Optimal PMU Placement using Best First Search Algorithm with Pruning

Transcription

1 Optimal PMU Placement using Best First Search Algorithm with Venkatesh.T Discipline of Electrical Engineering Indian Institute of Technology Indore Indore, India Abstract-This paper utilizes best first search (BFS) algorithm to determine the optimal placement of phasor measurement units (PMUs) for complete observability of a power system under normal operating conditions. The additional redundancy offered by this method has been removed by applying a pruning technique to further minimize the number of PMUs determined by BFS algorithm. The proposed method has been used to determine the optimal PMU placement solutions for the standard IEEE 14-bus system, IEEE 30-bus system and a practical 246-bus Indian system. The results obtained with the proposed method have been compared with the existing methods such as integer linear programming. It has been found that the proposed method is able to achieve the complete system observability with the minimum number of PMUs required. Keywords best first search (BFS); observability; optimal placement; phasor measurement unit (PMU); pruning I. INTRODUCTION The existing SCADA systems provide asynchronized measurements leading to inaccurate estimation of system states. Further, the slow data scan rate of about 2-10 seconds makes them inefficient to capture very short disturbances of the order of sub-seconds on the grid. These issues can be overcome by using the phasor measurement units, which utilize the global positioning system (GPS) receivers to accurately time-stamp each measurement. The capability of PMUs to measure 25 to 60 samples per second makes them suitable for analyzing the system under dynamic conditions. The deployment of PMU at each bus would facilitate direct measurement of all the states of the system. However, this is uneconomic and infeasible due to the higher installation cost of PMUs and limited communication facilities available. Thus, there is a need for strategic placement of PMUs across the power grid. The optimal location of PMUs is determined by minimizing the number of PMUs required to fully observe the system. A power system can be observed completely when sufficient measurements are available to estimate all the states of the system uniquely [1], [2]. The system can be observed numerically as well as topologically [3]. Full rank of the information (or gain) matrix or the measurement Jacobian matrix ensures numerical observability of the system. The huge computational burden in matrix manipulations, associated with numerical observability encourage the use of topological observability, which is based on graph theoretic approach, while optimally placing the PMUs. Though few Trapti Jain Discipline of Electrical Engineering Indian Institute of Technology Indore Indore, India traptij@iiti.ac.in methods based on numerical observability, such as Tabu search [4], genetic algorithm [5] and simulated annealing [6] have been reported in the literature for PMU placement. Most of the work reported utilizes topological observability based PMU placement, such as depth first search [3], spanning tree [3] and integer programming based methods [7]-[9]. Depth first search algorithm places PMUs at the buses having largest number of connected branches in order to achieve maximum number of observable buses with minimum PMUs. This method is simple and easy to implement but increases unwanted redundancy due to the overlapping of observational topologies while progressing from one node to the other [3]. Spanning tree based approach utilizes sequential placement of PMUs along the branches of a spanning tree of the power system such that the system is fully observed. This is an exhaustive method involving creation of large number of possible spanning trees and hence, large number of placement sets to obtain a minimal placement set. Integer programming based approach involves minimization of cost of PMUs subject to certain constraints which ensure system observability. This is a fast method but the optimal solution obtained is dependent on the initial guess chosen. Since the optimal PMU placement problem is NP-hard [10] and does not have a unique solution, heuristic algorithms have been applied to solve this problem. Tabu search [4], genetic algorithm [5], simulated annealing [6], binary search [11], particle swarm optimization [12], [13], ant colony optimization [14], immunity genetic algorithm [15], immunity particle swarm optimization [16] and differential evolution [17] have been utilized to achieve the optimal solution of PMU placement problem. These methods provide near optimal solutions but are computationally intensive for large scale practical power systems. This paper proposes an approach based on best first search algorithm, which utilizes an evaluation function to identify the most promising node. The additional redundancy of PMU locations obtained from BFS has been eliminated by a pruning technique such that the final solution contains minimum number of PMUs required for maintaining complete system observability. The proposed method has been tested on standard IEEE 14 bus, IEEE 30 bus and an Indian 246 bus system. The results obtained with the proposed method have been compared with the existing methods to find the effectiveness of the proposed algorithm in optimal PMU placement problem /14/$ IEEE

2 II. PMU BASED OBSERVABILITY ANALYSIS A PMU placed at a bus directly measures the voltage phasor at that bus and current phasors of some or all the branches connected to that bus depending upon the number of channels. It has been assumed in this paper that PMU with sufficient number of channels is installed at a bus so that the current phasors of all the branches incident to that bus can be measured in addition to the voltage phasor of that bus. The voltage phasors at the buses adjacent to the PMU installed bus can be determined using the measured branch current phasors, bus voltage phasor and known line parameters [11]. Thus, the placement of a PMU at a bus observes that bus as well as all the other buses directly connected to it as shown in Fig. 1 (a). Further, a power system always contain some buses with neither generation nor loads connected to them. Such buses are referred as zero injection buses (). If an observable is surrounded by all the observable buses except one, the unobservable bus can be observed by applying the KCL at as shown in Fig. 1 (b). However, if an unobservable is surrounded by all the observable buses, then the unobserved can be made observable by applying KCL at as shown in Fig. 1 (c). Thus, can be removed from the potential locations of PMU placement reducing the number of PMUs required for complete system observability. In addition to the above rules, the placement of PMU at a radial bus has been avoided as it will measure the voltage phasor at that bus and its adjacent bus. Instead, PMU placement is preferred at the bus adjacent to the radial bus so that the voltage phasor at the radial bus can be obtained by measurement of the current through the radial line. If a radial bus is connected to a as shown in Fig. 1 (d), there is no need to place a PMU at the adjacent to this radial bus. However, if two or more radial buses are connected to the same as shown in Fig. 1 (e), then a PMU must be placed at the to observe all the radial buses connected to it. (a) (b) III. BEST FIRST SEARCH BASED APPROACH Best first search (BFS) is a graph traversing algorithm which expands only the most promising nodes while reaching towards the goal. In order to identify these promising nodes, an evaluation function is used to assign a score to each node. This evaluation function comprises of an estimated cost as well as the actual cost of reaching towards the goal [18]. Depending upon the nature of the problem, a node with the highest or least score of an evaluation function can be considered as the best node. The objective in PMU placement problem is to make the system completely observable with minimum number of PMUs. Thus, the node having the higher degree of connections should be preferred for placing the PMUs. This concept has been used in depth first search (DFS) based PMU placement [3] but this results in increased number of PMU locations for maintaining complete system observability. In IEEE 14-bus system, nodes 4 and 6 are identified as potential PMU locations considering their degree of connectivity. However, the nodes 1, 10 and 14 are left unobserved and in order to make the system completely observable, three additional PMUs are required at these locations. These additional locations of PMU offer unwanted redundancy due to the overlapping of observational topologies as shown in Fig. 2. This problem has been overcome in the BFS based method, which utilizes an evaluation function to identify the appropriate node for PMU placement. The BFS algorithm searches only the most appropriate nodes in the system whereas the DFS explores every node in the system to find the possible PMU locations [19]. Further, the capability of the BFS algorithm to switch its path from the current path to the most promising path makes it superior to the DFS algorithm [20]. In this paper, an evaluation function (f i ) at a node i has been defined as the ratio of coverage value (Y i ) of a node to the cost of PMU placement (X). Mathematically, Yi fi = (1) X V (c) (d) IV (e) PMU Node Node Observed by a PMU at Adjacent Node Unobservable Node III II I Fig. 1. Topological observability of power system using PMU placement Fig. 2. Overlapping of observable regions in DFS method of PMU placement

3 The coverage value (Y i ) of a node is defined as a product of the number of nodes covered if a PMU is placed at a node i and a bias parameter (λ). The bias parameter (λ) helps in biasing the search of the goal state towards the unobserved state. The cost value (X) is defined as the product of number of buses (nb) in the system and the PMU count (p) obtained at the previous level of search. The BFS based proposed algorithm ranks each node of the power system using the evaluation function described in (1). The node with the highest value of the evaluation function is selected as the potential PMU location. Once, the PMU is placed on the chosen node, the algorithm checks for the system observability. In case, there are any unobserved buses in the system, the BFS algorithm updates the evaluation function at the unobserved nodes. The unobserved node with the highest value of the evaluation function is selected as the next PMU location. This process is continued till all the nodes in the system are observed. of PMU buses: Although, BFS based method is better in comparison to the DFS method for optimal PMU placement, it might have some observable islands intersecting each other. This increases the redundancy at certain buses which may be unnecessary. In order to eliminate this additional overlapping of observable islands, a pruning technique has been applied. This technique identifies the extra PMUs and removes them from the placement set. This is achieved by removing one PMU at a time from the set of PMUs and checking the system observability. If removing a PMU maintains complete observability, then that PMU can be removed from the placement set else, it is preserved. This process is repeated until all the extra PMUs are removed. IV. The zero injection buses are the virtual buses with neither generation nor loads connected to them. Since the observable or unobservable zero injection buses can be made observable under the conditions mentioned in section II, all the buses incident to the zero injection bus as shown in Fig. 3 (a) can be connected to one another resulting in the topology transformation of the given power system network. However, this topology transformation rule is not applicable, if any of the buses incident to a zero injection bus is either a radial or a zero injection bus or in case of two or more zero injection buses incident on the same zero injection bus as shown in Fig. 3 (b) and Fig. 3 (c) respectively. m k n m k n (a) MODELLING OF ZERO INJECTION BUSES m k n / (c) / Fig. 3. Single line diagram of three - bus system (b) / The above topology transformation rules can be summarised as mentioned below, t 1, if buses m and n is neither a radial nor a ZI bus t 1 T t 1, if buses m and n is either a radial or a ZI bus t 0 t 0, if buses m and n are radial nor a ZI buses t 0 Now, the topology of the given power system network is ' modified using, C ij = Cij + Tij, where C ij ' is the binary connectivity matrix after topology transformation considering zero injection buses, T ij is the transformation matrix that contains elements of the buses incident to the zero injection buses and C ij is the binary connectivity matrix before topology transformation which is defined as, 1, if node i and node j are connected C 1, if i j 0, if node i and node j are not connected 2 V. PROPOSED ALGORITHM The step by step procedure to obtain the optimal placement of PMUs for a system comprising of nb number of buses using the proposed approach is described below, 1. Assemble binary connectivity matrix C (nbxnb) as shown in (2). 2. Initialize the iteration count, b=1. 3. Calculate the coverage value at each node in the system as follows, Y b i nb = C j = 1 ij λ i nb where, the bias parameter, λ is one for the first iteration and for successive iterations, it is incremented by the largest coverage value obtained at the previous iteration. 4. Calculate the cost value as X b = nb p where, p represents the number of PMUs obtained at the previous level of search. Initially, the PMU count (p) is taken as one. 5. If b=1, then the evaluation function, f i at each node is computed using (1) and go to step 8 else, go to step The evaluation function, f l is updated only at node l using (1) where l is the unobserved node having the largest coverage value obtained from (3). 7. Compare the evaluation function, l of node l with the evaluation function, of the other nodes retained till the end of previous iteration. 8. Place a PMU on the node k having the highest value of evaluation function, if the node k is not a radial node else, place a PMU on the node adjacent to the k th node. 9. The evaluation function of the k th node is made zero so as to remove it from further consideration. However, the evaluation function at the other nodes is retained. 10. Form an observability vector, which stores information about observable and unobservable nodes by placing a PMU at (3)

4 the k th node. The elements of the observability vector, at the b th iteration are obtained as follows, 2, if node i is observed by PMU at node k 1, if a PMU is placed at node i or i k, i nb 0, if node i is unobserved by PMU at node k The true observability status of the nodes is obtained by adding the current observability vector with the true observability vector obtained at the m th iteration. The elements of total observability vector, are defined as, TO b m O b i = TO i + i i nb (5) where, 1, l 1, For the first iteration, the elements of TO m are assumed to be zero. 12. If the observability vector, does not contain any zero element, then go to step 16 else, go to step Reconstruct the binary connectivity matrix by making all the rows and the columns corresponding to the buses observed by a PMU placed at the k th node as zero. 14. Increment the iteration count, b=b Repeat the steps 3 to 14 till all the nodes in the system are observed. 16. Determine the optimal number of PMUs (N) obtained and store their locations in a location vector, L. 17. Initialize the iteration count for pruning, bp= Modify the location vector to L by eliminating one PMU at a time, thereby reducing the number of PMUs to (N-1). 19. Find the redundancy vector using, R j = C j nb (6) pj ' p L 20. Compute the product element, P bp using, nb bp P = R j (7) j= If bp=n, then go to step 22 else, increment the iteration count as bp=bp+1 and go to step The PMU nodes having the non zero element in the product vector, P N (vector consisting of product elements P bp ) are stamped as the additional PMU locations offering unwanted redundancy. Therefore, the PMUs placed at these locations can be removed, thereby offering complete system observability with a minimal placement set. VI. RESULTS AND DISCUSSION The effectiveness of the BFS based proposed approach for determining the optimal placement of PMUs have been demonstrated on IEEE 14-bus system [21], IEEE 30-bus system [22] and a Northern Regional Power Grid (NRPG) 246-bus Indian system [23]. The proposed algorithm excludes the radial buses from the potential locations of PMUs. Therefore nodes adjacent to these radial buses with higher degree of connectivity are chosen as the PMU placement sites [24]. It has been known that exclusion of zero injection buses from PMU placement sites reduces the number of PMUs required for complete system observability. However, to establish the effectiveness of the proposed approach, results for the various systems have been presented including and excluding zero injection buses from the PMU placement set. Table I and Table II lists the numbers as well as locations of radial and zero injection buses for the three test systems respectively. The effect of pruning on the optimal number of PMUs for complete observability obtained using proposed method is presented in Table III. It can be seen that the BFS algorithm gives some redundant PMU locations in both the cases of including and excluding from the PMU placement set. technique is found to be very effective especially in larger systems such as 246 Indian bus system. In this system, even after excluding from the PMU placement set, the minimum number of PMUs required for complete system observability is 65 which is reduced to 57 PMUs after applying pruning. The optimal number of PMUs and their locations obtained by the proposed BFS based algorithm with pruning after excluding zero injection buses from the PMU location set are listed in Table IV for all the test systems. The optimal number of PMU locations obtained using the proposed method has been compared with the results obtained using the standard ILP method and are shown in Table V. It can be observed that the total number of PMUs needed to maintain complete observability in the IEEE 14-bus and IEEE 30-bus systems are same as those obtained with the ILP method. However, in case of 246-bus Indian system, the proposed method yields significantly less number of PMUs as compared to the ILP method, in both the cases of including and excluding s. The proposed method gives only 57 PMUs to maintain complete system observability whereas the ILP method results in 70 PMUs. TABLE I. LOCATION OF RADIAL BUSES FOR THE TEST SYSTEMS Test No of Radial System Buses Location of Radial Buses IEEE IEEE , 13, 26 2, 4, 5, 12, 30, 31, 38, 41, 47, 51, 52, 53, 58, 76, 77, 112, 120, 123, 124, , 149, 153, 156, 159, 172, 178, 189, 208, 224, 242, 246 TABLE II. LOCATION OF ZERO INJECTION BUSES FOR THE TEST SYSTEMS Test System No of Zero Injection Location of Zero Injection Buses Buses IEEE IEEE , 9, 22, 25, 27, , 56, 59, 61, 62, 63, 69, 70, 71, 72, 73, 74, 75, 80, 81, 86, 102, 103, 104, 107, 122, 126, 129, 131, 147, 154, 155, 167, 175, 179, 180, 183, 209, 210, 211, 212, 213, 214, 215, 216, 217, 221, 222, 226, 229, 230, 231, 232, 233, 234, 236, 237, 238, 239, 240, 241, 243, 244

5 TABLE III. EFFECT OF PRUNING ON THE OPTIMAL PMU PLACEMENT USING THE PROPOSED METHOD Including Excluding Test System Before After Before After IEEE 14 Bus IEEE 30 Bus TABLE IV. OPTIMAL PMU PLACEMENT USING THE PROPOSED METHOD EXCLUDING Test Number System of PMUs Optimal PMU Locations IEEE , 6, 9 IEEE , 4, 10, 12, 18, 24, 29 3, 6, 11, 15, 21, 24, 29, 33, 34, 35, 36, 40, 44, 48, 54, 55, 63, 65, 82, 83, 88, 89, 91, 96, 97, 106, 109, 113, 116, 118, 121, 125, 132, , 140, 141, 142, 157, 158, 160, 165, 166, 168, 181, 185, 187, 190, 191, 194, 199, 201, 203, 205, 218, 219, 235, 245 TABLE V. COMPARISON OF THE PROPOSED METHOD RESULTS WITH ILP METHOD Including Excluding Test System Proposed Proposed ILP ILP Method Method IEEE 14 Bus [25] IEEE 30 Bus [25] [9] VII. CONCLUSION A new methodology based on best first search algorithm has been proposed in this paper to obtain the optimal placement of PMUs under normal operating conditions. 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