Coherency Detection and Network Partitioning Supported by Wide Area Measurement System Mohammad Hossein Rezaeian Koochi 1, Saeid Esmaeili 1, and Pooria Dehghanian2 2

1 Department of Electrical Engineering, Shahid Bahonar University of Kerman, Kerman, Iran Department of Electrical and Computer Engineering, Texas A&M University, College Station, Texas, USA [email protected]; [email protected]; [email protected]

Abstract— Due to the non-linearity and complexity of power systems, reduced model of these systems is used for dynamic and transient analysis. In this regard, a power system is partitioned into areas based on the similarity of dynamic responses of generators to the disturbances, which is called coherency. In this paper, a new method is presented for determining the degree of coherency between generators or non-generating buses using the data measured by wide area measurement system (WAMS). This method uses the amplitude as well as phase angles of frequency components that exist in the angular velocity signals after the disturbance has occurred. Those frequencies that are in the range of inter-area oscillation modes are selected to determine the coherency between each pair of generators using a correlation index. By expanding this index to the non-generating buses of the power system and using the tight coherency concept, the power system is partitioned into highly coherent areas. The effectiveness of this method for area detection is investigated on the 16machine, 68-bus test system. It is observed that accurate coherent areas can be identified using both the phase angle and amplitude of frequencies in the range of inter-area oscillation modes. Keywords— coherency; frequency component; network partitioning; phasor measurement unit; wide area measurement system

I.

INTRODUCTION

in power system dynamic is becoming more and more challenging due to the increasing number of complexity of the power and energy delivery systems. In emergency scenarios, portions of the electric power system would be adversely affected with a compromised electrical safety, exposing the network to risk under an unstable condition with some changes in the equilibrium point of the system. Thus, the importance of real-time monitoring and control of the power system infrastructure is undoubtable. This complexity and the nonlinearities that exist in power systems have made it difficult to perform those critical analyses that are essential for power system stability and dynamics [1], [2]. Since performing such analysis on a bulk power system is very time consuming, an efficient partition algorithm i.e. the reduced equivalent model of the power system is essential to overcome the communication complexity and reduce the computation. In so doing, the power system is separated into areas based on its dynamic response to a disturbance and then, each area is replaced with a reduced equivalent model.

C

OHERENCY analysis

The main steps of power system reduction methods are i) identification of groups of generators and buses to be reduced and, ii) aggregation and reduction of these groups. Among power system reduction methods, coherence-based approaches are mostly used in studies since they retain the dynamic characteristics of generators in the aggregation step. A group of generators are said to be coherent for a given disturbance if they have the same rotor angle or speed variations. Based on this concept, in the area identification step, the main idea is to find those buses with the same parameters’ variation (e.g. phase angle variations). Coherency detection methods can be 978-1-5386-1006-0/18/$31.00 ©2018 IEEE

categorized into two groups. The first group which is called model-based methods uses the eigenvalue and eigenvector of the power grid for coherency computation [3]-[8]. In the modal synchrony methods, a linearized model analysis of power system is applied to identify the coherent generators [8]. The methods employed in the second group are based on the disturbances and use time-domain simulations to find coherent clusters. Several studies have used Fourier spectrum [9], [10], principal component analysis [11], independent component analysis [12], hierarchical clustering methods [13], [14], subtractive clustering [15], Fuzzy c-medoids algorithm [16], wavelet [17], Hilbert-Huang transform [18] and decision trees [19] to find coherent generators. The instances of blackouts in recent years which resulted in major electricity outages proof that supervisory control and data acquisition system cannot provide the requirements for maintaining the security and resilience of bulk power systems [20], [21]. Fault diagnoses in power system infrastructure is a challenging task. Coherency analysis can be integrated with the fault detection algorithm to further expedite the decisionmaking by the operators in cases of emergencies [22]-[24]. Therefore, it is essential to go beyond traditional systems and use modern measurement systems with higher capabilities. In this regard, today, synchronized measurement technology (SMT) is used in the wide-area measurement systems (WAMS) for controlling and protecting power systems [25]. Phasor measurement units (PMUs) as one of the key elements of SMT are aimed to estimate and synchronize the voltage and current phasors. In the literature, there are several examples of methods that have used the data measured by PMUs to find coherent clusters, especially coherent generators [9]-[19]. Amongst, a Fuzzy c-medoids algorithm is implemented in [16] to find coherent areas with respect to the dissimilarity between the angular velocity deviation of voltage phasor at buses. However, in this method, it is needed to predefine the number of areas. Another clustering method is presented in [13] in which an agglomerative hierarchical clustering method is used to find coherent generators. Applying this method to area detection problem needs a modification since it is necessary to consider the connectivity of buses for calculating the dissimilarities. In this paper, a coherent area detection method is presented using the data measured by WAMS. In this method, the coherency between each pair of buses is assessed through calculating the correlation between the frequency components exist in angular velocity signals. To do that, the frequencies are extracted using Discrete Fourier Transform (DFT) and then, the frequencies that are in the range of inter-area oscillation modes are selected. It should be noted that in the method for calculating the correlation between generators, not only the amplitude, but also the phase angle of these frequencies is applied, which causes the proposed coherency detection method to find the degree of coherency accurately. Furthermore, this method is applied to both generating and non-generating buses of the power system to find the borders

of coherent areas, which is addressed in few research works. This is important for such model reduction applications such as dynamic analysis of power system and controlled islanding. II.

DYNAMIC COHERENCY DETECTION

In the occurrence of a disturbance, the generators’ rotor angles vary from their equilibrium points which actually represent the dynamic response of each generator to the disturbance. As a result of these variations, the phase angle of buses also varies. Therefore, the effect of the dynamic response of each generator can be determined by phase angles variation at the buses near to the generator. Considering și and șj as the phase angles at bus i and j respectively, one approach for evaluating the coherency between the buses is to measure the dissimilarity between the phase angle variations as in (1). T

si , j = ³ ( Δθ i ( t ) − Δθ j ( t ) )

(1)

0

where si , j denotes the dissimilarity index between bus i and j, and T is the observing time. The energy interaction between generators can be reflected by their speed deviations [10]. In the other words, the amount of energy absorbed or delivered by generators in the power grid is reflected by the generators’ speed variation or the rate of phase angles variation at buses. However, using the variation rate of phase angle or angular velocity as indices for representing dissimilarity is not appropriate. Analyzing the frequency components of the angular velocities can be more helpful for coherency identification since these frequencies represent the dynamic response of each generator following a disturbance. These components can be extracted using DFT as follows. N −1

Fi ( f ) = ¦ ωi ( k ) e

§ 2π fk · −¨ j ¸ © N ¹

f = 0,1,..., N − 1

Fig. 1.

Two typical waveforms and the their frequency spectrums.

their variations. For example, Fig. 1 shows two waveforms which don't have the same variation. But, as it can be seen, these waveforms include two frequencies 1 and 0.25 Hz with the same amplitudes. Therefore, if the coherency detection method uses only the amplitude of these frequencies, it will conclude that the generators that have the speed variation as in Fig. 1 are coherent. In this paper, to identify the coherent buses, the correlation matrix is formed as follows. ª c1,1 ! c1, N B º « » % # » C=« # «c » ¬ N B ,1 ! cN B , N B ¼

(2)

k =0

ωi ( k ) =

n

(3)

Δt

avg

ci , j =

T

(4)

where NB is the total number of buses in the power system. It is noteworthy to mention that, the number of frequencies in the vector Fi is equal to the number of samples of waveforms, as DFT transforms the N discrete time samples to the same number of frequency samples. Therefore, in order to obtain more frequency samples, it is essential to sample the angular velocities in higher rates, which is obtained by PMUs. Using DFT, the amplitude and phase angle of each frequency component exists in the waveform of angular velocity is obtained. As a result, the solution of DFT is a complex number for each frequency. In order to assess the coherence between two generators, both amplitude and phase angles of these frequencies should be considered. In fact, it is obvious that the similarity of the amplitudes of frequencies exists in two waveforms does not guarantee the similarity of

i

avg j

j

f =1 n

2 n

(6)

¦(F ( f ) − F ) ¦(F ( f ) − F ) avg

i

where Ȧi(k) denotes the angular velocity of generator i at time instant k, and ǻt is the time interval between two executive samples, which is constant for all simulations. Fi (f) represents the Fourier transform of the angular speed. The vector-space Fi and the NB × N dimension matrix F are then formed as follows.

F = ª¬ F1 ( f ) , ! Fi ( f ) , ! FN B ( f ) º¼

¦ ª¬( F ( f ) − F ) ( F ( f ) − F )º¼ i

θi ( k ) − θi ( k − 1)

(5)

f =1

i

j

avg 2 j

f =1

In (6) ci , j is the correlation coefficient between buses i and j. Fi avg represents the average of frequency components of bus i in the range of inter-area oscillation modes (0.1–1 Hz). n is also the number of frequency components in this range. Since the elements of F are complex numbers, the correlation between two buses will be a complex number too. Therefore, based on (6), two buses are called coherent when the value of (6) equals to 1 ∠ 0 (or 1 + j0). In the other words, the absolute value of real and imaginary parts of the correlation between two coherent buses should be 1 and 0 respectively.

III.

COHERENCE-BASED NETWORK PARTITIONING

Based on the concept of tight coherency, the phase angle of all buses in an area should have relatively the same deviation. This can be assessed by calculating the correlation between each pair of buses in the area using (6). As described in section II, the highest degree of coherency happens when the correlation between two buses becomes 1 + j 0. Therefore, for an area, the Euclidean distance between the correlation of each pair of buses located in the area and the point 1 + j 0 in the complex plane should be less than a predefined value of Į, as in (7).

Imaginary Cj,k Į

Ci,m

Ci,j

1

Real Ci,k

Fig. 2. Definition of coherency in the complex plane.

ci , j − 1 ≤ α

∀i , j ∈ S

(7) Fig. 3. The amplitude of frequency components of angular velocities (case 1)

where S is the set of buses located in the area. Fig. 2 shows the tight coherency between the buses in a typical area based on the correlations obtained by (6). In Fig. 2, the position of correlations between buses i, j and k are placed in the circle with the center 1 + j 0 and radius Į, which means that these buses are coherent with each other. On the other hand, the position of correlation between bus i and bus m is out of this circle, which means that bus m can’t be placed in the area including buses i, j and k. In order to find coherent areas, first, the correlations between all connected buses should be calculated using (6). To do that, voltage phasors from all over the system is calculated using the data measured by PMUs installed at selected buses of the power system. Today, these devices are used as the main part of wide area measurement system, which is fully installed or is under study in many countries [25], [26]. After calculating the correlation between phasor angles of incident buses, the area detection procedure is triggered. First, the two incident buses that have the higher correlation among all buses are identified. In the next step, the lines that connect these buses to the other buses of the power system are determined. After identifying the buses connected to the initial area, the correlation of each of these buses with all buses in the area is calculated. If the Euclidean distance between all of these correlations and the point 1 + j 0 in the complex plane are less than Į, this bus is coherent with the area and therefore can be added to the area. Otherwise, this bus remains in the rest of power system. This procedure will continue until it is not possible to find any other buses coherent with the first area. When the first area is identified, the procedure for identifying the second area is initiated using the rest of power system. It should be noted that Į should be 0 < Į < 1. It is also clear that choosing the lower value for Į will increase the number of areas, but such areas will include buses that are highly coherent with each other. In fact, the concept of coherency is relative. The lower the value of Euclidean distance between correlation ci , j and the point 1 + j 0, the higher the coherency between buses i and j. Therefore, choosing the appropriate value for Į is completely applicationbased. IV.

A. Case 1

In the first case, a line to line fault is applied on bus 42 at 91 % of maximum loadability of the test power system. The amplitude of frequency components exist in the angular velocity variation of voltage phasors at all buses is shown in Fig. 3. It can be realized that there is a significant difference between the amplitude of variations of buses 15 and 42, and other buses in the system at the frequency of 0.26 Hz. This means that buses 15 and 42 show a different response to the disturbance compared to other buses and should be placed in a distinct area. On the other hand, although, other buses of the system might seem to have the same variations, the degree of coherency between them must be assessed mathematically. To do that, the correlations between all generators as well as buses are calculated using (6). Fig. 4 shows a colorful plot to represent the degree of coherency using the Euclidean distance between all calculated correlations between generators as well as buses and the point 1 + j 0. It is obvious that the lower values of this distance show the more coherency between generators or buses. In Fig. 4 the dark blue points show the high coherency between buses and generators. It can be seen that there is a high coherency between generators 1 – 13, while generators 14, 15 and 16 are not coherent with each other nor with other generators in the system.

TEST CASES

The proposed method has been tested and verified through a case study on the reduced order equivalent model of New England Test System (NETS) and the New York Power System (NYPS) that has 68-bus, 16-machine. In order to simulate the data obtained by WAMS, it is assumed that voltage phasors are sent to the control center at the rate of 100 samples per second. Therefore, considering the Nyquist rate, waveforms with frequencies up to 50 Hz, including low oscillation modes, can be estimated by DFT.

Fig. 4. The coherency plot of generators and buses in Case 1.

53

Area 2

GM

49

42 15

50

56

Area 3

16

GM

63

45

44 39 43

37 13

GM

G M

6

21

GM

65

59 62 2 3

12

24

23 20

GM

5

68

64

36

GM 52

66

0 -10 0

22

67 61

10

GM

18

57 60 58

34 35

51

GM

11 30 32 33

9

17

GM

10

29

27 55

31 46 38

28

20 A m plitude

26

25

54

47

48

Area 2

20

19 4

5

GM

7

GM

GM

10 Time (s) Area 3

15

20

10 0 -10 -20 0

2

3

1

2

A m plitude

40

Area 1

GM

8

1

A m plitude

41

Area 4

GM

Area 1

A m plitude

GM 14

0 -1 -2 0

5

10 Time (s)

15

20

5

10 Time (s) Area 4

15

20

50

10 Time (s)

15

20

1 0 -1 0

Fig. 5. Areas Identified by the proposed method in Case 1.

Fig. 7. Angular velocity variation of buses in each area in Case 1.

After calculating the correlations between each pair of buses, the next step is to find the areas based on these correlations. In this paper, Į is considered to be 0.04. The areas identified by the proposed method are shown in Fig. 5.

It can be realized that the dominant frequencies are in the range of 0.1 – 1 Hz, (i.e. inter-area oscillation modes). As described in Section II, two buses are coherent if the absolute value of the real and imaginary part of their correlation is equal to 1 and 0 respectively. For example, the real and imaginary parts of correlation between generators 1 and 8 are 0.9941 and 0.0008, which means that these generators are highly coherent. Other highly coherent generators are listed in TABLE I. Generators that are not listed in this table are not highly coherent with any of generators in the system. However, there is a relative coherency between some of these generators, for example, between generators 12 and 13.

It should be noted that the order of areas is based on the results obtained by the proposed method. In this case, the best correlation is related to buses 43 and 44. Therefore, the process of constructing the first area is initialized using these buses. In the next step, the coherency of neighboring buses (i.e. buses 37, 39 and 45) is assessed. This procedure will stop if it is not possible to find any other buses coherent with the first area. When the first area is identified, the procedure for identifying the second area is triggered using the rest of power system. The position of correlations between the buses in each area in the complex plane is shown in Fig. 6. As it can be seen, in each area, the correlations between each pair of buses are placed in the circle centered at 1 + j 0 with radius Į = 0.04. Fig. 7 shows the variations of angular velocity signal of buses in each area after the disturbance has occurred. It can be seen that using these correlations, buses are grouped based on their dominant frequencies. B. Case 2 In the second case, a line to ground fault is applied on the transmission line between bus 22 and bus 23 at 86 % of maximum loadabg1ility of the power system. The amplitude of frequency components exist in the angular velocity variation of voltage phasors at all buses is shown in Fig. 8.

Fig. 6. The correlations between buses in the areas identified in Case 1

The correlation coefficients between buses that are connected through transmission lines are shown in Fig. 9. It can be understood that the coherency between some neighboring buses is very low. For example, buses 42 and 52 are weekly coherent and should not be placed in the same area. In the other words, these buses are the bordering buses of two distinct areas. In Fig 9 these lines are determined by red bars. The areas identified by the proposed method for Į = 0.04 are shown in Fig. 10. In this figure, the red lines are those which are determined in Fig. 9 by red bars. It should be noted that, here, the best correlation in the whole power system is between bus 38 and bus 46. As in Case 1, the position of correlations between the buses in each area shown in Fig. 11 demonstrates the high coherency between buses in each area. Note that, Fig. 10 shows that in this case, the power system is partitioned into fifteen areas using the proposed method, which is a high number for a power system with 16 generators.

Fig. 8. The amplitude of frequency components of angular velocities (case 2)

TABLE I THE CORRELATION BETWEEN HIGHLY COHERENT GENERATORS Highly coherent Highly coherent ci , j ci , j Generators Generators G1, G8 0.994 + j0.000 G6, G7 0.999 + j0.000 G2, G3 0.999 + j0.000 G10, G11 0.991 + j0.019 G4, G5 1.001 + j0.000

Area 1

Area 2

0.05 0 -0.05 0.95

1.05

Area No. 1 2 3 4 5 6 7 8

However, these areas are determined based on a high degree of coherency. This means that the coherency in each area is very high as confirmed by the average of correlations between the buses in each area shown in Table II. This parameter is calculated by averaging the correlation between each pair of buses in an area with n buses using (8).

§ § n · · n −1 n C AVG = ¨ 1 ¨ ¸ ¸ ¦ ¦ ci , j © © 2 ¹ ¹ i =1 j =i +1

(8)

Fig. 11 shows that there are several areas with no generators. In order to merge these areas with their neighboring areas, especially those that include generators, one way is to use the average linkage method as in (9). This method is used to find the degree of dissimilarity between different clusters.

Ca ,b =

1 N a Nb

Na

Nb

¦¦ c

i, j

-0.05 0.95

Area 4

1.05

-0.05 1.05 0.95

Area 7

1 Area 8

1 Area 10

1 Area 11

0.05

0

0

0

1 Area 13

-0.05 0.95

1 Area 14

0.05

0 -0.05 0.95

1.05

1

1.05

1.05

-0.05 0.95

1 Area 15

1.05

1

1.05

0.05

0 -0.05 1.05 0.95

1 Area 12

0.05

0.05

1.05

0 -0.05 1.05 0.95

0.05

-0.05 0.95

1 Area 9

0.05

0 -0.05 1.05 0.95

1.05

0 -0.05 1.05 0.95

0.05

0

1 Area 6

0.05

0

1

-0.05 0.95

Area 5

0.05

-0.05 0.95

0

1

0.05

0

TABLE II AVERAGE OF CORRELATIONS IN EACH AREA Average of correlations Area No. Average of correlations 0.9942 + j 0.0094 9 0.9980 + j 0.0030 1.0000 + j 0.0000 10 1.0023 + j 0.0002 0.9909 + j 0.0030 11 0.9993 + j 0.0014 0.9987 + j 0.0077 12 0.9935 + j 0.0061 0.9961 + j 0.0015 13 0.9992 + j 0.0108 1.0000 + j 0.0001 14 1.0080 + j 0.0003 0.9986 + j 0.0014 15 0.9741 + j 0.0164 0.9981 + j 0.0000

0.05

0

1

0.05

-0.05 0.95

Area 3

0.05

0

1

-0.05 1.05 0.95

Fig. 10. The correlations between buses in the areas identified in case 2.

where Na and Nb are the number of buses in areas A and B respectively. For example, areas 1, 3 and 12 are coherent with each other. The average linkage between area 1 and areas 12 and 3 are 0.976 + j 0.051 and 0.936 + j 0.053 respectively. Therefore area 12 and area 1 can be combined together. Furthermore, based on Fig. 9 there is a high coherency between areas 11, 14 and 7, which means that these areas can be combined together as well. Therefore, the number of areas is reduced to 12 and keeping on this procedure for lower degrees of coherency it will possible to reach the lower number of areas. On the other hand, as stated in Section III, the value of a can affect the number of areas. Therefore, instead of using (8), one can choose a higher value for a (higher than 0.04) to achieve lower number of areas.

(9)

i =1 j =1

Fig. 9. Real and imaginary parts of correlations between incident buses.

M G

Area 9

Area 12 M G

14 41

40

53 Area 2 38

46 Area 6 42 15

10

51

50

56 60

M G

52 Area 13

16 M G

57 58

66 64 63

44

37

39

13 43

Area 7

M G

M G

24

6

68 Area 14

21

M G

65

59 62 2 3

12

22

67

Area 1 61

36

45

M G

18

11 30 32 33 34 35

Area 3

M G

29 9

17

M G

M G

49

28

27 55

31

Area 8

26

25

54

47

48

Area 11

M G

8

1

23 19 4

20 M G

Area 5

Area 15

M G

5 M G

Area 10

7 M G

Area 4

Fig. 11. Areas Identified by the proposed method in case 2.

V.

CONCLUSION

In this paper, a new method for coherency detection and network partitioning was proposed. The requirements of this method were the phase angles measured by wide area measurement system over the time after the disturbance has occurred. This method used the dissimilarity between frequency components exist in the angular velocity of voltage phasors of buses to find coherent buses. The results of applying this method to a standard test power system showed that it could accurately find the coherent areas. It was seen that this method can calculate the degree of coherency between buses and generators accurately using both the phase angle and amplitude of the frequencies that are in the range of inter-area oscillation modes. It was also seen that, in this method, by defining different degrees of coherency for establishing areas, the power system can be partitioned into a different number of areas. REFERENCES [1]

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1 Department of Electrical Engineering, Shahid Bahonar University of Kerman, Kerman, Iran Department of Electrical and Computer Engineering, Texas A&M University, College Station, Texas, USA [email protected]; [email protected]; [email protected]

Abstract— Due to the non-linearity and complexity of power systems, reduced model of these systems is used for dynamic and transient analysis. In this regard, a power system is partitioned into areas based on the similarity of dynamic responses of generators to the disturbances, which is called coherency. In this paper, a new method is presented for determining the degree of coherency between generators or non-generating buses using the data measured by wide area measurement system (WAMS). This method uses the amplitude as well as phase angles of frequency components that exist in the angular velocity signals after the disturbance has occurred. Those frequencies that are in the range of inter-area oscillation modes are selected to determine the coherency between each pair of generators using a correlation index. By expanding this index to the non-generating buses of the power system and using the tight coherency concept, the power system is partitioned into highly coherent areas. The effectiveness of this method for area detection is investigated on the 16machine, 68-bus test system. It is observed that accurate coherent areas can be identified using both the phase angle and amplitude of frequencies in the range of inter-area oscillation modes. Keywords— coherency; frequency component; network partitioning; phasor measurement unit; wide area measurement system

I.

INTRODUCTION

in power system dynamic is becoming more and more challenging due to the increasing number of complexity of the power and energy delivery systems. In emergency scenarios, portions of the electric power system would be adversely affected with a compromised electrical safety, exposing the network to risk under an unstable condition with some changes in the equilibrium point of the system. Thus, the importance of real-time monitoring and control of the power system infrastructure is undoubtable. This complexity and the nonlinearities that exist in power systems have made it difficult to perform those critical analyses that are essential for power system stability and dynamics [1], [2]. Since performing such analysis on a bulk power system is very time consuming, an efficient partition algorithm i.e. the reduced equivalent model of the power system is essential to overcome the communication complexity and reduce the computation. In so doing, the power system is separated into areas based on its dynamic response to a disturbance and then, each area is replaced with a reduced equivalent model.

C

OHERENCY analysis

The main steps of power system reduction methods are i) identification of groups of generators and buses to be reduced and, ii) aggregation and reduction of these groups. Among power system reduction methods, coherence-based approaches are mostly used in studies since they retain the dynamic characteristics of generators in the aggregation step. A group of generators are said to be coherent for a given disturbance if they have the same rotor angle or speed variations. Based on this concept, in the area identification step, the main idea is to find those buses with the same parameters’ variation (e.g. phase angle variations). Coherency detection methods can be 978-1-5386-1006-0/18/$31.00 ©2018 IEEE

categorized into two groups. The first group which is called model-based methods uses the eigenvalue and eigenvector of the power grid for coherency computation [3]-[8]. In the modal synchrony methods, a linearized model analysis of power system is applied to identify the coherent generators [8]. The methods employed in the second group are based on the disturbances and use time-domain simulations to find coherent clusters. Several studies have used Fourier spectrum [9], [10], principal component analysis [11], independent component analysis [12], hierarchical clustering methods [13], [14], subtractive clustering [15], Fuzzy c-medoids algorithm [16], wavelet [17], Hilbert-Huang transform [18] and decision trees [19] to find coherent generators. The instances of blackouts in recent years which resulted in major electricity outages proof that supervisory control and data acquisition system cannot provide the requirements for maintaining the security and resilience of bulk power systems [20], [21]. Fault diagnoses in power system infrastructure is a challenging task. Coherency analysis can be integrated with the fault detection algorithm to further expedite the decisionmaking by the operators in cases of emergencies [22]-[24]. Therefore, it is essential to go beyond traditional systems and use modern measurement systems with higher capabilities. In this regard, today, synchronized measurement technology (SMT) is used in the wide-area measurement systems (WAMS) for controlling and protecting power systems [25]. Phasor measurement units (PMUs) as one of the key elements of SMT are aimed to estimate and synchronize the voltage and current phasors. In the literature, there are several examples of methods that have used the data measured by PMUs to find coherent clusters, especially coherent generators [9]-[19]. Amongst, a Fuzzy c-medoids algorithm is implemented in [16] to find coherent areas with respect to the dissimilarity between the angular velocity deviation of voltage phasor at buses. However, in this method, it is needed to predefine the number of areas. Another clustering method is presented in [13] in which an agglomerative hierarchical clustering method is used to find coherent generators. Applying this method to area detection problem needs a modification since it is necessary to consider the connectivity of buses for calculating the dissimilarities. In this paper, a coherent area detection method is presented using the data measured by WAMS. In this method, the coherency between each pair of buses is assessed through calculating the correlation between the frequency components exist in angular velocity signals. To do that, the frequencies are extracted using Discrete Fourier Transform (DFT) and then, the frequencies that are in the range of inter-area oscillation modes are selected. It should be noted that in the method for calculating the correlation between generators, not only the amplitude, but also the phase angle of these frequencies is applied, which causes the proposed coherency detection method to find the degree of coherency accurately. Furthermore, this method is applied to both generating and non-generating buses of the power system to find the borders

of coherent areas, which is addressed in few research works. This is important for such model reduction applications such as dynamic analysis of power system and controlled islanding. II.

DYNAMIC COHERENCY DETECTION

In the occurrence of a disturbance, the generators’ rotor angles vary from their equilibrium points which actually represent the dynamic response of each generator to the disturbance. As a result of these variations, the phase angle of buses also varies. Therefore, the effect of the dynamic response of each generator can be determined by phase angles variation at the buses near to the generator. Considering și and șj as the phase angles at bus i and j respectively, one approach for evaluating the coherency between the buses is to measure the dissimilarity between the phase angle variations as in (1). T

si , j = ³ ( Δθ i ( t ) − Δθ j ( t ) )

(1)

0

where si , j denotes the dissimilarity index between bus i and j, and T is the observing time. The energy interaction between generators can be reflected by their speed deviations [10]. In the other words, the amount of energy absorbed or delivered by generators in the power grid is reflected by the generators’ speed variation or the rate of phase angles variation at buses. However, using the variation rate of phase angle or angular velocity as indices for representing dissimilarity is not appropriate. Analyzing the frequency components of the angular velocities can be more helpful for coherency identification since these frequencies represent the dynamic response of each generator following a disturbance. These components can be extracted using DFT as follows. N −1

Fi ( f ) = ¦ ωi ( k ) e

§ 2π fk · −¨ j ¸ © N ¹

f = 0,1,..., N − 1

Fig. 1.

Two typical waveforms and the their frequency spectrums.

their variations. For example, Fig. 1 shows two waveforms which don't have the same variation. But, as it can be seen, these waveforms include two frequencies 1 and 0.25 Hz with the same amplitudes. Therefore, if the coherency detection method uses only the amplitude of these frequencies, it will conclude that the generators that have the speed variation as in Fig. 1 are coherent. In this paper, to identify the coherent buses, the correlation matrix is formed as follows. ª c1,1 ! c1, N B º « » % # » C=« # «c » ¬ N B ,1 ! cN B , N B ¼

(2)

k =0

ωi ( k ) =

n

(3)

Δt

avg

ci , j =

T

(4)

where NB is the total number of buses in the power system. It is noteworthy to mention that, the number of frequencies in the vector Fi is equal to the number of samples of waveforms, as DFT transforms the N discrete time samples to the same number of frequency samples. Therefore, in order to obtain more frequency samples, it is essential to sample the angular velocities in higher rates, which is obtained by PMUs. Using DFT, the amplitude and phase angle of each frequency component exists in the waveform of angular velocity is obtained. As a result, the solution of DFT is a complex number for each frequency. In order to assess the coherence between two generators, both amplitude and phase angles of these frequencies should be considered. In fact, it is obvious that the similarity of the amplitudes of frequencies exists in two waveforms does not guarantee the similarity of

i

avg j

j

f =1 n

2 n

(6)

¦(F ( f ) − F ) ¦(F ( f ) − F ) avg

i

where Ȧi(k) denotes the angular velocity of generator i at time instant k, and ǻt is the time interval between two executive samples, which is constant for all simulations. Fi (f) represents the Fourier transform of the angular speed. The vector-space Fi and the NB × N dimension matrix F are then formed as follows.

F = ª¬ F1 ( f ) , ! Fi ( f ) , ! FN B ( f ) º¼

¦ ª¬( F ( f ) − F ) ( F ( f ) − F )º¼ i

θi ( k ) − θi ( k − 1)

(5)

f =1

i

j

avg 2 j

f =1

In (6) ci , j is the correlation coefficient between buses i and j. Fi avg represents the average of frequency components of bus i in the range of inter-area oscillation modes (0.1–1 Hz). n is also the number of frequency components in this range. Since the elements of F are complex numbers, the correlation between two buses will be a complex number too. Therefore, based on (6), two buses are called coherent when the value of (6) equals to 1 ∠ 0 (or 1 + j0). In the other words, the absolute value of real and imaginary parts of the correlation between two coherent buses should be 1 and 0 respectively.

III.

COHERENCE-BASED NETWORK PARTITIONING

Based on the concept of tight coherency, the phase angle of all buses in an area should have relatively the same deviation. This can be assessed by calculating the correlation between each pair of buses in the area using (6). As described in section II, the highest degree of coherency happens when the correlation between two buses becomes 1 + j 0. Therefore, for an area, the Euclidean distance between the correlation of each pair of buses located in the area and the point 1 + j 0 in the complex plane should be less than a predefined value of Į, as in (7).

Imaginary Cj,k Į

Ci,m

Ci,j

1

Real Ci,k

Fig. 2. Definition of coherency in the complex plane.

ci , j − 1 ≤ α

∀i , j ∈ S

(7) Fig. 3. The amplitude of frequency components of angular velocities (case 1)

where S is the set of buses located in the area. Fig. 2 shows the tight coherency between the buses in a typical area based on the correlations obtained by (6). In Fig. 2, the position of correlations between buses i, j and k are placed in the circle with the center 1 + j 0 and radius Į, which means that these buses are coherent with each other. On the other hand, the position of correlation between bus i and bus m is out of this circle, which means that bus m can’t be placed in the area including buses i, j and k. In order to find coherent areas, first, the correlations between all connected buses should be calculated using (6). To do that, voltage phasors from all over the system is calculated using the data measured by PMUs installed at selected buses of the power system. Today, these devices are used as the main part of wide area measurement system, which is fully installed or is under study in many countries [25], [26]. After calculating the correlation between phasor angles of incident buses, the area detection procedure is triggered. First, the two incident buses that have the higher correlation among all buses are identified. In the next step, the lines that connect these buses to the other buses of the power system are determined. After identifying the buses connected to the initial area, the correlation of each of these buses with all buses in the area is calculated. If the Euclidean distance between all of these correlations and the point 1 + j 0 in the complex plane are less than Į, this bus is coherent with the area and therefore can be added to the area. Otherwise, this bus remains in the rest of power system. This procedure will continue until it is not possible to find any other buses coherent with the first area. When the first area is identified, the procedure for identifying the second area is initiated using the rest of power system. It should be noted that Į should be 0 < Į < 1. It is also clear that choosing the lower value for Į will increase the number of areas, but such areas will include buses that are highly coherent with each other. In fact, the concept of coherency is relative. The lower the value of Euclidean distance between correlation ci , j and the point 1 + j 0, the higher the coherency between buses i and j. Therefore, choosing the appropriate value for Į is completely applicationbased. IV.

A. Case 1

In the first case, a line to line fault is applied on bus 42 at 91 % of maximum loadability of the test power system. The amplitude of frequency components exist in the angular velocity variation of voltage phasors at all buses is shown in Fig. 3. It can be realized that there is a significant difference between the amplitude of variations of buses 15 and 42, and other buses in the system at the frequency of 0.26 Hz. This means that buses 15 and 42 show a different response to the disturbance compared to other buses and should be placed in a distinct area. On the other hand, although, other buses of the system might seem to have the same variations, the degree of coherency between them must be assessed mathematically. To do that, the correlations between all generators as well as buses are calculated using (6). Fig. 4 shows a colorful plot to represent the degree of coherency using the Euclidean distance between all calculated correlations between generators as well as buses and the point 1 + j 0. It is obvious that the lower values of this distance show the more coherency between generators or buses. In Fig. 4 the dark blue points show the high coherency between buses and generators. It can be seen that there is a high coherency between generators 1 – 13, while generators 14, 15 and 16 are not coherent with each other nor with other generators in the system.

TEST CASES

The proposed method has been tested and verified through a case study on the reduced order equivalent model of New England Test System (NETS) and the New York Power System (NYPS) that has 68-bus, 16-machine. In order to simulate the data obtained by WAMS, it is assumed that voltage phasors are sent to the control center at the rate of 100 samples per second. Therefore, considering the Nyquist rate, waveforms with frequencies up to 50 Hz, including low oscillation modes, can be estimated by DFT.

Fig. 4. The coherency plot of generators and buses in Case 1.

53

Area 2

GM

49

42 15

50

56

Area 3

16

GM

63

45

44 39 43

37 13

GM

G M

6

21

GM

65

59 62 2 3

12

24

23 20

GM

5

68

64

36

GM 52

66

0 -10 0

22

67 61

10

GM

18

57 60 58

34 35

51

GM

11 30 32 33

9

17

GM

10

29

27 55

31 46 38

28

20 A m plitude

26

25

54

47

48

Area 2

20

19 4

5

GM

7

GM

GM

10 Time (s) Area 3

15

20

10 0 -10 -20 0

2

3

1

2

A m plitude

40

Area 1

GM

8

1

A m plitude

41

Area 4

GM

Area 1

A m plitude

GM 14

0 -1 -2 0

5

10 Time (s)

15

20

5

10 Time (s) Area 4

15

20

50

10 Time (s)

15

20

1 0 -1 0

Fig. 5. Areas Identified by the proposed method in Case 1.

Fig. 7. Angular velocity variation of buses in each area in Case 1.

After calculating the correlations between each pair of buses, the next step is to find the areas based on these correlations. In this paper, Į is considered to be 0.04. The areas identified by the proposed method are shown in Fig. 5.

It can be realized that the dominant frequencies are in the range of 0.1 – 1 Hz, (i.e. inter-area oscillation modes). As described in Section II, two buses are coherent if the absolute value of the real and imaginary part of their correlation is equal to 1 and 0 respectively. For example, the real and imaginary parts of correlation between generators 1 and 8 are 0.9941 and 0.0008, which means that these generators are highly coherent. Other highly coherent generators are listed in TABLE I. Generators that are not listed in this table are not highly coherent with any of generators in the system. However, there is a relative coherency between some of these generators, for example, between generators 12 and 13.

It should be noted that the order of areas is based on the results obtained by the proposed method. In this case, the best correlation is related to buses 43 and 44. Therefore, the process of constructing the first area is initialized using these buses. In the next step, the coherency of neighboring buses (i.e. buses 37, 39 and 45) is assessed. This procedure will stop if it is not possible to find any other buses coherent with the first area. When the first area is identified, the procedure for identifying the second area is triggered using the rest of power system. The position of correlations between the buses in each area in the complex plane is shown in Fig. 6. As it can be seen, in each area, the correlations between each pair of buses are placed in the circle centered at 1 + j 0 with radius Į = 0.04. Fig. 7 shows the variations of angular velocity signal of buses in each area after the disturbance has occurred. It can be seen that using these correlations, buses are grouped based on their dominant frequencies. B. Case 2 In the second case, a line to ground fault is applied on the transmission line between bus 22 and bus 23 at 86 % of maximum loadabg1ility of the power system. The amplitude of frequency components exist in the angular velocity variation of voltage phasors at all buses is shown in Fig. 8.

Fig. 6. The correlations between buses in the areas identified in Case 1

The correlation coefficients between buses that are connected through transmission lines are shown in Fig. 9. It can be understood that the coherency between some neighboring buses is very low. For example, buses 42 and 52 are weekly coherent and should not be placed in the same area. In the other words, these buses are the bordering buses of two distinct areas. In Fig 9 these lines are determined by red bars. The areas identified by the proposed method for Į = 0.04 are shown in Fig. 10. In this figure, the red lines are those which are determined in Fig. 9 by red bars. It should be noted that, here, the best correlation in the whole power system is between bus 38 and bus 46. As in Case 1, the position of correlations between the buses in each area shown in Fig. 11 demonstrates the high coherency between buses in each area. Note that, Fig. 10 shows that in this case, the power system is partitioned into fifteen areas using the proposed method, which is a high number for a power system with 16 generators.

Fig. 8. The amplitude of frequency components of angular velocities (case 2)

TABLE I THE CORRELATION BETWEEN HIGHLY COHERENT GENERATORS Highly coherent Highly coherent ci , j ci , j Generators Generators G1, G8 0.994 + j0.000 G6, G7 0.999 + j0.000 G2, G3 0.999 + j0.000 G10, G11 0.991 + j0.019 G4, G5 1.001 + j0.000

Area 1

Area 2

0.05 0 -0.05 0.95

1.05

Area No. 1 2 3 4 5 6 7 8

However, these areas are determined based on a high degree of coherency. This means that the coherency in each area is very high as confirmed by the average of correlations between the buses in each area shown in Table II. This parameter is calculated by averaging the correlation between each pair of buses in an area with n buses using (8).

§ § n · · n −1 n C AVG = ¨ 1 ¨ ¸ ¸ ¦ ¦ ci , j © © 2 ¹ ¹ i =1 j =i +1

(8)

Fig. 11 shows that there are several areas with no generators. In order to merge these areas with their neighboring areas, especially those that include generators, one way is to use the average linkage method as in (9). This method is used to find the degree of dissimilarity between different clusters.

Ca ,b =

1 N a Nb

Na

Nb

¦¦ c

i, j

-0.05 0.95

Area 4

1.05

-0.05 1.05 0.95

Area 7

1 Area 8

1 Area 10

1 Area 11

0.05

0

0

0

1 Area 13

-0.05 0.95

1 Area 14

0.05

0 -0.05 0.95

1.05

1

1.05

1.05

-0.05 0.95

1 Area 15

1.05

1

1.05

0.05

0 -0.05 1.05 0.95

1 Area 12

0.05

0.05

1.05

0 -0.05 1.05 0.95

0.05

-0.05 0.95

1 Area 9

0.05

0 -0.05 1.05 0.95

1.05

0 -0.05 1.05 0.95

0.05

0

1 Area 6

0.05

0

1

-0.05 0.95

Area 5

0.05

-0.05 0.95

0

1

0.05

0

TABLE II AVERAGE OF CORRELATIONS IN EACH AREA Average of correlations Area No. Average of correlations 0.9942 + j 0.0094 9 0.9980 + j 0.0030 1.0000 + j 0.0000 10 1.0023 + j 0.0002 0.9909 + j 0.0030 11 0.9993 + j 0.0014 0.9987 + j 0.0077 12 0.9935 + j 0.0061 0.9961 + j 0.0015 13 0.9992 + j 0.0108 1.0000 + j 0.0001 14 1.0080 + j 0.0003 0.9986 + j 0.0014 15 0.9741 + j 0.0164 0.9981 + j 0.0000

0.05

0

1

0.05

-0.05 0.95

Area 3

0.05

0

1

-0.05 1.05 0.95

Fig. 10. The correlations between buses in the areas identified in case 2.

where Na and Nb are the number of buses in areas A and B respectively. For example, areas 1, 3 and 12 are coherent with each other. The average linkage between area 1 and areas 12 and 3 are 0.976 + j 0.051 and 0.936 + j 0.053 respectively. Therefore area 12 and area 1 can be combined together. Furthermore, based on Fig. 9 there is a high coherency between areas 11, 14 and 7, which means that these areas can be combined together as well. Therefore, the number of areas is reduced to 12 and keeping on this procedure for lower degrees of coherency it will possible to reach the lower number of areas. On the other hand, as stated in Section III, the value of a can affect the number of areas. Therefore, instead of using (8), one can choose a higher value for a (higher than 0.04) to achieve lower number of areas.

(9)

i =1 j =1

Fig. 9. Real and imaginary parts of correlations between incident buses.

M G

Area 9

Area 12 M G

14 41

40

53 Area 2 38

46 Area 6 42 15

10

51

50

56 60

M G

52 Area 13

16 M G

57 58

66 64 63

44

37

39

13 43

Area 7

M G

M G

24

6

68 Area 14

21

M G

65

59 62 2 3

12

22

67

Area 1 61

36

45

M G

18

11 30 32 33 34 35

Area 3

M G

29 9

17

M G

M G

49

28

27 55

31

Area 8

26

25

54

47

48

Area 11

M G

8

1

23 19 4

20 M G

Area 5

Area 15

M G

5 M G

Area 10

7 M G

Area 4

Fig. 11. Areas Identified by the proposed method in case 2.

V.

CONCLUSION

In this paper, a new method for coherency detection and network partitioning was proposed. The requirements of this method were the phase angles measured by wide area measurement system over the time after the disturbance has occurred. This method used the dissimilarity between frequency components exist in the angular velocity of voltage phasors of buses to find coherent buses. The results of applying this method to a standard test power system showed that it could accurately find the coherent areas. It was seen that this method can calculate the degree of coherency between buses and generators accurately using both the phase angle and amplitude of the frequencies that are in the range of inter-area oscillation modes. It was also seen that, in this method, by defining different degrees of coherency for establishing areas, the power system can be partitioned into a different number of areas. REFERENCES [1]

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[3]

[4]

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[6]

[7]

[8]

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