Optimal placement of PMUs to maintain network observability using a modified BPSO algorithm

Electrical Power and Energy Systems 33 (2011) 28–34

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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes

Optimal placement of PMUs to maintain network observability using a modified BPSO algorithm Mahdi Hajian a,⇑, Ali Mohammad Ranjbar b, Turaj Amraee b, Babak Mozafari b a b

Department of Electrical and Computer Engineering, University of Calgary, Alberta, Canada T2N 1N4 Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran

a r t i c l e

i n f o

Article history: Received 23 November 2007 Received in revised form 10 April 2010 Accepted 13 August 2010

Keywords: Power system observability Phasor Measurement Unit Optimal placement Binary Particle Swarm Optimization

a b s t r a c t This paper presents a novel approach to optimal placement of Phasor Measurement Units (PMUs) for state estimation. At first, an optimal measurement set is determined to achieve full network observability during normal conditions, i.e. no PMU failure or transmission line outage. Then, in order to consider contingency conditions, the derived scheme in normal conditions is modified to maintain network observability after any PMU loss or a single transmission line outage. Observability analysis is carried out using topological observability rules. A new rule is added that can decrease the number of required PMUs for complete system observability. A modified Binary Particle Swarm Optimization (BPSO) algorithm is used as an optimization tool to obtain the minimal number of PMUs and their corresponding locations while satisfying associated constraint. Numerical results on different IEEE test systems are presented to demonstrate the effectiveness of the proposed approach. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction Phasor Measurement Units (PMUs) were introduced in early 1990 as devices capable of measuring synchronous real-time voltage and current phasors in power systems [1]. Synchronous measurements of PMUs located in different buses are achieved via same-time sampling of voltage and current waveforms using a common synchronizing signal. The most common used synchronizing signal is obtained from Global Positioning System (GPS) reference which is able to provide an accuracy down to 1 ls [2]. Real-time phasor measurement at different nodes can improve the performance of monitored control systems in different applications such as state estimation, fault location, transient and small signal stability analysis [3–6]. One of the applications of phasor measurements in power systems is state estimation which is performed in a control center to provide a platform for monitoring, and security applications such as contingency analysis and optimal power flow [1]. The first step in state estimation is to gather measured data from different substations in a power network. These measurements must be sufficient to make the system observable [7]. Assuming all nodes of the system are equipped with PMUs, all system state variables can be directly monitored and there is no need to estimate other variables. However, due to installation cost of PMUs or non-existence of communication facilities, an ubiquitous placement of ⇑ Corresponding author. Tel.: +1 403 210 5467; fax: +1 403 282 6855. E-mail address: [email protected] (M. Hajian). 0142-0615/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2010.08.007

PMUs is rarely conceivable. Therefore, the problem is to find the minimum number of PMUs to achieve full observability of the network. In addition, a reliable PMU placement set should be robust enough to maintain power network observability anticipating possible contingencies such as PMU failure or a transmission line outage. The PMU placement problem has been addressed in several literatures. The problem was first introduced in [8]. A modified bisecting search and simulated annealing optimization algorithm are used to optimally select locations of PMUs. Topological observability rules are employed in order to examine observability of a network. In [9], an integer programming based formulation is used to find an optimal PMU placement scheme considering conventional measurements. In [10], the PMU placement problem is solved via tree search method considering complete and incomplete observability. In [11], the Tabu search algorithm is used to optimize the number of required PMUs for full network observability. Augmented incidence matrix is proposed to analyze observability of a candidate PMU set. The possibility of contingencies such as measurement loss or a transmission line outage in the placement problem is also considered in the literature. Optimal placement methods considering contingencies for conventional measurements are addressed in [12,13]. In [14], an optimal PMU placement set is found using a nondominated sorting genetic algorithm and topological observability analysis considering single line contingencies. In [15], a heuristic approach is proposed to obtain a PMU placement set robust to a single measurement loss and branch outage. In [16], an

M. Hajian et al. / Electrical Power and Energy Systems 33 (2011) 28–34

integer quadratic programming is used in order to optimally find a PMU placement set considering PMU failure or a branch outage. However, in this work the effect of zero-injection buses is ignored. In [17], integer linear programming is proposed for solving the optimal PMU placement problem anticipating PMU loss or a line outage. In this paper, the optimal PMU placement considering PMU loss or a line outage is addressed using a modified Binary Particle Swarm Optimization (BPSO) technique. Topological observability analysis is used to assess observability of each placement candidate. In order to attain maximum utilization of existing data, such as characteristics of zero-injection buses, a new rule is added to the previous rules of topological analysis. Using the modified observability analysis rules and the modified BPSO, the optimal PMU placement is accomplished during normal conditions, i.e. assuming no contingency. In order to consider PMU loss or a line outage, the topological observability analysis is adopted to examine network observability anticipating contingencies. Using the modified BPSO and the adopted topological observability rules, the derived placement scheme in normal conditions is updated to maintain system observability during contingency conditions. The remainder of this paper is organized as follows. The observability analysis using PMU data is presented in Section 2. An overview of Particle Swarm Optimization technique is described in Section 3. The proposed PMU placement problem in normal and contingency conditions is addressed in Section 4. Numerical results of the proposed algorithm are provided in Section 5. Finally, Section 6 concludes the paper.

2. Observability analysis using PMU data Given a N-bus network provided with m measurements of voltage and current phasors, the linear equations relating measurements and the state vector are

z¼Hxþe

ð1Þ

where the vector z is linearly related to the n-dimensional state vector x containing N-bus voltage phasors (i.e. n = 2N  1 ). H is the (m  n) matrix, and e is the (m  n) additive measurement error vector. The observability of a system can be examined considering network topology, types, and locations of measurements. Generally, two different observability concepts are defined for a linear system model (Eq. (1)), namely numerical and topological analysis. Numerical observability is defined as the ability of a system model to be solved for state estimation. The full rank of matrix H (i.e. 2N  1) is considered to be the criteria to declare full observability of a system [18]. However, due to high calculation burden of verifying the rank of matrix H, this approach would not be preferred for practical applications. In addition, for any placement scheme in which the corresponding matrix H is not of full rank, this method is not able to specify locations of unobservable buses. Topological observability analysis is defined as the existence of at least one spanning measurement tree of full rank in a network [8]. This tree connects all observable nodes and branches which can be observed by direct measurements or calculations. The following rules are commonly used to assess the existence of this tree. (1) Buses with PMU are assigned with direct voltage phasor measurement and direct current phasor measurement of incident lines. The vector of direct measurement of voltage and current phasors are denoted as DV and DI, respectively. (2) If voltage and current phasors at one end of a line are known, then the unknown voltage phasor at the other end of the line can be calculated (assuming known impedance of transmission lines).

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(3) If voltage phasors of both ends of a line are known, then the current phasor of this line can be calculated. Voltage and current phasors of buses and lines observed by rules 2 and 3 are defined as pseudo-measurements and indicated with PV and PI, respectively. (4) If all line current phasors incident to a zero-injection bus are known except one, the current phasor of the unknown one can be calculated through KCL equations. A zero-injection bus is a bus in which the net power injection is zero. (5) If there is an unknown zero-injection bus and voltage phasors of its adjacent buses are all known, then the voltage phasor of the zero-injection bus can be obtained by the node equations. Topological observability analysis based on a network graph is normally accomplished by the above rules. However, another rule can be introduced which enhances the performance of observability analysis. This rule is a generalization of rule 4 which was first introduced in [19]. (6) Considering a group of unknown zero-injection buses, as shown in Fig. 1, with all known voltage of adjacent buses, the node equations for each of the buses in the group can be written as follows: N X

Y ij V j ¼ 0;

for i ¼ 1; . . . ; k

ð2Þ

j¼1

where k is the number of zero-injection buses in the group, Vj is the voltage of the jth bus, and Yij is the ijth element of the admittance matrix. In Eq. (2), there is k complex equations and given that the voltage of the adjacent buses are all known, the number of unknown complex variables is exactly equal to k. Hence, the zero-injection buses in the group are all observable. Voltages and currents observed by the rules 4–6 are defined as extended measurements and represented by EV and EI, respectively. Finally, Fig. 2 shows the flowchart of the observability analysis based on aforementioned rules. In this flowchart, OV and OI represent the vectors of observed buses and lines, respectively. The input of this algorithm is a candidate placement scheme. After the assessment of topological observability rules, the number and location of observed buses and lines (i.e. components of OV and OI vectors which are greater than 1) are deduced. 3. Particle swarm optimization algorithm 3.1. Continuous version of PSO Particle Swarm Optimization (PSO) was first introduced in [20]. It provides a population-based search procedure in which individ-

Fig. 1. A group of zero-injection buses.

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M. Hajian et al. / Electrical Power and Energy Systems 33 (2011) 28–34

the continuous equations to the binary mode. However, the effect of previous velocity of particles in updating velocity vectors was not introduced. In this paper, a modified BPSO is proposed in which velocities and positions of particles are updated based on following equations: k

d1;i ¼ pbest i  xki k d2;i

v

¼ gbest 

kþ1 i

xkþ1 i

¼ v þ c1  k i

¼

xki

þv

ð3Þ

xki

ð4Þ k d1;i

þ c2 

k d2;i

kþ1 i

ð5Þ ð6Þ

In the above equations, , , and + are xor, and, and or operators, respectively. xki and v ki are the binary position and velocity vectors corresponding to particle i at iteration k, respectively. pbesti is the best experience of particle i and gbest is the best experience in the whole population in finding the optimal point of the problem. k d1;i is the ‘‘cognition” part, which shows private thinking of a partik cle itself. d2;i is the ‘‘social” part, which reflects cooperation among particles. c1 and c2 are random binary vectors generated at each iteration of the algorithm. Contrary to the real-value PSO, in BPSO a better performance is achieved by setting a single Vmax to the whole of a particle instead of imposing the maximum velocity to each dimension of the particle [22]. In BPSO, Vmax is defined as the maximum number of ones in a particle. This limit is checked after updating velocities of particles using Eq. (5). In the case of violating the limit, the number of ones in the velocity vector is reduced random until it becomes less than Vmax. 3.3. BPSO approach for a discrete optimization problem The PMU placement problem leads to a discrete binary optimization problem that can be expressed as the problem described in Eqs. (7) and (8).

Minimize JðxÞ subject to FðxÞ ¼ 0

ð7Þ ð8Þ

where JðxÞ and FðxÞ are the objective function and equality constraint and x is the vector of variables to be optimized. In order to better demonstrate the BPSO algorithm, the steps required for solving this discrete binary optimization problem are elaborated as follows:

Fig. 2. Flowchart of topological observability analysis.

uals, called particles, fly around in a multidimensional search space and change their positions with time. Position changes of particles is based on their individual experience, and experience of neighboring particles in finding the optimal point of an optimization problem.

3.2. Discrete binary version of PSO (BPSO) In BPSO, the search space is discrete and variables can only take on values of 0 and 1. The BPSO was first introduced by Kennedy and Eberhart [21]. That BPSO was achieved with a simple modification to the real-value particle swarm optimization. In [22], the authors proposed a modified BPSO with a better performance than the previous BPSO in finding optimal solution of an optimization problem. The modification was performed through transforming

Step 1: A random population is created based on the dimension size of the problem. In the case of optimal PMU placement, each individual consists of a number of bits corresponding to existence (1) or nonexistence (0) of PMU in each bus of the system. Step 2: The problem is transformed into an unconstrained one by creating an augmented objective function incorporating penalty factors for any value violating the constraint:

Minimize J  ðxÞ ¼ JðxÞ þ aFðxÞ

ð9Þ

where a is a penalty factor corresponding to the equality constraint Eq. (8). Step 3: The augmented objective function is evaluated for each individual. The current position, xi, is stored in pbesti, if J(xi) < J(pbesti). The best experience among pbests is denoted as gbest. Step 4: The member velocity v of each particle is modified according to Eq. (5) considering the maximum velocity limit. Step 5: Positions of particles are updated according to Eq. (6). Step 6: If maximum number of iterations is not reached yet, the algorithm is pursued from step 3, otherwise it is stopped and gbest is announced as the global optima of the problem.

M. Hajian et al. / Electrical Power and Energy Systems 33 (2011) 28–34

4. Optimal PMU placement considering normal and contingency conditions The proposed approach for the optimal PMU placement is implemented in two sequences. At first, the optimal placement is carried out with the goal of minimizing the total number of required PMUs for complete system observability during normal conditions, i.e. no PMU failure or line outage. Then, the derived placement set is modified so that the network regains its observability during the two types of contingency, (i.e. PMU loss or a branch outage). Both of the problems are solved using the described modified BPSO algorithm. 4.1. Optimal PMU placement in normal conditions The objective of this stage is to minimize the total number of PMUs required for complete system observability assuming normal conditions. The optimization problem can be described mathematically as:

Minimize

N X

PMU i

ð10Þ

i¼1

subject to Nnobs ¼ 0

ð11Þ

where PMUi indicates the installation of PMU at bus i, it is 1, if a PMU is installed on bus i and 0 vice versa. Nnobs represents the number of unobserved buses in the network. Before implementation of the modified BPSO to solve the problem, several definitions are pointed out as follows: 4.1.1. Initial placement As an initial point of the optimization problem, a PMU placement is created by graph theoretic search procedure. A reasonable starting point can greatly accelerate the convergence speed of the solution and also increase the possibility of finding the global optima in complex optimization problems [8]. The initial placement is generated based on the following steps. Step 1: A PMU is placed at a bus located in an unobservable region which has the maximum number of incident lines. Step 2: Topological observability rules (Fig. 2) are verified, and numbers and locations of unobserved buses, if any, are determined. Step 3: If there is an unobservable region remaining, step 1 is repeated, otherwise the procedure is stopped. This placement scheme results in a PMU placement set, which satisfies complete system observability. However, as shown in simulation results, this scheme is not necessarily the optimal one, and an optimization tool is required to derive the minimum number of required PMUs. 4.1.2. Reducing search space In order to decrease the number of PMU candidate locations, a set of buses is identified to be eliminated from search space. These buses are considered to be those that have only one incident line (radial buses), and buses with zero-injection power (zero-injection buses). This elimination is attributed to the fact that placing a PMU on a radial bus provides minimum advantage in obtaining phasor voltages of adjacent buses. Also, placing a PMU on a zero-injection bus leads to covering of data from these buses and consequently the loss of their benefit in observability analysis (i.e. rules 3–5). 4.1.3. PMU placement procedure The described BPSO is used as the optimization tool and the augmented objective function described in Section 3.3 is formed based

31

on the objective function and the equality constraint presented in Eqs. (10) and (11). An initial population of BPSO is created based on the graph theory described in Section 4.1.1. The augmented objective function for each particle is calculated based on the number of PMUs and the number of unobservable buses corresponding to that particle. The number of unobservable buses is determined by assessing topological observability rules described in Section 2. 4.2. Optimal PMU placement in contingency conditions In this stage, an algorithm is developed in which the placement configuration derived in normal conditions is modified to maintain network observability during a single PMU loss or line outage. The objective function of this part can be expressed mathematically as follows:

Minimize

N X

PMU i

ð12Þ

i¼1

subject to Nnobs ¼ 0jsingle PMU loss or branch outage

ð13Þ

The flowchart of topological observability analysis described in Section 2 is adopted to consider the effect of PMU loss or topology change and identify the total number of unobserved buses during contingency conditions. An initial starting point is also generated to improve the performance of the optimization algorithm in finding the global optima of the problem. The details of PMU placement in contingency conditions are described in the following sections. 4.2.1. Loss of a single PMU In order to consider loss of a single PMU, the observability analysis of a placement scheme is performed as follows. For each PMU located in a placement scheme, the topological observability analysis of the network (Fig. 2) is carried out omitting the selected PMU from the placement scheme. The total number of unobservable buses is declared as the sum of unobserved buses during loss of each PMU in the placement scheme. 4.2.2. Loss of a single branch In this part, the observability analysis is performed to consider the impact of a branch outage on network observability. In order to maintain network observability during a line outage, each bus of the system must be observable from two paths. It is clear that if one of the paths is lost (single line outage), that bus is still observable through the other path. Note that observability of buses with one incident line is not of interest during single line outage, because those buses become isolated from the network after the line outage. In summary, the observability analysis algorithm in contingency conditions is described by the flowchart depicted in Fig. 3. 4.2.3. Initializing a starting point A starting point is generated based on a modification to the placement scheme derived in normal conditions. The procedure of creating the initial point is described as follows: Step 1: The base scheme is considered and the first PMU is eliminated. Step 2: By adding the minimum number of PMUs to the adjacent buses of the eliminated PMU, the observability of the network is recovered. Step 3: If all the PMUs in the base scheme are considered go to the next step, otherwise the next PMU is chosen and step 2 is repeated. Step 4: Topological observability analysis is carried out and buses which are not observable through two paths are identified. A PMU is then added to each of these buses.

32

M. Hajian et al. / Electrical Power and Energy Systems 33 (2011) 28–34 Table 1 The numbers of zero-injection and radial buses in the IEEE test systems. Test system

No. of zero-injection buses

No. of radial buses

IEEE IEEE IEEE IEEE IEEE

1 6 12 15 10

1 3 9 1 7

14-bus 30-bus 39-bus 57-bus 118-bus

ber in BPSO is selected to be 1000 with the population size of 100. Since heuristic algorithms, such as PSO, are based on a random search in the search space of the problem, the result of each execution of these algorithms might be different from another one. Therefore, they must be run several times to ensure that the optimal point of the problem is found. In the simulations presented in this paper, the best solution of the modified BPSO is found after 50 runs of the algorithm. The performance analysis of the modified BPSO, and the PMU placement results in normal and contingency conditions are presented in the following sections.

5.1. Performance analysis of the modified BPSO

Fig. 3. Observability analysis considering contingency conditions.

Step 5: The base placement scheme plus the added PMUs are considered as the initial placement scheme. This placement scheme leads to a PMU configuration, which maintains complete system observability after loss of a PMUs or branch outage. However, as mentioned in normal placement, this scheme is not necessarily the optimal one and the minimum number of required PMUs should be obtained through the optimization procedure. 4.2.4. PMU placement procedure The described BPSO is used as the optimization algorithm to derive the minimum number of required PMUs for system observability during contingency conditions. The augmented function is formed based on Eqs. (12) and (13). In order to calculate the amount of objective function for each particle the modified topological observability analysis (Fig. 3) is employed. 5. Simulation results Simulations are carried out on IEEE 14-, 30-, 39-, 57-, and 118bus test systems [23]. The numbers of zero-injection and radial buses in the test systems are shown in Table 1. The simulations are carried out in MATLAB environment on an Intel Pentium III (1.6 GHz) with 512 MB RAM. The following parameters are used in the simulations: The value of a in Eq. (9) is 2. This is to place emphasis on the equality constraint to ensure full network observability. The maximum velocity of particles, Vmax, is 1/4 of dimension size of each particle in normal-condition placement and 2/3 in contingency-condition placement. The maximum iteration num-

In order to demonstrate the effectiveness of the modified BPSO in finding the optimal solution of the problem, the performance of the algorithm is examined for the case of IEEE 39-bus test system. A comparison of the results in normal-condition placement between the modified BPSO, BPSO presented in [22], and the original BPSO in [21] is shown in Table 2 in terms of the best, average, and worst results after 50 runs of each algorithm with the same population size and iteration number as described before. As it can be observed from this table, a better performance is achieved by using the modified BPSO algorithm. The average run time of the modified BPSO in this case is 13 min which is almost the same as the other two methods. The objective values of the best solutions through out the iterations for the three algorithms are shown in Fig. 4. The modified BPSO can reach the optimal point of the problem in relatively lower number of iterations. The modified BPSO finds the optimal solution after 26 generations, while the proposed BPSO in [22] and the original BPSO in [21] reach the optimal point after 38 and 55 generations, respectively. A sensitivity analysis is also performed on the results to examine the effect of changing the parameters of the modified BPSO. The results showed that the average execution time of the algorithm increases almost linearly with the number of population size while the average number of required PMUs decreases as the population size increases. For example, changing the population size of the algorithm to 50, and 150 results in the average execution time of 7.2 and 22 min and the average number of required PMUs of 8.5 and 8.15. Therefore, selection of the population size of the problem is a compromise between execution time and the quality of the solution. The same trend is also observed regarding the effect of maximum iteration number of algorithm on the results.

Table 2 A comparison among the results of different BPSO algorithms for the case of IEEE-39 bus test system in normal-condition placement. Algorithm

The modified BPSO The proposed BPSO in [22] The original BPSO in [21]

Number of required PMUs Best

Average

Worst

8 8 8

8.26 8.37 8.93

9 9 10

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M. Hajian et al. / Electrical Power and Energy Systems 33 (2011) 28–34

scheme which contains 8 PMUS installed is: 3, 8, 11, 16, 20, 23, 25, and 29, and in IEEE 57-bus test system the other placement scheme, corresponding to 12 PMUs, is: 1, 6, 9, 15, 19, 25, 27, 32, 38, 41, 50, and 53. Since from the point of view of full network observability, these schemes are similar, only one of them is presented in this section and the following sections.

14 The modified BPSO The proposed BPSO in [22] The original BPSO in [21]

Objective function

13 12 11

5.2.2. PMU placement including the new observability rule The new topological observability rule is considered in observability analysis in this case. The number of the required PMUs is decreased to 11, and 28 in IEEE 57-bus, and 118-bus systems, respectively. The new rule does not affect the number of the required PMUS for the other cases. Table 4 presents the number and locations of the required PMUs for full network observability in IEEE 57-bus, and IEEE 118-bus systems. A comparison between the number of the required PMUs in the proposed approach and the other approaches used in optimal PMU placement is performed in Table 5. It can be observed that the proposed approach outperforms the most of the other techniques, especially in IEEE 57 and IEEE 118 bus system.

10 9 8 7 6 0

10

20

30

40

50

60

70

80

90

100

Iteration number Fig. 4. A comparison among the best solutions of the modified BPSO, the proposed BPSO in [22], and the original BPSO in [21] for the case of IEEE-39 bus test system.

5.2. PMU placement in normal conditions

5.3. PMU placement in contingency conditions

Assuming that there is no PMU failure or line outage, the placement problem is solved using the modified BPSO algorithm. The initial placement using graph theoretic search procedure leads to placing 5, 11, 12, 19, and 38 PMUs in IEEE 14-bus, 30-bus, 39bus, 57-bus, and 118-bus test systems, respectively. To decrease the search space of the BPSO, radial and zero-injection buses are eliminated from the search space of the problem. Therefore, referring to Table 1, the dimension size of the BPSO is 12, 21, 18, 41, 101 in IEEE 14-bus, 30-bus, 39-bus, 57-bus, and 118-bus systems, respectively. In order to highlight the significance of the new observability rules, i.e. rule 6, the PMU placement problem is solved in the following sections excluding and including this rule. In both cases, the average execution times of the algorithm are 1, 6, 15, 43, and 85 min in IEEE 14-bus, 30-bus, 39-bus, 57-bus and 118bus, respectively. Note that, in order to compute the average execution time, the criteria of reaching the maximum iteration number is changed to the one in which the algorithm is stopped after reaching the predefined objective value.

Considering the possibility of PMU failure or a single line outage, a placement scheme is modified to be robust enough to maintain network observability following those contingencies. The initial placement as described in Section 4.2.3 leads to placing 8, 20, 30, 42, and 95 PMUs in IEEE 14-bus, 30-bus, 39-bus, 57-bus, and 118-bus test systems, respectively. However, these configurations are not the optimal one, and the optimization routine as described in Section 4.2 is applied to obtain the minimum number of the required PMUs. The number and locations of the required PMUs for complete system observability is presented in Table 6 for different IEEE test cases. The average simulation times of the modified BPSO are 4, 14, 35, 80, and 225 min in IEEE 14-bus, 30bus, 39-bus, 57-bus and 118-bus, respectively. Table 7 compares the number of the required PMUs in the proposed approach and the other relevant approaches used in optimal PMU placement. From this table, it can be observed that the proposed approach maintains the network observability with a lower number of PMUs compared to the other techniques.

5.2.1. PMU placement excluding the new observability rule In this case, the new topological observability rule is not considered in observability analysis. Table 3 shows the number and locations of the required PMUs for full network observability in each of IEEE test cases. As can be seen from Table 3, by using the modified BPSO the number of the required PMUs for full system observability is decreased significantly compared with the number derived in initial placement. Note that, PMU locations corresponding to the minimum number of the required PMUs might not be a unique scheme. For example, in IEEE 39-bus system, the other placement Table 3 The number and locations of the required PMUs obtained excluding the new observability rule. Test system

Number of required PMUs

Locations of required PMUs

IEEE IEEE IEEE IEEE IEEE

3 7 8 12 29

2, 6, 9 2, 3, 10, 12, 18, 24, 27 3, 8, 12, 16, 20, 23, 25, 29 1, 6, 9, 15, 19, 25, 27, 32, 38, 50, 53, 56 2, 8, 11, 12, 15, 19, 21, 27, 31, 32, 34, 40, 45, 49, 52, 56, 62, 65, 72, 75, 77, 80, 85, 86, 90, 94, 101, 105, 110

14-bus 30-bus 39-bus 57-bus 118-bus

Table 4 The number and locations of the required PMUs obtained in IEEE 57-bus, and IEEE 118-bus system including the new observability rule. Test system

Number of required PMUs

Locations of required PMUs

IEEE 57-bus IEEE 118-bus

11 28

1, 5, 13, 19, 25, 29, 32, 38, 41, 51, 54 2, 8, 11, 12, 17, 21, 25, 28, 33, 34, 40, 45, 49, 52, 56, 62, 72, 75, 77, 80, 85, 86, 90, 94, 101, 105, 110, 114

Table 5 Comparison of results among different methods in PMU placement problem considering normal conditions.

*

Method/system

14 bus

30 bus

39 bus

57 bus

118 bus

Proposed method Dual search [8] Integer programming [9] Tree search [10] Tabu search [11] Integer quadratic programming [16] Integer linear programming [17]

3 3 3 3 3 N/A

7 N/A* N/A 7 N/A 10

8 8 N/A N/A 10 N/A

11 N/A 12 11 13 17

28 29 29 N/A N/A 32

3

7

8

11

28

N/A is due to the unavailability of the result.

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M. Hajian et al. / Electrical Power and Energy Systems 33 (2011) 28–34

Table 6 The number and locations of the required PMUs obtained considering a PMU failure or line outage.

scheme in normal and contingency conditions that can compete with the other techniques used in this problem.

Test system

Number of required PMUs

Locations of required PMUs

References

IEEE 14-bus IEEE 30-bus

7 15

1, 2, 4, 6, 9, 10, 13 2, 3, 4, 8, 10, 12, 13, 15, 16, 18, 20, 22, 24, 27, 30

IEEE 39-bus

17

3, 7, 8, 12, 13,16, 20, 21, 23, 25, 26, 29, 30, 34, 36, 37, 38

IEEE 57-bus

22

1, 2, 4, 9, 12, 15, 18, 19, 25, 28, 29, 30, 32, 33, 38, 41, 47, 50, 51, 53, 54, 56

IEEE 118-bus

62

1, 3, 7, 8, 10, 11, 12, 15, 17, 19, 21, 22, 24, 25, 27, 28, 29, 32, 34, 35, 40, 41, 44, 45, 46, 49, 50, 51, 52, 54, 56, 59, 62, 66, 68, 72, 73, 74, 75, 76, 77, 78, 80, 83, 85, 86, 87, 89, 90, 92, 94, 96, 100, 101, 105, 107, 109, 110, 111, 112, 115, 117

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Table 7 Comparison of results between different methods with regard to placement in normal state. Method/system

14 bus

30 bus

39 bus

57 bus

118 bus

Proposed method Integer quadratic programming [16] Integer linear programming [17]

7 N/A

15 21

17 N/A

22 33

62 68

8

17

22

26

65

N/A is due to the unavailability of the result.

6. Conclusion This paper presented a new approach for the optimal placement of phasor measurement units. At first, the PMU placement was solved with the goal of minimizing the total number of required PMUs for the complete system observability. Then, the effect of PMU loss or a branch outage was taken into consideration, and a placement scheme was obtained which maintains complete system observability during the contingency conditions. A modified discrete binary version of particle swarm algorithm was used as an optimization tool in finding the minimal number of the required PMUs for the complete system observability in both cases. Also, an improved topological observability analysis was proposed using a new rule based on observability analysis of zero-injection buses. Numerical results on the IEEE test systems indicated that the proposed placement method is capable of providing a placement