also be testing on larger systems for a more robust comparison.

Fig. 11: Top 10 clustering methods for scenario F3, average Js values and standard deviation.

VI.

ACKNOWLEDGEMENTS

This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under contract number DE-AC05-76RL01830. R EFERENCES [1]

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Fig. 12: Top 10 clustering methods for scenario F4, average Js values and standard deviation.

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