Optimal placement of phasor measurement units using clustered gravitational search algorithm

Abstract

This paper presents a new multi-objective problem formulation based optimal placement of Phasor Measurement Units (PMUs) considering the power system observability and reliability conditions. In this proposed method, Optimal Placement of PMUs (OPP) are determined to satisfy two objectives simultaneously such as minimizing the number of PMUs for achieving the complete observability and maximizing the reliability for better operation of power systems. Since the above two objectives are conflicting in nature, Fuzzified Clustered Gravitational Search Algorithm (FCGSA) is proposed to solve Multi-objective Optimal Placement of PMUs (MOPP) problem to provide a good tradeoff solution between the competing objectives. The fuzzy membership for each objective function is designed and proposed to determine the best solution of MOPP problem. Conventional rules are applied to minimize the number of PMUs and a new rule is developed to maximize the observability as well as reliability of the power systems. It helps the system operators to make necessary remedial actions to prevent the outage of the low reliability bus. The proposed method is validated on IEEE 14, 30 and 57 bus systems. The most effective strategy of allocating the optimal number of PMUs and their locations is demonstrated by comparing its performance with other methods reported in the available literatures.

Keywords

Phasor measurement unit optimal placement clustered gravitational search algorithm fuzzy set theory multi-objective problem

1 Introduction

The PMU is a precise measuring device that provides fast and smart communications, which was initially, introduced in the early 1980s, using Global Positioning System (GPS) and Digital Signal Processing (DSP) techniques [1]. The major outputs of PMUs are voltage and current phasors, power injections and power flows, frequency and the rate of change of frequency. In recent years, many research activities have been carried out by both utilities and research institutions to find the minimum number of PMUs and their optimal locations in order to guarantee the complete observability of power systems. It is one of the main recommendations by USA-Canada Task Force in April 5, 2004 [2].

The Energy Policy Act of 2005 (EPAct) addressed the reliability issues of the larger power systems in a number of ways. Not only in USA, but also other countries like China, Australia, Brazil, etc., also recognized the importance of the WAMS system and they started implementing various norms on reliability standards. Recently, deployment of PMU technology has been considered by the Government of India in the Eleventh Plan of the National Electricity Policy [3]. This compels power utilities to consider reliability issues in the optimal placement of PMUs (OPP) problem formulation.

In this paper, the aim is to incorporate the reliability of the power system as an objective function with the existing OPP problem formulation. Since the observability and reliability level objectives are conflicting in nature, a Fuzzified based AI techniques is proposed to solve the multi-objective OPP problem. Hence, in the proposed formulations of MOPP problem, the optimal number of PMUs and their locations are determined in order to maximize both observability and reliability levels of the power system. Here, the reliability level of the load buses is determined using capacity outage probability table (COPT) [4, 5]. The optimal number of PMUs and their locations are selected in such a manner that it guarantees the complete system observability and also monitors the low reliability buses (i.e., having more probability of fault occurrence), either directly or indirectly (by calculations). The bus measurement redundancy is defined as the number of times a bus is observed by PMUs set more than once. In all test systems, the low reliability bus is observed by more than one PMU. It adds to the advantage of increasing the reliability of observing low reliability buses, even if one of PMUs, fail. Therefore, whenever any fault occurs on it, then the remedial actions can be taken quickly to avoid the outage of these buses.

Recently, Artificial Intelligence (AI) techniques like Genetic Algorithm have been employed to solve the optimal PMUs placement problem maintaining the network observability [6]. Abhinav et al. [7] developed a particle swarm algorithm which was used as an optimization tool for finding the minimal number of required PMUs for the complete system observability. Bo Wang et al. [8] presented an Advance Ant Colony Optimization approach for optimal PMU placement problem using Global Positioning System. Biography based optimization algorithm is used for obtaining optimal location of meters in [9]. This solution technique is found to be good for searching the near global optimal solution and can be considered successful to a certain extent.

In [10, 11], in the multi-objective optimal placement of PMUs, voltage stability level of the load buses have been incorporated in the MOPP problem. In recent years, a new optimization method known as Gravitational Search Algorithm was successfully developed by Rashedi to multimodal optimization [12]. This new algorithm has been successfully applied to the various nonlinear functions and results demonstrated that it has high performance and is flexible enough to enhance exploration abilities. Because of its characteristics, this new approach seems to be a good candidate to solve difficulties linked to the real world optimal placement of PMUs problems and as long as authors know it has not been tested on this kind of problems before. In addition to maintain right balance between exploration and exploitation, the Clustered GSA (CGSA) is proposed. Hence, in this paper, a new algorithm based on the Fuzzified CGSA to solve multi-objective OPP problem is proposed and different test systems are selected to verify the accuracy and efficiency of the proposed method.

This paper is organized as follows. Section 2 covers the proposed methodology with the problem constraints. Section 3 explains the fuzzy membership value of each objective function for solving the MOPP problem. Section 4 deals with overview of CGSA and implementation of FCGSA, for solving the MOPP problem and Section 5 presents the case studies with IEEE 14, 30 and 57 bus systems. Concluding remarks have been outlined in Section 6.

2 Multi-objective OPP problem formulation

A conventional OPP Problem is defined as finding the minimum number of PMUs and their optimal locations to guarantee complete power system observability. In the proposed approach, the optimal solution is obtained by simultaneously considering the two conflicting objectives, such as minimizing the number of PMUs for complete observability and maximizing the reliability level of the power system.

2.1 Objective functions

2.1.1 Minimizing the number of PMUs (F₁)

Minimizing the number of PMUs for achieving the complete observability of power systems can be expressed as follows: $F_{1} = Min \sum_{i = 1}^{N} p_{i}$ (1) where N is the total number of buses in a system, p_i is the binary variable which defines the possibility of PMU on a bus.

Let, PMUs placement vector, P = [p₁ p₂ … p_N] ^T having binary variable elements of p_i, defines the possibility of PMU on a bus. It is given by, $p_{i} = {\begin{matrix} 1, & if a PMU is installed at bus i; \\ 0, & otherwise; \end{matrix}$ (2)

2.1.2 Minimizing the reliability index (F₂) or (Maximizing the system reliability)

The reliability issues are the major concerns in daily power system planning and operation. Hence, it is necessary to consider the reliability function in OPP problem. In the proposed reliability model, smaller values of reliability index means higher reliability level in the buses and vice versa. Minimizing the reliability index [13] can be expressed as follows: $F_{2} = Min \sum_{i = 1}^{N} u_{i} * {EENS}_{i}$ (3) where u_i is a binary variable which defines the possibility of PMU on a bus, EENS is Expected Energy Not Served, which defines the reliability level of load bus. $u_{i} = {\begin{matrix} 1, & if a PMU is installed at bus i; \\ 0, & otherwise; \end{matrix}$ (4) $\begin{matrix} EENS = pr {U (P_{j})} * L_{j} \\ = \sum_{j = LC} {PR}_{j} * L_{j} (Mwhr / yr) \end{matrix}$ (5) where U (P_j) is the unavailability of generator state, PR_j is the probability of unavailability of generator state obtained from Capacity Outage Probability Table (COPT), LC is the total number of contingencies leading to load curtailment and L_j is the load curtailment due to the contingency of generator j. Here, a high value of EENS indicates low reliability level and vice versa.

2.2 Constraints

2.2.1 Constraint for minimizing the number of PMUs (i.e. Zero injection bus constraint)

Zero injection bus is a bus from which no current is being injected into the system. In OPP problem, total number of PMUs can be further minimized by considering the zero injection bus constraint. The detailed explanation can be found in [21]. The zero injection bus constraint is incorporated into OPP problem which is expressed using an Equation (6). $C * P \geq v$ (6) where C is the connectivity matrix of the system, P is the optimal PMUs placement vector, size of N * N, having binary variable c_i,j and v is the binary vector, I is the observability vector with a size of N * 1 are defined as: $c_{i, j} = {\begin{matrix} 1, if i = j; \\ 1, if i and j are connected buses; \\ 0, otherwise; \end{matrix}$ (7) $v = {\begin{matrix} unit vector, if zero injection bus \\ constraint is not considered; \\ not unit vector, if zero injection bus \\ constraint is considered; \end{matrix}$ (8) when buses, which are incident to the zero-injection bus, all are observable except one, the unobservable bus will be observed by applying the KCL at zero-injection bus (i.e., zero injection bus effect) [21].

2.2.2 Constraint for maximizing the system observability (Bus Observability Index)

The OPP problem does not have a unique solution using this basic objective function (F1). The optimization algorithms may result in different sets of solutions with the same minimum number of PMUs. Hence, System Observability Redundancy Index (SORI) may be considered as another criterion in solving the OPP problem. A bus is topologically observable when there is at least one measurement on that bus either directly or indirectly, through a PMU.

For solving the conventional single objective OPP problem, SORI_s index is used to calculate the total system observability by adding the Bus Observability Index (BOI) of all the buses in a system and it is maximized randomly [6]. ${SORI}_{s} = \sum_{i = 1}^{N} {BOI}_{i}$ (9) where BOI_i represents the number of PMUs which can monitor the bus i. It always lies between one and the number of branches connected to it (τ_i). It is considered for maximizing the total system observability in all cases. The limitation of the BOI is given as: $1 \leq {BOI}_{i} \leq (τ_{i} + 1)$ (10)

2.2.3 Constraint for maximizing both the system observability and reliability (Proposed)

In the multi-objective OPP problem formulation, we proposed an index is used to maximize both the observability and reliability levels of the systems. The Low Reliability Bus Observability Index (LRBOI) is used as a performance indicator to improve the reliability of observing low reliability buses. It should be monitored by at least two PMUs in order to prevent the outage of these buses, because the probability of fault occurrence is more, compared to other buses in a system. Therefore, it is possible to monitor these buses, even one of PMUs fail.

For solving the MOPP problem, the SORI_m index is proposed in order to maximize both the total system observability and reliability levels and it is given as: ${SORI}_{m} = (\sum_{i \in h} {LRBOI}_{i} + \sum_{i \in N - h} {BOI}_{i})$ (11) where LRBOI_i represents the number of PMUs which can monitor the low reliability bus i, h is the number of low reliability buses. Here, LRBOI always lies between two and the number of branches connected to it (τ_i). The limitation of the LRBOI is given as: $2 \leq {LRBOI}_{i} \leq (τ_{i} + 1)$ (12)

In the Equations (10 and 12), one is added with τ_i because a PMU can measure the voltage phasor of all the adjacent buses connected to the bus and itself.

2.3 Formulation of Multi-objective function

Most of the power system optimization problem involves simultaneous optimization of several objective functions. Generally, these objective functions are non-commensurable, competing and conflicting. The practical multi-objective optimization problem is presented as follows: $F (x) = Min [F_{1} (x), F_{2} (x), \dots F_{n} (x)]$ (13)

Subject to $g_{e} (x) \leq 0, e = 1, 2, \dots I_{n}$ (14) $h_{ie} (x) = 0, ie = 1, 2, \dots E_{n}$ (15) where F₁ (x) , F₂ (x) … F_n (x) are different objective functions, I_n is the number of inequality constraint and E_n is the number of equality constraints.

3 Formulation of fuzzy membership functions

In this proposed method, two objective functions such as minimizing the number of PMUs (F₁) and maximizing the reliability level (F₂) are considered simultaneously. Here, choosing a best compromise solution is important in a multi objective problem. Fuzzy gives a good platform for integrating analytical and heuristic approaches in order to obtain a more realistic problem formulation. Hence, in the proposed method, the fuzzy membership methodology is used to find the best optimum solution. The best solution is the solution that has a maximum fitness value (Fit_p) which is calculated using an Equation (16). ${Fit}_{p} = \frac{(μ_{m}^{p} + μ_{r}^{p})}{\sum_{p = 1 to m} (μ_{m}^{p} + μ_{r}^{p})}$ (16) where m is the total number of non-dominated solutions or populations of the host nest. Here, $μ_{m}^{p}$ and $μ_{r}^{p}$ are the fuzzy membership values pth string of population that indicate the degrees of satisfaction of the objective functions F₁ and F₂, respectively. To determine the membership value of each objective function, it is required to formulate a fuzzy membership function for each objective function.

3.1 Formulation of fuzzy membership function for minimizing the number of PMUs (F₁)

OPP problem is solved where the membership function chosen for objective function F₁ is shown in Fig. 1.

Fig.1

Fuzzy membership for function F₁.

The design of the membership function implies that for any solution, if the objective function in the fuzzy domain (F_m) is more than or equal to F_m,max, then the associated fuzzy membership function value is zero. On the other hand, if the objective function is less than or equal to F_m,min, then the associated fuzzy membership function value is assigned to be one. If the objective function in the fuzzy domain is between F_m,min and F_m,max, then the associated fuzzy membership function value is computed using Equation (17), and such solutions will participate in the optimization process depending on the membership value. $μ_{m}^{p} = {\begin{matrix} 1, & for F_{m} \leq F_{m, min} \\ \frac{(F_{m, max} - F_{m})}{F_{m, max} - F_{m, min}}, & for F_{m, min} < F_{m} < F_{m, max} \\ 0, & for F_{m} \geq F_{m, max} \end{matrix}$ (17) where $μ_{m}^{p}$ is the membership function value for the objective function, F_m is the degree of the objective function in the fuzzy domain, F_m,min and F_m,max are the minimum and maximum values of objective function among all non-dominated solutions, respectively.

3.2 Formulation of fuzzy membership function for the reliability index-EENS (F₂)

The highest level of reliability is achieved only by placing PMUs in all the buses in a system. But it increases the required number of PMUs and directly weakens the basic objective function (F₁). The reliability function is handled as another objective function in addition to the fuel cost and emission function. Hence a novel membership model for reliability function is developed and is shown in Fig. 2.

Fig.2

Fuzzy membership for function.

The design of the membership function implies that, if an objective function EENS in the fuzzy domain is less than or equal to F_r,min, then it will affect the objective function F₁ proportionally. Hence the fuzzy membership function value is assigned to be zero. Such solutions do not participate in the optimal solution set. On the other hand, when the reliability function in fuzzy domain is greater than or equal to F_r,min, it will affect the system reliability and hence to avoid such solution, the fuzzy membership function value is assigned to be zero. If the value of the objective function in the fuzzy domain is between F_r,min and F_r,max, then the associated fuzzy membership function value is computed using Equation (18) and such solutions will participate in the optimization process depending on the fitness value. $μ_{r}^{p} = {\begin{matrix} 0, & for F_{r} \leq F_{r, min} \\ \frac{(F_{r, avg} - F_{r})}{(F_{r, avg} - F_{r, min})}, & for F_{r, min} < F_{r} < F_{r, avg} \\ 1, & for F_{r} = F_{r, avg} \\ \frac{(F_{r, max} - F_{r})}{(F_{r, max} - F_{r, avg})}, & for F_{r, avg} < F_{r} < F_{r, max} \\ 0, & for F_{r} \geq F_{r, max} \end{matrix}$ (18)

where $μ_{r}^{p}$ is the membership function value for the objective function EENS; F_r is the degree of the objective function of EENS in the fuzzy domain; F_r,min, F_r,avg and F_r,max are the minimum, average and maximum value of EENS among all non-dominated solutions, respectively.

4 Gravitational search algorithm

GSA is constructed based on the law of gravity and the notion of mass interactions. The GSA algorithm uses the theory of Newtonian physics and its searcher agents are the collection of masses. Using the gravitational force, every mass in the system can see the situation of other masses. The gravitational force is therefore a way of transferring information between different masses. The pseudo code of the Gravitational Search Algorithm was developed using these four idealized rules:

In GSA, agents are considered as objects and their performance is measured by their masses.

All these objects attract each other by a gravity force, and this force causes a movement of all objects globally towards the objects with heavier masses.

The heavy masses correspond to good solutions of the problem.

The position of the agent corresponds to a solution of the problem, and its mass is determined using a fitness function.

By lapse of time, masses are attracted by the heaviest mass and it would present an optimum solution in the search space.

The general steps of the gravitational search algorithm is given below.

Step 1. Search space identification.

Step 2. Generate initial population.

Step 3. Fitness evaluation of agents.

Step 4. Update G(t) using Equations (19–20) $G (t) = G (G_{0}, t)$ (19) $G (t) = G_{0} e^{- α \frac{t}{T}}$ (20)

Step 5. Update best(t), worst(t) and M_i (t) for i = 1, 2, …, m using Equations (21–22) given below. $m_{i} (t) = \frac{{fit}_{i} (t) - worst (t)}{best (t) - worst (t)}$ (21) $M_{i} (t) = \frac{m_{i} (t)}{\sum_{j = 1}^{N} m_{j} (t)}$ (22)

Step 6. Calculation of the total force in different directions using Equations (23–24) given below $F_{i}^{d} (t) = \sum_{j = 1, j \neq i}^{N} {rand}_{j} F_{ij}^{d} (t)$ (23) $F_{ij}^{d} (t) = G (t) \frac{M_{pi} (t) \times M_{aj} (t)}{R_{ij} + ɛ} (x_{j}^{d} (t) - x_{i}^{d} (t))$ (24)

Step 6. Calculation of acceleration and velocity using Equations (25–26) given below $a_{i}^{d} (t) = \frac{F_{i}^{d} (t)}{M_{ii} (t)}$ (25) $v_{i}^{d} (t + 1) = {rand}_{i} \times v_{i}^{d} (t) + a_{i}^{d} (t)$ (26)

Step 7. Updating agents’ position. $x_{i}^{d} (t + 1) = x_{i}^{d} (t) + v_{i}^{d} (t + 1)$ (27)

Step 8. Repeat step 3 to step 7 until the stop criteria is reached.

Step 9. Stop

Where

$x_{i}^{d}$ - Position of ith agent in the dth dimension

G (t) - Gravitational constant at time t

ɛ - Small constant

R_ij (t) - Distance between the particles i and j

rand_j - Random number (0-1)

M_ii - Inertial mass of the ith particle or agent

G₀ - Initial value of G (t)

4.1 Clustered gravitational search algorithm

In Gravitational Search Algorithm (GSA), it was noticed that, as the iterations go on, all the agents (objects) accumulate near the best particle and in this way the exploration is lost. Though this increases exploitation and works very well for low modality functions, but this comes out to be a disadvantage in case of multi dimensional functions. The algorithm must have the right balance between the two.

To keep the exploration in GSA alive without killing the exploitation, the cluster method is proposed. In this method, the whole population is divided into three basic groups and the search process is carried out which is explained in Fig. 3. Similar to other artificial intelligence (AI) techniques, GSA starts with an initial fixed number of randomly generated objects as shown in Fig. 3(a). In the search space, the quality of solution is measured by their masses. At the end of first iteration, based on the fitness value of each object, the objects are arranged in ascending/descending order based on the objective function. Then the entire objects are divided into three basic groups: namely the Leader, the follower and the freelancer as shown in Fig. 3(b).

Fig.3

Clustered GSA methodology.

The Leaders (shown in red) are the object which is heavier one than the other object obtained at the end of the first iteration which is shown in Fig. 3(c). Each leader object shall lead a group of followers. The Leader and the optimizer group together shall work like a simple GSA population thereafter.

In this way there would be some independent GSA populations led by their leader that will search for the optimum solution. The last group, the freelancers (black) shall be randomly initiated every iteration and in this way they shall keep the search alive. The sub groups have one leader and few followers as shown in Fig. 3(c). Each sub group those led by a leader and the freelancers shall have a best solution. The best out of these bests solution by each group shall be the final best solution (objects with heavier masses) of the iteration.

Depending on the requirements of the function, the ratio of population of Leader, follower and the freelancer can be adjusted.

5 Results and discussion

In this work, all programs are developed using MATLAB code. The system configuration is Pentium IV processor with 3.2 GHz speed and 1 GB RAM.

5.1 System specification

The applicability and validity of the CGSA for the proposed method have been tested on standard IEEE test systems. Table 1 shows the specifications of IEEE standard 14, 30 and 57 bus test systems [14 –18]. It consists of a number of zero injection buses and their locations, numbers of load buses and their locations and the number of branches connected to a bus in the test systems.

Table 1
Specifications of IEEE standard systems

IEEE system Total no. of branches No. of zero injection buses Locations of zero injection buses Max. no. of branches Connected to a bus No. of load buses Locations of load buses

14 bus 20 1 7 5 11 2, 3, 4, 5, 6, 9, 10, 11, 12, 13, 14

30 bus 41 6 6, 9, 22, 25, 27, 28 7 21 2, 3, 4, 5, 7, 8, 10, 12, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 26, 29, 30

57 bus 80 15 4, 7, 11, 21, 22, 24, 26,34, 36, 37, 39 40, 45, 46, 48 6 42 1, 2, 3, 5, 6, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 23, 25, 27, 28, 29, 30, 31, 32 33, 35, 38, 41, 42, 43, 44, 47, 49, 50, 51,52, 53, 54, 55, 56, 57

IEEE system	Total no. of branches	No. of zero injection buses	Locations of zero injection buses	Max. no. of branches Connected to a bus	No. of load buses	Locations of load buses
14 bus	20	1	7	5	11	2, 3, 4, 5, 6, 9, 10, 11, 12, 13, 14
30 bus	41	6	6, 9, 22, 25, 27, 28	7	21	2, 3, 4, 5, 7, 8, 10, 12, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 26, 29, 30
57 bus	80	15	4, 7, 11, 21, 22, 24, 26,34, 36, 37, 39 40, 45, 46, 48	6	42	1, 2, 3, 5, 6, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 23, 25, 27, 28, 29, 30, 31, 32 33, 35, 38, 41, 42, 43, 44, 47, 49, 50, 51,52, 53, 54, 55, 56, 57

5.2 Parameter settings of CGSA for different test system

The control parameters of CGSA such as G0, α and Population size are to be determined, before its implementation. The setting of these parameters optimally would yield better solutions in a lesser computational time. By default setting of the parameters taken initially, one of the parameters is varied and the other parameters are kept constant. It has been tested for each parameter taking several values within a boundary limit. Ten simulations for each setting and they are performed in order to achieve some statistical information about the average evolution.

Based on the above guidelines, statistical analysis is carried out to select the best parameter values of CGSA for IEEE 14 bus system is given in Table 2. The variation of G0, α and Population size with respect to the objective function is shown in Table 2. Here, for the different values of G0 the variation of objective function is given in Table 2 (column 1–3). The minimum number of PMU (i.e. 5) is obtained for two different G0 values i.e. 100 & 110. But the SORI is maximum for the G0 value 100. Hence the 100 is considered as best value for G0. Similarly the best value α and Population size is determined from the Table 2 (i.e. α = 20 and Population size = 30).

Table 2
Parameter settings of CGSA for IEEE 14 bus system

G0 F1 SORI_s α F1 SORI_s Population size F1 SORI_s

90 6 24 15 6 27 10 7 32

100 5 20 16 6 26 20 5 22

110 5 19 17 6 28 30 6 25

120 7 35 18 6 21 50 5 24

19 9 37 100 6 28

20 5 21

21 8 35

22 7 31

G0	F1	SORI_s	α	F1	SORI_s	Population size	F1	SORI_s
90	6	24	15	6	27	10	7	32
100	5	20	16	6	26	20	5	22
110	5	19	17	6	28	30	6	25
120	7	35	18	6	21	50	5	24
			19	9	37	100	6	28
			20	5	21
			21	8	35
			22	7	31

Similarly, the best combination of control parameters of the CGSA is determined for IEEE 30 and 57 bus systems and is tabulated in Table 3.

Table 3

Control parameters of CGSA for IEEE test systems

Control parameter	14 bus	30 bus	57 bus
G0	100	100	100
α	20	21	21
Population size	30	50	100

5.3 Case studies

Different test cases are considered to validate the proposed CGSA methodology. Table 4 shows the different cases of OPP problems considered. It should be noted that in Single objective OPP (SOPP) problems, maximizing measurement redundancy is considered on the random buses. Maximizing the measurement redundancy is considered mainly on the low reliability buses and also on the random buses.

Table 4
Different cases of OPP problems considered

Test Case number Description

1 To validate the CGSA for SOPP problem, without considering the zero injection bus constraint

2 To validate the CGSA for SOPP problem, with zero injection bus constraint

3 To validate the FCGSA algorithm for MOPP problem, with zero injection bus constraint

Test Case number	Description
1	To validate the CGSA for SOPP problem, without considering the zero injection bus constraint
2	To validate the CGSA for SOPP problem, with zero injection bus constraint
3	To validate the FCGSA algorithm for MOPP problem, with zero injection bus constraint

5.3.1 Case 1 (SOPP - without considering zero injection bus)

In this case, the conventional single objective OPP problem is solved using a binary CGSA considering the observability constraints and without considering the zero injection bus constraint. By using CGSA, optimal number of PMUs and their locations are obtained from 30 trials and it is given in Table 5 for different test system. It should be noted that the obtained results are equal to the optimal number reported in the available literatures [19]. It justifies that the proposed CGSA can produce a better quality solution for the OPP problem.

Table 5
Solutions of OPP problem- Case 1

IEEE system Min. no. of PMUs Locations of PMUs % between the min. no. of PMUs and total no. of buses SORI_s Avg. BOI

14 bus 4 2, 6, 7, 9 28% 19 1.36

30 bus 10 2, 4, 6, 9, 10, 12, 15, 18, 25, 27 33% 52 1.73

57 bus 17 1, 4, 6, 9, 15, 20, 24, 25, 27, 32 36, 38, 39, 41, 46, 50, 53 29% 72 1.26

IEEE system	Min. no. of PMUs	Locations of PMUs	% between the min. no. of PMUs and total no. of buses	SORI_s	Avg. BOI
14 bus	4	2, 6, 7, 9	28%	19	1.36
30 bus	10	2, 4, 6, 9, 10, 12, 15, 18, 25, 27	33%	52	1.73
57 bus	17	1, 4, 6, 9, 15, 20, 24, 25, 27, 32 36, 38, 39, 41, 46, 50, 53	29%	72	1.26

5.3.1.1 Result analysis - IEEE 14 bus system

A. Bus observability analysis

To validate the proposed algorithm, IEEE 14 bus system is considered. Table 6 shows the advantage of considering the bus measurement redundancy in IEEE 14 bus system. It is observed that different solutions are obtained from thirty trials. From the Table 6, it is clear that, the fifth solution has given a higher observability of the system than others. The maximizing bus measurement redundancy value has an advantage that a larger portion of the system will remain observable, in case, one of the PMUs fail. It is proved that, the locations of PMUs, i.e., 2, 6, 7 and 9 are more desirable to guarantee the complete system observability and also to enhance it.

Table 6
Different solutions for IEEE 14 bus system

Solution Locations of PMUs Number of times for each bus is observed (BOI) SORI_s Avg. BOI

1 2 3 4 5 6 7 8 9 10 11 12 13 14

1 2, 7, 10, 13 1 1 1 2 1 1 1 1 2 1 1 1 1 1 16 1.14

2 2, 6, 8, 9 1 1 1 2 1 1 2 1 1 1 1 1 1 1 16 1.14

3 2, 7, 11, 13 1 1 1 2 1 2 1 1 1 1 1 1 1 1 16 1.14

4 2, 8, 10, 13 1 1 1 1 1 1 1 1 1 1 1 1 1 1 14 1

5 2, 6,7, 9 1 1 1 3 2 1 2 1 2 1 1 1 1 1 19 1.36

Solution	Locations of PMUs	Number of times for each bus is observed (BOI)	SORI_s	Avg. BOI
1	2, 7, 10, 13	1	1	1	2	1	1	1	1	2	1	1	1	1	1	16	1.14
2	2, 6, 8, 9	1	1	1	2	1	1	2	1	1	1	1	1	1	1	16	1.14
3	2, 7, 11, 13	1	1	1	2	1	2	1	1	1	1	1	1	1	1	16	1.14
4	2, 8, 10, 13	1	1	1	1	1	1	1	1	1	1	1	1	1	1	14	1
5	2, 6,7, 9	1	1	1	3	2	1	2	1	2	1	1	1	1	1	19	1.36

B. Robustness and computation efficiency of CGSA

In order to compare the robustness and computation efficiency of the CGSA, the OPP problem is solved for the IEEE 14 bus system using standard GA and PSO algorithm. Therefore, the initial random generated population is kept same for all the three techniques (GA, PSO and CGSA). The comparison of the computation efficiency of the GA, PSO and CGSA for the IEEE 14 bus system is given in Table 7.

Table 7

Comparison of results for IEEE 14 bus system-Case 1

Solution technique	Mean time (in sec)	Frequency of achieving minimum number of PMUs with best location
GA	120	11
PSO	40	15
CGSA	38	23

It is concluded that the CGSA can offer an optimal number of PMUs and their locations with less computation time than GA and PSO. It should be noted that the placement of PMUs problem is an off-line procedure, so the execution time is not considered here. The Fig. 4 shows the distribution of optimal number of PMUs and their locations obtained from 30 trial runs.

Fig.4

Distribution of PMUs in optimal locations for IEEE14 bus system.

In y axis, 1, 2 and 3 represents the optimal number of PMUs and their best locations, optimal number and not best locations, not optimal number and not best locations, respectively. It should be noted that the frequency of achieving the minimum number of PMUs with the best locations is high in case of CGSA when compared with GA and PSO. It shows the superiority of the CGSA over other solution techniques.

5.3.2 Case 2 (SOPP - considering zero injection bus)

In this case, the conventional single objective OPP problem is solved using a binary CGSA, considering the zero injection bus and observability constraints. Table 8 shows the different solutions of OPP problem obtained from 30 trials and the corresponding values of SORI_s and average BOI values.

Table 8
Different sol. of OPP problem during 30 trails- Case 2

IEEE system Min.no. of PMU Locations of PMUs SORI_s Avg. BOI

14 bus 3 2, 6, 9 16 1.14

30 bus 7 1, 5, 10, 12, 18, 23, 27 36 1.2

7 1, 5, 10, 12, 18, 24, 27 37 1.23

7 1, 2, 10, 12, 18, 24, 27 39 1.3

7 2, 4, 10, 12, 15, 18, 27 41 1.36

57 bus 11 1, 6, 13, 19, 25, 29, 32, 38, 41, 51, 54 57 1

11 1, 4, 13, 19, 25, 29, 32, 38, 41, 51, 54 58 1.01

11 1, 4, 13, 20, 25, 29, 32, 38, 51, 54, 56 59 1.03

IEEE system	Min.no. of PMU	Locations of PMUs	SORI_s	Avg. BOI
14 bus	3	2, 6, 9	16	1.14
30 bus	7	1, 5, 10, 12, 18, 23, 27	36	1.2
	7	1, 5, 10, 12, 18, 24, 27	37	1.23
	7	1, 2, 10, 12, 18, 24, 27	39	1.3
	7	2, 4, 10, 12, 15, 18, 27	41	1.36
57 bus	11	1, 6, 13, 19, 25, 29, 32, 38, 41, 51, 54	57	1
	11	1, 4, 13, 19, 25, 29, 32, 38, 41, 51, 54	58	1.01
	11	1, 4, 13, 20, 25, 29, 32, 38, 51, 54, 56	59	1.03

Similar to the section 5.3.1.A, the best solution of OPP problem is obtained for the test systems and given in Table 9. The efficiency of zero injection bus effect is proved when compared with Case 1 and it is shown in Table 10.

Table 9

Solutions of OPP problem - Case 2

IEEE system	Min. no. of PMUs	Locations of PMUs	% between the min. no. of PMUs and total no. of buses	SORI_s	Avg. BOI
14 bus	3	2, 6, 9	21%	16	1.14
30 bus	7	2, 4, 10, 12, 15, 18, 27	23%	41	1.36
57 bus	11	1, 4, 13, 20, 25, 29, 32, 38, 51, 54, 56	19%	59	1.03

Table 10

Efficiency of considering the zero injection bus constraint-Case 2

IEEE system	Min. no. of PMUs		Reduction in no. of PMUs
	Case 1	Case 2
14 bus	4	3	1
30 bus	10	7	3
57 bus	17	11	6

Table 11 shows the comparison results of Case 2 with other techniques available in earlier articles. This shows that the proposed CGSA is able to make a better quality solution to the OPP problem. The optimal numbers of PMUs and their locations, observability area of each PMU are shown in Fig. 5 for IEEE 14 bus system.

Fig.5

IEEE 14 bus system with PMU placement-Case 2.

Table 11

Performance comparison of CGSA - Case 2

Method	Min. no. of PMUs
	14 bus	30 bus	57 bus
Dual search and SA [20], ILP [21], PSO [22], IGA [23], ILP [24], RTS [19], BILP [15], BICA [25] and CLA [16]	3	7	11
ILP [26]	3	7	12
ILP [27]	3	–	12
Binary Search [28]	3	7	–
GA [4]	3	7	12
ILP [14]	3	–	14
BILP [25]	3	–	11
Tabu Search [29]	3	–	13
BPSO [30]	3	7	12
Proposed method (CGSA)	3	7	11

5.4 Case 3 (MOPP – Considering zero injection bus and Proposed constraint)

In this case, Multi-objective OPP Problem (MOPP) with reliability function is solved using Fuzzified CGSA, considering the zero injection bus constraint for minimizing the number of PMUs and proposed constraint for maximizing the system reliability. Here, two objective functions such as minimizing the number of PMUs (F₁) for complete system observability and maximizing the system reliability (F₂) are satisfied simultaneously. The proposed results are justified with Case 2 which forms the main contribution of this paper.

5.4.1 IEEE 14 bus system

To demonstrate the importance of incorporation of the reliability function with existing OPP problem, the IEEE 14 bus system is individually solved for the two objective functions and the corresponding variations in other objective is evaluated and it is given in Table 12. In Case 2, when the PMUs are placed to achieve the complete observability with minimum number of PMUs (F1), three numbers of PMUs are required on buses 2, 6 and 9. Then the corresponding system reliability index is 0.8713 (low reliability of the system). On the other hand, when the PMUs are placed to maximize the reliability of the system, then eleven PMUs are required to monitor the load buses. Here, the reliability of observing low reliability buses and other buses are very high but it increases the total number of PMUs required. Since the cost of the PMU is high, it is unable to adapt 11 numbers of PMUs in 14 bus system. But it is clearly observed that the variation in the system reliability level is significant with respect to different objective functions and it should not be neglected. Also these objectives are conflicting in nature; the improvement of one of them weakens the other. Therefore, a new multi-objective OPP problem is formulated with reliability function to obtain a best compromise solution.

Table 12
Solution for the two different objectives for IEEE 14 bus system

Solution Min. no. of PMUs (F₁) Max. of reliability level (F₂)

Min. no. of PMUs 3 11

Locations of PMUs 2, 6, 9 3, 4, 9, 2, 14, 13, 6, 10, 5, 12, 11

System observability 100% 100%

SORI 16 43

Avg. BOI 1.14 3.71

Reliability index (EENS) 0.8713 (i.e., low reliability) 0 (i.e., high reliability)

Solution	Min. no. of PMUs (F₁)	Max. of reliability level (F₂)
Min. no. of PMUs	3	11
Locations of PMUs	2, 6, 9	3, 4, 9, 2, 14, 13, 6, 10, 5, 12, 11
System observability	100%	100%
SORI	16	43
Avg. BOI	1.14	3.71
Reliability index (EENS)	0.8713 (i.e., low reliability)	0 (i.e., high reliability)

The load bus reliability index is determined by using an Equation (5) and arranged in descending order for the IEEE 14 bus system as shown in Table 13. It is noted that the reliability level is greatly improved in Case 3 (i.e., EENS = 0.4535) compared with Case 2 (i.e., EENS = 0.8713). Here, the Fuzzy decided the optimal number of PMUs and their locations, based on the two factors such as observing the maximum number of buses and low reliability buses. Based on observability and reliability of the buses, the optimal number of PMUs and their locations are determined using CGSA from 30 trials and the results with observability indices and reliability index are given in Table 14.

Table 13

Reliability level of load buses - IEEE 14 bus system

Rank	Load bus	Reliability index	Rank	Load bus	Reliability index
1	3	0.4174	7	6	0.0496
2	4	0.2118	8	10	0.0398
3	9	0.1307	9	5	0.0336
4	2	0.0961	10	12	0.0270
5	14	0.0660	11	11	0.0155
6	13	0.0598

Table 14

Solution of MOPP problem for IEEE 14 bus system- Case 3

Min. no. of PMUs	Locations of PMUs	Observability index				Reliability index (EENS)
		BOI	LRBOI	SORI_m	Avg. BOI
4	2, 4, 6, 9	13	9	22	1.58	0.4535

5.4.1.1 Result Analysis - IEEE 14 bus system

A. Bus observability analysis

Table 15 shows the comparison of bus observability analysis for IEEE 14 bus system in between Cases 2 and 3. Here, D, ID and ZIBE represent the buses are directly monitored, indirectly monitored and monitored by applying the zero injection bus effect, respectively. The advantage of direct monitoring is that it avoids errors in calculation of voltage phasor of adjacent buses of a PMU bus. The advantage of indirect monitoring or the monitoring by zero injection bus effect is that it minimizes the required number of PMUs.

Table 15
Comparison of bus observability analysis for IEEE 14 bus system

Bus no. based on reliability rank Method of monitoring No. PMUs monitored the load bus (BOI)

Case 2 Case 3 Case 2 Case 3

3 ID ID 1 2

4 ID D 1 3

9 D D 1 2

2 D D 1 2

14 ID ID 1 1

13 ID ID 1 1

6 D D 1 1

10 ID ID 1 1

5 ID ID 1 1

12 ID ID 1 1

11 ID ID 1 1

Bus no. based on reliability rank	Method of monitoring	No. PMUs monitored the load bus (BOI)
3	ID	ID	1	2
4	ID	D	1	3
9	D	D	1	2
2	D	D	1	2
14	ID	ID	1	1
13	ID	ID	1	1
6	D	D	1	1
10	ID	ID	1	1
5	ID	ID	1	1
12	ID	ID	1	1
11	ID	ID	1	1

From the Table 15, it is clear that in Case 3, the load buses 4, 9, 2 and 6 are directly observed and the number of PMUs monitoring the low reliability load buses 3, 4, 9 and 2 are higher than Case 2. The remaining buses, except the load buses, bus 8 is observed by applying zero injection bus effect and buses 1 and 2 are observed by indirect and direct measurement, respectively. Thus, proposed method ensures the complete observability of the IEEE 14 bus system.

B. Reliability analysis

The reliability of the system is increased by placing PMUs directly on the low reliability buses 4, 9 and 2. When the PMU is placed on bus 4, it monitors the low reliability buses 3, 4, 9 and 2, load bus 5 and zero injection bus 7. When the PMU is placed on bus 9, it monitors the low reliability buses 4 and 9, buses 10, 14 and 7. When the PMU is placed on bus 2, it monitors the low reliability buses 2, 3 and 4, buses 5 and 1. Therefore, the low reliability buses 3, 4, 9 and 2 are monitored 2, 3, 2 and 2 times, respectively (i.e., maximizing low reliability bus measurement redundancy). Therefore, it is possible to monitor continuously the low reliability bus even when, one of the PMUs fail.

Even though, bus 6 is high reliability bus, than buses 14 and 13, but PMU is placed on bus 6, because the placement of PMU on bus 14 or 13, leads to additional PMU to monitor bus 11. Hence in order to avoid this scenario, fuzzy made the decision to place a PMU on bus 6 to monitor the load buses 6, 11, 12 and 13.

Therefore, number of PMUs is minimized and then reliability of the system is compromised. The bus 8 is observed by applying the zero injection effect. Thus, minimizing the number of PMUs and maximizing the system reliability are achieved simultaneously, in the proposed MOPP problem. The optimal number of PMUs and their locations, the oservability area of each PMU are shown in Fig. 6.

Fig.6

IEEE 14 bus system with PMU placement for Case 3 - MOPP problem.

C. Comparative analysis

Table 16 shows the comparison of the observability indices and reliability index in between Cases 2 and 3 for IEEE 14 bus system. It is clear that, the system observability and reliability levels are improved in Case 3. There is a great improvement in observability and reliability levels with a little increase in the number of PMUs from 3 to 4, as shown in Table 16. It shows the superiority of the proposed method (Case 3) over conventional single objective OPP method (Case 2).

Table 16

Comparison of results- IEEE 14 bus system

Case	Min. no. of PMUs	Loc. of PMUs	Observability index				EENS
			BOI	LRBOI	SORI_m	Avg. BOI
2	3	2, 6, 9	11	5	16	1.14	0.8709
3	4	2, 4, 6, 9	13	9	22	1.58	0.4535

Table 17

Reliability level of load buses-IEEE 30 bus system

Rank	Load bus	EENS	Rank	Load bus	EENS
1	5	0.4047	12	4	0.0327
2	8	0.1289	13	14	0.0266
3	7	0.0980	14	10	0.0249
4	2	0.0932	15	16	0.0150
5	21	0.0752	16	26	0.0150
6	12	0.0481	17	18	0.0137
7	30	0.0455	18	23	0.0137
8	19	0.0408	19	3	0.0103
9	17	0.0387	20	29	0.0103
10	24	0.0374	21	30	0.0095
11	15	0.0352

5.4.2 IEEE 30 bus system

The load bus reliability index is determined by using an Equation (5) and arranged in descending order for the IEEE 30 bus system as shown in Table 17. Based on the observability and reliability of buses, the optimal number of PMUs and their locations are determined using CGSA from 30 trials and the corresponding observability indices and reliability index are given in Table 18. The fuzzy decided the optimal solutions to simultaneously maximize the system observability and reliability levels.

Table 18
Solution of MOPP problem- IEEE 30 bus system- Case 3

Min. no. of PMUs Locations of PMUs Observability index Reliability index (EENS)

BOI LRBOI SORI_m Avg. BOI

10 2, 4, 5, 6, 8, 10, 12, 19, 24, 30 42 11 53 1.77 0.3962

Min. no. of PMUs	Locations of PMUs	Observability index	Reliability index (EENS)
10	2, 4, 5, 6, 8, 10, 12, 19, 24, 30	42	11	53	1.77	0.3962

5.4.2.1 Result Analysis - IEEE 30 bus system

A. Bus observability analysis

Table 19 shows the comparison of bus observability analysis for IEEE 30 bus system in between Cases 2 and 3. It is clear that the load buses 5, 8, 2, 4, 10, 12, 19, 24 and 30 are directly observed by placing PMUs on these buses. The load buses 5, 8, 7, 2, 12, 4, 10 and 20 are observed by 2, 2, 3, 4, 2, 4, 2 and 2 times, respectively. The bus 26 is observed by applying zero injection bus effect. The remaining buses, except the load buses are observed, either by direct or indirect measurement. Thus, the proposed method ensures the complete observability of the IEEE 30 bus system.

Table 19
Comparison of bus observability analysis IEEE for 30 bus system in Cases 2 and 3

Bus no. based on reliability rank Method of monitoring No. PMUs monitored the load bus (BOI) Bus no. based on reliability rank Method of Monitoring No. PMUs monitored the load bus (BOI)

Case 2 Case 3 Case 2 Case 3 Case 2 Case 3 Case 2 Case 3

5 ID D 1 2 4 D D 3 4

8 ZIBE D 1 2 14 ID ID 2 1

7 ZIBE ID 1 3 10 D D 1 2

2 D D 1 4 16 ID ID 1 1

21 ID ID 1 1 26 ZIBE ZIBE 1 1

12 D D 2 2 18 D ID 1 1

30 ID D 1 1 23 ID ID 1 1

19 ID D 1 1 3 ID ID 1 1

17 ID ID 1 1 29 ID ID 1 1

24 ZIBE D 1 1 20 ID ID 1 2

15 D ID 2 1

Bus no. based on reliability rank	Method of monitoring	No. PMUs monitored the load bus (BOI)	Bus no. based on reliability rank	Method of Monitoring	No. PMUs monitored the load bus (BOI)
5	ID	D	1	2	4	D	D	3	4
8	ZIBE	D	1	2	14	ID	ID	2	1
7	ZIBE	ID	1	3	10	D	D	1	2
2	D	D	1	4	16	ID	ID	1	1
21	ID	ID	1	1	26	ZIBE	ZIBE	1	1
12	D	D	2	2	18	D	ID	1	1
30	ID	D	1	1	23	ID	ID	1	1
19	ID	D	1	1	3	ID	ID	1	1
17	ID	ID	1	1	29	ID	ID	1	1
24	ZIBE	D	1	1	20	ID	ID	1	2
15	D	ID	2	1

B. Reliability analysis

The reliability of the system is increased by placing PMUs directly on the low reliability buses 5, 8, 7 and 2, the load buses 12, 30, 19, 24, 4 and 10. Therefore, the low reliability buses 5, 8, 7 and 2, are observed by 2, 2, 3 and 4 times, respectively. The load buses 12, 4, 10 and 20 are observed by 2, 4, 2 and 2 times, respectively.

Out of 21 load buses in IEEE 30 bus system, totally 11 load buses 5, 8, 7, 2, 12, 30, 19, 24, 4, 10 and 20 including the low reliability buses 5, 8, 7 and 2 are monitored. Therefore, it is possible to monitor the low reliability buses in order to prevent outage of these buses, even if one of the PMUs, fail. Even though the reliability of the buses 10, 4, 12 are higher than buses 7 and 21, fuzzy made the decision to place PMUs on buses 10, 4 and 12, in order to minimize the number of PMUs because these buses are having high observability of the system and then the reliability of the system is compromised. This proves that the proposed approach is more desirable to determine the optimal number of PMUs and their locations, for simultaneously maximizing the system observability and reliability levels.

C. Comparative analysis

Table 20 shows the comparison of the observability indices and reliability index for IEEE 30 bus system in between Cases 2 and 3. It is clear that, the system observability and reliability levels are improved in Case 3 by adding the number of PMUs from 7 to 10, as shown in Table 20. It shows the advantage of the proposed method over conventional single objective OPP method.

Table 20

Comparison of results-IEEE 30bus system

Case	Min. no. of PMUs	Loc. of PMUs	Observability index				EENS
			BOI	LRBOI	SORI_m	Avg. BOI
2	7	2,4,10, 12,15, 18, 27	36	5	41	1.37	0.9686
3	10	2, 4, 5, 6, 8, 10, 12, 19, 24, 30	42	11	53	1.77	0.3962

5.4.3 IEEE 57 bus system

In this section, the proposed MOPP problem is solved for IEEE 57 bus system, which is the largest system available in the literature. In this method, the buses 18, 23, 29, 43, 44, 46, 22, 37, 39 and 46 are observed by applying zero injection bus effect in order to minimize the number of PMUs. The load buses 12, 8, 9, 6, 1, 15, 50, 53, 49, 56, 25, 54, 20 and 32 are directly observed by placing PMUs on these buses. The load buses 12, 6, 1, 16, 13, 15, 50, 49 and 10 are observed by 2 times, respectively and load buses 8, 9 and 13 are observed by 3 times, respectively.

Out of 42 load buses, totally 18 load buses 12, 8, 9, 6, 1, 16, 15, 50, 53, 13, 49, 27, 56, 25, 35, 10, 20 and 32 including the low reliability load buses 1, 6, 8, 9, 12 and 16 are monitored in addition to the complete observability of the power system.

Out of thirty trials, the optimal number and locations of PMUs are determined using CGSA and the results are given in Table 21. Here too, the system observability and reliability levels are enhanced over conventional single objective OPP method (Case 2), which justifies the proposed method.

Table 21
Solution of MOPP problem- IEEE 57 system- Case 3

Min. no.of PMUs Loc. of PMUs Observability index EENS

BOI LRBOI SORI_m Avg. BOI

15 1, 6, 8, 9, 12, 15, 20, 25, 27, 32, 35, 49, 50, 53, 56 61 14 75 1.32 1.386

Min. no.of PMUs	Loc. of PMUs	Observability index	EENS
15	1, 6, 8, 9, 12, 15, 20, 25, 27, 32, 35, 49, 50, 53, 56	61	14	75	1.32	1.386

5.5 Improvement achieved by proposed problem

Table 22 shows the comparison of the number of PMUs, observability and reliability levels of the test systems in between Cases 2 and 3. It should be noted that the system observability and reliability levels are improved by using the proposed method and from the numerical analysis, the system reliability level is greatly enhanced by little increase in number of PMUs in Case 3.

Table 22
Comparison of the number of PMUs, observability and reliability indices between Cases 2 and 3

IEEE system Case 2 (Single objective) Case 3 (Multi- objective) Difference in ratio of no. of PMUs to the total no. of buses in between Case 2 and Case 3 (in %) Improvement in reliability level in Case 3 than Case 2 (in %)

Min. no. of PMUs SORI_s Avg. BOI EENS Min. no. of PMUs SORI_m Avg. BOI EENS

14 bus 3 16 1.14 0.8709 4 22 1.5 0.4535 7% 47%

30 bus 7 41 1.36 0.9686 10 53 1.57 0.3962 10% 59%

57 bus 11 59 1.03 4.3148 15 71 1.31 1.3861 7% 68%

IEEE system	Case 2 (Single objective)	Case 3 (Multi- objective)	Difference in ratio of no. of PMUs to the total no. of buses in between Case 2 and Case 3 (in %)	Improvement in reliability level in Case 3 than Case 2 (in %)
14 bus	3	16	1.14	0.8709	4	22	1.5	0.4535	7%	47%
30 bus	7	41	1.36	0.9686	10	53	1.57	0.3962	10%	59%
57 bus	11	59	1.03	4.3148	15	71	1.31	1.3861	7%	68%

6 Conclusion

This paper presented the application of CGSA for solving the conventional single objective OPP and proposed multi-objective OPP problems. The following points listed below summarize the proposed work.

The CGSA, when applied to the conventional single objective OPP problem (Cases 1 and 2), is able to offer the minimum number of PMUs and their optimal locations, when compared with other conventional and non conventional techniques reported in the earlier literatures.

In Case 3, the problem has been formulated as a multi-objective OPP with two conflicting objective functions such as minimization of number of PMUs and maximization of reliability level of the systems. The proposed multi-objective OPP problem has been solved using Fuzzy Adapted Binary CGSA and the optimal number of PMUs and their locations are obtained for the different test systems. From the comparative analysis, it is observed that, the proposed methodology is efficient in determining the minimum numbers of PMUs and their optimal locations in order to maximize both the observability and reliability levels of the systems.

It is concluded that the observability and reliability levels of the systems are enhanced with little increase in number of PMUs employing proposed method. The feasibility and performance of the proposed methodology are demonstrated on IEEE 14, 30 and 57 bus systems. The results presented in this paper will encourage the researchers to apply confidently the CGSA approach for larger power systems.

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Optimal placement of phasor measurement units using clustered gravitational search algorithm

Abstract

Keywords

1 Introduction

2 Multi-objective OPP problem formulation

2.1 Objective functions

2.1.1 Minimizing the number of PMUs (F1)

2.2.1 Constraint for minimizing the number of PMUs (i.e. Zero injection bus constraint)

5.1 System specification

Table 2 Parameter settings of CGSA for IEEE 14 bus system G0 F1 SORIs α F1 SORIs Population size F1 SORIs 90 6 24 15 6 27 10 7 32 100 5 20 16 6 26 20 5 22 110 5 19 17 6 28 30 6 25 120 7 35 18 6 21 50 5 24 19 9 37 100 6 28 20 5 21 21 8 35 22 7 31

5.4.1 IEEE 14 bus system

Table 18 Solution of MOPP problem- IEEE 30 bus system- Case 3 Min. no. of PMUs Locations of PMUs Observability index Reliability index (EENS) BOI LRBOI SORIm Avg. BOI 10 2, 4, 5, 6, 8, 10, 12, 19, 24, 30 42 11 53 1.77 0.3962

Table 21 Solution of MOPP problem- IEEE 57 system- Case 3 Min. no.of PMUs Loc. of PMUs Observability index EENS BOI LRBOI SORIm Avg. BOI 15 1, 6, 8, 9, 12, 15, 20, 25, 27, 32, 35, 49, 50, 53, 56 61 14 75 1.32 1.386

References

2.1.1 Minimizing the number of PMUs (F₁)

Table 2
Parameter settings of CGSA for IEEE 14 bus system

G0 F1 SORI_s α F1 SORI_s Population size F1 SORI_s

90 6 24 15 6 27 10 7 32

100 5 20 16 6 26 20 5 22

110 5 19 17 6 28 30 6 25

120 7 35 18 6 21 50 5 24

19 9 37 100 6 28

20 5 21

21 8 35

22 7 31

Table 18
Solution of MOPP problem- IEEE 30 bus system- Case 3

Min. no. of PMUs Locations of PMUs Observability index Reliability index (EENS)

BOI LRBOI SORI_m Avg. BOI

10 2, 4, 5, 6, 8, 10, 12, 19, 24, 30 42 11 53 1.77 0.3962

Table 21
Solution of MOPP problem- IEEE 57 system- Case 3

Min. no.of PMUs Loc. of PMUs Observability index EENS

BOI LRBOI SORI_m Avg. BOI

15 1, 6, 8, 9, 12, 15, 20, 25, 27, 32, 35, 49, 50, 53, 56 61 14 75 1.32 1.386