A Community-Based Partitioning Approach for PMU Placement in Large Systems

Transcription

1 1 A Community-Based Partitioning Approach for PMU Placement in Large Systems Anamitra Pal 1,*, Gerardo A. Sánchez-Ayala 2, James S. Thorp 3, and Virgilio A. Centeno 3 1 Network Dynamics and Simulation Science Laboratory, Virginia Bioinformatics Institute, Virginia Tech, Blacksburg, Virginia-24061, USA 2 Quanta Technology, Raleigh, North Carolina-27607, USA 3 Bradley Department of Electrical and Computer Engineering, Virginia Tech, Blacksburg-24061, USA * Corresponding Author. Address: 1880 Pratt Drive (0477), Blacksburg, Virginia-24061, USA. anam86@vbi.vt.edu Abstract A phasor measurement unit (PMU) placement scheme is developed in this paper that ensures complete observability of large systems while reducing the computational burden of the optimization. Redundancy in measurement of the critical buses of the network that are identified based on system studies and/or topologies is also provided by the proposed methodology. The community-based islanding approach initially partitions the system into smaller islands. Placement of PMUs in these islands is then computed using integer linear programming (ILP). A bound is also developed to find the maximum error from a global optimal solution. The proposed technique is applied to standard IEEE systems as well as on more realistic power system networks. The results indicate that the proposed technique optimizes the benefits of having PMUs at strategic locations of a large power system network without the associated computational burdens. Keywords Community-based islanding, Integer linear programming (ILP), Large systems, Observability, Phasor measurement unit (PMU), Redundancy. 1. INTRODUCTION Now that a synchrophasor-only state estimator is becoming a reality, it is clear that more and more phasor measurement units (PMUs) will be installed in the power grid [1]. It is believed that more than one thousand PMUs will be introduced in the US power system in the next few years [2]. Six countries in Central America are already installing PMUs in their inter-connected networks [3]. India and China also plan to add many more PMUs to their respective grids [4], [5]. The reason for this expedited growth in

2 2 synchrophasor investment is the numerous benefits that PMUs provide. Some of these benefits are robust state estimation, wide area control, and adaptive protection [6]. With this over-abundance of PMUs in the network, it is essential to develop a technique that can efficiently compute PMU placements for large systems. Since the inception of PMUs, a lot of research has been done to find an optimal PMU placement (OPP) scheme for generic systems. References [7]-[12] used a variety of mathematical tools to perform the optimizations. But the two primary problems that have led to the downfall of most of these techniques as far as implementation in large systems is concerned are: 1. The diversity of applications to be addressed using PMUs, and 2. The computational burden of the optimization itself Reference [13] states that on the basis of applications, there are two methods followed by power engineers for placing PMUs in a system: (a) Development of a prioritized list of placement sites based on observability, and (b) Measurements are placed to correctly represent critical dynamics of the system. The first method is concerned with state estimation and so does not take into account transient and dynamic stability of the system. The second approach does not consider complete observability as one of its priorities. Reference [14] used ILP to reconcile these two approaches. It placed PMUs on the critical buses of the network identified on the basis of system studies/topologies and then found the optimal number of PMUs required for complete system observability. But the OPP problem is known to be NPcomplete [15]. Therefore, although ILP gave optimal solutions, its applicability to large systems (> 500 buses) was limited by the computational burden of the integer programming-based optimization [16]. A number of metaheuristic approaches were also proposed to address the computational burden of the OPP problem [10]-[12], [17], and [18]. However, for generic systems, metaheuristic methods do not provide an upper bound on how close the solution obtained by them is to the optimal solution [19]. Hence they were also not suitable for application to large systems. This paper addresses the issue of PMU placement in big systems by introducing a community-based islanding approach [20] that partitions a large network into smaller islands. The islands in a power system can be different utilities operating under an independent system operator (ISO) or geographically separate

3 3 grids that share power through tie-lines. The critical bus based ILP technique proposed in [14] is then used to find the OPP set in the resulting islands. A bound is also developed to compute the maximum error from a global optimal solution. The proposed technique is initially applied to standard IEEE systems [21] and then tested on larger and more realistic power system networks, such as a 127-bus model of the WECC system [22], a 283-bus model of the Central American system [23], a 750-bus and 1133-bus models of the Indian system [24], and a 1443-bus model of the Brazilian power system [25]. The rest of the paper is structured as follows. Section 2 describes the concept of community-based islanding. The bound of the maximum error from a global optimal solution is computed in Section 3. The methodology developed to eliminate branches and place PMUs in the created islands is illustrated with an example in Section 4. The results obtained on applying the proposed algorithm on different systems is described in details in Section 5. The conclusions are provided in Section COMMUNITY BASED ISLANDING An interesting analysis was done in [20] for finding and evaluating community structure in general networks. This paper shows how the methodology developed in [20] can be used to address the OPP problem in large systems. Figure 1 shows how the computational burden can increase as the size of the system increases. In Figure 1, X-axis denotes the number of buses present while the Y-axis depicts the time required for computing the optimization using ILP. Power system networks ranging from 14 buses to more than 1400 buses were analyzed to create the plot. The reason for the time vs. number of buses plot having an exponential shape is that ILP tries to find the optimal solution to the OPP problem (which is NP-complete [15]) by searching through all possible combinations. Since there are two choices for every bus of the network, for a n bus system a brute force approach will have a worst-case time complexity of O(2 n ). Therefore, as the system size increases, the required computational effort increases exponentially. A pre-processing method to counter this growth was developed in [26]. But [26] did not consider the possibility that some buses could be more important than others from system stability perspectives. The technique developed in this paper shows how PMU placement can be done in large

4 4 systems without compromising on the real-time monitoring and protection of key regions of the system. The community-based partitioning logic is illustrated as follows. Let us have a network of n nodes that can be split into k islands having at most m nodes each by removing a certain number of edges. In such a scenario, the optimization problem for a single island will have a worst-case time complexity of O(2 m ), while the worst-case time complexity for computing the optimization for all the islands would be O(k 2 m ). Now, increasing k will make m n, which will then considerably reduce the worst-case time complexity. An example of this is shown in Figure 2 where k is varied from one to five and m = n/k. Figure 1: Indication of the computational burden of the optimization 3. COMPUTATION OF BOUND Some of the characteristics of the PMU placement problem that have been considered in this paper are summarized as follows: a. When placed at a bus of the system, a PMU observes the voltage of that bus and all of the branch currents that emerge from that bus. b. Zero injection (ZI) buses have not been modeled in this study. This is because in real-life buses are primarily substations and the internal consumption of the substation is something that cannot be neglected. Moreover, by treating ZI buses as normal buses, the problem of

5 5 topological observability not ensuring numerical observability is also avoided [27]. c. Redundancy is provided to the phasor measurements of only the critical buses of the system. Figure 2: Variation of worst-case time with number of nodes for different values of k The community-based islanding approach provides a computationally elegant solution to the PMU placement problem that has the characteristics defined above. However, this benefit comes at the cost of achieving a near-optimal solution instead of an optimal solution. This is because when a system is partitioned to form islands, the partitioning is done by removing edges from the network. Removing an edge may result in loss of observability of the nodes between which the edge initially lay. Now, as more and more edges are removed to form islands, it is possible that the total number of PMUs required for complete observability of the individual islands will be more than the number required for complete observability of the original system. This increase in the number of PMUs required is defined as the error from an optimal solution. This section computes a bound on the maximum error from an optimal solution that will be achieved by partitioning the system. Preliminaries: Let the power network be depicted by an undirected connected graph G(V, E) where V is the set of nodes and E is the set of edges. Let E be the set of edges that must be removed to partition the graph G into multiple islands. Let G (V, E E ) denote the resulting undirected disconnected graph.

6 6 Furthermore, let OPT(H) denote the optimal number of PMUs required for complete observability of an undirected power network H. Lemma 1: OPT(G ) OPT(G) + E where the symbol denotes cardinality. Proof of Lemma 1: Let the set S denote an optimal PMU placement solution for G, that is S = OPT(G). Then, we will prove Lemma 1 by showing that for a feasible optimal PMU placement solution set S for G, the following holds true: S S + E. Let i and j be any two nodes such that the edge {i, j} E. For each edge {i, j} E that is removed, let G 1 and G 2 describe the power network graphs before and after that edge was removed. Then, by definition, before the first element in E is removed, G 1 = G, after every element of E is removed, the new G 1 becomes equal to the current G 2, and after the last element in E is removed, G 2 = G. Also, let S 1 and S 2 denote the optimal PMU placement solution sets for G 1 and G 2, respectively. Then, by definition, before the first element in E is removed, S 1 = S, after every element of E is removed, the new S 1 becomes equal to the current S 2, and after the last element in E is removed, S 2 = S. Now, as each edge {i, j} E is removed, one of the four mutually exclusive scenarios, numbered as (a)-(d) below, can occur. (a) If i S 1 and j S 1, then S 2 = S 1. This is true because since neither node i nor node j has PMUs on them, both nodes i and j are observed by PMUs placed at other locations in S 1. Therefore, the edge between nodes i and j is redundant as far as observability is concerned. Hence, when G 2 is created from G 1 by removing such an edge, it will be completely observed by S 1. Thus, for this scenario S 2 = S 1 which further implies that S 2 = S 1. (b) If i S 1 and j S 1, then S 2 = S 1. This is true because since both nodes i and j already have PMUs on them, even if the edge between them is removed, the observability of the system will not be altered in any way. Therefore, when the edge between nodes i and j is removed, G 2 will continue to be completely observed by S 1. Thus, for this scenario also S 2 = S 1 which further implies that S 2 = S 1. (c) If i S 1 and j S 1, then S 2 S This is true because when the edge between nodes i

7 7 and j is removed, node j may become unobservable. The reason we say may is because even after edge {i, j} is removed, j might still be observed by one or more elements of S 1. In such a case, G 2 will still be completely observable by S 1 and S 2 will be equal to S 1, thereby implying that S 2 = S 1. However, if j does become unobservable, then in order to make S 2 completely observe G 2, S 2 = S 1 {j} and S 2 = S (d) If i S 1 and j S 1, then S 2 S This is true because when the edge between nodes i and j is removed, node i may become unobservable. The reason we say may is because even after edge {i, j} is removed, i might still be observed by one or more elements of S 1. In such a case, G 2 will still be completely observable by S 1 and S 2 will be equal to S 1, thereby implying that S 2 = S 1. However, if i does become unobservable, then in order to make S 2 completely observe G 2, S 2 = S 1 {i} and S 2 = S Repeating this process E times to account for all the edges in E, it can be easily shown that S S + E, and this completes the proof. In practice the number of extra PMUs required was found to be much less than the number of branches that were removed (See Figure 12). This is because when an optimal solution was found for the individual islands, it often resulted in S = S. Thus, as long as the number of branches removed to create the islands is kept to a minimum, the community-based partitioning approach is guaranteed to give veryclose-to-optimal results. In accordance with this realization, an algorithm is developed in the next section that places PMUs in created islands by eliminating least number of branches. 4. BRANCH ELIMINATION AND PMU PLACEMENT (BEPP) SCHEME In the previous section, a bound was developed on the maximum error from an OPP solution. However, in order for the bound to be effective, an algorithm is needed that can identify the minimum number of branches that must be removed to create the islands. Based on the logic of [20], an algorithm is developed here that not only creates islands by removing minimum number of branches but also computes the optimal placement of PMUs in the created islands while catering to practical constraints.

8 8 The proposed BEPP scheme comprises of three stages. The first stage involves assigning weights to the different edges of the network. The logic followed for doing so is that branches with the highest probability of connecting nodes from two different communities should have the highest weights. Although a variety of ways are possible for assigning weights to the edges the one used in [20] was used here because of its ability to create evenly-sized partitions while removing minimal number of edges. Once the branch with the highest weight is identified and removed, a clustering algorithm is used to check for presence of islands. The clustering algorithm identifies nodes that belong to the same community and groups them together. If all the nodes can be clustered under the same group, then it means that no island has been formed by the removal of the highest weight branch and the weighting scheme needs to be repeated to identify the next branch with the highest weight. Once the desired number of islands has been formed, the third stage involves using ILP to find an OPP set in the created islands. Constraints such as redundancy in measurement of the critical buses of the network are imposed at this stage. A flowchart of the BEPP scheme is provided in Figure 3. The three stages are described in more details in the following sub-sections with their application to the IEEE 14-bus system being the subject of the fourth sub-section. A. Weighting Scheme The first stage of the proposed BEPP scheme involves identifying and eliminating the least number of branches to create the partitions. Since the partitioning of a network into islands requires the removal of the inter-community edges, for their successful identification and elimination, these edges must have the highest weights. One way to ensure that is by defining a measure for the traversability of an edge when going by the shortest path from node i to node j for all i and j. On doing so, the following two cases arise: 1. Nodes i and j are in the same community 2. Nodes i and j are in different communities

9 Figure 3: Flowchart of the BEPP Scheme 9

10 10 In the first case, for most networks, the inter-community edges will not be traversed. The reason being that inside a community it is more likely that the shortest path between two nodes will be through edges that lie within the community itself. However, for the second case, it is necessary that the intercommunity edges are traversed. Now, as this process is repeated for all the nodes in the network, the inter-community edges will be traversed more number of times than the intra-community edges. The reason for this is that whenever nodes from two different communities would need to be connected, they would be done so via the inter-community edges. Therefore, by definition, the measure that is developed based on the traversability of an edge will be highest for the inter-community edges. The above-mentioned concept is implemented in a three step process. In the first step, a symmetric matrix called the Depth matrix D is created whose element D(i, j) denotes the depth of node j from node i and is equal to the minimum number of edges that must be traversed to reach node j starting from node i. If node j cannot be reached from node i, then D(i, j) is set to zero. After the computation of the Depth matrix, the next step is to define a Vertex Number matrix V whose element V(i, j) denotes the minimum number of shortest paths by which node j can be reached from node i. The Vertex Number matrix is also a symmetric matrix and is computed using the elements of the Depth matrix. The third step is the creation of the Weight matrix which provides a measure for comparing one branch to another. With the aid of Depth and Vertex Number matrices, the Weight matrix is created using the script given in Figure 4. In Figure 4, A denotes the n-dimensional adjacency matrix of the network, i denotes the source node, j denotes the destination node, and k denotes a node which lies in between i and j. The ratio V(k, i)/v(j, i) is the index that measures the traversability of an edge. The variable counter ensures that the weight of the node furthest from i is the least and it increases progressively as it gets closer to i. The gradual increase is quantified at subsequent depths by the sum of all the weights of the downstream edges, increased by one to account for the decreasing depth, and finally multiplied by the ratio of the vertex numbers. Once the Weight matrix is computed, the edge having the highest weight is removed from the network and the resulting adjacency matrix is set as an input to the clustering algorithm described next.

11 11 For i, j, k n if(d(j, i) == max(d(:, i)) if(a(j, k) 0) W(k, j) = V(k, i) V(j, i) endif endif counter = 1 if(counter max(d(:, i))) if(d(j, i) == max(d(:, i)) counter) if(a(j, k) 0) if(d(j, i) > D(k, i)) W(k, j) = (sum(w(j, : ) + 1)) V(k, i) V(j, i) endif endif endif counter = counter + 1 endif endfor Figure 4: Script for computing the Weight matrix B. Clustering Algorithm The second stage of the BEPP scheme is the clustering algorithm. Its function is to detect and identify created islands. It starts with the first element of the adjacency matrix obtained from Section 4A and identifies the non-zero elements present in the first row (or column). Once it comes across a non-zero entry, it travels along the corresponding column (or row) to search for other non-zero entries. In this way, if all the nodes can be reached from the first node, it means that no islands have been formed by the removal of the branch identified in Section 4A and that the weighting scheme has to be repeated to identify the next branch to be eliminated. However, if there are nodes that cannot be reached from the first node by following the above-mentioned process, then it means that an island is present. In order to identify the nodes present in the island, the smallest numbered node that could not be reached by the first node is set as the first node of the new island and the clustering approach is repeated to identify other nodes which can then be reached from this new first node. This process is repeated until all the islands have been identified. If the number of islands formed is less than the number of islands desired, then the weighting scheme is run again to identify more edges that must be removed. However, if the number of

12 12 islands formed is greater than or equal to the desired number, then the created islands along with the nodes present in each of them are set as inputs to the ILP technique described next. C. Binary Integer Programming A methodology was developed in [14] that ensured observability of the most important buses of the network under N 1 contingency criterion using ILP. This was achieved by providing more weights to the nodes that were critical. High voltage buses, high connectivity buses, buses relevant to transient/dynamic stability, and/or potential small signal control buses were recognized as critical nodes for this study. Since the weighting scheme described in Section 4A does not affect the relative importance of the nodes of the system, the technique developed in [14] can be smoothly integrated into the proposed scheme. This is done by following a five-step process as shown below: Step 1: Identify critical nodes of the system. Step 2: For the i th island, identify the critical nodes present in it. Define a null vector X init having the i same length as the number of nodes in the i th island. Set the locations of the critical buses as one in X init i. Step 3: Define a vector g i such that, g i = f i wx init i (1) In (1), f i is a vector of ones having the same length as X init and w is a scalar having any value greater i than one. The scalar weight w ensures priority in placement of PMUs on the critical nodes. Step 4: The optimization criteria is defined as, min X i (g i T. X i ) (2) The constraints imposed on (2) are A i X i f i T. X i = nnz(x init i ) (3) X init i In (3), A i is the incidence matrix of the i th island and nnz(y) is the number of non-zero elements in vector y. The incidence matrix is the same as the adjacency matrix except that its diagonal elements are unity instead of zeros.

13 13 Step 5: If a critical node is not connected to any other node that has PMU on it, then define the node immediately next to that critical node as a critical node and repeat Steps 2 to 4. By following these five steps for each of the created islands, an OPP scheme can be realized that guarantees redundancy in measurement of all the critical buses of the network. More details about the mathematical formulation of the binary integer programming based ILP can be found in [14]. Since the community-based partitioning technique has a worst-case time complexity of O(n 3 ) for sparse graphs [20], the worst-case time complexity for partitioning a system into k equal islands is O (n 3 + k 2 (n k ) ), which is considerably less than O(2 n ) for all n > 10 and k 2. Thus, by combining ILP with the community-based islanding approach, a simple PMU allocation technique is realized that provides real-time monitoring of critical buses of the network as well as ensures complete system observability with reduced computational burden. A variety of power system networks were analyzed to assess the utility of this scheme. Its application to the IEEE 14-bus system is described in the next section. D. Illustration of BEPP Scheme on IEEE-14 Bus System In this section, the application of the BEPP scheme is demonstrated on the IEEE 14-bus system. This system consists of 14 buses and 20 lines [21]. The objective is to create two islands in this network based on the proposed scheme. The adjacency matrix of the original network is shown in Figure 5. When the adjacency matrix is fed into the weighting scheme, the Depth, the Vertex Number and the Weight matrices obtained as outputs are shown in Figures 6-8. From the Weight matrix obtained in Figure 8, it is realized that branch 5-6 has the highest weight (highlighted in red), and that it should be the first branch that must be removed. After removing this branch from the adjacency matrix, the modified adjacency matrix is fed into the clustering algorithm. On checking for islands, it is realized that no islands have been formed by the removal of branch 5-6. Therefore, the modified adjacency matrix is fed back into the weighting scheme for identifying new branches that must be eliminated in order to create the desired number of islands (in this case, two).

14 14 Bus Numbers Figure 5: Original Adjacency matrix of the IEEE 14-bus system Bus Numbers Figure 6: Original Depth matrix of the IEEE 14-bus system

15 15 Bus Numbers Figure 7: Original Vertex Number matrix of the IEEE 14-bus system Bus Numbers Figure 8: Original Weight matrix of the IEEE 14-bus system On repeating this process two more times, branches 4-9 and 7-9 are identified as the branches with successively highest weights as shown in Figures 9 and 10 (highlighted in red, along with the zero weights of the eliminated lines), and are subsequently removed. The new adjacency matrix obtained is

16 16 shown in Figure 11 (with the changes from the original adjacency matrix highlighted in red). When this adjacency matrix is fed into the clustering algorithm, it identifies two islands. The first island consists of buses 1, 2, 3, 4, 5, 7, and 8 while the second island consists of buses 6, 9, 10, 11, 12, 13, and 14. Next, the adjacency matrices of the two islands are set as inputs to the binary integer programming section of the BEPP scheme. When none of the buses are considered critical, the numbers and locations obtained for placement of PMUs are shown in Table 1 (first row). Next, when bus 4 is considered to be a critical bus (with bus 2 identified to be the bus providing redundancy to the measurement of the critical bus), the new numbers and locations are again found out (as seen in second row of Table 1). Bus Numbers Figure 9: Weight matrix of the IEEE 14-bus system after line 5-6 is removed TABLE 1: Illustration of BEPP Scheme for computing PMU placements by partitioning IEEE 14-bus system into two islands Using Technique developed in [14] Using BEPP Scheme # PMUs Location # PMUs Location When no bus is considered critical 4 2, 6, 7, 9 4 2, 7, 10, 13 When bus 4 is considered critical 5 2, 4, 7, 10, , 4, 7, 10, 13

17 17 Bus Numbers Figure 10: Weight matrix of the IEEE 14-bus system after line 5-6 and line 4-9 are removed Bus Numbers Figure 11: Adjacency matrix of the IEEE 14-bus system after removal of lines 5-6, 4-9, and 7-9 In Table 1, the results obtained using the proposed scheme is compared with the optimal results obtained through the technique developed in [14]. The comparison indicated that when no buses were identified to be critical, although the locations differed slightly, the numbers of PMUs required remained

18 18 the same for the two approaches. This is because even though three lines were removed to create the islands in accordance with the BEPP scheme, when ILP was applied to the individual islands, the total number of PMUs required remained the same. In the second case, when bus 4 was identified to be a critical bus, it was observed that the numbers and locations of PMUs required remained the same for both the approaches. The conclusion drawn from this illustration is that the proposed BEPP scheme is a good choice for reaching a near-optimal (if not an optimal) solution for PMU placement in power system networks, especially when used in conjunction with the concept of critical buses. 5. SIMULATIONS AND RESULTS A. Results Obtained for Different Test Systems The effect of partitioning on the number of extra PMUs needed was analyzed in the first set of simulations. Since this paper focused more on large systems, the test systems having more than 100 buses were selected for this simulation, with none of the buses being identified as critical. The results obtained are shown in Figure 12. From the figure it becomes clear that the ratio of the number of extra PMUs needed to the number of branches eliminated, as more and more partitions are made, is very small. Only for the 127-bus system did the ratio equal 1 (when it was partitioned into 2 and 3 islands, respectively), and even then, when more partitions were made, the ratio for that system dropped quickly. The reason for the ratios being low in general was that of the four possibilities identified in Section 3, only two had the potential of adding more PMUs. Therefore, although many branches may have been removed to create the islands, their net effect on the total number of extra PMUs required for full observability was very small. Another inference that was drawn from Figure 12 was that the extra number of PMUs required did not increase with increase in the size of the system. On the contrary, it was realized that the ratio was lower for the larger systems (> 500 buses) than it was for the smaller systems.

19 19 Figure 12: Ratio of number of extra PMUs required to the number of braches eliminated for different power system networks as the number of islands is increased In the second set of simulations, the computation times required for performing the optimization using the BEPP scheme was compared with the traditional ILP technique developed in [28]. For the BEPP scheme, the test systems were partitioned into minimum number of islands greater than or equal to two, such that the size of the individual islands was less than 500 buses. For this set simulations also none of the buses were identified as critical. The results are provided in Table 2. The computations were performed using MATLAB on a machine having Intel (R) Core i5 Processor with a speed of 2.40 GHz and an installed memory (RAM) of 5.86 GB. From the table, it becomes clear that for systems with more than 500 buses, the computation times using the BEPP scheme is significantly less in comparison to the computation times using the ILP technique. Of special emphasis are the computation times for the bus system and the 1443-bus system, for which optimal PMU placements could not be computed using ILP even after letting it run for 150,000 seconds. A more powerful machine (Intel (R) Core i7 Processor having a speed of 3.40 GHz and an installed memory (RAM) of 64 GB) was used to find the optimal number of PMUs for those two systems. However, using the BEPP scheme, the original machine was able to come up with a near-optimal solution at a fraction of the time.

20 20 TABLE 2: Comparison of the proposed BEPP Scheme with the traditional ILP based PMU placement algorithm Using ILP [28] Using BEPP Scheme System Number of PMUs Time (in seconds) Island Details Number of Size Partitions Number of PMUs Time (in seconds) IEEE 14-bus system IEEE 30-bus system IEEE 57-bus system IEEE 118-bus system 127-bus WECC system 283-bus Central American system IEEE 300-bus system 750-bus Indian system 1133-bus Indian system >150, bus Brazilian system 443 >150,

21 21 In the final set of simulations, the critical buses were identified and redundancy under the proposed scheme is provided to them. The critical buses were chosen based on their voltage levels and connectivity, as well as on the basis of the transient and dynamic stability studies that were performed on the systems. The number of partitions made was based on system topology and/or computational ease. The 118 and the 300 bus systems were split into three islands. Since the 283-bus system represented the Central American Power Transmission Network comprising of six countries (Guatemala, Nicaragua, Honduras, El Salvador, Costa Rica, and Panama), it was split into six islands. The 750 and the 1133-bus systems represented the Northern-and-Eastern Power Grids of India and were partitioned such that each of the islands formed had less than 500 buses. The results obtained for the test systems are shown in Table 3. From the table it becomes clear that a near-optimal solution is obtained using the BEPP scheme even after considering some buses as critical. Thus, using the technique proposed in this paper, a PMU placement scheme is developed that optimizes the benefits of having PMUs at strategic locations of a large power system network without the associated computational burdens. TABLE 3: Number of PMUs required after considering critical buses System Number of islands created Number of critical buses Optimal number of PMUs using [14] Number of PMUs using BEPP 118 Bus System Bus System Bus System Bus System Bus System B. Analysis of the Results Some inferences that were drawn from the results obtained in Section 5A are summarized below: 1. The clustering algorithm is so designed that it will terminate the outer loop of Figure 3 when

22 22 the number of islands formed is greater than or equal to the desired number. Alternately, it can also terminate (if the user so desires) if the number of buses inside each of the created islands is less than a specific threshold (in the simulations, that threshold was set at 500 buses). 2. The BEPP scheme is not a purely topographical approach because it takes into account the dynamics of the power system (during the selection of the critical buses). However, considering the fact that it can partition the system so as to match existing geographic divisions (for example, the six-nation Central American system can be split very effectively into six islands), use of the BEPP scheme to study some of the physical or geographical properties of power networks can be explored in the future. 3. The BEPP scheme ensures that the only locations where more PMUs would get added to the network is at the boundary buses. But since boundary buses are the locations where market prices change, it makes practical sense to have those buses monitored in real-time by PMUs. 4. Since most metaheuristic approaches do not provide an upper-bound on the gap between their solution and the optimal solution, the results of the BEPP scheme were compared with ILP techniques that are guaranteed to give optimal results. 5. Most of the realistic power system data used in the simulations were obtained directly from the agencies who were interested in the results of the study. The data primarily consisted of load flow cases and dynamic files that were solved using the PSS/E software. For performing the transient and dynamic stability studies on the test systems (so as to identify critical buses), the power flow was solved using the Newton-Raphson method. To conclude, the BEPP scheme developed in this paper is a complete package: it can give optimal results for small systems (by setting desired number of islands = 1), and can give very-close-to-optimal results for larger systems in which the direct application of ILP-based approaches (such as those developed in [14] and [28]) could not give results within reasonable time. 6. CONCLUSIONS The core contribution of this paper is the introduction of a partitioning logic that divides a large power

23 23 system network into small islands for facilitating PMU placement. The main advantage of using this technique is that PMU locations are found for large networks without performing any type of topology modification/reduction. Thus, the results produced are more accurate even for models that have many low voltage inter-connections. The partitioning scheme identifies communities in the system by analyzing the connections between buses. After splitting the network into islands, a critical bus based PMU allocation technique is used to find the optimal locations. The proposed method is found to provide genuinely good results for large systems at reduced computational effort. The community-based partitioning approach appears to have a lot of potential applications in power systems. Its use in the development of a partitioned linear state estimator is currently being considered. 7. REFERENCES [1] Jones, K. D., Thorp, J. S., and Gardner, R. M., Three-phase linear state estimation using phasor measurements, Proc. IEEE Power Eng. Soc. Gen. Meeting, Vancouver, BC, Canada, pp. 1-5, July [2] Department of Energy, U. S., Smart grid investment grant program progress report, July [Online]. Available: %20Progress%20Report%20July% pdf [3] Sanchez, G. A., Pal, A., Centeno, V. A., and Flores, W. C., PMU placement for the Central American power network and its possible impacts, Proc. IEEE Power Eng. Soc. Conf. Innovative Smart Grid Technol., Medellin, Colombia, pp. 1-7, October [4] Samantaray, S. R., Letter to the editor: smart grid initiatives in India, Electr. Power Compon. Syst., Vol. 42, No. 3-4, pp , February [5] Xu, Z., Xue, Y., and Wong, K. P., Recent advancements on smart grids in China, Electr. Power Compon. Syst., Vol. 42, No. 3-4, pp , February [6] Phadke, A. G., and Thorp, J. S., Synchronized Phasor Measurements and their Applications. New York: Springer, 2008.

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