Optimal power quality monitor placement using ... - Semantic Scholar

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Optimal power quality monitor placement using genetic algorithm and Mallow’s Cp A. Kazemi ⇑, A. Mohamed, H. Shareef, H. Zayandehroodi Department of Electrical, Electronic and Systems Engineering, Universiti Kebangsaan Malaysia, Bangi 43600, Selangor, Malaysia

a r t i c l e

i n f o

Keywords: Power quality monitor placement Genetic algorithm Mallow’s Cp

a b s t r a c t This study presents a method to determine the optimal number and placement of power quality monitors (PQMs) in power systems by using genetic algorithm (GA) and Mallow’s Cp which is a statistical criterion for selecting among many alternative subset regressions. This procedure helps to avoid the dependency of set voltage sag threshold values of PQMs in the conventional monitor reach area based (MRA) method. In the proposed GACp method, the fitness function for problem modeling aims to minimize allocated monitors and minimize the difference between the Mallow’s Cp and the number of variables used for the multivariable regression model during estimation of unmonitored buses. After obtaining the optimal placements of PQMs by using the GACp method, the observability and redundancy of the monitors are tested to further reduce the redundant PQMs. The IEEE 30 bus test system is simulated using the DIGSILENT power factory software to validate the proposed method. The simulated results show that the GACp method requires only two PQMs to observe all voltage sags that may appear at each bus in the test system without redundancy.

1. Introduction Significant economic losses associated with industrial equipment failure caused by voltage sags have raised concerns for utilities and their customers for the past decades. Voltage sag is defined as a decrease in RMS voltage or current at the power frequency for durations from 0.5 cycles to 1 min [1]. It is considered as one of the most common power quality disturbances that cause equipment malfunction and process interruption [2]. Voltage sags occur because of short-circuit faults and the large motor starting in power systems [3]. Information about the actual cause and source of voltage sags can help power engineers to decide on the resumption of systems to normal operation [4]. The implementation of power quality monitoring system in power supply networks is a main step in obtaining information about voltage sag disturbances [5]. Ideally, an entire power system should be monitored by a power quality monitor (PQM) at each bus through a communication facility. However, such system produces huge amount of redundant data, making it cost inefficient and economically unreasonable. Therefore, methods that can select the number and location of monitored sites should be developed to minimize the number of

monitors and to reduce monitoring costs without missing essential voltage sag information. In recent years, several studies have been attempted to solve the PQM placement problem by determining the optimal number and locations of PQMs [6,7]. A primary requisite in selecting the location of monitors is the guaranteed observability of the entire system to ensure the capture of any voltage sag event by at least one PQM [8]. In such case, the PQM placement methods can be divided into four main methods, namely, monitor reach area (MRA), covering and packing, graph theory, and multivariable regression (MVR). The concept of MRA was introduced to determine the optimal location of PQMs [9]. MRA is defined as the area of network that can be observed from a given monitor position. In [6,7], an improved optimal monitoring program is presented to optimize the MRA expression by using genetic algorithm (GA) for identifying optimal meter location [10,11]. Integer programming and fuzzy logic have been applied to determine the optimal placement of PQMs in large transmission networks for the assessment of voltage sags [12]. The method expressed in [8] deals with uncovered line faults, which are ignored by the original MRA method. In [9], an approach based on monitor reach matrix (MRM) obtained from the solution of analytical expressions was presented for the determination of the optimal location of voltage sag monitors. Another optimal PQM placement method uses severity index, MRA matrix, and GA [13]. The covering and packing has been developed to determine the optimum number and location of PQMs by minimizing the cost of PQMs via the integer linear programming technique [4]. Simi-

Edited by Foxit PDF Editor Copyright (c) by Foxit Software Company, 2004 - 2007 For Evaluation Only. larly, a monitor positioning algorithm has been used to determine the optimal number and locations of PQMs for a given distribution system [14]. In this algorithm, graph theory is applied and system topology is considered to form the coverage matrix. This paper presents a new algorithm for optimal PQM placement based on genetic algorithm (GA) and the Mallow’s Cp, and it is named as the GACp method. With regards to the optimization problem, the objective function or the fitness function in GA aims to minimize the number of allocated monitors, which is defined as the minimum difference between the Mallow’s Cp value and the number of variables used in the multivariable regression model for estimating the unmonitored buses. In the GACp method, the concepts of the multivariable regression (MVR) theory, the Mallow’s Cp, observability, redundancy and GA are applied to determine the appropriate locations for installing the PQMs. The proposed method is compared with previous methods based on the concepts of MRA and sag severity index (SSI) to validate its accuracy [15,16]. All methods are tested on the IEEE 30 bus test system.

The best estimation for B can be considered as the one which minimizes the sum of the squared errors. To minimize the vector of least squares estimate, consider

L¼

n X

e2i ¼ e0 e ¼ ðY  XBÞ0 ðY  XBÞ

ð3Þ

i¼1

Expanding (3),

L ¼ Y 0 Y  B0 X 0 Y  Y 0 XB þ B0 X 0 XB ¼ Y 0 Y  2B0 X 0 Y þ B0 X 0 XB

ð4Þ

Differentiate (4) with respect to B and setting to zero, the minimum square estimate has to obey the following condition,

 @L  ¼ 2X 0 Y þ 2X 0 XB ¼ 0 @Bb

ð5Þ

where X0 Xb = X0 Y is a function of minimum square normal to the solution that gives the value of minimum square estimate, b which is written as,

b ¼ ðX 0 XÞ1 X 0 Y

ð6Þ

The estimated regression model now becomes,

2. Theoretical background The background theory of the MVR, Mallow’s Cp, GA, observability and redundancy are described to illustrate the theory behind the proposed PQM placement method. 2.1. Multivariable regression theory

b i ¼ b0 þ Y

k X

bj xij

i ¼ 1; 2; . . . ; n

b iÞ The difference between the observed (Yi) and estimated ð Y variables is the error that is given by [17].

b E¼YY

Regression analysis was first considered in the 18th century for navigation purposes. Later, with the advent of high speed computing, regression analysis developed rapidly and the scope of analysis has expanded from logistic regression analysis to position regression. Regression analysis has several possible applications including prediction of future observations, assessment of the effect of relationship and general description of data structure. Generally, regression analysis is used for explaining or modeling the relationship between a single output variable called dependent variable, and one or more predictor or independent variables [17]. When there is only one independent variable it is called as a simple regression but when there are more than one independent variable it is called as multiple regression or sometime multivariable regression. Multiple regression finds a set of partial regression coefficients, bj such that the dependent variable, Y can be approximated by a linear combination of the ‘k’ independent variables, x. A predicted value, denoted by Y dependent variable is given by,

ð7Þ

j¼1

ð8Þ

2.2. The Mallow’s Cp The Mallow’s Cp is a criterion for the total mean square error (MSE) of the regression model. The Mallow’s Cp is given by,

C p ¼ ðSSEðPÞ=MSEðPÞÞ  n þ 2P

ð9Þ

where SSE(P) is the residual sum of square error for the model with P  1 variables which is represented by:

SSEðPÞ ¼

n X b i Þ2 ðY i  Y

ð10Þ

i¼1

and MSE(P) is the residual MSE when all the available variables are used and is expressed as:

MSEðPÞ ¼ SSEðPÞ=ðn  PÞ ¼

k X b i Þ2 =ðn  PÞ ðY i  Y

ð11Þ

i¼1

where bj(j = 0, 1, 2, . . . , k) are unknown parameters of regression coefficients (B) and e is a random error. For ‘n’ number of observations, (1) can be written in matrix form as,

where n is the number of observations and P is the number of variables used for the model plus one. One represents the intercept constant. Mallow’s Cp help to find adequate models by plotting Cp value against number of independent variables plus one which is known as P [18].

Y ¼ XB þ e

2.3. Relationship between Mallow’s Cp and P

Y ¼ b0 þ b1 x1 þ b2 x2 þ    þ bk xk þ e

ð1Þ

ð2Þ

in which,

2

Y1

3

6Y 7 6 27 Y ¼ 6 7; 4 : 5 Yn

2

3

1 x11

x12

: x1k

61 x 21 6 X¼6 4: :

x22

: x2k 7 7 7; : : 5

1 xn1

xn2

:

: xnk

3 b0 6b 7 6 17 B ¼ 6 7; 4 : 5 2

bk

3

2

e1 6e 7 6 27

e¼6

7 4 : 5

en

Here, Y is a (n  1) vector for observation, X is a (n  p) matrix corresponding to k number of independent variable (p = k + 1), B is a (p  1) vector of regression coefficients, and e is a (n  1) vector for random errors.

Usually Cp is plotted against P for the collection of subset models of various sizes under consideration as shown in Fig. 1. Acceptable models in the sense of minimizing the total bias of the predicted values are those models for which Cp approaches the value of P. Thus, only those subset models that have Cp values close to P must be considered if un bias is a desired criterion for selection of a subset model [18]. For instance, if the independent variables (k) (this study measured voltages from PQM) = 5 and the total number of parameters (P) = 6 representing 5 independent variables and 1 intercept, then if a choice between a Mallow’s Cp of 5.1 and a Mallow’s Cp of 2

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Initial population using random generation

Mallow s Cp

10

i=1

8 6

most adequate model Set population of individuals

4

2

Individuals 1

2

6

Evaluate fitness

10

P = number of variable +1 Fig. 1. Relationship between Cp and P.

Parents selection

i=i+1

is to be considered, the Cp value of 5.1 which is very close to P = 6 should be chosen instead of Cp value of 2. This is because the model with small bias fall near or below the Cp = P line, whereas model with larger bias fall far below the Cp = P line.

Crossover + mutation + migration

2.4. Bus voltage estimation using MVR For estimating the unmonitored bus voltages using the MVR, it is necessary to identify most adequate combination of PQM buses (independent variables), where PQM should be installed in the system. According to Mallow’s Cp criterion, the most suitable combination of PQM buses provides the Cp value very close to P as highlighted before. Therefore, considering a fault at a specific bus, training data for each bus (which include voltages of all system buses) are recorded to calculate various regression coefficient (B) matrices based on Eq. (6). The B matrix calculated with most appropriate combination of PQM buses provides the best estimate of the unmonitored bus voltages if a fault occurs at this particular bus. Similarly, by generating training data for other buses and following the same procedure, B matrices with optimal PQM buses can be obtained. However, there are many combinations to explore and the best PQM locations identified for individual bus data may not be the same. Hence, optimal PQM placement problem deals with finding the best common PQM buses that can give suitable B matrices which can accurately estimate unmonitored bus voltages. One of the techniques that can be used to obtain the optimum combination of PQM buses is genetic algorithm (GA). 2.5. Genetic algorithm The basic GA is composed of four main stages, namely, specifying an initial population, evaluation of fitness function, selection and crossover and mutation [19]. Fig. 2 shows how the stages of GA are related to one another and illustrates the use of GA in the PQM placement problem. Basically, GA comprises a population of individuals in different phases or generations. In each phase, operations which include selection, crossover, and mutation similar to the biological process of natural selection are applied to the population. This application is performed to make all individuals evolve and attain the best fitted solution that will satisfy the problem requirements. Thus, each GA individual is formed by a string of bits, which carries information that defines its own features and parameters related to a possible solution. This process is called the decoding process which allows individual evaluations and possible solutions to attain the objectives and constraints related to the problem. The GA operators change the individuals in each gen-

No

i = Maximum generation

Yes Stop Fig. 2. Flowchart of basic GA.

eration by manipulating the bits in their strings in a suitable way that leads to the convergence of such a process. The initial population is randomly created, and then a loop is performed, where the fitness function, selection, crossover and mutation in each generation are conducted until the convergence criterion is satisfied [19].

3. PQM placement problem formulation Finding the optimum combination of PQM buses that can provide a suitable B matrix for a particular bus for estimating the unmonitored bus voltages is formulated as a binary optimization problem which consists of three common elements, namely decision vectors, objective function and optimization constraints. The optimization process explores the solution space as defined in the objective function through the bits manipulation of decision vector subject to the system constraints. 3.1. Decision vector In order to satisfy the solution process, the Monitor Placement (MP) vector is introduced to represent the binary decision vector (x) in the optimization process. The bits of this vector indicate the need of the monitor to be installed at a particular bus in a power system. The dimension of the vector corresponds to the number of buses in the test system. A bit with a value of ‘0’ in the MP (n) indicates that no monitor is required at bus n while a

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Start

GA execution number (j) =1 and number generation (m) =1

Create a MP vector with length equal to number of system buses

Randomly initialize all entries of the string in the first generation

Evaluate performance of each MP vector based on fitness function (13)

Training data matrix for bus i

Select 25% of the best strings in current generation

m=m+1

Select randomly crossover the parent string

Mutate the off - spring MP vector

Reject the new MP vectors which dose not fulfil the optimization constraints

No m=50 Yes Save results of optimal MP vector and number of PQMs required

j=j+1

No j=30 Yes Stop Fig. 3. Flowchart of GA optimization procedures for determining optimum MP vectors.

bit with a value of ‘1’ shows that a monitor should be installed at bus n. The MP vector can be described by,

MPðnÞ ¼



1; if PQM is required at bus n 0;

otherwise

8n

ð12Þ

mine the optimal combination of number of monitors and locations to install the monitors. The fitness function in this case is to minimize the values of Mallow’s Cp for each MVR model relative to P which is given by the MP vector. To accomplish this objective, the fitness function is formulated as follows:

Fitness function : Min jCp  Pj

ð13Þ

3.2. Objective function and constraints

Subject to : 0 6 Cp 6 P; Cp ! P

The purpose of the optimization is to determine a suitable combination of PQMs needed for optimal placement while obtaining the unmonitored bus voltages during any fault occurrences at a particular bus which may lead to voltage sag events in a power system. Therefore, the objective function is formulated so as to deter-

The constraint in (13) indicates the occurrence of minimum |Cp – P| when the Mallow’s Cp is close to P. If Mallow’s Cp is greater than P, it is considered as an infeasible solution. As indicated previously, it should be noted that optimum MP vector is required to evaluate the fitness function for individual training bus data.

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Start

Arrange 30 optimum MP vectors as described in section 4.1

Establish groups of Mp vectors from previous step with same optimal number of PQMs

Select the groups of MP vectors with the maximum number of frequency of occurrence

Is frequency of occurrence of selected groups same ?

Yes

No

Calculats Cp

Select the group of MP vectors with minimum number of PQMs

Select optimization MP vector which gives maximum Mallow ’s Cp close to P

End Fig. 4. Flowchart to select best MP vector for a particular bus using Mallow’s Cp.

4. Application of GACp method for optimal PQM placement This section explains how GA is used to evaluate the optimum MP vector for each bus and application of the Mallow’s Cp to obtain PQM placement from all the optimum MP vectors for the entire power system. 4.1. Finding the best model for each bus using GA According to the Mallow’s Cp concept, establishing and testing all combinations between PQM buses (independent variables) and checking Mallow’s Cp are difficult and time consuming. Therefore, GA which is a useful optimization tool is used for finding the best MVR model for each bus in a test system. The steps taken by GA to get the optimal MP vector for an individual bus are described as follows: Step 1. Create a MP vector with size equal to the number of system buses. Each bit in the string corresponds to a bus number and each bit can be a possible candidate for a PQM location. Step 2. Randomly initialize all the entries of the MP vector (individual chromosome) in the first generation within feasible

arrangement. The bits with logic 1 in each string (corresponding PQM location) are considered as a feasible model to be evaluated. Step 3. Evaluate performance of each MP vector based on the formulated fitness function given in (13). Step 4. Select 25% of the best strings in the current generation as parents’ chromosomes. Step 5. Randomly crossover the parent strings of a randomly chosen pair for creating a new offspring string vector. Step 6. Mutate the offspring string vector by changing randomly a few of its entries. Step 7. Reject the new MP vector(s) which do not fulfill the optimization constraint. Step 8. Repeat steps 3–7 until the generation size reaches 50. Step 9. Save results of the optimal MP vector. Step 10. Repeat the GA optimization procedure for 30 times to avoid the solutions that converge in the local minimum. This gives 30 optimum MP vectors indicating 30 possible combinations of PQM buses that can be used to estimate the unmonitored bus voltages if a fault occurs at a particular bus. The selected optimal GA parameters through experimentation get the best tradeoff between the accuracy and speed of the optimization algorithm. Therefore, in this algorithm the generation size and execution number are selected 50 and 30 respectively. The whole GA optimization procedure for determining the optimum MP vectors is illustrated in Fig. 3. 4.2. Use of Cp to find the best MP vector from the optimum selections Variable selection techniques are important in statistical modeling because they seek to simultaneously reduce the chances of data over fitting and to minimize the effects of omission bias. For regression models, variable selection is particularly useful because of its association with certain optimality criterions. One of these is the Mallow’s Cp which evaluates the fit of a regression model by the squared distance between its predictions and the true values [18]. Therefore, according to the concept of Mallow’s Cp and Fig. 1, if there are many optimum selections with similar PQM buses (independent variables), one can look into the calculated Mallow’s Cp values of MP vectors. The MP vector with the lowest Cp value close to P is considered as the most adequate MP vector representing the best PQM placement. The following subsections explains the procedure to select the best MP vector from the 30 optimum solutions obtained from GA for specific bus and selecting the common MP vector for the whole system. 4.3. Use of Cp for optimum selection from GA solutions Due to random nature of GA optimization for various execution of GA, the final solutions are not always the same. Therefore, as explained in Section 4.1, 30 MP vectors for individual bus are calculated for 30 runs. The procedures to obtain a suitable MP vector from the 30 GA solutions are described as follows: Step 1. Arrange 30 optimum MP vectors as described in Section 4.1 in ascending order according to the number of required PQM buses. Step 2. Establish groups of MP vectors from step 1 with the same optimal number of PQMs. Step 3. Select the groups of MP vectors that give maximum number of frequency of occurrence (out of 30). In this case, the frequency of occurrence refers to the number of times recorded for groups of MP vectors with similar optimal number of PQMs.

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Create 30 optimum MP vectors using GA method for training data set of bus1

Create 30 optimum MP vectors using GA method for training data set of bus 2

Select optimal MP vector according to Mallow ’s Cp for bus1

Create 30 optimum MP vectors using GA method for training data set of bus n

Select optimal MP vector according to Mallow ’ s Cp for bus n

Select optimal MP vector according to Mallow ’s Cp for bus2

Best selected MP vector for bus1

Best selected MP vector for bus2

Best selected MP vector for bus n

Select optimal MP vector according to Mallow ’s Cp for the whole test system

End Fig. 5. Flowchart of the proposed optimum PQM placement using the GACp method.

GACp Method with Redundancy 29

30

MRA Method GACp Method with the lowest Redundancy

28 25

Brown

27

Blue

26

24

23

22 15

14

18

8

21

19

16 20

12 C

G

13

17

C

10 11

4

3

9 6

1

7 2

5 G

C

Fig. 6. One-line diagram of the IEEE 30 bus test system.

Edited by Foxit PDF Editor Copyright (c) by Foxit Software Company, 2004 - 2007 For Evaluation Only. Table 1 Thirty optimal MP vectors obtained by GA using training data set for a fault at bus 12.

Table 4 Optimal MP vectors for faults at all buses in the IEEE 30 bus system.

GA solution

Mp vector

PQM at bus no.

Bus

Mp vector

PQM no.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

100000010011000010100000100100 100100010000110001100000000001 111110001110010011100011011100 001010111010011100000001111111 101000010000010000000100010101 101000110100110000010111001111 010100000100001100010001000001 001100010010000001010011000000 000001010000001010001110100000 101000000000100110000010100010 100100101000000101010001000000 001001000010010010000100011000 000010010110011100000000000001 010000001100100010000000001101 000001011000011101110001010111 000010110000000110000101001000 010111000110111011101101101010 111110101010110000101001011011 010011101110100001101101111001 101000000000101100000100101000 001000110110000001000000010100 000111110100101100001101100110 000101000000000110000010110001 111100101110011100011001110000 110100100110101110100011100100 110100100110101110100011100100 001010111010011100000001111111 000010011100000000010001000101 101000110100001111111010001110 111011010000001111111010001110

8 8 17 16 8 15 8 8 8 8 8 8 8 8 14 8 18 17 27 8 8 15 8 16 15 17 16 8 15 17

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

000010001110000000000011000101 100011001000100001001000010000 010000011000111111010100111111 111000111100010010000100100101 011001010111101000110011100111 000000000000011000001110000111 010010010101010000000010001000 100001100000100110000100010000 000000010001001000101100010001 100101011010100100001011111000 000000010001111000000101000001 001000110110000001000000010100 001000100101000010010001000001 011100010000100000001010100000 000000100010000000101101101000 111011000001111000101101011110 000010010100100001001000100010 100100011100110000000000000100 000001000000001001000100111010 101011100110111100001101111100 101010010110010111100001011010 001100110001110010000000000000 011000000001011100000000010001 101110100000100100111110100100 000000100000011000010110010001 110100011011011000101101100101 100101000110101001111100100110 010000010000100110010000010010 010111011000001001010010110101 001010010011100011101100111110

8 8 17 13 17 8 8 8 8 14 8 8 8 8 8 17 8 8 8 18 15 8 8 14 8 16 15 8 14 16

Table 2 Maximum number of occurrences of GA solutions for bus 12 training data. The group with no. of PQMs No. of occurrence

8 16

14 1

15 4

16 3

17 5

18 1

Table 3 Example of calculated Cp and selection of optimal MP vector with maximum Cp close to P for a fault observed at bus 12. GA solution

Cp

GA solution

Cp

1 2 5 7 8 9 10 11

1.06E06 5.16E05 4.64E06 1.90E07 2.15E05 2.13E05 1.17E06 6.15E05

12 13 14 16 20 21 23 28

0.00036 8.88E06 1.98E05 3.27E05 6.68E06 6.35E05 9.26E07 3.49E06

Step 4. Find groups with similar number of frequency of occurrence from step 3. If there is more than one group with the same frequency of occurrence, go to step 5, else go to step 6. Step 5. Select the group of MP vectors with minimum number of PQMs. Step 6. Calculate the Cp values for the MP vector group described in step 3 or 5. Step 7. Select the best MP vector which gives the maximum Mallow’s Cp close to P. The whole procedure to select the best MP vector for a particular bus using Mallow’s Cp is illustrated in Fig. 4. This algorithm is also used for selecting the best MP vectors for other buses. 4.4. Use of Cp to for optimum selection from best MP vectors The previous section shows how to obtain N best MP vectors for a N bus system. However, it still requires finding a common MP vector that is suitable for the whole system. One possible way to

Table 5 The maximum number of PQM occurrences in the IEEE 30 bus test system. The group with no. of PQMs No. of occurrence

8 18

13 1

14 3

15 2

16 2

17 3

18 1

Table 6 Cp values of all 18 combinations with 8 PQMs. Run/itr. no. 1 2 6 7 8 9 11 12 13

Cp

Run/itr. no.

Cp

9.6E08 1.0E04 1.7E02 1.3E04 7.2E05

14 15 17 18 19 22

1.0E02 1.5E05 7.8E08 6.9E06 9.0E05 1.1E04

23 25 28

5.6E06 7.7E09 2.0E04

2.1E02 2.3E03 6.4E05 1.1E03

obtain the common MP vector is again to use Mallow’s Cp considering N MP vectors as described in Fig. 4. The overall procedure is shown in Fig. 5. 5. Observability and redundancy after PQM placement The optimum PQM allocation model determines the positions where PQMs should be installed to maximize the monitored area over a power system under investigation. A monitor should be installed on a bus that would allow it to observe the largest possible number of voltage sags that may occur in a power network. This observation characteristic of an installed monitor at a certain bus within a power network defines the observability concept used in the current study [8]. Moreover, the determination of the minimum number of PQMs aims to establish the number of PQMs required monitoring the entire power network with the lowest possible redundancy. In other words, each possible voltage sag that

Edited by Foxit PDF Editor Copyright (c) by Foxit Software Company, 2004 - 2007 For Evaluation Only. may occur in the power network should be observed by at least one of the installed PQMs. However, the placement of monitors in a power system results in different overlaps in the monitors’ coverage areas for different arrangements. These overlaps indicate the number of monitors that record the same fault occurrence in a power system, and thus, have to be minimized [20]. The MRA introduced in Olguin and Bollen [9] is used to check for observability and redundancy. The MRA matrix for each simulated fault is defined by the value of the fault voltages at each bus. Each row in such a matrix relates to a specific fault in a power network and each column refers to a specific bus in a power network. Thus, a voltage value is stored in each matrix position related to the bus (column) and simulated fault (row). The procedure to compose the MRA matrix can be expressed mathematically as,

MRA ¼



1 if V 6 threshold voltage 0

otherwise

ð14Þ

The observability can be calculated by multiplying the monitor reach area (MRA) matrix with the transposed MP vector as follows:

Observability ¼ MRA  MPT

ð15Þ

The observability information in (15) can be used to further reduce the number of PQMs in a power system. It is important to note that the multiplication of the MRA matrix by the transposed MP vector gives the number of monitors that can detect sags due to fault at specific bus. If one of the resulting matrix element is 0 (zero) then it means no monitor is capable to detect sag caused by the fault at the particular bus, whereas if the value is greater than 1 (one), that means more than one monitor observed the fault at the same bus [13]. 6. Test system The IEEE 30 bus system shown in Fig. 6 is used to evaluate the performance of the proposed methodology. In the figure, bus 1 is the slack bus, bus 2 is the voltage controlled bus, buses 5, 11 and 13 are installed with synchronous condenser, and the remaining 24 buses are loads. In addition, the test system has three different voltage levels, including 11 kV at buses 11 and 13, 132 kV at buses 1–9, and 28, and 33 kV at the remaining buses. The placement of the PQMs is verified by performing power flow simulations using the DIGSILENT Power Factory 14.0.516, and all the required statistical indices and GA programs are calculated using the MATLAB software. Several short-circuit tests, including single-phase-toground (LG), double-phase-to-ground (LLG) and three-phase (LLL) faults are required to determine the relationship between the estimated and the monitored bus voltages. In addition, GA is programmed considering 50 generations, 30 individuals, 80% crossover, and 1% mutation. 6.1. Determine optimum MP vectors using GA optimization procedures for an individual bus According to the proposed methodology, the fitness function and the constraints related to the problem, as well as the bus faults data are considered in the GA program. The GA optimization procedure is repeated for 30 times to avoid the solutions that converge in the local minimum. Due to GA random nature, the final solutions are not always the same in GA optimization at various executions For example, in the IEEE 9 bus system, the 30 MP vectors suggested by GA to observe the whole system when a fault occurs at bus 12, is shown in Table 1. From the table, the ‘0’ values in the MP vectors indicate that no monitor should be installed at the respective buses, whereas the values of ‘1’ indicate that a monitor should be installed at the respective buses. As shown in Table 1, since the

GA procedures do not always provide the same solution, it is necessary to adopt Mallow’s Cp criterion to determine which GA solution, out of the 30 solutions gives the most appropriate MP vector for PQM placement. The use of Mallow’s Cp to obtain the MP vector for each bus is given next. 6.2. Determination of optimum MP vector using Mallow’s Cp from GA selected solutions for a bus To determine the optimum MP vector, initially it is required to group the 30 GA solutions by arranging them in ascending orders based on the number of PQMs and finding the PQM groups with highest frequency of occurrence. Table 2 shows the MP vector groups with similar optimal number of PQMs when a fault occurs at bus 12. From the table, it is noted that the MP vector group with 8 PQMs occurs sixteen times out of the 30 GA solutions. This means that GA suggests 8 PQM as optimum PQM number irrespective of their positions in the IEEE 30 bus system. Once the number of PQMs is determined, the next step is to obtain the best MP vector out of the selected group; in this case it is out of the 16 GA solutions. Thus, to find the best PQM locations, Mallow’s Cp value is calculated for all the selected MP vectors as shown in Table 3 and the MP vector with maximum Cp close to P is chosen as the optimal MP vector to observe the whole system for a fault at bus 12. In this case, the GA solution number 21 is considered as optimal with MP vector equals to 001000110110000001000000010100. For this GA solution, the obtained Cp value is 6.35E05 which is the closest to the P value of 8. To obtain optimum MP vectors for faults occurring at other buses in the system, similar procedure is repeated using the training data for faults at other buses in the IEEE 30 bus system. Table 4 shows the obtained optimum MP vectors considering faults occurring at all the buses in the test system. However, repeating this procedure gives a total of 30 optimum MP vectors corresponding to each bus in the IEEE 30 bus system. Therefore, it is necessary to find a common optimum MP vector that is suitable to observe the fault occurrences in any bus in the system as exhibited in the next section. 6.3. Selecting the best optimal MP vector using Mallow’s Cp The common MP vector is obtained by using the Mallow’s Cp and the results are shown in Table 5. From the table, for all the bus faults in the IEEE 30 bus system it can be observed that the number of 8 PQMs gives the highest number of occurrences (18). The Mallow’s Cp values for all the 18 MP vectors with 8 PQMs

Table 7 Best optimal MP vector observed in the IEEE 30 bus system. Bus

PQMs

Bus

PQMs

1 2 3 4

0 0 0 0

16 17 18 19

0 0 0

5 6

0 0

20 21

7

0

22

8

1 0 0 0

23

1 0 0

27

9 10 11 12 13 14 15

1

24 25 26 28 29 30

1 0 1 1 0 0 0 1 0 0 0 1

Table 8 The fault matrix of the IEEE 30 bus for single phase to ground fault. 1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

0.17 0.17 0.04 0.04 0.05 0.04 0.05 0.04 0.07 0.83 0.98 0.81 0.96 0.84 0.82 0.84 0.84 0.85 0.86 0.85 0.84 0.84 0.85 0.85 0.89 0.93 0.91 0.06 0.93 0.93

0.17 0.17 0.04 0.04 0.05 0.04 0.05 0.04 0.07 0.83 0.98 0.81 0.96 0.84 0.82 0.84 0.84 0.85 0.86 0.85 0.84 0.84 0.85 0.85 0.89 0.93 0.91 0.06 0.93 0.93

0.17 0.17 0.03 0.03 0.05 0.04 0.04 0.04 0.07 0.83 0.98 0.81 0.96 0.84 0.82 0.84 0.84 0.85 0.85 0.85 0.84 0.84 0.85 0.84 0.89 0.93 0.91 0.05 0.93 0.93

0.17 0.17 0.04 0.03 0.05 0.04 0.04 0.04 0.07 0.83 0.98 0.81 0.96 0.84 0.82 0.84 0.84 0.85 0.85 0.85 0.84 0.84 0.85 0.84 0.89 0.93 0.91 0.05 0.93 0.93

0.17 0.17 0.05 0.04 0.03 0.04 0.04 0.04 0.07 0.83 0.98 0.81 0.96 0.84 0.82 0.84 0.84 0.85 0.86 0.85 0.84 0.84 0.85 0.85 0.89 0.93 0.91 0.06 0.93 0.93

0.17 0.17 0.04 0.04 0.05 0.03 0.04 0.04 0.06 0.83 0.98 0.81 0.96 0.84 0.82 0.84 0.84 0.85 0.85 0.85 0.84 0.84 0.85 0.84 0.89 0.93 0.91 0.05 0.93 0.93

0.17 0.17 0.04 0.04 0.04 0.04 0.03 0.04 0.07 0.83 0.98 0.81 0.96 0.84 0.82 0.84 0.84 0.85 0.85 0.85 0.84 0.84 0.85 0.85 0.89 0.93 0.91 0.05 0.93 0.93

0.17 0.17 0.04 0.04 0.05 0.04 0.04 0.03 0.07 0.83 0.98 0.81 0.96 0.84 0.82 0.84 0.84 0.85 0.85 0.85 0.84 0.84 0.85 0.84 0.89 0.93 0.91 0.05 0.93 0.93

0.17 0.17 0.04 0.04 0.05 0.03 0.04 0.04 0.03 0.83 0.98 0.81 0.96 0.84 0.82 0.84 0.84 0.85 0.85 0.85 0.84 0.84 0.85 0.84 0.89 0.93 0.91 0.05 0.93 0.93

0.73 0.73 0.62 0.62 0.63 0.62 0.63 0.62 0.61 0.17 0.94 0.26 0.91 0.38 0.27 0.26 0.20 0.31 0.29 0.27 0.20 0.20 0.33 0.29 0.50 0.66 0.57 0.63 0.66 0.66

0.83 0.83 0.77 0.77 0.77 0.77 0.77 0.77 0.77 0.92 0.73 0.90 1.06 0.94 0.91 0.93 0.93 0.94 0.94 0.94 0.93 0.93 0.94 0.94 0.99 1.02 1.00 0.78 1.02 1.02

0.73 0.73 0.63 0.63 0.63 0.63 0.63 0.63 0.62 0.35 0.94 0.19 0.91 0.33 0.26 0.31 0.35 0.37 0.40 0.40 0.38 0.38 0.37 0.41 0.59 0.72 0.66 0.64 0.73 0.73

0.80 0.80 0.73 0.73 0.74 0.73 0.74 0.74 0.74 0.85 1.06 0.79 0.81 0.86 0.82 0.84 0.85 0.87 0.88 0.87 0.86 0.86 0.87 0.87 0.93 0.99 0.96 0.74 0.99 0.98

0.73 0.73 0.63 0.63 0.63 0.63 0.63 0.63 0.62 0.33 0.94 0.20 0.91 0.16 0.22 0.30 0.33 0.34 0.36 0.37 0.35 0.35 0.33 0.38 0.57 0.71 0.64 0.63 0.72 0.71

0.73 0.73 0.63 0.63 0.63 0.63 0.63 0.63 0.62 0.30 0.94 0.21 0.91 0.30 0.18 0.30 0.32 0.29 0.33 0.33 0.33 0.33 0.29 0.35 0.54 0.69 0.62 0.63 0.70 0.69

0.73 0.73 0.63 0.63 0.63 0.63 0.63 0.63 0.62 0.26 0.94 0.22 0.91 0.35 0.27 0.17 0.24 0.35 0.35 0.34 0.30 0.30 0.35 0.36 0.55 0.69 0.62 0.63 0.70 0.70

0.73 0.73 0.62 0.62 0.63 0.62 0.63 0.62 0.61 0.19 0.94 0.25 0.91 0.37 0.27 0.23 0.16 0.32 0.31 0.29 0.22 0.23 0.33 0.31 0.51 0.67 0.59 0.63 0.68 0.67

0.73 0.73 0.62 0.62 0.63 0.62 0.63 0.62 0.61 0.25 0.94 0.22 0.91 0.32 0.20 0.29 0.27 0.15 0.20 0.23 0.28 0.28 0.30 0.33 0.53 0.68 0.60 0.63 0.69 0.68

0.73 0.73 0.62 0.62 0.63 0.62 0.62 0.62 0.61 0.22 0.94 0.24 0.91 0.34 0.22 0.28 0.25 0.19 0.15 0.17 0.25 0.25 0.31 0.31 0.52 0.67 0.59 0.63 0.68 0.67

0.73 0.73 0.62 0.62 0.63 0.62 0.62 0.62 0.61 0.20 0.94 0.24 0.91 0.35 0.24 0.27 0.24 0.22 0.18 0.15 0.24 0.24 0.31 0.31 0.51 0.67 0.59 0.63 0.68 0.67

0.73 0.73 0.62 0.62 0.63 0.62 0.62 0.62 0.61 0.18 0.94 0.26 0.91 0.38 0.27 0.27 0.21 0.31 0.30 0.28 0.16 0.17 0.31 0.26 0.47 0.64 0.55 0.63 0.65 0.64

0.73 0.73 0.62 0.62 0.63 0.62 0.62 0.62 0.61 0.18 0.94 0.26 0.91 0.38 0.27 0.27 0.21 0.31 0.30 0.28 0.17 0.16 0.31 0.26 0.47 0.63 0.55 0.63 0.64 0.64

0.73 0.73 0.62 0.62 0.63 0.62 0.63 0.62 0.61 0.27 0.94 0.23 0.91 0.33 0.20 0.30 0.29 0.31 0.33 0.32 0.28 0.28 0.15 0.25 0.47 0.63 0.55 0.63 0.65 0.64

0.73 0.73 0.62 0.62 0.62 0.62 0.62 0.62 0.61 0.23 0.94 0.26 0.91 0.37 0.25 0.30 0.26 0.33 0.33 0.31 0.23 0.22 0.24 0.16 0.37 0.57 0.47 0.63 0.58 0.57

0.73 0.73 0.62 0.62 0.62 0.62 0.62 0.62 0.61 0.27 0.95 0.31 0.92 0.42 0.30 0.35 0.31 0.37 0.37 0.36 0.27 0.26 0.29 0.19 0.12 0.36 0.23 0.63 0.38 0.37

0.73 0.73 0.62 0.62 0.62 0.62 0.62 0.62 0.61 0.27 0.95 0.31 0.92 0.42 0.30 0.35 0.31 0.37 0.37 0.36 0.27 0.26 0.29 0.19 0.12 0.08 0.23 0.63 0.38 0.37

0.73 0.73 0.62 0.62 0.62 0.62 0.62 0.62 0.61 0.30 0.95 0.34 0.92 0.45 0.33 0.38 0.34 0.40 0.40 0.39 0.30 0.29 0.33 0.22 0.14 0.39 0.10 0.63 0.26 0.24

0.17 0.17 0.04 0.04 0.05 0.04 0.04 0.04 0.07 0.83 0.98 0.81 0.96 0.84 0.82 0.84 0.84 0.85 0.85 0.85 0.84 0.84 0.85 0.84 0.89 0.93 0.91 0.03 0.93 0.93

0.73 0.73 0.62 0.62 0.62 0.62 0.62 0.62 0.61 0.30 0.95 0.34 0.92 0.44 0.33 0.38 0.34 0.40 0.40 0.39 0.30 0.29 0.33 0.22 0.14 0.39 0.10 0.63 0.08 0.16

0.73 0.73 0.62 0.62 0.62 0.62 0.62 0.62 0.61 0.30 0.95 0.34 0.92 0.44 0.33 0.38 0.34 0.40 0.40 0.39 0.30 0.29 0.32 0.22 0.14 0.39 0.10 0.63 0.18 0.08

Edited by Foxit PDF Editor Copyright (c) by Foxit Software Company, 2004 - 2007 For Evaluation Only.

S.Cnbus

Edited by Foxit PDF Editor Copyright (c) by Foxit Software Company, 2004 - 2007 For Evaluation Only. Table 9 The MRA matrix of the IEEE 30 bus system for single phase to ground fault. S.Cnbus

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0

1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0

1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0

1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0

1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0

1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0

1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0

1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0

1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 0 0 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0

0 0 0 0 0 0 0 0 0 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0

0 0 0 0 0 0 0 0 0 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0

0 0 0 0 0 0 0 0 0 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0

0 0 0 0 0 0 0 0 0 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0

0 0 0 0 0 0 0 0 0 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0

0 0 0 0 0 0 0 0 0 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0

0 0 0 0 0 0 0 0 0 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 0

0 0 0 0 0 0 0 0 0 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0

0 0 0 0 0 0 0 0 0 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 0

0 0 0 0 0 0 0 0 0 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1

0 0 0 0 0 0 0 0 0 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1

0 0 0 0 0 0 0 0 0 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1

1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 0 0 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1

0 0 0 0 0 0 0 0 0 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1

Table 10 Observability of the single phase to ground fault. PQMs/bus

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

8 12 15 19 21 22 26 30

1 0 0 0 0 0 0 0

1 0 0 0 0 0 0 0

1 0 0 0 0 0 0 0

1 0 0 0 0 0 0 0

1 0 0 0 0 0 0 0

1 0 0 0 0 0 0 0

1 0 0 0 0 0 0 0

1 0 0 0 0 0 0 0

1 0 0 0 0 0 0 0

0 1 1 1 1 1 1 1

0 0 0 0 0 0 0 0

0 1 1 1 1 1 1 1

0 0 0 0 0 0 0 0

0 1 1 1 1 1 1 1

0 1 1 1 1 1 1 1

0 1 1 1 1 1 1 1

0 1 1 1 1 1 1 1

0 1 1 1 1 1 1 1

0 1 1 1 1 1 1 1

0 1 1 1 1 1 1 1

0 1 1 1 1 1 1 1

0 1 1 1 1 1 1 1

0 1 1 1 1 1 1 1

0 1 1 1 1 1 1 1

0 0 1 1 1 1 1 1

0 0 0 0 0 0 1 1

0 0 0 0 0 1 1 1

1 0 0 0 0 0 0 0

0 0 0 0 0 0 1 1

0 0 0 0 0 0 1 1

Table 11 Observability of double phase to ground fault. PQMs/bus

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

8 12 15 19 21 22 26 30

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

0 1 1 1 1 1 1 1

0 0 0 0 0 0 0 0

0 1 1 1 1 1 1 1

0 0 0 0 0 0 0 0

0 1 1 1 1 1 1 1

0 1 1 1 1 1 1 1

0 1 1 1 1 1 1 1

0 1 1 1 1 1 1 1

0 1 1 1 1 1 1 1

0 1 1 1 1 1 1 1

0 1 1 1 1 1 1 1

0 1 1 1 1 1 1 1

0 1 1 1 1 1 1 1

0 1 1 1 1 1 1 1

0 1 1 1 1 1 1 1

0 1 1 1 1 1 1 1

0 0 0 1 1 1 1 1

0 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

0 0 0 0 1 1 1 1

0 0 0 0 0 0 1 1

Table 12 Observability of unbalanced three phase faults. PQMs/bus

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

8 12 15 19 21 22 26 30

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

0 1 1 1 1 1 1 1

0 0 0 0 0 0 0 0

0 1 1 1 1 1 1 1

0 0 0 0 0 0 0 0

0 1 1 1 1 1 1 1

0 1 1 1 1 1 1 1

0 1 1 1 1 1 1 1

0 1 1 1 1 1 1 1

0 1 1 1 1 1 1 1

0 1 1 1 1 1 1 1

0 1 1 1 1 1 1 1

0 1 1 1 1 1 1 1

0 1 1 1 1 1 1 1

0 1 1 1 1 1 1 1

0 1 1 1 1 1 1 1

0 1 1 1 1 1 1 1

0 0 0 0 0 1 1 1

0 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

0 0 0 0 0 1 1 1

0 0 0 0 0 0 1 1

Edited by Foxit PDF Editor Copyright (c) by Foxit Software Company, 2004 - 2007 For Evaluation Only. Table 13 Comparison between the GACp and the MRA methods.

Transmission and radial systems Balanced and unbalanced systems Threshold voltage (a) Optimal number of PQMs

7. Comparison of PQM placement methods

GACp method

MRA method

U

U

U

U

 Independent of

U Depend on a

a Placement of PQMs

8 and 26

Observability for a = 0.55

2 PQMs

4, 7, 11, 15, 17, 20, 26, 29 8 PQMs

are then calculated and tabulated as shown in Table 6. From the table, the MP vector with the highest Cp value is selected. In this case, the MP vector equals to 000000010001001000101100010001 gives the highest Cp value. Thus, for the IEEE 30 bus test system, to observe voltage sag requires 8 PQMs which should be located at buses 8, 12, 15, 19, 21, 22, 26, and 30 as shown in Fig. 6 and Table 7.

6.4. Observability and redundancy after the best optimal MP vector The MRA introduced in [1,21,22] is used to check for observability and redundancy. The MRA matrix for each simulated fault is defined by the value of the fault voltages in each bus. Each row in such matrix relates to a specific fault in the power network and each column refers to a specific power network bus. Thus, a voltage value is stored in each matrix position related to the bus (column) and simulated fault (row). Table 8 shows the fault matrix of the IEEE 30 bus system for single phase to ground fault, which is used to obtain the MRA. With regard to Eq. (14), each element of the fault matrix is compared to a fixed value (the threshold voltage) to obtain the MRA and trigger the PQMs to store the waveforms and characteristics of the voltage sags. In this study, the threshold voltage was adjusted to 0.55 pu, and the MRA matrix elements were obtained. Therefore, each element of the MRA was filled with zero when the fault voltage was higher than the threshold voltage and is filled with one, otherwise. Table 9 shows the MRA matrix of the IEEE 30 bus for single phase to ground fault. Eq. (15) is used to check observability and redundancy and need to the MP vector. Therefore, for the IEEE 30 bus test system, the meter placement (MP) vector is defined in Table 10 that buses 8, 12, 15, 19, 21, 22, 26, and 30 are the optimal PQM locations and are represented by the values of one. Then, according to procedure expressed in Section 5, Eq. (15) is implemented for IEEE 30 bus. The results of the observability of the PQMs at buses 8, 12, 15, 19, 21, 22, 26, and 30 in the IEEE 30 bus test system with single phase to ground (LG), double phase to ground (LLG), and unbalanced three phase faults (LLL) are shown in Tables 10–12 respectively. The results of Tables 10–12 are combined to show the redundancy in the test system, in which the PQM at bus 8 can observe buses 1–9 and 28. Moreover, the PQMs at bus 26 can observe buses 10, 12, 29, 30, and 14–27. In Fig. 6, the positions of the optimal installation sites of the PQMs and observability area are represented with brown1 and blue contours, respectively. Therefore, the proposed methodology can completely observe the IEEE 30 bus system with two PQMs when the observability is checked with 0.55 pu as the PQM threshold value.

The proposed GACp method is compared with the MRA method in Ibrahim et al. [13] which is based on the MRA and particle swarm optimization (PSO) and is validated on the IEEE 30 bus test system. Using the same threshold voltage value, a, of 0.55 pu, the MRA method identified 8 PQMs as the optimal number of PQMs. These PQMs are suggested to be placed at buses 4, 7, 11, 15, 17, 20, 26, and 29, as shown in the blue square dots in Fig. 6. Based on the results shown in Table 13, the proposed GACp method identified only 2 PQMs to observe all the LG, LLG, and LLL faults in the test system. The GACp method does not need the threshold voltage during the optimization process. Moreover, redundancy can be reduced using the observability test. For example, redundancy check with threshold voltage value equal to 0.55 pu suggests that only two PQMs (at buses 8 and 26) which can be used to observe all faults (LG, LLG, and LLL) in the test system. 8. Conclusion This study presented a new method based on GA and Mallow’s Cp for PQM placement. In the proposed method, the optimal number and locations of PQMs are obtained using GA. After obtaining the optimal placements of PQMs by using the GACp method, the observability and redundancy of the monitors were checked to further reduce the redundant PQMs. The proposed method was tested on the IEEE 30 bus test system for validation and comparison. The results show that the proposed method do not need the threshold voltage in the optimization process and requires only two PQMs to observe all faults (LG, LLG and LLL) in the test system. References [1] IEEE Std.1159. In: IEEE recommended practice for monitoring electric power quality. New York: IEEE Press; 1995. [2] Xuemeng YY, Yonghai X, Danyue LY. Analysis and calculation on indices of voltage sag. IEEE Power Energy Eng 2009:1–5. [3] Ghosh A, Lubkeman D. The classification of power system disturbance waveforms using a neural network approach. IEEE Trans Power Delivery 1995;10:109–15. [4] Eldery MA, El-Saadany EF, Salama MMA, Vannelli A. A novel power quality monitoring allocation algorithm. IEEE Trans Power Delivery 2006;21:768–77. [5] Hurtgen M, Maun JC. Optimal PMU placement using iterated local search. Int J Electr Power Energy Syst 2010;32:857–60. [6] Eslami M, Shareef H, Mohamed A, Khajehzadeh M. An efficient particle swarm optimization technique with chaotic sequence for optimal tuning and placement of PSS in power systems. Int J Electr Power Energy Syst 2012;43:1467–78. [7] Sirjani R, Mohamed A, Shareef H. Optimal allocation of shunt Var compensators in power systems using. Int J Electr Power Energy Syst 2012;43:562–72. [8] Almeida CFM, Kagan N. Allocation of power quality monitors by genetic algorithms and fuzzy sets theory. In: International intelligent system applications to power systems, Sao Paulo; 2009. p. 1–6. [9] Olguin G, Bollen MHJ. Optimal dips monitoring program for characterization of transmission system. In: IEEE power engineering society general meeting, Toronto, Canada; 2003. p. 2484–90. [10] Mazlumi K, Askarian Abyaneh H, Gerivani Y, Rahimi Pordanjani I. A new optimal meter placement method for obtaining a transmission system indices. In: IEEE power tech, Lausanne; 2007. p. 1165–9. [11] Olguin G, Vuinovich F, Bollen MHJ. An optimal monitoring program for obtaining voltage sag system indexes. IEEE Trans Power Delivery 2006;21:378–84. [12] Haghbin M, Farjah E. Optimal placement of monitors in transmission systems using fuzzy boundaries for voltage sag assessment. In: IEEE power tech, Bucharest, Romania; 2009. p. 1–6. [13] Ibrahim AA, Mohamed A, Shareef H. Optimal placement of voltage sag monitors based on monitor reach area and sag severity index. In: IEEE student conference on research and development (SCOReD), Putrajaya, Malaysia; 2010. p. 467–70. [14] Won D-J, Moon S-I. Optimal number and location of power quality monitors considering system topology. IEEE Trans Power Delivery 2008;23:288–95. [15] Ibrahim AA, Mohamed A, Shareef H. Optimal power quality monitor placement in power systems based on particle swarm optimization and artificial immune system. In: IEEE data mining and optimization (DMO), Selangor, Malaysia; 2011. p. 141–5.

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