Fault Diagnosis with Wavelet Packet Transform and Principal Component Analysis for Multi-terminal Hybrid HVDC Network

Tao Li; Yongli Li; Xiaolong Chen

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Fault Diagnosis with Wavelet Packet Transform and Principal Component Analysis for Multi-terminal Hybrid HVDC Network PDF

- ORCID：
Tao Li (Student Member, IEEE)
✉
- ORCID：
Yongli Li (Senior Member, IEEE)
✉
- ORCID：
Xiaolong Chen
✉

School of Electrical Automation and Information Engineering, Tianjin University, Tianjin 300072, China

Updated：2021-11-23

DOI：10.35833/MPCE.2021.000362

Abstract

In view of the fact that the wavelet packet transform (WPT) can only weakly detect the occurrence of fault, this paper applies a fault diagnosis algorithm including wavelet packet transform and principal component analysis (PCA) to the inverter-side fault diagnosis of multi-terminal hybrid high-voltage direct current (HVDC) network, which can significantly improve the speed and accuracy of fault diagnosis. Firstly, current amplitude and current slope are used to sample the data, and the WPT is used to extract the energy spectrum of the signal. Secondly, an energy matrix is constructed, and the PCA method is used to calculate whether the squared prediction error (SPE) statistics of various signals that can reflect the degree of deviation of the measured value from the principal component model at a certain time exceed the limit to judge the occurrence of the fault. Further, its maximum value is compared to determine the fault types. Finally, based on a large number of MATLAB/Simulink simulation results, it is shown that the PCA method using the current slope as the sampled data can detect the occurrence of a ground fault with small transition resistance within 2 ms, and identify the fault types within 10 ms, without being affected by the sampling frequency.

Keywords

Fault diagnosis; hybrid high-voltage direct current (HVDC); wavelet packet transform (WPT); principal component analysis (PCA).

I. Introduction

LINE commutated converter based high-voltage direct current (LCC-HVDC) transmission [

1] has been applied for transferring electricity from bulk power generation to load center through long-distance overhead lines. When the AC system in the load center suffers from severe short-circuit faults, commutation failure [2], [3] of the LCC inverter may occur, resulting in active power impact during the fault and reactive power impact during the recovery process. If the LCC inverters are located very close to each other, simultaneous commutation failures of several LCC inverters may occur, which is a great threat to system stability. The voltage sourced converter based HVDC (VSC-HVDC) [4]-[8] uses a fully controlled switching device such as the insulated gate bipolar transistor (IGBT). It works without commutation failure and does not need the support of the alternating current (AC) system. Therefore, it has been widely used in wind power integration and system interconnection. However, the disadvantages of VSC-HVDC system, such as high cost and inability to effectively deal with DC faults, restrict its application in long-distance and high-power transmission occasions. Considering the characteristics of the LCC-HVDC and VSC-HVDC comprehensively, by flexibly combining their advantages, a converter station can be formed containing both LCC and VSC hybrid multi-terminal HVDC transmission system [9]-[13], which will inevitably be the main development trend of HVDC transmission.

Fault diagnosis technology has always been one of the research hotspots of hybrid HVDC transmission, and some researchers have conducted in-depth research on it in HVDC transmission networks. Reference [

14] uses discrete wavelet transform (WT) and phase-mode transformation to extract the modulus maxima and criterion factor of the signal that can be used as the criterion of fault time and fault types, which can realize fault detection and fault classification within 1 ms. However, the difference in criterion factors between different types of faults is relatively small. Taking A-phase-to-ground (A_g) fault and B-phase-to-ground (B_g) fault as examples, the criterion factors of these two ground faults are 0.0732 and 0.0735, respectively, which inevitably increases the difficulty in fault classification. Reference [15] introduces a semi-supervised machine learning (SSML) method that uses co-training of decision tree as an eager learner and k-nearest neighbor as a lazy learner to handle the presence of unlabeled data. It is applied to fault classification, which can accurately realize the classification of 11 types of faults such as A_g, B_g, C_g, and so on. However, the realization of this fault detection method requires auxiliary tools such as discrete WT, harmony search algorithm, co-training of decision tree and k-nearest neighbor, which obviously increases the complexity of the detection algorithm. Meanwhile, this detection algorithm does not accurately indicate the time of fault detection. Reference [16] uses Renyi wavelet packet energy entropy (RWPEE) and Renyi wavelet packet time entropy (RWPTE) to extract the fault information from the modular multi-level converter (MMC) upper-arm current signal when the AC-side fault occurs, and uses the maximum value of RWPEE and RWPTE to judge the fault classification. However, the maximum values of RWPEE for AB-phase-to-ground (AB_g) and ABC-phase-to-ground (ABC_g) faults are about 0.2 and 0.21, respectively. Such a small difference undoubtedly increases the difficulty of fault classification. In addition, the fault algorithm proposed in this paper makes the maximum values of RWPEE and RWPTE appear at about 0.1 s when the fault occurs, which indicates that the detection algorithm takes 0.1 s to detect the fault. Reference [17] proposes a novel technique called wavelet singular entropy, incorporating the advantages of the WT, singular value decomposition, and Shannon entropy, which is immune to the noise in the fault transient and is not affected by the transient magnitude. Therefore, it can be used for fault detection and classification. However, this method needs to detect the fault within 200 ms, and uses the singular entropy with a small gap as the basis for fault classification. For example, the singular entropies of A-phase and AB-phase are 0.9789 and 0.9158, respectively. Reference [18] applies the power component fault detection algorithm with reliability factors to the LCC-HVDC network. When

p_{o n} > K_{P} p_{o n}

u_{o n} > K_{U} u_{M 0}

, and

i_{o n} > K_{I} i_{M 0}

, the single-phase-to-ground fault is detected; when

Δ p_{t h n} > K_{P t h} Δ p_{N t h}

u_{t h n} < K_{U t h} u_{N t h}

, and

i_{t h n} > K_{I t h} i_{N t h}

, the three-phase-to-ground fault is detected, which will have more favorable characteristics than other detection methods, taking only the characteristics of voltage or current under the fault condition into consideration. However, this method can only detect single-phase-to-ground fault and three-phase-to-ground fault.

Based on the shortcomings of the above-mentioned AC-side fault diagnosis algorithm, this paper proposes a fault diagnosis algorithm including wavelet packet transform (WPT) [

19], [20] and principal component analysis (PCA) [21] algorithm, and applies them to the inverter-side fault diagnosis of hybrid HVDC network. Firstly, the amplitude and slope of the d-axis component of the AC current on the inverter side are sampled, and WPT is used to perform dB3 wavelet packet decomposition on the sampled signal to obtain the energy spectrum value of the signal. Secondly, the energy matrix is constructed, the energy matrices of the health and fault signals are taken as the training set and test set, respectively, and the mean of the training set is subtracted from the test set and divided by the variance of the training set as the input of the PCA method. Finally, it is calculated whether squared prediction error (SPE) statistics of various signals exceed the limit to judge the occurrence of fault, and its maximum value is used as the criterion for fault classification. Furthermore, a large number of simulation results show that the method proposed in this paper can effectively solve the problems of inaccurate fault detection criteria, slow detection speed, and unclear fault type judgment in some fault detection algorithms. The significance of the proposed method is mainly reflected in three aspects: ① it improves the speed and accuracy of fault detection; ② it lays a foundation for the subsequent research on the coordinated fault ride-through control strategy of multi-VSC subsystem; ③ it perfects the fault handling capacity of multi-terminal hybrid HVDC transmission system.

The remaining sections of this paper are arranged as follows. In Section II, a multi-terminal hybrid LCC-VSC-HVDC network and its AC-side fault characteristics analysis are introduced. Section III introduces the principles of WPT and PCA in detail, and combines the two algorithms to put forward the flow of the fault diagnosis algorithm in this paper. In Section IV, a multi-terminal hybrid LCC-VSC-HVDC network is built in MATLAB/Simulink software, and a large number of simulation experiments are carried out to verify the practicability of the fault diagnosis algorithm proposed in this paper. Finally, Section V concludes this paper.

II. TOPOLOGY AND Fault Characteristics ANALYSIS

As mentioned in the previous section, this section mainly has three parts, including topology introduction, analysis of fault current on the inverter side, and analysis of AC transient voltage on the rectifier side.

A. Topology

A typical back-to-back HVDC system connecting with three AC system is studied in this subsection. Figure 1 shows the system outline, where one AC grid is connected to others via three HVDC converters including the rectifier-side LCC1 and inverter-side VSC1 and VSC2. Among them, LCC1 obtains the 600 MW active power provided by the power supply on one side through a 220 kV/150 kV step-down transformer, and VSC2 and VSC3 send the obtained power to the grid on the other side through two 150 kV/220 kV step-up transformers. U_s is the voltage of the bus with an AC filter; L and R are the equivalent inductance and resistance of the transmission line, respectively; L_dc is the current-limiting reactor of the DC line; and VT represents the voltage transformer.

Fig. 1 Three-port transmission network based on LCC-VSC-HVDC.

B. Analysis of Fault Current on Inverter Side

The structure of inverter station VSC1 and VSC2 in Fig. 1 is basically similar, so this paper takes VSC1 as the object to study the fault characteristics of inverter side. Since the output voltage of VSC1 is an AC voltage, the adjacent circuit output by VSC1 can be equivalent to an AC electromotive force ${\dot{E}}_{1}$ with internal impedance Z_s₁, as shown in Fig. 2. In addition, the line impedance is equivalent to Z_L₁ and Z_L₂ with fault point k, and the grid-side topology is equivalent to the equivalent circuit with electromotive force ${\dot{E}}_{2}$ and internal impedance Z_s₂.

Fig. 2 Equivalent circuit on inverter side.

Assuming that the fault resistance is R_f, the equivalent ectromotive force ${\dot{E}}_{e q}$ and equivalent impedance Z_eq in the topology shown in Fig. 2 are given as:

\{\begin{array}{l} {\dot{E}}_{e q} = \frac{{\dot{E}}_{2} (Z_{s 1} + Z_{L 1}) + {\dot{E}}_{1} (Z_{L 2} + Z_{s 2})}{Z_{s 1} + Z_{L 1} + Z_{L 2} + Z_{s 2}} \\ Z_{e q} = \frac{(Z_{s 1} + Z_{L 1}) (Z_{L 2} + Z_{s 2})}{Z_{s 1} + Z_{L 1} + Z_{L 2} + Z_{s 2}} \end{array}

(1)

Assuming that a three-phase short-circuit fault occurs at k, the equivalent circuit shown in Fig. 3(a) can be obtained, and the fault current can be obtained as:

I_{k f}^{3} = \frac{{\dot{E}}_{e q}}{R_{f} + Z_{e q}}

(2)

Fig. 3 Equivalent circuit under fault condition. (a) Three-phase short-circuit fault. (b) Single-phase-to-ground fault. (c) Two-phase-to-ground fault.

In the same way, the equivalent circuits of the single-phase-to-ground fault and the two-phase-to-ground fault shown in Fig. 3(b) and 3(c) can be obtained. Among them, the fault current of the single-phase-to-ground fault is:

I_{k f}^{1} = \frac{3 {\dot{E}}_{e q}}{3 R_{f} + Z_{e q (1)} + Z_{e q (2)} + Z_{e q (0)}}

(3)

where Z_eq₍₁₎, Z_eq₍₂₎, and Z_eq(0) are positive-, negative-, and zero-equivalent impedances, respectively.

The positive-, negative-, and zero-sequence currents of a two-phase-to-ground fault are:

\{\begin{array}{l} I_{k f (1)}^{(1.1)} = \frac{{\dot{E}}_{e q}}{Z_{e q (1)} + \frac{Z_{e q (2)} (Z_{e q (0)} + 3 R_{f})}{Z_{e q (2)} + Z_{e q (0)} + 3 R_{f}}} \\ I_{k f (2)}^{(1.1)} = - \frac{Z_{e q (0)} + 3 R_{f}}{Z_{e q (2)} + Z_{e q (0)} + 3 R_{f}} I_{k f (1)}^{(1.1)} \\ I_{k f (0)}^{(1.1)} = - \frac{Z_{e q (2)}}{Z_{e q (2)} + Z_{e q (0)} + 3 R_{f}} I_{k f (1)}^{(1.1)} \end{array}

(4)

Therefore, the fault phase current are:

\{\begin{array}{l} I_{k f b}^{(1.1)} = [a^{2} - \frac{Z_{e q (2)} + a (Z_{e q (0)} + 3 R_{f})}{Z_{e q (2)} + Z_{e q (0)} + 3 R_{f}}] I_{k f (1)}^{(1.1)} \\ I_{k f c}^{(1.1)} = [a - \frac{Z_{e q (2)} + a^{2} (Z_{e q (0)} + 3 R_{f})}{Z_{e q (2)} + Z_{e q (0)} + 3 R_{f}}] I_{k f (1)}^{(1.1)} \end{array}

(5)

\{\begin{array}{l} a = - \frac{1}{2} + j \frac{\sqrt[]{3}}{2} \\ a^{2} = - \frac{1}{2} - j \frac{\sqrt[]{3}}{2} \end{array}

(6)

In summary, (2), (3), and (5) can be used to analyze the fault current when different types of faults occur on the inverter side. For stationary components, there is $Z_{e q (1)} = Z_{e q (2)}$ . Take single-phase-to-ground and three-phase-to-ground faults as examples to analyze the fault current characteristics. Figure 3 shows the distribution of fault current under different positive- and zero-sequence impedances. Among them, Fig. $4 (a)$ is the front three-dimensional waveform, and Fig. $4 (b)$ and 4(c) are detailed waveforms on the left and right sides of Fig. 4(a), respectively.

Fig. 4 Relationship between single-phase fault current and positive- and zero-sequence impedances. (a) Front view. (b) Left-side view. (c) Right-side view.

As can be observed in Fig. 4, the magenta surface and the green surface are the single-phase and three-phase fault current waveforms, and the curve at the intersection of the magenta surface and the green surface satisfies $Z_{e q (1)} = Z_{e q (0)}$ , i.e., the single-phase short-circuit fault is equal to the three-phase short-circuit current based on (2) and (3). Meanwhile, we can see that when $Z_{e q (1)} > Z_{e q (0)}$ , i.e., the red surface in Fig. $4 (b)$ , the single-phase short-circuit current is greater than the three-phase short-circuit current. On the contrary, when $Z_{e q (1)} < Z_{e q (0)}$ , i.e., the blue surface in Fig. 4(c), the single-phase short-circuit current is less than the three-phase short-circuit current. Therefore, there are certain shortcomings in selecting the fault phase only by current amplitude.

C. Analysis of AC Transient Voltage on Rectifier Side

Through the above analysis, it can be known that the inverter-side current rises after a fault on the inverter side, which in turn causes the DC side current to increase rapidly. Under this situation, the rectifier side will rapidly increase the firing angle to suppress the increase of DC current, which will cause the reactive power consumed by the rectifier station to rise rapidly. Under the adjustment of the controller, the DC current will rapidly decrease to the limit value, and the reactive power consumed by the sending-end converter station will also decrease. However, a large number of AC filters are still in normal operation, resulting in a large amount of reactive power being sent to the sending-end power grid, which further causes the transient over-voltage of the power grid at the sending-end [

22].

To facilitate the analysis, the simplified DC equivalent circuit of the sending-end shown in Fig. 5 can be obtained. Among them, E_s is the AC electromotive force of the sending-end system; X_s is the equivalent reactance of the system; U_s is the AC bus voltage; P and Q are the transmitted active and reactive power, respectively; and Q_c is the reactive power compensated by the AC filter.

Fig. 5 Equivalent circuit on rectifier side.

The AC bus voltage at steady state is:

U_{s} = \sqrt[]{{(E_{s} - \frac{Q X_{s}}{E_{s}})}^{2} + {(\frac{P X_{s}}{E_{s}})}^{2}}

(7)

If the loss of system impedance is not considered, $E_{s} = U_{s}$ can be considered. Therefore, based on (7), we can get:

Q = [P^{2} + {(\frac{P^{2} X_{s}}{E_{s}^{2}})}^{2}] \frac{X_{s}}{E_{s}^{2}}

(8)

The HVDC transmission systems usually use AC filters as reactive power compensation devices, and its compensation capacity is mainly related to bus voltage, i.e., when the voltage rises, the reactive power compensation capacity of the AC filter will also increase. Under this situation, the reactive power delivered by the DC system to the AC system can be expressed as [

23]:

\hat{Q} = {(\frac{{\hat{U}}_{s}}{U_{s}})}^{2} Q_{c} - Q_{m i n}

(9)

where ${\hat{U}}_{s}$ is the transient voltage of the AC bus; and Q_min is the minimum reactive power consumed by the rectifier station.

Meanwhile, the transient voltage of the AC bus satisfies the following relationship:

{\hat{U}}_{s} = \sqrt[]{(E_{s} + Δ E_{s})^{2} + (δ E_{s})^{2}}

(10)

where $Δ E_{s}$ is the longitudinal component of the voltage drop; and $δ E_{s}$ is the transverse component of the voltage drop.

\{\begin{array}{l} Δ E_{s} = \hat{Q} \frac{X_{s}}{E_{s}} \\ δ E_{s} = P \frac{X_{s}}{E_{s}} \end{array}

(11)

In the transient process, the converter station and the AC system only have a small amount of active power transmission. To simplify the calculation, the influence of the transverse component of the voltage drop can be ignored, and the equation is solved:

{\hat{U}}_{s} = \frac{1 - \sqrt[]{1 - 4 \frac{Q_{c} X_{s}}{U_{s}^{2} E_{s}} (E_{s} - \frac{Q_{m i n} X_{s}}{E_{s}})}}{2 \frac{Q_{c} X_{s}}{U_{s}^{2} E_{s}}}

(12)

Equation (12) shows that the transient voltage of the sending-end AC system is mainly related to Q_c and Q_min when a fault occurs on the inverter side. Therefore, the following study will focus on analyzing the relationship between them.

Figure 6 shows the relationship among the transient voltage of the sending-end AC system, Q_c, and Q_min. As can be observed, when the inverter-side system fails, excessive reactive power provided by the AC filters will increase the bus voltage of the sending-end AC system, which will inevitably affect the normal operation of the equipment connected to the sending-end AC system.

Fig. 6 Relationship among transient voltage of sending-end AC system, Q_c, and Q_min.

III. Fault Diagnosis Algorithm

The analysis in Section II shows that the current of the receiving-end AC system and the bus voltage of the sending-end AC system will increase when the inverter-side system fails, which will have an important impact on the entire power transmission system. Therefore, there is an urgent need for fast and effective fault detection algorithms to improve the fault handling speed of the HVDC system. Based on this, this section will apply the WPT and PCA methods to detect the fault.

Although WT [

24] can achieve good time- frequency analysis, it only decomposes the low-frequency components of the signal and ignores its high-frequency components. As a natural extension of WT, WPT can further decompose the unprocessed high-frequency signal of WT, so as to realize more refined signal analysis. The frequency band is divided into multiple levels, and it can also be selected adaptively according to the characteristics of the signal to be studied, so that it matches the signal spectrum, thereby improving the corresponding time-frequency resolution. For the purpose of improving the frequency resolution, the wavelet subspace needs to be subdivided in frequency according to the binary fraction. The common method is to use the new subspace U_d^j to uniformly characterize the scale subspace V_d and the wavelet subspace W_d.

\{\begin{array}{l} U_{d}^{0} = V_{d} \\ U_{d}^{1} = W_{d} \end{array} d \in Z

(13)

Then the orthogonal decomposition V_d=V_d+₁⊕W_d₊₁ can be unified with the decomposition of U_d^j as:

U_{d}^{0} = U_{d + 1}^{0} \oplus U_{d + 1}^{1} d \in Z

(14)

Define the subspace U_d^j as the closed space of the function u_j(t) and $U_{d}^{2 j}$ as the closed space of the function u₂_j(t), and make u_j(t) satisfy the following two-scale equation:

\{\begin{array}{l} u_{2 j} (t) = \sqrt[]{2} \sum_{k \in Z} h (k) u_{j} (2 t - k) \\ u_{2 j + 1} (t) = \sqrt[]{2} \sum_{k \in Z} g (k) u_{j} (2 t - k) \end{array}

(15)

where h(k) and g(k) are the orthogonal mirror filter banks associated with the scaling function U_d⁰ and the mother wavelet function $U_{d}^{1}$ , respectively.

g (k) = {(- 1)}^{k} h (1 - k)

(16)

The main purpose of applying WPT to a certain signal is to extract feature information through wavelet packet decomposition and reconstruction. The decomposition and reconstruction algorithm of wavelet packet is as follows.

Assuming $g_{d}^{j} (t) \in u_{d}^{j}$ , g_d^j(t) is expressed as:

g_{d}^{j} (t) = \sum_{l \in Z} q_{l}^{d, j} u_{j} (2^{j} t - l)

(17)

The wavelet packet decomposition is to use {q_l^d^+1,^j} to solve {q_l^d^,2^j} and {q_l^d^,2^j+¹}:

\{\begin{array}{l} q_{l}^{d, 2 j} = \sum_{k \in Z} h_{k - 2 l} q_{k}^{d + 1, j} \\ q_{l}^{d, 2 j + 1} = \sum_{k \in Z} g_{k - 2 l} q_{k}^{d + 1, j} \end{array}

(18)

On the contrary, the wavelet packet reconstruction algorithm uses ${{q_{l}}^{d, 2 j}}$ and ${{q_{l}}^{d, 2 j + 1}}$ to solve ${{q_{l}}^{d + 1, j}}$ :

q_{l}^{d + 1, j} = \sum_{k \in Z} (h_{l - 2 k} q_{k}^{d, 2 j} + g_{l - 2 k} q_{k}^{d, 2 j + 2})

(19)

When decomposing the dyadic orthogonal wavelet, the length of the high- and low-frequency signals obtained by each decomposition is half the length of the signal before decomposition, and the sum of the length of the high- and low-frequency signals is equal to that of the signal before decomposition. WPT is similar to this, except that the same decomposition is performed on the high-frequency components. For discrete digital signal x_n, $n = 1,2, \dots, N$ , a total of 2^j wavelet packets can be obtained in the j^th layer after wavelet packet decomposition.

In the AC-side fault detection of the hybrid HVDC network, the detection signal is first subjected to wavelet packet decomposition, and 2^j decomposition vectors can be obtained. These vectors contain the information of different frequency bands, and the fault information is among them. The energy spectrum algorithm based on WPT is to express the decomposition result of the wavelet packet in the form of energy. The energy can be marked by the square sum of each frequency band signal, which provides a prerequisite for the implementation of fault detection. The following takes three-layer wavelet packet decomposition as an example to illustrate the method of extracting energy spectrum features.

1) The signal is subjected to three-layer wavelet packet decomposition to obtain a characteristic signal composed of decomposition coefficients containing eight frequency band components from low frequency to high frequency.

2) Reconstruct the decomposition coefficients obtained above, and extract the signal S₃^j ( $j = 0,1, \dots, 7$ ) of each frequency band.

3) Calculate the energy value of each sub-band signal, and its expression is as follows:

E_{3}^{j} = \sum_{k = 1}^{N} ω_{j k}^{2}^{}

(20)

where $ω_{j k}$ ( $j = 0,1, \dots, 7, k = 0,1, \dots, 7$ ) is the amplitude of the discrete point of the reconstructed signal $S_{3}^{^{j}}$ .

4) Use energy as an element to construct a feature vector, as shown below.

T = [\begin{matrix} E_{3}^{0} & E_{3}^{1} & \dots & E_{3}^{7} \end{matrix}]

(21)

The WPT is used to extract the energy spectrum value of the health signal and the fault signal, which can be used to detect the occurrence of a fault. However, for similar faults, taking single-phase-to-ground fault or two-phase-to-ground fault as an example, the wavelet energy spectrum partially overlaps, which may lead to some mistakes for this method when distinguishing fault types. Based on this, this paper introduces the PCA method into the fault detection based on the WPT, which can amplify the fault signals with weak gaps and facilitate the identification of different types of faults.

IV. PCA Method

As a multivariate statistical method, PCA [

25]-[28] can realize that the information in a few variables contains most of the critical information of the original data by reducing the data dimension. In other words, PCA can realize the dimensionality reduction processing of high-dimensional data space. The principle is as follows.

Sampling a column vector x consisting of m variables for n times, make the sample vector obtained from each sampling as x_i ( $i = 1,2, \dots, n$ ), and take $X = [x_{1}, x_{2}, \dots, x_{n}] \in R^{n \times m}$ . The covariance matrix is:

S = c o v (x) \approx \frac{1}{n - 1} X^{T} X S \in R^{m \times n}

(22)

The PCA model decomposes X as:

T = X p

(23)

where $T = [t_{1}, t_{2}, \dots, t_{m}]$ ( $T \in R^{n \times m}$ , $t_{i} \in R^{n \times 1}$ ) is the score matrix, each column of which is the principal element formed by the projection of the process variable on the main hyperplane variable; and $p = [p_{1}, p_{2}, \dots, p_{m}]$ $(p \in R^{m \times m}, p_{i} \in R^{m \times 1})$ is the load matrix.

The following steps of the PCA method are used to detect faults.

1) Take the standardized health signal as the training set and the fault signal as the test set, and the test set is used to subtract the mean value of the training set and to divide the variance of the training set:

X_{i j} = (x_{i j} - {\bar{y}}_{i j}) / S_{i j}

(24)

where x_ij is the test set sample; y_ij is the training set sample; ${\bar{y}}_{i j}$ is the mean value of the training set; and S_ij is the variance of the training set ( $i, j = 1,2, \dots, m$ ).

2) Calculate the correlation coefficient matrix R between the data variables obtained in the above steps:

R = [\begin{matrix} r_{11} & r_{12} & \dots & r_{1 m} \\ r_{21} & r_{22} & \dots & r_{2 m} \\ ⋮ & ⋮ & ⋮ \\ r_{m 1} & r_{m 2} & \dots & r_{m m} \end{matrix}]

(25)

where r_ij is the correlation coefficient between the variables X_i and X_j, and $r_{i j} = r_{j i}$ .

r_{i j} = \frac{\sum_{k = 1}^{n} (X_{k i} - {\bar{X}}_{i}) (X_{k j} - {\bar{X}}_{j})}{\sqrt[]{\sum_{k = 1}^{n} (X_{k i} - {\bar{X}}_{i})^{2} \sum_{k = 1}^{n} (X_{k j} - {\bar{X}}_{j})^{2}}} i, j = 1,2, \dots, m

(26)

3) Solve the eigenvalues of the matrix R and arrange them in ascending order, denoted as $λ_{1}, λ_{2}, \dots, λ_{m}$ , and their corresponding eigenvectors are $p_{1}, p_{2}, \dots, p_{m}$ .

4) Calculate the principal elements:

t_{i} = X p_{i}

(27)

where t_i is the projection of the data matrix X in the direction of the corresponding eigenvector p_i. Among them, the longer the projection length, the greater its range of variation.

5) Use the following equation to calculate the cumulative contribution rate of each principal component:

η = \frac{\sum_{k = 1}^{i} λ_{k}}{\sum_{k = 1}^{m} λ_{k}} i = 1,2, \dots, m

(28)

Usually the value range of $η$ is 85%-95%.

PCA, as a statistical analysis method, often uses multivariate statistical hypothesis testing to monitor abnormal processes, of which the commonly used are the SPE and Hotelling T2 statistics.

The SPE statistic is also called the Q statistic, which characterizes the deviation degree of the measured value from the principal component model. The definition of SPE statistics is:

S P E (x) = x (I - \bar{p} {\bar{p}}^{T}) x^{T}

(29)

where x is the x^th data sample in the data matrix; and I is the identity matrix. The control limits of the SPE statistics are calculated as:

S P E = θ_{1} {[\frac{C_{α} \sqrt[]{2 θ_{2} h_{0}^{2}}}{θ_{1}} + \frac{θ_{2} h_{0} (h_{0} - 1)}{θ_{1}^{2}}]}^{\frac{1}{h_{0}}}

(30)

\{\begin{array}{l} h_{0} = 1 - \frac{2 θ_{1} θ_{3}}{3 θ_{2}^{2}} \\ θ_{i} = \sum_{j = i + 1}^{m} λ_{j}^{i} i = 1,2, 3 \end{array}

(31)

where $λ_{i}$ is the eigenvalue in the covariance matrix; and $C_{α}$ is the limit value when the normal distribution is satisfied and the confidence level is $α$ .

The T2 statistic represents the fluctuation of the principal component vector within the principal component model, and is defined as:

T 2 (x) = x \bar{p} Λ^{- 1} {\bar{p}}^{T} x^{T}

(32)

where $Λ$ is the diagonal matrix formed by the eigenvalues corresponding to the first k principal elements. The calculation formula for the control limit of T2 statistics is expressed as:

T 2 = \frac{l (n - 1)}{n - l} F (l, n - l, α)

(33)

where $F (l, n - l, α)$ is the critical point of F distribution when the test level is $α$ and the degrees of freedom are l and $n - l$ .

The use of T2 and SPE statistics for fault detection will produce four detection results.

1) T2 and SPE both exceed the control limit.

2) T2 exceeds the control limit, while SPE does not.

3) SPE exceeds the control limit, while T2 does not.

4) Neither T2 nor SPE exceeds the control limit.

When the detection result is the case 4, it means that there is no fault in the system. When changes in the system cause correlation damage to some variables, the value of SPE will change significantly, corresponding to the test results of (1) and (3). When a certain change occurs in the system with little effect on the correlation between variables, the value of T2 will have a large change, and the value of SPE will be within the normal range, corresponding to the test result of (2). Therefore, when a fault occurs in the system, the first three detection results are likely to appear, which means that as long as one of the three detection results occurs, it can be determined that there is a fault in the system.

In summary, the flow chart of the fault diagnosis algorithm are shown in Algorithm 1. It mainly includes the following two major steps: the first step is to use WPT to obtain the energy spectrum value of the signal, and the second step is to use the PCA method to calculate the SPE and T2 statistics of the energy spectrum value. Furthermore, from Table I, the state variables of the PCA method are $X = (T_{f a u l t} - X_{m e a n}) / X_{s t d}$ with variance X_std, which can obviously amplify the characteristic information of the fault signal.

TABLE I Simulation Parameters

Device	U_dc (kV)	P (MW)	L_dc (H)	R_line	L_line	R (mΩ)	L (mH)	C (mF)	I_dcref (kA)	η	α
LCC1	300 (1 p.u.)	600 (2 p.u.)	0.05	0.0139 $Ω / k m \times 150$ km	0.159×10^-3 $H / k m \times 150$ km	75	25	2	2 (2 p.u.)	0.85	0.95
VSC1	300 (1 p.u.)	300 (1 p.u.)	0.05	0.0139 $Ω / k m \times 150$ km	0.159×10^-3 $H / k m \times 150$ km	75	25	2	1 (1 p.u.)	0.85	0.95
VSC2	300 (1 p.u.)	300 (1 p.u.)	0.05	0.0139 $Ω / k m \times 150$ km	0.159×10^-3 $H / k m \times 150$ km	75	25	2	1 (1 p.u.)	0.85	0.95

Algorithm 1: flow chart of fault diagnosis algorithm
1 Sampling S1
2 $w p t = w p d e c (S 1, d b 3)$
3 $ω_{j} = w p r c o e f (w p t, [3, j])$
4 $E_{3}^{j} = \sum_{k = 1}^{N} ω_{j k}^{2}$
5 $T_{h e a l t h y / f a u l t} = [E_{3}^{0}, E_{3}^{1}, \dots, E_{3}^{7}]$
6 $X_{m e a n} = m e a n (T_{h e a l t h y})$
7 $X_{s t d} = s t d (T_{h e a l t h y})$
8 $X = (T_{f a u l t} - X_{m e a n}) / X_{s t d}$
9 $[P, T, L A T E N T, T S Q U A R E D] = p r i n c o m p (X)$
10 $k = 0$
11 $p e r c e n t = i n p u t$ (‘ $η$ ’)
12 for $i = 1 : s i z e (L A T E N T, 1)$
13 $η (i) = s u m (L A T E N T (1 : i)) / s u m (L A T E N T)$
14 if $η (i) \geq p e r c e n t$
15 $k = i$
16 break
17 end
18 end
19 $X_{p} = z e r o s (s i z e (X))$
20 for $j = 1 : k$
21 $X_{p} = X_{p} + T (:, j) P (:, j)$ ’
22 end
23 $b e t a = i n p u t$ (‘α’)
24 $θ = z e r o s (4,1)$
25 for $i = 1 : 3$
26 for $j = k + 1 : s i z e (X, 2)$
27 $θ (i) = θ (i) + L A T E N T {(j)}^{(i)}$
28 end
29 end
30 $h_{0} = 1 - 2 θ (1) θ (3) / [3 (θ {(2))}^{2}]$
31 $S P E_{b e t a} = θ (1) {(n o r m i n v (b e t a) \frac{(2 θ (2) h_{0}^{2})^{\frac{1}{2}}}{θ (1)} + 1 + \frac{θ (2) h_{0} (h_{0} - 1)}{(θ {(1))}^{2}})}^{\frac{1}{h_{0}}}$
32 $T 2_{b e t a} = \frac{f i n v (b e t a, k, (s i z e (X, 1) - k)) k ((s i z e {(X, 1))}^{2} - 1)}{s i z e (X, 1) (s i z e (X, 1) - k)}$

V. VERFICATION

To verify the effectiveness of the fault diagnosis algorithm proposed in this paper, a three-port 300 kV LCC-VSC-HVDC transmission network shown in Fig. 1 is built based on MATLAB/Simulink software. The detailed parameter settings are shown in Table I.

The following subsections will present the research conducted from four aspects: fault diagnosis based on current amplitude, fault diagnosis based on current slope, the influence of different sampling frequency and different ground resistances on diagnosis method, and comparison with other methods.

A. Fault Diagnosis Based on Current Amplitude

In the simulation, the active power from LCC1-side is given as 600 MW and the power of the inverter stations VSC1 and VSC2 are both set to be 300 MW. Furthermore, the DC voltage is set to be 300 kV (1 p.u.), and then the DC current of the rectifier stations LCC1 is 2 kA, and the DC currents of the inverter stations VSC1 and VSC2 are both 1 kA. Meanwhile, the inverter stations VSC1 and VSC2 are symmetrical, so this paper takes VSC1 station as the object to study the fault diagnosis algorithm. For the purpose of detecting AC faults quickly and accurately, this paper uses the d-axis component of the AC current (I_d₁) output by the inverter station VSC1 as the sampling data.

We select 4000 sets of current data (I_d₁) in two cycles (0.04 s) at the moment of the fault occurrence (0.5 s) as the samples detected by the algorithm (sampling frequency is 100 kHz). By decomposing and reconstructing the sampled signal by dB3 layer wavelet packet, we can obtain the reconstruction coefficients in eight frequency intervals of 0-12.5 kHz, 12.5-25 kHz, 25-37.5 kHz, 37.5-50 kHz, 50-62.5 kHz, 62.5-75 kHz, 75-87.5 kHz, and 87.5-100 kHz. Figure 7 shows the reconstruction coefficients of different signals after dB3 layer decomposition. As can be observed, the original signal is basically consistent with the reconstruction coefficient curve in the frequency range of 0-12.5 kHz. To study the high-frequency information of the fault signal in depth, this paper uses the reconstruction coefficient in the frequency range of 12.5-100 kHz to form the energy spectrum shown in Fig. 8.

Fig. 7 Reconstruction coefficient of dB3 layer decomposition of current signal. (a) Healthy. (b) A_g. (c) B_g. (d) C_g. (e) AB_g. (f) BC_g. (g) AC_g. (h) ABC_g.

Fig. 8 Energy spectrum based on current amplitude. (a) Healthy. (b) A_g. (c) B_g. (d) C_g. (e) AB_g. (f) BC_g. (g) AC_g. (h) ABC_g.

From Fig. 8, the maximum energy level of the health signal is 10^-7, the maximum energy level of the single-phase-to-ground signal is 10^-6, and the maximum energy level of the two-phase-to-ground and three-phase-to-ground signals is 10^-5, which is impossible to accurately judge the fault type through the value of the energy spectrum. Meanwhile, the difference of the same type of fault signals is small, which makes the work of fault identification more difficult.

Therefore, this paper constructs the energy matrix shown in (34) and uses it as the input of the PCA method, and further uses whether T2 and SPE statistics in this method exceed the limit to judge the occurrence of fault.

T = [\begin{matrix} E_{3}^{1} & E_{3}^{2} & \dots & E_{3}^{7} \end{matrix}]

(34)

Figure 9 shows the T2 and SPE statistics curves of various signals.

Fig. 9 T2 and SPE statistics based on current amplitude. (a) T2. (b) SPE.

In Fig. 9, the area below the red curve indicates that no failure occurs, and the area above the red curve indicates that a failure occurs. In order to quickly detect faults and identify fault types, we select a 10 ms simulation waveform after the fault occurs for analysis.

As can be observed, the curve of T2 statistic is basically below the red line in the selected period, which indicates that the fault cannot be detected by using T2 statistic as the fault criterion. Therefore, SPE statistic is selected as the criterion of fault diagnosis. However, when the current amplitude is used as the sampling data in Fig. 9, there may be some types of faults that cannot be detected. Take A_g fault as an example. The curve of SPE statistics within 10 ms is basically below the red line, which indicates that the system has no fault. Based on this, this paper selects the current slope as the sampling data. To summarize, we will explore whether the fault can be accurately detected by selecting the current slope as the sampling data.

B. Fault Diagnosis Based on Current Slope

To compensate for inaccurate fault detection caused by the use of current amplitude as sampling data, we further use the current slope as sampling data to verify the fault diagnosis algorithm proposed in this paper.

Figure 10 shows the energy spectrum curves. Similarly, it is difficult to judge the same type of faults by using the energy spectrum. Therefore, consistent with the above method, PCA method is also used to calculate T2 and SPE statistics.

Fig. 10 Energy spectrum based on current slope. (a) Healthy. (b) A_g. (c) B_g. (d) C_g. (e) AB_g. (f) BC_g. (g) AC_g. (h) ABC_g.

Figure 11 and Table II show the curves and values of the SPE statistics using the current slope as the sampling data. As can be observed, the difference of the T2 statistics of various signals is small. Therefore, consistent with the above analysis, we also use the SPE statistics as the fault diagnosis criterion. Furthermore, the difference from Fig. 9 is that the SPE statistics curve in Fig. 11 is obviously above the red curve within the selected 10 ms, and can basically surpass the red curve obviously within 2 ms. Meanwhile, the maximum value of SPE statistics of various signals within 10 ms has the following relationship: $S P E_{A B g} > S P E_{B C g} > S P E_{A B C g} > S P E_{C g} > S P E_{A C g} > S P E_{A g} > S P E_{B g}$ . The above analysis can show that using the current slope as sampling data to apply the fault diagnosis method proposed in this paper can detect the fault and distinguish the fault type obviously within 2 ms and 10 ms, respectively.

Fig. 11 T2 and SPE statistics based on current slope. (a) T2. (b) SPE.

TABLE II SPE Statistics Based on Current Slope

Phase	Samples at the initial time of detection	SPE_max	Samples (SPE_max)
A_g	296	$1.357 \times 10^{4}$	817
B_g	232	$9.262 \times 10^{3}$	993
C_g	0	$1.085 \times 10^{5}$	995
AB_g	152	$1.351 \times 10^{6}$	665
BC_g	0	$8.073 \times 10^{5}$	363
AC_g	0	$5.955 \times 10^{4}$	1000
ABC_g	0	$3.687 \times 10^{5}$	350

To summarize, the fault diagnosis process is shown in Fig. 12. For the purpose of exploring whether the fault diagnosis method proposed in this paper is applicable, the following subsection will focus on the influence of sampling frequency and ground resistance on the method proposed in this paper.

Fig. 12 Flow chart of fault diagnosis.

C. Influence of Sampling Frequency and Ground Resistance

Figure 13 and Appendix A show the simulation diagrams and values of SPE statistics based on current slope at different sampling frequencies. As can be observed, the fault can be detected in 5 ms when the sampling frequency is 10 kHz, 20 kHz, 40 kHz, and 50 kHz, especially for two-phase-to-ground and three-phase-to-ground faults. This method can detect the fault in 2 ms. Meanwhile, the maximum values of SPE statistics of various signals under four sampling frequencies satisfy the following relationship: $S P E_{A B g} > S P E_{B C g} > S P E_{A B C g} > S P E_{C g} > S P E_{A C g} > S P E_{A g} > S P E_{B g}$ , which is consistent with the results of the above analysis. In summary, the fault diagnosis method proposed in this paper is not affected by the sampling frequency, and the corresponding threshold can be set in order by the maximum value of the SPE statistics of different signals to determine the fault type.

Fig. 13 SPE statistics based on current slope under different sampling frequencies. (a) Waveform of the maximum SPE statistics when $f_{s} = 10$ kHz. (b) Waveform of SPE statistics at detected fault occurrence time when $f_{s} = 10$ kHz. (c) Waveform of the maximum SPE statistics when $f_{s} = 20$ kHz. (d) Waveform of SPE statistics at detected fault occurrence time when $f_{s} = 20$ kHz. (e) Waveform of the maximum SPE statistics when $f_{s} = 40$ kHz. (f) Waveform of SPE statistics at detected fault occurrence time when $f_{s} = 40$ kHz. (g) Waveform of the maximum SPE statistics when $f_{s} = 50$ kHz. (h) Waveform of SPE statistics at detected fault occurrence time when $f_{s} = 50$ kHz.

Figure 14 and Appendix B show the simulation curves and values of SPE statistics for different ground resistances. From them, when $R_{f} \leq 1$ $Ω$ , the method proposed in this paper can detect the fault within about 2 ms, and the maximum value of the SPE statistics of various signals within 10 ms also satisfies the following relationship: $S P E_{A B g} > S P E_{B C g} > S P E_{A B C g} > S P E_{C g} > S P E_{A C g} > S P E_{A g} > S P E_{B g}$ . When $R_{f} = 10$ $Ω$ , a single-phase-to-ground fault can be detected within 10 ms, and a two-phase and three-phase-to-ground fault can also be detected within 2 ms. However, the maximum value of SPE statistics has the following relationship in this situation: $S P E_{A B C g} > S P E_{A C g} > S P E_{B C g} > S P E_{A B g} > S P E_{C g} > S P E_{B g} > S P E_{A g}$ , which is inconsistent with the results of the above analysis. When $R_{f} = 50$ $Ω$ , $100$ $Ω$ , and $300$ $Ω$ , the method in this paper needs to detect the fault within 20 ms, and the maximum value of SPE statistics cannot be used to distinguish the fault type.

Fig. 14 SPE statistics based on current slope under different ground resistances. (a) Waveform of the maximum SPE statistics when $R_{f} = 0.01$ $Ω$ . (b) Waveform of SPE statistics at detected fault occurrence time when $R_{f} = 0.01$ $Ω$ . (c) Waveform of the maximum SPE statistics when $R_{f} = 0.1$ $Ω$ . (d) Waveform of SPE statistics at detected fault occurrence time when $R_{f} = 0.1$ $Ω$ . (e) Waveform of the maximum SPE statistics when $R_{f} = 1$ $Ω$ . (f) Waveform of SPE statistics at detected fault occurrence time when $R_{f} = 1$ $Ω$ . (g) Waveform of the maximum SPE statistics when $R_{f} = 10$ $Ω$ . (h) Waveform of SPE statistics at detected fault occurrence time when $R_{f} = 10$ $Ω$ . (i) $R_{f} = 50$ $Ω$ . (j) $R_{f} = 100$ $Ω$ . (k) $R_{f} = 300$ $Ω$ .

ⅤI. CONCLUSION

Taking $R_{f} = 300$ $Ω$ as an example, the waveform curve in Fig. 14(k) is above the red curve for less time, which indicates that there may be some disturbances in the system instead of failure, which further increases the difficulty in fault detection. Furthermore, the difference between the maximum values of the SPE statistics of various signals is small, and it cannot be used to accurately detect the fault type.

Based on the above analysis, the method proposed in this paper can detect and distinguish a ground fault with small transition resistance ( $R_{f} \leq 1$ $Ω$ ) within 2 ms and 10 ms, respectively. For high resistance faults, the method proposed in this paper has certain drawbacks.

In summary, Table III shows the threshold settings for different fault types. Combining the threshold setting results in Table III and the division of fault types in Fig. 12, the corresponding fault types can be roughly judged. In addition, for a mature system, the server backend should have substantial test data. As long as these detailed data are combined with the method proposed in this paper, the corresponding fault type can be accurately judged.

TABLE III Threshold Settings for Different Fault Types

Fault	SPE_max
A_g and B_g	$S P E_{m a x} < 5 \times 10^{4}$
C_g and AC_g	$5 \times 10^{4} < S P E_{m a x} < 5 \times 10^{5}$
AB_g, BC_g, and ABC_g	$5 \times 10^{5} < S P E_{m a x} < 5 \times 10^{6}$

D. Comparison with Other Methods

To further highlight the advantages of the proposed method in this aspect of fault classification, Table IV provides comparative data of several methods. From them, the method proposed in [

14] shows that the threshold criteria for single-, two-, and three-phase-to-ground faults are greater than 0.05, less than 0.05, and equal to 0, respectively. Compared with the method proposed in this paper, the threshold criterion of the method in [14] is too small, and it may not be able to accurately determine the fault type. Meanwhile, this method has smaller differences in the criteria for the same type of fault. Taking the A_g fault and B_g fault as examples, the threshold criteria are 0.0732 and 0.0735, respectively. Similarly, the threshold criterion of the method proposed in [17] is also relatively small, and it is difficult to use the threshold rule to judge the fault type.

TABLE IV Comparison with Other Methods

Fault	This paper	Reference [14]	Reference [17]
A_g	$1.357 \times 10^{4}$	0.0732	0.9789
B_g	$9.262 \times 10^{3}$	0.0735	0.7952
C_g	$1.085 \times 10^{5}$	0.0774	1.7580
AB_g	$1.351 \times 10^{6}$	0.0142	0.9158
BC_g	$8.073 \times 10^{5}$	0.0037	1.7589
AC_g	$5.955 \times 10^{4}$	0.0433	1.7593
ABC_g	$3.687 \times 10^{5}$	0	1.7586

In view of the defect that the fault diagnosis algorithm based on WPT can only weakly distinguish healthy operation and fault status, this paper uses current slope as sampling data, and combines WPT and PCA method to propose a fast and accurate fault diagnosis algorithm to improve the fault detection ability of HVDC system. This paper lays a foundation for the subsequent research on the coordinated fault ride-through control strategy of multi-terminal hybrid HVDC network. The following conclusions can be drwan from the simulation.

1) The proposed method can detect the ground fault with small transition resistance ( $R_{f} \leq 1$ $Ω$ ) within 2 ms, and it is not affected by the sampling frequency.

2) The method proposed in this paper can judge the fault type within 10 ms by using the threshold value in Table III and the order of the maximum values of the following SPE statistics: $S P E_{A B g} > S P E_{B C g} > S P E_{A B C g} > S P E_{C g} > S P E_{A C g} > S P E_{A g} > S P E_{B g}$ . However, this method has certain defects in detecting high-resistance faults, i.e., the larger the ground resistance is, the less obvious the fault detection and discrimination will be.

3) Compared with the other method, the advantage of the method proposed in this paper is that the threshold criterion has certain rules and is more obvious.

Appendix

APPENDIX A

TABLE AI Schedule A: SPE Statistics Based on Current Slope with

f_{s} = 10

kHz

Fault	Samples at the initial time of detection	SPE_max	Samples (SPE_max)
A_g	72	$3.442 \times 10^{4}$	81
B_g	30	$2.183 \times 10^{4}$	97
C_g	80	$2.721 \times 10^{5}$	97
AB_g	24	$3.780 \times 10^{6}$	65
BC_g	0	$2.255 \times 10^{6}$	33
AC_g	0	$1.691 \times 10^{5}$	99
ABC_g	0	$1.049 \times 10^{6}$	33

TABLE AII Schedule A: SPE Statistics Based on Current Slope with

f_{s} = 20

kHz

Fault	Samples at the initial time of detection	SPE_max	Samples (SPE_max)
A_g	61	$2.712 \times 10^{4}$	161
B_g	56	$2.169 \times 10^{4}$	200
C_g	12	$1.970 \times 10^{5}$	193
AB_g	32	$2.762 \times 10^{6}$	129
BC_g	0	$1.599 \times 10^{6}$	65
AC_g	0	$1.030 \times 10^{5}$	194
ABC_g	0	$7.541 \times 10^{5}$	65

TABLE AIII Schedule A: SPE Statistics Based on Current Slope with

f_{s} = 40

kHz

Fault	Samples at the initial time of detection	SPE_max	Samples (SPE_max)
A_g	112	$1.656 \times 10^{4}$	329
B_g	104	$1.140 \times 10^{4}$	393
C_g	0	$1.391 \times 10^{5}$	393
AB_g	64	$1.716 \times 10^{6}$	257
BC_g	0	$1.023 \times 10^{6}$	137
AC_g	0	$7.150 \times 10^{4}$	393
ABC_g	0	$4.728 \times 10^{5}$	137

TABLE AIV Schedule A: SPE Statistics Based on Current Slope with

f_{s} = 50

kHz

Fault	Samples at the initial time of detection	SPE_max	Samples (SPE_max)
A_g	110	$1.584 \times 10^{4}$	410
B_g	113	$1.057 \times 10^{4}$	498
C_g	0	$1.279 \times 10^{5}$	490
AB_g	72	$1.636 \times 10^{6}$	329
BC_g	0	$9.755 \times 10^{5}$	177
AC_g	0	$7.251 \times 10^{4}$	497
ABC_g	0	$4.471 \times 10^{5}$	169

APPENDIX B

TABLE BI Schedule B: SPE Statistics Based on Current Slope with

R_{f} = 0.01

Ω

Fault	Samples at the initial time of detection	SPE_max	Samples (SPE_max)
A_g	72	$3.437 \times 10^{4}$	81
B_g	32	$2.192 \times 10^{4}$	97
C_g	80	$2.720 \times 10^{5}$	98
AB_g	24	$3.756 \times 10^{6}$	65
BC_g	0	$2.245 \times 10^{6}$	33
AC_g	0	$1.735 \times 10^{5}$	97
ABC_g	0	$1.046 \times 10^{6}$	33

TABLE bII Schedule B: SPE Statistics Based on Current Slope with

R_{f} = 0.1

Ω

Fault	Samples at the initial time of detection	SPE_max	Samples (SPE_max)
A_g	72	$3.008 \times 10^{4}$	81
B_g	30	$2.156 \times 10^{4}$	97
C_g	80	$2.760 \times 10^{5}$	98
AB_g	18	$3.529 \times 10^{6}$	65
BC_g	0	$2.154 \times 10^{6}$	33
AC_g	0	$2.512 \times 10^{5}$	97
ABC_g	0	$1.018 \times 10^{6}$	33

TABLE biII Schedule B: SPE Statistics Based on Current Slope with

R_{f} = 1

Ω

Fault	Samples at the initial time of detection	SPE_max	Samples (SPE_max)
A_g	72	$2.857 \times 10^{4}$	81
B_g	26	$1.898 \times 10^{4}$	97
C_g	80	$2.242 \times 10^{5}$	97
AB_g	17	$1.852 \times 10^{6}$	65
BC_g	0	$1.229 \times 10^{6}$	33
AC_g	0	$2.176 \times 10^{5}$	97
ABC_g	0	$7.762 \times 10^{5}$	33

TABLE bIv Schedule B: SPE Statistics Based on Current Slope with

R_{f} = 10

Ω

Fault	Samples at the initial time of detection	SPE_max	Samples (SPE_max)
A_g	26	$2.116 \times 10^{2}$	89
B_g	24	$9.256 \times 10^{2}$	57
C_g	80	$9.282 \times 10^{3}$	97
AB_g	24	$1.933 \times 10^{4}$	97
BC_g	0	$9.031 \times 10^{4}$	97
AC_g	0	$3.660 \times 10^{5}$	74
ABC_g	0	$7.867 \times 10^{5}$	97

TABLE BV Schedule B: SPE Statistics Based on Current Slope with

R_{f} = 50

Ω

Fault	Samples at the initial time of detection	SPE_max	Samples (SPE_max)
A_g	184	47.44	185
B_g	144	77.35	146
C_g	144	142.50	145
AB_g	56	101.60	73
BC_g	0	155.70	9
AC_g	40	73.14	49
ABC_g	0	199.50	9

TABLE BVI Schedule B: SPE Statistics Based on Current Slope with

R_{f} = 100

Ω

Fault	Samples at the initial time of detection	SPE_max	Samples (SPE_max)
A_g	No	7.092	218
B_g	152	15.270	153
C_g	144	16.850	145
AB_g	64	6.969	65
BC_g	176	19.360	178
AC_g	192	60.080	201
ABC_g	160	82.040	177

TABLE BVII Schedule B: SPE Statistics Based on Current Slope with

R_{f} = 300

Ω

Fault	Samples at the initial time of detection	SPE_max	Samples (SPE_max)
A_g	145	4.501	146
B_g	144	20.140	145
C_g	144	7.042	161
AB_g	No	6.240	281
BC_g	No	11.420	201
AC_g	176	7.635	177
ABC_g	88	5.634	189

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