A data-driven approach for fault time determination and fault area location using random matrix theory

Electrical Power and Energy Systems 116 (2020) 105566

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

A data-driven approach for fault time determination and fault area location using random matrix theory

T

⁎

Yongxin Xionga, Wei Yaoa, , Weibiao Chenb, Jiakun Fanga, Xiaomeng Aia, Jinyu Wena a

State Key Laboratory of Advanced Electromagnetic Engineering and Technology, School of Electrical and Electronic Engineering, Huazhong University of Science and Technology, Wuhan 430074, China b Guangdong Power Grid Development Research Institute Co., Ltd., Shantou, 515031, China

A R T I C LE I N FO

A B S T R A C T

Keywords: Data-driven approach Phasor measurement unit (PMU) Wide-area measurement system (WAMS) Fault diagnosis Random matrix theory (RMT) Bad data

This paper proposes a wide-area measurement-based data-driven approach for fault time determination and fault area location in the power system. To avoid the influence of bad data, the random matrix theory (RMT) is applied to the proposed approach. The measurements obtained from wide-area measurement system (WAMS) are used to form the raw data matrix, its mean spectral radius (MSR) index and the correlation among entries are employed to determine the fault time and locate the fault area using RMT. Since the proposed approach only requires the data obtained from WAMS and is a data-driven method, no physical model and topology information is needed compared with the existing model-based methods. Case studies are carried out on the IEEE 39-bus power system and a practical provincial power grid of China respectively, and a decision-support tool is programmed using the proposed method to help operators obtain fault information more timely and accurately. Simulation results show that the proposed method can determine the fault time precisely and locate the fault line, whether there is bad data in wide-area measurements or not.

1. Introduction Fault diagnosis plays an important role in ensuring reliable and stable operation of power system. The accurate and quick fault diagnosis can help operators deal with faults timely to avoid large-scale blackouts, especially when the power system is confronted with extreme weather and equipment failures [1]. The application of PMUs in power system provides a high dimensional electrical information with wide-area common time frame for fault diagnosis, which promotes the research field. Because of the unified accurate time stamp of the global positioning system(GPS) and high sampling frequency [2,3], PMU data have been more frequently employed to detect faults and provide operators with accurate fault analysis information in recent year. Some of existing fault diagnosis methods analyze the traveling wave to realize fault diagnosis [4–7]. Fault areas are located by analyzing transient voltage waveform using the continuous wavelet transformation (CWT)[8] and discrete wavelet transformation (DWT)[9]. Furthermore, on-line and off-line stages are used to solve the problem that DWT cannot detect faults occurring on transmission lines precisely in [10]. However, the influence of erroneous measurements has not been studied in the literature, which will affect the accuracy of fault diagnosis [11–13]. With the premise of knowing the precise model of power

⁎

system, fundamental impedance-based fault location methods have been presented in [14,15] by analyzing voltage and current phasors at each end of transmission lines. Similar fault location techniques for three-terminal and N-terminal transmission lines based on PMU data has been presented in [16,17]. And a method for fault area detection based on conventional methods with synchronous phasors is proposed in [18]. Nevertheless, precise impedance of power system is hard to get. In computation aspect, several approaches based on intelligence algorithm for fault diagnose have been proposed in [19–23]. The support vector machine is used to form the equivalent voltage phasor angle and the equivalent current phasor angle in [24]. The wavelet computation is analyzed using fuzzy inference system and Monte Carlo simulation to locate the fault area on transmission lines in [25]. The main limitation of these approaches based on intelligence algorithm is the requirement for favorable parameter setting and algorithm training. To the best knowledge of the authors, most of the existing fault diagnosis methods based on PMU data mainly rely on electrical abruptness as the starting criterion, therefore, its accuracy is sensitive to bad data and required for the precise network impedance [26–29], which are difficult to be obtained in a practical power system. Consequently, the fault diagnosis method, which is robust to erroneous measurements and independent of practical power system network

Corresponding author. E-mail address: [email protected] (W. Yao).

https://doi.org/10.1016/j.ijepes.2019.105566 Received 4 February 2019; Received in revised form 17 September 2019; Accepted 23 September 2019 0142-0615/ © 2019 Elsevier Ltd. All rights reserved.

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Nomenclature

λi   × μ (X ) σ (X ) Xũ X̃ H X̃ x i,̃ j Z̃ c

D′e (i) dMSR (i) (t ) Dp (i) N, T Ni rMSR SMSR (i) (t ) X xi Xt XA (i) XAref (i)

the i th eigenvalue of a matrix. N × N dimensional complex space. the mean value of X . the standard value of X . a random matrix’s singular equivalent matrix. a random matrix. a random matrix’s transpose. an entry of a random matrix. the standard matrix product. the row column ratio of a random matrix.

data. Section 3 introduces the method of model construction using random matrix theory briefly. The method of fault time determination and fault area location is shown in Section 4. In Section 5, the case study using the IEEE 39-bus system is carried out to demonstrate the effectiveness of the proposed fault diagnose method in the first part. And the comparison between the proposed method and one of the existing fault diagnose method is also carried out in this section in the second part. Another case study is carried out using real PMU data collected from a practical provincial power grid of China in Section 6. After that, the characteristic and limitation of the proposed method are discussed in Section 7. Finally, conclusions are drawn in Section 8.

models, is needed to be developed. The big data technology is a new and widespread concern technology with the ‘Volume, Variety, Value, Velocity (4 V)’ characteristics in recent year[30]. Since the PMU data has the basic characteristics of big data, analysis methods for big data can be adapted to analyze the PMU data. Random matrix theory (RMT) is one of the useful analysis methods for complex systems, and the analysis methods based on RMT for fault diagnose has been applied in power system preliminarily [31,32]. This theory is mainly used to analyze the correlation of the factors affecting the operation of the power system[33–35]. The correlation between the influence factors and the system status is analyzed based on a data-driven method in [36]. A high dimensional complex grid recognition method based on RMT has been proposed in [37]. In view of this, this paper proposes a data-driven approach for fault time determination and fault area location based on PMU data using RMT. The proposed approach is physical model-free and robust to bad data. Case studies are respectively carried out on the IEEE 39-bus system and a practical provincial power grid of China to verify the efficiency of the proposed method. It should be pointed out that because the proposed approach is an assistant method to help operators determine the fault time and fault area timely and accurately, the fault area is located in a rough area at the first time. After that, the precise fault point location can be realized, which is the future research and not involved in this paper. The main contributions of this paper can be listed as follows:

2. Introduction of multidimensional information data in power system If the fault cannot be handled timely and accurately in power system, they will sometimes develop into serious accidents, resulting in a continuous deterioration of the system, and even a large-scale blackout accident. Therefore, how to quickly and accurately diagnose faults in power system is one of the most concerns of operators [38,39]. To the best knowledge of the authors, existing fault diagnosis methods mainly use the information provided by the SCADA system and FIS at present in China. Due to the variable time scales, when a complex fault occurs in the power system, a large amount of information (including correct alarm information, mis-transmission information, duplicate information, and irrelevant information) will flood into the dispatch center, which will increase the difficulty for operators to identify faults in an emergency situation. Therefore, it is very necessary to propose an accurate and reliable fault diagnosis approach, which can analyze various fault information acquired by the dispatching center in real time, and provide operators with auxiliary fault diagnosis information. With the application of wide-area measurement system (WAMS) in power systems, its wide coverage and good time synchronization provide a new and reliable source of data for power system fault diagnosis. Due to different design concepts, the data collected by different equipments have different emphasis in terms of type, accuracy, sampling time, uploading mechanism, and so on. How to integrate the information of multiple data sources, apply it on fault diagnosis, and further improve the accuracy of fault diagnosis, is one of the development directions of fault diagnosis field. However, the fault diagnosis methods, which use the data collected by SCADA and FIS system, are not on the same time scale. Therefore, the different upload speed and erroneous measurements will affect the accuracy of fault diagnosis. Since the WAMS data has the characteristics of accurate global positioning system (GPS) time scale and location information, it is an important issue to make full use of WAMS data assisting operators in finding and handling faults, and prevent further development of faults. The characteristics of different data sources are described as follows:

• This data-driven approach is proposed using the random matrix

• •

the influence factor matrix. the MSR difference of XA (i) and XAref (i) . the state data matrix. the row and column number of the split-window. the noise matrix. the mean spectral radius. the integral of dMSR (i) (t ) . the extracted PMU data. a vector of a random matrix. the raw data matrix. the augmented matrix. the reference augmented matrix.

theory, which analyzes the correlation among entries of the random matrix structured by PMU data. When faults occur in the power system, all entries of this random matrix are relevant. However, if there are no faults but bad data in the power system, bad data entries of this data matrix are uncorrelated with normal entries, which have no influence on the mean spectral radius (MSR). Therefore, the proposed method can be robust to erroneous measurements. As a data-driven approach, the proposed method only requires for practical PMU data and analyzes the correlation among entries of random matrix structured by PMU data. Therefore, it has no requirement for physical power system topologies and the impedance data of practical power system. The proposed method is tested using not only IEEE-39 bus system but also a practical provincial power grid of China, and a decisionsupport tool is programmed using the proposed method to help operators obtain fault information more accurately. Simulation results show that the proposed data-driven approach is effective to determine fault time and fault area under both simulation conditions and practical conditions.

The remainder of this paper is arranged as follows. Section 2 introduces the characteristics of multidimensional data information in the power system, including the supervisory control and data acquisition (SCADA) data, the fault information system (FIS) data and the WAMS 2

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for 60 Hz system [42]. A PMU can measure 50/60 Hz AC waveforms (voltages and currents) typically at a rate of 48 samples per cycle (2400 samples per second for 50 Hz systems/2880 samples per second for 60 Hz systems). WAMS can collect real-time data such as voltage, current, rotor speed and excitation voltage of generators and lines equipped with PMU, and calculate indirect data such as voltage and current phasors, frequency and its change rate, power of the generators and lines. WAMS mainly has the following advantages: (1) WAMS data is synchronized in time. The PMU data of the whole network is unified in same time scale, and the WAMS master station can obtain the data with the same time scale of the entire network. (2) WAMS data is spatially wide. The extensive arrangement of PMUs ensures the source of wide-area grid data and enables real-time collection and monitoring of wide-area information. (3) WAMS can provide voltage and current phase angle data. This is one of the advantages compared to SCADA data. (4) WAMS has high sampling frequency and good real-time performance. Therefore, WAMS can provide real-time cross-section information of the whole network at the same time scale, providing a solid database for various practical applications. In addition to the advantages of good real-time performance, the on-line fault diagnosis based on WAMS is better than that based on SCADA data because of its high sampling frequency (10, 25, 50 frames per second for 50 Hz system) and fast uploading speed. The structure of the WAMS system is shown as Fig. 1. The structure of the SCADA, FIS, and WAMS system is shown as Fig. 1. And the characteristics of SCADC, FIS, and WAMS data are shown as Table 1. Because WAMS data has the characteristics of the synchronous time scale and location information, it is an important research to propose fault diagnosis methods based on WAMS data or multi-dimensional data information, which will improve the speed of locating faults, and prevent further development. The whole thought of authors’ study is shown as Fig. 2. This paper presents a data-driven approach for fault time determination and fault area location using WAMS data, to help operators distinguish alarm information accurately and make decisions quickly, which are the research contents in red dotted boxes in Fig. 2. Moreover, WAMS data can be also used to integrate the fault information provided by SCADA and FIS to form multidimensional information for power fault diagnosis, which is the future work of the authors’ study.

2.1. The characteristics of SCADA data As one of the most important subsystems of the energy management system (EMS), SCADA system can monitor the operation state of the power system and assist in fault diagnosis. It is an indispensable tool in power dispatching. The SCADA/EMS data is collected by remote terminal units (RTUs) at each station. The characteristics of SCADA data include the full layout, the low sampling frequency (0.25–0.5 Hz) [40], and the fast uploading speed. The SCADA data contains event sequence action signals, protection action signals, and total station accidents data. The use of SCADA data to assist online fault diagnosis has the advantages of better real-time performance and comprehensive monitoring range. However, the low reporting frequency and poor time synchronization make it difficult to provide detailed analysis results of faults, but only faults’ summary information. 2.2. The characteristics of FIS data The FIS data can collect the information from various protection devices, the protection action information and the waveform data of the fault collected by relay protection devices and fault recorders, so it can record the transient development process of the fault, providing the detailed protection information. However, the FIS data has the following problems: the protection action information is complicated, and because of the limited transmission capacity of communication channels, the recorded fault data with high sampling frequency has slow uploading speed and poor stability [41]. Therefore, only the switching information of FIS data is extracted for fast fault diagnosis, and the fault record data is mainly used in detailed fault diagnosis analysis after the faults. 2.3. The characteristics of WAMS data The WAMS data is synchronously collected by the PMU to reflect the real-time operating parameters of the wide-area power system, and transmitted to the data processing center via the high-speed communication network to obtain the dynamic operation of the grid under the unified time scale. It consists of three parts: PMU or Data Collectors distributed in each station, high-speed communication part covering the whole network, and main data processing and application part set on the dispatching station. According to IEEE Standard C37. 118, the reporting rate of phasors for PMU data is typically 10, 25, 50 frames per second for 50 Hz system, and 10, 12, 15, 20, 30, 60 frames per second

3. Model construction using random matrix theory The two basic concepts of the random matrix theory (RMT) are

Fig. 1. The structure of SCADA, FIS, and WAMS system. 3

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Table 1 The characteristics of SCADA, FIS and WAMS data (for 50 Hz system). Data Source

SCADA

WAMS

FIS

Data Type Parameters Reporting rate Uploading Speed

Analog and binary Switch and electrical Slow (0.25–0.5 Hz) Slow

Analog and binary Only electrical Medium (10,25,50 frames per second) Medium

Analog and binary Switch and electrical Fast (over 1000 Hz) Fast

f (x ) =

⎧ 1 λ πcL ⎨0 ⎩

2 −2 L

L

(1 − c ) 2 ⩽ λ ⩽ 1 other value of λ

(4) ∼ where λ is the eigenvalue of matrix Z , c is the row column ratio, and c = N / T ∈ (0, 1]. According to the Ring Law, the eigenvalues of matrix ∼ Z are roughly distributed in a ring with outer ring radius r1 = 1 and inner ring radius r2 = (1 − c ) L /2 . 3.2. The linear eigenvalue statistic

∼ For a random matrix X , the trace of the matrix can reflect the statistical properties of matrix elements, but a single eigenvalue of the matrix cannot reflect such properties because of randomness. As a statistical description of eigenvalues, linear eigenvalue statistic (LES) can reflect the eigenvalues distribution of a random matrix. For the ∼ random matric X , its linear eigenvalue statistic is defined as [36] N

Fig. 2. The logic diagram of overall research thought.

Nn (φ) =

∑

φ (λi )

(5) ∼ where λi (i = 1, 2, 3, …, N ) are eigenvalues of matrix X , φ (x ) is a test function, different forms of LES can be obtained with different test functions. The MSR, as a form of LES, can be used to represent the statistical properties of random matrices. The calculation equation of MSR is showed as Eq. (6), and its physical meaning is the average distances of all eigenvalues to center point in the complex plane. Therefore, this paper will carry out the following calculation indicators on the basis of the MSR. i=1

empirical spectral distribution (ESD) function and limit spectrum distribution (LSD) function. When the row column ratio of a random matrix is constant, and its row number and column number tend to infinity, the LSD function has many characteristics, for instance, the Marchenko-Pastur (M-P) Law and the Ring Law. Taking the fact that the Ring Law can accurately describe the LSD function of random matrices into consideration, this paper will be introduced based on the Ring Law [30,31].

rMSR =

3.1. The ring law

∼ Assuming that X = {x i,̃ j} N × T is a random matrix with non-Hermitian ∼ features, each entry of X is independent distributed. The expectation ∼ and variance of X are satisfied by E (x (̃i, j) ) = 0, E (|x (̃i, j) |2 ) = 1. The product Z of L random matrices is called the Matrix Product, which is represented as [36,37].

∏

Xu

i=1

Xu =

∼∼ H XX U

zi ⎞ ⎛ ⎜i = 1, 2, …, N ⎟ N σ (z i ) ⎝ ⎠

λi (6)

i=1

In a power system with WAMS, PMUs installed in different power plants, substations and important line nodes will have GPS unified time stamp synchronous phasor, and the PMU data will be transmitted to the dispatching center in real time. The temporal and spatial characteristics of PMU data can be extracted by building a random matrix model. Supposing that n PMUs are equipped in a power system, m phasors are chosen from each PMU as reference (a total of n × m = N variables), and T is the length of study time, therefore, a random matrix can be constituted using PMU data as

(1)

(2)

where Xu ∈  N × N is the singular equivalent matrix of the random ∼ matrix X ,  is a set of complex numbers. Xu can be calculated by Eq. (2), where U ∈  N × N is a Haar unitary matrix.

zi ̃ =

N

∑

3.3. Construction of random matrix model using PMU data

L

Z=

1 N

x1,1 x1,2 ⎡ ⋯ ⋯ ⎢ x m,1 x m,2 ⎢ ⋮ ⋮ X=⎢ ⎢ x (n − 1) × m + 1,1 x (n − 1) × m + 1,2 ⎢ ⋯ ⋯ ⎢ x n × m,1 x n × m,2 ⎣

(3)

According to Eq. (3), Z is transformed into a standard matrix product ∼ Z = {z (ĩ , j) } N × T , whose elements satisfy the follow equations: E (x (̃i, j) ) = 0, E ( x (̃i, j) 2 ) = 1/ N . When the row number and column ∼ number of Z tend to infinity and the row column ratio remains con∼ stant, the eigenvalues’ ESD characteristic of Z almost converges to the ∼ Ring Law. The probability density function of matrix Z is showed as

⋯ x1, T ⎤ ⋯ ⋯ ⎥ ⋯ x m, T ⎥ ⋮ ⋮ ⎥ ⋯ x (n − 1) × m + 1, T ⎥ ⎥ ⋯ ⋯ ⋯ x n × m, T ⎥ ⎦

(7)

where x i, j is the value of the PMU data i at the sample point j, i = 1, 2, …, N , j = 1, 2, …, T . The real-time split-window showed in Fig. 3 is used to generate the raw data matrix Xt . The length of the split-window is Nw , and the width is Tw . The split-window will move a sampling point after each sampling, therefore, it is supposed to contain the current data and Tw − 1 adjacent 4

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4.1. Fault time determination method based on MSR index The detailed steps of the fault time determination method are showed as Fig. 5 and Table 2. In Fig. 5, the change of MSR with time is studied to determine the exact fault time (see Table 3). Fig. 3. The schematic of split-window of raw data matrix.

4.2. The fault area location method based on PMU data correlation analysis

historical values at time t. The raw data matrix Xt is formed as Eq. (8) in the real-time split-window.

Xt = [x t − Tw + 1, x t − Tw + 2, ⋯, x t ]

In order to locate the suspicious fault area after the fault time is determined, the augmented matrix method mentioned in [36] is introduced, and the fault area is located by analyzing MSR index inference of the augmented matrix XA (i) and the reference augmented matrix XAref (i) . It should be noted that, in a practical power system, usually more than one PMU is equipped at each substation or power plant, thus the augmented matrix is constructed with each electrical node as the unit. The detailed processes of fault areas location method is shown as Fig. 6. Supposing that there are na electrical nodes equipped with PMUs in a power system, and the number of PMUs in each electrical node is np1, np2, …, npi , …, n pna , and the total number of PMUs in this power n system can be remarked as ∑i =a1 npi = np . Electrical parameters, for example, the voltage amplitudes of all buses, are chosen to form the corresponding state data matrix Dp (i) ∈  (npi × ng ) × T . Supposing that there are nfi influencing factors at each node, and the summary of all influn encing factors can be represented as ∑i =fi 1 = nf , and the influence factor × T n fi is formed. matrix Df (i) ∈  In order to indicate the effect of influence factors, Df (i) needs to be expanded by copying k times as follows:

(8)

where x t is a column vector consisting of PMU measurement data at time t. The choice of the split-window follows two principles: (1)In the premise of ensuring the speed of calculation, including all or partition chosen measurement data (such as the three-phase voltage and current data); (2)keeping the row column ratio constant. In order to analyze the correlation among entries, Xt should be ∼ converted into a standard non-Hermitian matrix X by Eq. (9)

x i,̃ j = [x i, j − μ (x i )]·

σ (x i )̃ + μ (x i )̃ ⎛⎜i = 1, 2, ⋯, Nw;j = 1, 2, ⋯, Tw ⎞⎟ σ (x i ) ⎠ ⎝ (9)

x i = (x i,1, x i,2 , …, x i, Tw );x i ̃ = (x i,1 μ and where ̃ , x i,2 ̃ , …, x i,̃ Tw ) , (x i ), σ (x i ), μ (x i ), ̃ σ (x i )̃ are mean values and standard values of x i and x i ,̃ moreover, μ (x i )̃ =0, σ (x i )̃ =1. 4. The method of fault time determination and fault area location The processes of proposed method in this paper is shown as Fig. 4, which are demonstrated as follows:

De (i)

• The PMU data is extracted to form the raw data matrix X, which is a large random matrix; • Calculate the standard matrices product Z∼, the augmented matrix X and the reference augmented matrix X ; Calculate the mean spectral radius (MSR) indexes and their r • integral S using Eq. (6) and Eqs. (14) and (15); • Analyze on MSR indexes and their integral to determine the fault A (i)

⎡ Df (i) ⎤ ⎢ Df (i) ⎥ (k × nfi) × T =⎢ ⎥∈ ⎢ ⋮ ⎥ D ⎢ ⎦ ⎣ f (i) ⎥

(10)

where k = [(npi × ng )/ nfi], [g] is represented the maximum integer value less than g. In order to reduce the correlation of repeated data, random noise is introduced to form the new influence factor matrix D′e (i) [36], which can be represented as follows:

Aref (i)

MSR (i)

D′e (i) = De (i) + mag (i) × N(i)

MSR (i)

(11)

where mag(i) is the noise amplitude, and N(i) ∈  (k × nfi) × T is the noise matrix with elements obeying standard normal distribution. The augmented matrix XA (i) and the reference augmented matrix XAref (i) [36] are formed as Eqs. (12) and (13) [36]:

time and fault area.

The concrete steps of the fault time determination and the fault area location method will be illustrated in the following subsections.

Fig. 4. The detailed processes of fault diagnose using big data theory. 5

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Fig. 5. The steps of the fault time determination method based on MSR index.

busi is related to the disturbance, and vise versa. At the same time, the suspicious buses corresponding to the maximum value of SMSR (i) (t ) has the biggest influence to the system at time t.

Table 2 The processes of fault time determination method. the sampling begin time t and the total sampling time T (k = 1) ; • Determine the 3-phase voltage amplitudes of PMUs to generate the raw data matrix X • Extract at time t = t ; the width of the real-time split-window, obtaining the real-time spilt• Determine window matrix X and normalizing it by Eq. (9); ∼ Calculate the standard matrix product Z , and obtaining its eigenvalues and the • corresponding rings by the Ring Law; ∼ r of the matrix Z according to Eq. (6); • Calculate MSR curve and compare r with reference value r . If • rDraw(tthe )
X

t

SMSR (i) (t ) =

MSR

MSRref (t )

MSRref

fault time can be determined.

Table 3 The processes of fault area determination method. PMU data is extracted to form the raw data matrix X, which is a large random • The matrix; ∼ Calculate the standard matrix product Z , the augmented matrix X and the • reference augmented matrix X ; the mean spectral radius (MSR) indexes r and their integral S • Calculate according to Eqs. (6), (14) and (15); the MSR indexes and their integral to determine the fault time and fault • Analyze area. A (i)

Dp (i) ⎤ XA (i) = ⎡ ⎢ D′e (i) ⎥ ⎣ ⎦ Dp (i) ⎤ XAref (i) = ⎡ ⎢ N(i) ⎥ ⎣ ⎦

tT

1

dMSR (i) (t ) dt

(15)

5. Case Study I: IEEE New England 39-bus system To verify the effectiveness of the proposed method, case studies are carried out based on the IEEE 39-bus system shown in Fig. 7 [43]. As for the simulation system, the simulation step size is Δt = 0.01 s, the measurement data is obtained from the simulation results with Gauss noise, whose standard deviation of amplitude is 1% and standard deviation of phase angle is 1 degrees [44,45].

Aref (i)

MSR (i)

∫t

(14)

The fault area location method based on PMU data correlation analysis can be summarized as Fig. 6. If ∃ dMSR (i) (t ) > dMSRref (i) , it indicates that there is fault occurring at the suspicious buses. Then, sort all integral values of MSR SMSR (i) (t ) from big to small, the larger the value of SMSR (i) (t ) is, the more significant the corresponding influence factors are, and the corresponding buses or lines are more likely to be the fault area.

MSR (t )

MSR (t )

X

Aref (i) A (i) dMSR (i) (t ) = rMSR (t ) − rMSR (t )

k

MSR (i)

5.1. The efficiency of the proposed method

(12)

In this case, all bus voltage amplitudes of the test system are selected to form the raw data matrix X, whose sampling rate is 100 Hz, [46] therefore, the length of the split-window is Nw = 39, and the width is Tw = 80 . An instantaneous three-phase short-circuit fault occurs on line 9–39 close to bus 9 at time t = 5.00 s, and the fault is cleared at t = 5.10 s. The system responses of partial bus voltage amplitudes and the MSR indexes are showed in Fig. 8, and the distribution of eigenvalues at different moments is showed in Fig. 9. In order to verify the immune ability to bad data of the proposed method, in this case, the voltage amplitude of bus 7 is set to zero at the time t = 3.00–3.50 s and t = 5.00–5.50 s, and the simulation results are

(13)

The difference of MSR between XA (i) and XAref (i) is defined as dMSR (i) (t ) in Eq. (14), and the integral value of the difference in the splitwindow is defined as the MSR integral SMSR (i) (t ) in Eq. (15). By observing the change of dMSR (i) (t ) curve with time, which corresponding factors are related to the system disturbance can be determined, and SMSR (i) (t ) is used to present the correlation degree of the buses to the disturbance. If dMSR (i) (t ) > dMSRref (i) , (dMSRref (i) is a reference value, which is determined by realities), it is shown that the influence factor of 6

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Fig. 6. The steps of the fault area determination method based on MSR index.

found from Fig. 10 that the MSRs of bus 7, 8, 9, 39 satisfy the equation dMSR (i) (t ) > dMSRref (i) at t = 5.00 s, and from Fig. 10, it can be concluded that the most possible fault area is near bus 9 because its MSR integral is the biggest. However, the proposed method is mainly providing the fault area judgement and not affected by bad data in time, the accurate fault location is the main direction of future research. It can be found from Figs. 11 and 12 that the bad data has no influence on the results and the proposed method can determine the fault time precisely. The reason is explained as follows, the RMT index represents the correlation between random variables in the matrix. When faults occur, the change between electrical parameters is correlated. But if there is bad data existing in PMU measurements, bad data is often not correlated with other normal data, which cannot satisfy the RMT. Consequently, the MSR curve does not change and the robustness of the proposed method is verified when there is bad data in measurements. In order to verify the reliability of the method thoroughly, 4 more situations of different fault areas are considered. Every fault occurs at time t = 5:00s, and the fault is cleared at t = 5:10s. The sampling rate is 100 Hz.

• 3-phase short short-circuit grounding fault at Line 27–26, fault point near Bus 27 • 3-phase short short-circuit grounding fault at Line 11–12, fault point near Bus 11 • 3-phase short short-circuit grounding fault at Line 24–16, fault point near Bus 24 • 3-phase short short-circuit grounding fault at Line 4–5, fault point

Fig. 7. IEEE 39-bus system.

shown in Figs. 11 and 12. At t = 5.00 s, rMSR (t ) < rMSRref , it is indicated from Fig. 8 that there is a fault occurred in power system at this time. Fault recovery time tr is corresponding to the time tr′ that rMSR (t ) recovering to normal value. In this case, rMSR (t ) is coming back to normal value at t = 5.89 s, according to that, the fault recovery time can be calculated as tr′ − (Tw − 1)·Δt = 5.89 − (80 − 1)·0.01 = 5.10 s, which is satisfied with the settlement. The Ring Law is also verified in Fig. 9. In order to locate suspicious fault area, dMSR (i) (t ) and its integer value SMSR (i) (t ) are calculated. In this case, the reference value dMSRref (i) = 0.05, which is a specifical value determined on the basis of simulation test of a large number of WAMS measured data. It can be

near Bus 4

The bus voltage amplitude, MSR curve and the MSR integral of different situations are showed as 13 (see 14): From the results of different situations we can conclude that the proposed method can determine the fault time precisely, and can locate the fault area to exact line.

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Fig. 8. Partial bus voltage amplitudes and the MSR curve.

Fig. 9. The distribution of eigenvalues before and after the fault time.

Fig. 10. The difference between the MSRs of the augmented matrix and its max integral. Fig. 11. Partial bus voltage amplitude curve and MSR curve with bad data at t = 3.00–3.50 s.

focus on the change of impedance matrix, the similar methods are carried out to calculate pre-fault and post-fault impedance matrix and then locate the fault area in [26–29]. Therefore, specific topology of power system is required as the basis of these existing methods. In order

5.2. The comparison about the proposed fault diagnose method and existing fault diagnose method The existing methods based on PMU data to locate fault area mainly 8

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Fig. 12. Partial bus voltage amplitude curve and MSR curve with bad data at t = 5.00–5.50 s.

Fig. 13. The bus voltage amplitude, MSR curve and the MSR integral of 4 different situations.

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Fig. 14. Comparison with exiting fault diagnose method based on PMU data. Table 4 The fault area diagnose result of the existing method (Stage 2: Exact fault location). Line

Distance

The matching degree

Line 24–16

59.9% from Bus 24

0.00320

Table 6 The main difference between the proposed method and the existing methods in given references.

to verify the efficiency of the proposed method in fault diagnose objectively, the comparison about the proposed fault diagnose method and the existing fault diagnose method is crucial to be carried out. Taking the fault area location approach in literature [26] as an example, the compared method uses phasor measurement unit (PMU) voltage measurements where the injected current at a fault point can be calculated by using the voltage change and its relevant transfer impedance on any bus. A two-stage fault-location optimization model is proposed in the compared approach. The first stage is the fault region identification stage, which uses the matching degree index to determine the suspicious fault region in order to reduce the search area. The second stage is used to identify the exact fault line and fault distance. The compared method from literature [26] can be repeated as Table 4:

Characteristics

Need for impedance data

Toleration for bad data

Tested in a practical grid

Reference No. The proposed approach

[47–54] No requirement

[48,50] Yes

No given Yes

Stage 1: Fault region identification.

• Establish the pre-fault impedance matrix Z ; • When there is a fault occurs, select r PMUs’ voltage measurements to calculate the matching degree delta at all buses; • Estimate the possible fault buses k , accept the lines connected with 0

k

∗ i

ki∗ as the possible fault lines.

Stage 2: Exact fault location.

Table 5 The characteristics and limitation of proposed method and the existing method. Method The proposed method

The compared method

Characteristics

Limitation

requirement for physical model of • Without system determine fault time and avoid the • Can influence of bad data locate the fault area to exact fault line • Can locate the exact fault distance to nearest • Can bus station

locate the exact fault distance to nearest bus station, need to take other method to • Cannot locate the exact fault point, which is the concern of future work

• The method is carried out under the premise of knowing the fault time 10

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Fig. 15. The simulation results for different fault type, fault impedance and fault inception angle in L9–39 near Bus 9.

branch length to bus 24 in 39-bus system is used as an example. The fault diagnose results of the proposed method and the existing method are showed as follows: From the results we can see that the proposed method can determine fault time precisely and locate the fault area to Line 24–16. However, the contrast method can select the lines 16–21, 16–19, 16–17, 16–15, and 16–24 as the most likely fault lines in its stage 1, for more exact fault area location result, with the premise of knowing the power network structure, more calculation is required in its stage 2. The comparison results of the two fault diagnose method can be concluded as Table 5. From Table 5 we can see that the proposed method is a decision-support tool for operators to determine the fault time and location at the first time. The proposed method can work without requirement for physical model of system, and can determine fault time and avoid the influence of bad data. Although it cannot locate the exact fault distance to nearest bus station, the fault location result to exact fault line is precisely enough to alarm operators of related sub-stations and take actions. More steps are needed to locate the exact fault point, which is the concern of future work. The existing method can locate the exact fault distance to nearest bus station, however, it cannot determine the exact fault time, and didn’t discuss the influence of bad data. Compared with other existing fault detection methods proposed in [47–54], the main difference between the proposed method and these methods are depicted in Table 6. In computation complexity aspect, in order solve iterative and computation demanding, a non-iterative method for wide-area fault location by taking advantage of the substitution theorem is carried out in literature [47]. The sparse estimation

Fig. 16. An actual provincial power grid topology of China.

• Select one fault line in suspicious fault region, set fault location variable x = 0 ; • Update the post-fault impedance matrix, and calculate the matching degree ∆k . Continue this process at every foot step x until x = 1; • Continue the last process at every suspicious fault line until all suspicious lines have been tested; • Estimate the fault line and location by minimizing ∆k . To compare the efficiency of the proposed method with the existing method, a three-phase short-circuit fault at branch 24–16 with 60% 11

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with bad measurements. A novel method for wide-area fault location incorporating both synchronized and unsynchronized voltage measurements is proposed in [53], which can improve fault location estimation when the synchronized measurements are sparse. However, all methods in given references require for network impedance, and the test in a practical grid is not given. It can be seen from Table 6 that the proposed method has no requirement for precise network impedance data, and is tolerated for erroneous measurement data. The proposed approach is also tested and applied in a practical grid of South China. 5.3. The influence of different fault impedance and fault inception angle The voltage phasors are considered for fault identification, in order to identify the effectiveness of the proposed method for high impedance faults, a lot more studies are carried out. These cases are with different fault impedance (0Ω, 50Ω, 100Ω) and fault inception angle, while taking the AG, AB, ABG, ABC, ABCG fault types into consideration. All short-circuit fault occurs on line 9–39 close to bus 9 at time t = 5.00 s, and the fault is cleared after 0.01s. The results of simulation are shown as Fig. 15. According to Fig. 15, the proposed method can locate the fault area precisely for different faults in Line 9–39 near Bus 9, when the fault impedance is less than 100Ω, even the fault inception angle and the fault impedance are different. However, when the fault impedance is 100Ω, although the change of voltage and current phasors is small, the max MSR integral index for other buses (such as Bus7, Bus 27 and Bus 39 which is connected to Bus 9) increase, the fault area can still be detected, the max MSR integral indexes for Bus1, Bus 7, Bus 9, Bus 27 and Bus 39 under different case situations are shown as Fig. 15. It can be seen from Fig. 15 that the proposed method can detecte fault area under different situation of different fault impendence and fault inception angle when the fault impedance is less than 100Ω. The reason is that the RMT index represents the correlation between random variables in the matrix, although fault impendence and fault inception angle are different, the proposed method can also determine the fault time and fault area precisely.

Fig. 17. PMU waveforms collected from a practical provincial power grid.

6. Case Study II: a practical provincial power grid In this section, the proposed method is verified by the PMU data recorded in a practical provincial power grid of China shown in Fig. 16. The PMU waveforms collected from a practical provincial power grid of the actual power grid is shown as Fig. 17, and there are 21 nodes in this power grid, the number of PMUs installed in each node is also shown in Fig. 16. In this case, three-phase voltage and current amplitudes of all PMUs (totally 74 sets of data, each include 6 components) are chosen to form the raw data matrix X. And the reporting rate of these data is 100 Hz. Moreover, the width of split-window in this case is chosen by Tw = 3Nw . It is worth mentioning that all the PMU data used in this section are obtained from the practical recorded measurement of the practical provincial power grid of China. It can be used for validating the feasibility of the proposed method applied in a practical power grid. A real trip event occurred on the transmission line between bus 1 and bus 2 at t = 14: 15: 52.640 , the recorded PMU waveforms are shown in Fig. 17. the MSR curve of bus 1 calculated by the proposed method is shown in Fig. 18, which include the PMUs three-phase voltage and current amplitudes data at bus 1 and bus 2. From Fig. 19, rMSR (t ) < rMSRref at t = 14: 15: 52.640 , it indicates that there is a fault occurring at this time, which is consistent with actual fault time. In order to locate the fault area, three-phase voltage and current amplitudes of all nodes in this system are chosen as the raw data, and the state data matrix Dp (i) , the influence factor matrix De (i) ′, the augmented matrix XA (i) , and the reference augment matrix XAref (i) are formed with the data, the difference of MSR dMSR (i) (t ) and its integral SMSR (i) (t ) of node i at time t is calculated by Eqs. (14) and (15). The MSR curve of practical power system’s PMU data is shown in Fig. 18. In this case, dMSRref (i) = 0.05, and the three-phase voltage attitudes of node 1

Fig. 18. The MSR curve of practical provincial power grid’s PMU data.

Fig. 19. The difference between the MSRs of the augmented matrix (Node 1).

approach and the sparse-data-driven approach are proposed for fault location in [51,54], respectively. In erroneous measurements tolerance aspect, the fault location methods in [48,50] can locate fault line even 12

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Table 7 The list of the maximum mean spectral radius integral and corresponding name of some influencing factors (Node 1). No. of influence factor

Corresponding measurement name

The maximum integral of MSR

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

Va of Line I in double circuit Line 1–2 Vb of Line I in double circuit Line 1–2 Vc of Line I in double circuit Line 1–2 Va of Line II in double circuit Line 1–2 Vb of Line II in double circuit Line 1–2 Vc of Line II in double circuit Line 1–2 Va of Line I in double circuit Line 1–4 Vb of Line I in double circuit Line 1–4 Vc of Line I in double circuit Line 1–4 Va of Line II in double circuit Line 1–4 Vb of Line II in double circuit Line 1–4 Vc of Line II in double circuit Line 1–4

80.161 0.127 0.071 1.478 0.059 0.085 1.252 0.173 0.078 1.411 0.091 0.097

it is indicated that the most possible fault area is near Bus 1. In Table 7, the maximum MSR integral of influence factor 1 and 13, the A-phase voltage amplitudes of Line 1–2 among the 6 factors is far larger than other factors, therefore, the fault area is determined at Aphase of Line 1–2, which is satisfied with the real fault situation. Consequently, the effectiveness of the proposed method can be verified. To verify the robustness of the proposed method to bad data, the bad data of WAMS recorded in a power plant is showed in Fig. 20. The proposed time determination method is used to calculate the MSR of data from 5 PMUs installed in this power plant (4 PMUs installed on the generator and 1 PMU installed on the outlet bus of the power plant). The MSR curve is showed in Fig. 21. In addition, the proposed method is also applied to other actual fault accidents in practical power system, and the results are shown in Table 8. The results of fault time determination and fault area location are consistent with the real fault recording data, which can indicate the effectiveness of the proposed method in a practical power system. Besides, a decision-support tool, which combined SCADA, FIS and WAMS data shown as Fig. 22, is programmed using the proposed method to help dispatchers obtain fault information more accurately, especially in emergency situations. In this software, the fault and protective devices’ action information is shown in the left column, dispatchers can see when and where the fault occurs in this column. The brief fault information(fault style, fault time, fault area, and fault current), SCADA date, FIS data, and WAMS data is shown in the second column. Dispatchers can see the waveforms(e.g. fault current and fault voltage waveforms and so on) by clicking the corresponding subsystems. Because the WAMS data is unified in a time scale, the fault information analyzed by SCADA, FIS, and WAMS system can be listed by time. Comparing to the separate information provided by each system, dispatchers can know and handle the fault information more accurately to prevent further development.

Fig. 20. The actual recorded PMU waveforms with bad data.

7. Discussion The proposed data-driven approach analyzes the correlation among entries of random matrix structured by PMU data, which is robust to erroneous measurements and has no requirement for precise network impedance. As a data driven method, the proposed method analyzes the relevance of the PMU measurements, and the time consuming is a little high. However, by adopting parallel computation, the consuming time of the proposed method can be reduced to an acceptable range in practical projects. Besides, when there is erroneous measurements in the PMU data, the proposed approach can identify erroneous measurements, after that, instead of recovering the PMU data, the proposed method can tolerate and distinguish it with the real fault measurements. Consequently, the proposed method can locate fault area without hypothesis testing, even though there are erroneous measurements. Furthermore, the proposed fault time and fault area location method is the first step for fault detection, it is used to assist operators detect the fault time and fault area roughly and quickly while

Fig. 21. The MSR curve with bad data.

are showed in Fig. 19. And Tab. I is the maximum MSR integral and corresponding name of some influence factors. From Fig. 18, it is indicated that the fault occurred at time t = 14: 15: 52 , and from Fig. 19, 13

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Table 8 The fault time and fault location results with practical provincial power grid PMU data in China using proposed method. Record time

Scheduling log records

The results using the proposed method

2015.04.21

21:51, No.3 Generator of Power Plant A tripped

2015.07.09

09:54, A-phase of the double circuit line II between Station A and B tripped and reclosed successfully. 09:56, A-phase of the double circuit line II between Station A and B tripped, and reclosed unsuccessfully, all phases tripped 13:23:43, C-phase of the Second Line between Station C and Station D tripped, and reclosed successfully. 13:32:02, C-phase of the Second Line between Station C and node Station tripped and reclosed successfully.

2015.07.09 2015.10.04 2015.10.04

Fault Time: 21:51:32.200, Fault Area: No.3 Generator of Power Plant A Fault Time: 09:54:14.800, Fault Area: A-phase of the Second Line between Station A and B Fault Time: 09:56:22.0, Fault Area: A-phase of the Second Line between Station A and B Fault Time: 13:23:44.160, Fault Area: C-phase of the Second Line between Station C and D Fault Time: 13:32:03.160, Fault Area: C-phase of the Second Line between Station C and D

Fig. 22. The user interface of the decision-support tool based on SCADA, FIS and WAMS data.

information floods into the dispatching center. After that, further steps and methods are adopted to determine exact fault location, which is not involved in this paper.

precisely under both simulation conditions and real power grid conditions, even though there is erroneous measurements in the data matrix. By studying the MSR curve and its integral of the augmented matrix and the reference augmented matrix, the fault area location method based on correlation analysis of PMU data can determine the fault line. In summary, The proposed data-driven fault time determination and fault area location method is robust to bad data without specific power system topologies and impedance.

8. Conclusion To overcome the limitation that existing faults diagnosis methods mainly rely on precise physical topologies and models of practical power grids and are sensitive to erroneous measurements, this paper proposes a data-driven fault time determination and fault area location method using RMT, and the MSR index is applied to analyze the correlation of the data obtained from WAMS. By analyzing MSR index and its integral of random matrix structured by PMU data, fault time can be determined precisely and fault line can be detected. Case studies are carried out in IEEE 39-bus system and a real provincial power grid of China. Simulation results show that the proposed fault time determination method using the MSR index can determine the fault time

Declaration of Competing Interest The authors claim no conflicts of interest. Acknowledgments This work was supported by National Natural Science Foundation of China under Grant No. U1866602 and 51577075.

Appendix A. Sample data sets for SCADA, PMU and FIS 1. The sample data sets for SCADA include the analogue and binary information. They contains the sequence of event (report), terminal station signals (analog and binary) and relay protection action signal (binary); 2. The sample data sets for PMU only include the analogue information. They contains the waveform of electrical quantity, such as voltage/current amplitude/angle, active/reactive power and so on; 3. The sample data sets for FIS include the analogue and binary information. They contains the waveform of electrical quantity (analog), and the relay protection action signal (binary). The sample data sets for SCADA, PMU and FIS in the practical power system can be expressed as follows:

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Fig. A.1. The sample data sets for SCADA, PMU and FIS.

A.1. The real PMU data when single-pole tripped and reclosed successfully and unsuccessfully The proposed method is based on the PMU data, and the real PMU data in a single-pole tripping with successfully reclosing and with unsuccessfully reclosing is different, which can be showed as follows: (see Fig. A.1). From Fig. A.2 (a) we can see that the fault occurs at the A-phase, and it tripped at T = 01:52:24:400, at T = 01:52:25:560, A-phase is reclosed unsuccessfully, which leads to 3-phase tripped at this time. From Fig. A.2 (b) we can see that the fault occurs at the B-phase, and it tripped at T = 16:43:57:960, at T = T = 16:43:58:920, B-phase is reclosed successfully, and the voltage of A and C phase change very slightly at the two time point. From the above analysis, we can see that the real PMU data in a single-pole tripping with successfully reclosing and with unsuccessfully 15

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Fig. A.2. The sample data sets for SCADA, PMU and FIS.

reclosing is different, therefore, the proposed method can analyze the correlation between the 3-phase-voltage measurements specify these events with raw PMU data.

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