Open AccessArticle

A Method of Discriminating Between Power Swings and Faults Based on Principal Component Analysis

Power Internet of Things Key Laboratory of Sichuan Province, State Grid Sichuan Electrical Power Research Institute, Chengdu 610041, China

State Grid Sichuan Electrical Power Company, Chengdu 610041, China

Author to whom correspondence should be addressed.

Appl. Sci. 2025, 15(5), 2867; https://doi.org/10.3390/app15052867

Submission received: 2 January 2025 / Revised: 3 March 2025 / Accepted: 4 March 2025 / Published: 6 March 2025

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Browse Figures

Figure 1
Schematic diagram of PCA processing results. "> Figure 2
Projection trajectories under different disturbances. (a) Three-phase shorting fault; (b) power swing. "> Figure 3
The value of C. "> Figure 4
Flow chart of blocking protection relay. "> Figure 5
Flow chart of deblocking the protection relay. "> Figure 6
Sketch of simulation system. "> Figure 7
Trajectory of measured impedance during power swing. "> Figure 8
The waveform of the three-phase voltage under a power swing of 2 Hz. "> Figure 9
The value of C under a power swing of 2 Hz. "> Figure 10
Sketch of 400 kV system. "> Figure 11
The waveform of the three-phase voltage under a fault. "> Figure 12
The value of C under a fault. "> Figure 13
Waveforms of voltage and current. "> Figure 14
ΔP under different sampling frequencies of power swings. "> Figure 15
ΔP under different sampling frequencies for faults. ">

Versions Notes

Abstract

Distance protection is widely applied in AC transmission systems. It may operate incorrectly under power swings, so a power swing blocking unit (PSBU) is needed to work with the distance protection relay. Such a unit should not only block the protection relay in time when a power swing occurs, but also deblock the protection relay after detecting a fault during the power swing. In this paper, a method that satisfies these requirements is proposed. To discriminate between power swings and faults, the characteristics of three-phase voltage under a power swing and fault situation are used. Principal Component Analysis (PCA) is applied to extract and quantify the characteristics. To detect faults during power swings, an index is proposed, and the change rate of the index is used to form the criterion. Simulations for different kinds of power swing and fault situations are conducted based on a two-end system and a nine-bus system in PSCAD/EMTDC. The simulation test results indicate that the proposed method can block the protection relay reliably under a power swing and deblock the relay quickly after detecting a fault during the power swing. Moreover, the proposed method is compared with other methods. The comparison results show that the proposed method has an advantage in terms of response speed and is less affected by measurement noise.

Keywords:

power swing; three-phase shorting fault; power swing blocking and deblocking; principal component analysis

1. Introduction

Distance relays are widely applied in AC transmission systems because of their simple operating principles, which utilize the measuring impedance. During power swings, the measured impedance oscillates. It may also enter into the operational zone of the distance relay, leading to undesirable tripping. Therefore, a power swing blocking unit (PSBU) is needed to work with a distance relay [1,2,3]. A complete PSBU takes on two tasks: blocking the distance relay after detecting a power swing (PSB), and then unblocking it after detecting a fault during the power swing (PSD). In order to accomplish these tasks, extensive studies have been conducted.

In asymmetric fault situations, asymmetric faults and power swings can be readily discriminated, due to the presence of negative- and zero-sequence components. Therefore, many studies have aimed to discriminate between three-phase symmetrical faults and power swings. Conventional methods are based on the concept that there is no sudden change in the voltage or current of a transmission line during a power swing. The authors of [4] utilized the change rate of the measured impedance to discriminate power swings and faults. To accomplish this discrimination, two concentric impedance characteristics, separated by impedance

Δ Z

, were placed on the impedance plane, and a timer was used to time the duration of the impedance locus as it traveled between them. However, complex studies are needed to obtained a suitable time set and value of

Δ Z

. Tracking of the change rate of the swing center voltage (SCV) was conducted in [5,6]. However, these methods assumed a 90° impedance angle for each component in the system during derivation, which typically requires compensation in practical applications, complicating the process. The change rates of active and reactive power were used in [7]. However, this method required an extended period of time to differentiate between power swings and faults, particularly in the case of three-phase symmetrical faults.

Apart from conventional methods, applications of novel techniques in PSBUs have also been reported in the literature. Signal processing techniques, such as the wavelet transform [8,9,10], wavelet singular entropy [11], the Hilbert–Huang transform [12], the Prony method [13], and the fast Fourier transform [14], have been introduced for identifying power swings. Although the reported results are satisfactory, some of these techniques require high sampling frequencies or complex calculations. Some training-based techniques have also been applied for the identification of power swings. The adaptive fuzzy neural inference system was used in [15,16], support vector machine (SVM) was applied in [17], and a discrimination scheme based on a decision tree (DT) was shown in [18]. In [19], a method based on a combination of the S-transform and a probabilistic neural network (PNN) was proposed. In [20], another combination of a multi-resolution morphological gradient and SVM was used. Test results were used to validate the effectiveness of the methods mentioned above, but they also required some training procedures to be employed.

In addition, some other novel methods have also been proposed for PSBUs. A mathematical morphology (MM) with a low computational burden was applied in [21,22]. The characteristics of the Lissajous figure were introduced in [23]. In [24], a combination of the Lissajous figure and the auto-regression technique was used. Although the test results indicated that the auto-regression model performed well, the requirements of finding suitable parameters for it and using a relatively high sampling frequency (200 sampling points in one cycle) make the method less attractive. Estimation- and error-calculation-based methods were introduced in [25,26,27]. The authors of [25] tracked the difference between the predicted and actual power using the auto-regression technique. In [26], the values of the current dynamic phasor were estimated to discriminate between faults and power swings. In [27], the difference between the estimated value and the measured actual value of the current was calculated to form a criterion. The test results showed that these methods performed well. However, the researchers did not consider the effect of measurement noise. In fact, measurement noise reduces the reliability of these methods. In [28], the magnitude of the positive-sequence current and its phase difference with the positive-sequence voltage were used to discriminate between faults and power swings. However, due to power swings, accurate phase measurements cannot be obtained, which may lead to misjudgment. In [29], the covariance of current signals was calculated to discriminate between power swings and faults. Although the results of different tests indicated that the method proposed in [29] performed well, its reliability and operational speed are limited by communication, as it needs current information for both sides. In [30], the application of Principal Component Analysis (PCA) for distinguishing between line faults and power swings was explored. Based on different fault and power swing cases, PCA was used to extract features and form criteria for classifying faults and power swings, thereby achieving differentiation between faults and power swings. However, the method proposed in [30] required a large number of fault and power swing cases.

A novel method, based on the characteristics of three-phase voltage and current, that can accomplish the PSB and PSD tasks of a PSBU is proposed in this study. Given the ability of PCA to simplify data complexity while retaining key features, PCA is utilized to process three-phase current and voltage, thereby enabling a simpler and more intuitive extraction of current and voltage characteristics under different scenarios to form corresponding criteria. For the PSB function, PCA is used to reduce the dimensionality of three-phase voltage, and an index C_avr is proposed, based on the processed voltage data, to distinguish power swings and faults. For the PSD function, three-phase voltage and current data are processed by using PCA in the same way. Based on the processed voltage and current data, the index P_UI is proposed to detect a fault occurring during a power swing. In contrast to the method described in [30], which also employs PCA, the method proposed here does not necessitate extensive cases. The performance of the proposed methods for PSB and PSD functions are tested by using the simulation data obtained from test systems built in PSCAD/EMTDC. The test results indicate that the proposed method can distinguish power swings and faults reliably and quickly detect a fault during a power swing, being minimally impacted by noise.

The remainder of the paper is organized as follows: PCA is introduced in Section 2. The proposed method is introduced in Section 3, including its basic principles and criteria and the flow charts of PSB and PSD. Simulation tests of the proposed method are shown in Section 4. Finally, conclusions are drawn in Section 5.

2. Principal Component Analysis

PCA is designed to transform the data into a reduced form and to keep most of the original variance present in the initial data. In mathematical terms, PCA transforms the original data into a set of uncorrelated representations with fewer dimensions through a linear transformation, thereby extracting the main characteristic components of the data. From a geometric perspective, PCA projects the original data from the original coordinate system onto a new coordinate system with smaller dimensions by maximizing the divergence of the original data along the orthogonal directions in the new coordinate system. The main calculation steps of PCA are as follows [31].

Define X as an

n \times m

matrix which contains m number of n-dimensional vectors:

X = [x_{1}, x_{2}, \dots, x_{m}]

(1)

where

x_{k} = {[x_{1 k}, x_{2 k}, \dots, x_{n k}]}^{T}

, k = 1, 2, …, m.

Define

\bar{X}

as follows:

\bar{X} = [\begin{matrix} x_{11} - μ_{1} & x_{12} - μ_{1} & \dots & x_{1 m} - μ_{1} \\ x_{21} - μ_{2} & x_{22} - μ_{2} & \dots & x_{2 m} - μ_{2} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ x_{n1} - μ_{n} & x_{n 2} - μ_{n} & \dots & x_{nm} - μ_{n} \end{matrix}]

(2)

where

μ_{l} (L = 1, 2, \dots, n)

is the average value of the data in the lth row of matrix X.

Define matrix M as follows:

M = \frac{1}{m} \bar{X} {\bar{X}}^{T}

(3)

Calculate the eigenvalues of M and arrange them in descending order. Take the first l eigenvalues for calculating the corresponding eigenvectors and orthogonalize them to obtain the orthogonal unit vectors

ξ_{1}, ξ_{2}, \dots, ξ_{l}

. These l unit vectors can be called the principal components (PCs), and

x_{k}

can be expressed as follows:

x_{k} \approx q_{1 k} ξ_{1} + q_{2 k} ξ_{2} + \dots + q_{lk} ξ_{l}

(4)

where

q_{ik} (i = 1, 2, \dots, l)

is the coefficient.

In addition, define matrix P as follows:

P = {[ξ_{1}, ξ_{2}, \dots, ξ_{l}]}^{T}

(5)

The coordinates of the original data in the new coordinate system which applies these l PCs as the coordinate axes can be obtained by pre-multiplying matrix X by matrix P.

In order to display the relevant calculation steps and the effect of PCA on data more conveniently, consider an example of three points in the rectangular coordinate system shown in Figure 1.

As Figure 1 shows, the coordinates of the three points are A(1, 1), B(1, −1), and C(−1, −1). These three points cannot be discriminated by using the abscissa or ordinate alone. A matrix X of size

2 \times 3

is formed by using the coordinates of these three points, and the eigenvalues (

λ_{1} = 4 / 3, λ_{2} = 4 / 9

) are calculated based on (2), (3), and (5). The eigenvector corresponding to

λ_{1}

is calculated as

ξ_{1} = {[\sqrt{2} / 2, \sqrt{2} / 2]}^{T}

. In the one-dimensional coordinate system with

ξ_{1}

as the basis vector, the corresponding coordinates of the three points are obtained as

\sqrt{2}

, 0, and

- \sqrt{2}

, respectively. From the calculated results, PCA retains the characteristics of the original data when it reduces the dimensionality of the original data, and it makes the difference between the three points easier to distinguish.

3. Power Swing Blocking Method Using PCA

3.1. Basic Principle

For the scenarios of normal system operation, power swings, and line faults, the waveform of the voltage of the transmission line exhibits different characteristics under each of these scenarios. In other words, the three-phase voltage of the transmission line, to some extent, reflects the current state of the transmission line. If the instantaneous values of three-phase voltage are considered as the coordinates of a point in a three-dimensional coordinate system, then the set of points represented by a series of instantaneous three-phase voltage values under different scenarios will have different characteristics in their distribution within the three-dimensional coordinate system. However, it is not easy to directly observe the distribution of the points corresponding to the three-phase voltage in a three-dimensional coordinate system. In order to better observe the different distribution characteristics of the points corresponding to the three-phase voltage under different operating scenarios, a data processing method is needed to reduce the dimensionality of the three-dimensional data corresponding to the three-phase voltage. As analyzed in the second part of our study, PCA can reduce the dimensionality of the data while preserving the main features of the original data. PCA is employed here to process the three-phase voltage data, projecting the three-dimensional coordinate points represented by the three-phase voltage data onto a two-dimensional coordinate system for subsequent analysis.

To obtain the desired two-dimensional coordinate system, the three-phase voltage of the transmission line during normal operation is used. During normal operation, the three-phase voltage in the time domain can be expressed as follows:

\{\begin{array}{l} u_{A} = M \cos (ω_{0} t + φ) \\ u_{B} = M \cos (ω_{0} t + φ + \frac{4}{3} π) \\ u_{C} = M \cos (ω_{0} t + φ + \frac{2}{3} π) \end{array}

(6)

where M is the amplitude of the voltage,

ω_{0}

is the power frequency, and

φ

is the initial phase of phase A voltage.

Taking the three-phase voltage sampling points over one cycle as three row vectors, matrix X can be constructed as follows:

X = [\begin{matrix} x_{A}^{T} \\ x_{B}^{T} \\ x_{C}^{T} \end{matrix}]

(7)

where the superscript “T” denotes the transpose of the matrix, and

x_{A}

x_{B}

, and

x_{C}

are three column vectors, and can be expressed as follows:

x_{A} = [\begin{matrix} M \cos φ \\ M \cos (ω_{0} Δ t + φ) \\ ⋮ \\ M \cos [ω_{0} (N - 1) Δ t + φ] \end{matrix}]

(8)

x_{B} = [\begin{matrix} M \cos (φ + \frac{4}{3} π) \\ M \cos (ω_{0} Δ t + φ + \frac{4}{3} π) \\ ⋮ \\ M \cos [ω_{0} (N - 1) Δ t + φ + \frac{4}{3} π] \end{matrix}]

(9)

x_{C} = [\begin{matrix} M \cos (φ + \frac{2}{3} π) \\ M \cos (ω_{0} Δ t + φ + \frac{2}{3} π) \\ ⋮ \\ M \cos [ω_{0} (N - 1) Δ t + φ + \frac{2}{3} π] \end{matrix}]

(10)

In the expressions of

x_{A}

x_{B}

, and

x_{C}

Δ t

is the sampling interval and N is the number of sampling points over a cycle. According to (7)–(10), each column of matrix X represents the sampling values of three-phase voltage at a certain moment, which also corresponds to the coordinates of a point in a three-dimensional coordinate system.

Since the data are sampled over a cycle during normal operation of the power system, the average value of each row in X can be regarded as 0. From the definition of matrix

\bar{X}

shown in (2), matrix X in (7) can be used directly as

\bar{X}

in the PCA calculation process.

\bar{X} {\bar{X}}^{T}

, used in PCA calculation process, can be expressed as follows:

\bar{X} {\bar{X}}^{T} = X X^{T} = [\begin{matrix} x_{A}^{T} x_{A} & x_{A}^{T} x_{B} & x_{A}^{T} x_{C} \\ x_{B}^{T} x_{A} & x_{B}^{T} x_{B} & x_{B}^{T} x_{C} \\ x_{C}^{T} x_{A} & x_{C}^{T} x_{B} & x_{C}^{T} x_{C} \end{matrix}]

(11)

According to (8)–(10),

X X^{T}

is a symmetric matrix, and

x_{A}^{T} x_{A} = x_{B}^{T} x_{B} = x_{C}^{T} x_{C}

, where

x_{A}^{T} x_{A}

can be calculated as follows:

x_{A}^{T} x_{A} = \frac{M^{2}}{2} \sum_{k = 0}^{N - 1} [\cos (2 ω_{0} k Δ t) + 1]

(12)

Since the value of

M^{2} / 2 \sum_{k = 0}^{N - 1} \cos (2 ω_{0} k Δ t)

can be regarded as 0, it is found that

x_{A}^{T} x_{A}

is equal to

N M^{2} / 2

Similarly,

x_{A}^{T} x_{B}

x_{A}^{T} x_{C}

, and

x_{B}^{T} x_{C}

can be calculated as follows:

x_{A}^{T} x_{B} = x_{A}^{T} x_{C} = x_{B}^{T} x_{C} = - N M^{2} / 4

(13)

Therefore, the eigenvalues of M are

λ_{1, 2} = 3 / 2

and

λ_{3} = 0

. The eigenvectors corresponding to

λ_{1, 2}

are

ξ_{1} = \sqrt{2} / 2 {[- 1, 0, 1]}^{T}

and

ξ_{2} = \sqrt{6} / 6 {[- 1, 2, - 1]}^{T}

, respectively.

Vectors

ξ_{1}

and

ξ_{2}

are the vector basis required for constructing a new two-dimensional coordinate system. Using the directions of

ξ_{1}

and

ξ_{2}

, respectively, as the directions of the X and Y axes, a new coordinate system, denoted by XOY_U, is established.

Define matrix P as follows:

P = [\begin{matrix} ξ_{1}^{T} \\ ξ_{2}^{T} \end{matrix}] = [\begin{matrix} - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \\ - \frac{\sqrt{6}}{6} & \frac{\sqrt{6}}{3} & - \frac{\sqrt{6}}{6} \end{matrix}]

(14)

The corresponding coordinates of the three-dimensional points, whose coordinate values are represented by the column vectors of matrix X shown in (7), in XOY_U can be calculated as follows:

[\begin{matrix} x_{U} \\ y_{U} \end{matrix}] = P X

(15)

where

x_{U}

and

y_{U}

are Nth-order row vectors, and the expressions for each element of vector

x_{U}

and

y_{U}

are as follows:

\{\begin{matrix} x_{U} (k) = \frac{\sqrt{6}}{2} M \sin (ω_{0} k Δ t + \frac{4}{3} π) \\ y_{U} (k) = \frac{\sqrt{6}}{2} M \cos (ω_{0} k Δ t + \frac{4}{3} π) \end{matrix}

(16)

where

k = 0, 1, 2, \dots, N - 1

It is found from (16) that during normal operation, the distance from the coordinate point to the coordinate origin in XOY_U remains unchanged. Moreover, the distance is proportional to the amplitude of the voltage.

Considering the fact that the amplitude of the voltage will become small within a short time after a three-phase shorting fault occurs, but will change slowly during a power swing, the distance mentioned above will become small in a short period of time after a three-phase fault occurs, but will change slowly during a power swing. The projection results shown in Figure 2 also confirm the analysis. In this figure, the blue line represents normal operation conditions and the red line represents disturbance conditions. Figure 2a shows the results of a three-phase shorting fault, and Figure 2b shows the results of a power swing with an oscillation frequency of 5 Hz. Thus, the distance can be used to quantify the difference between the power swing and fault conditions.

3.2. Criterion for PSB and PSD Function

Suppose that the instantaneous values of the three-phase voltage at time t are

u_{A} (t)

u_{B} (t)

, and

u_{C} (t)

. Based on the expressions of

ξ_{1}

and

ξ_{2}

in section A of this chapter, the coordinates of the three-phase voltage at this moment in XOY_U can be calculated as follows:

\{\begin{array}{l} x_{U} = \frac{\sqrt{2}}{2} [u_{C} (t) - u_{A} (t)] \\ y_{U} = \frac{\sqrt{6}}{6} [2 u_{B} (t) - u_{A} (t) - u_{C} (t)] \end{array}

(17)

where

x_{U}

and

y_{U}

represent the abscissa and ordinate, respectively.

Define C as follows:

C = m (x_{U}^{2} + y_{U}^{2})

(18)

where m is a coefficient which makes C equal to 100 during normal operation, and it can be calculated as

200 / (3 M^{2})

in advance, according to (16).

As the above analysis shows, the value of C reduces to a small value within a short time after a three-phase shorting fault occurs, but it changes slowly during a power swing. Therefore, by comparing the threshold value with the value of C, the three-phase shorting fault and the power swing can be discriminated. The criterion for the PSB function is as follows:

C_{avr} > C_{set}

(19)

where

C_{set}

is the threshold, and

C_{avr}

is the average value of C, corresponding to three sampling points after a half-cycle delay. If the calculated result satisfies (19), the disturbance should not be regarded as a fault, and the protection relay should be blocked.

Considering the two tasks of the PSBU, another criterion for the PSD function is also needed. Based on the above analysis, the change rate of C can be used to detect most faults under a power swing. However, in some special cases, the effect of using the change rate of C is not good. For the following case under consideration, the fault occurs at the oscillation center, whose voltage amplitude becomes close to 0 when the fault occurs. At this moment, there is no sudden change in the voltage amplitude, nor in the value of C shown in Figure 3.

Therefore, in order to detect a three-phase shorting fault during a power swing more reliably, another index is proposed here. Similarly, the current sampling data are processed by using PCA to obtain the coordinates, denoted by

x_{I}

and

y_{I}

. The index P_UI(k) is defined as follows:

P_{U I} (k) = x_{U} (k) x_{I} (k) + y_{U} (k) y_{I} (k)

(20)

The criterion for the PSD function is as follows:

Δ P = |P_{U I} (k) - P_{U I} (k - 1)| > P_{set}

(21)

where

P_{set}

is the threshold. If (21) is satisfied, the PSBU will deblock the protection relay.

3.3. Flow Chart of PSB and PSD Function

The flow chart of the PSB function, using the criterion mentioned in Section 3.2., is shown in Figure 4. As shown in the figure, the starting unit uses the instantaneous value of three-phase current to judge the starting time. The criterion of starting unit is as follows:

|i_{ψ}| > I_{set}

(22)

where

ψ = A, B, C

. As Figure 4 shows, after the starting unit operates, the value of

C_{avr}

is calculated after a half-cycle delay. If (19) is satisfied, the PSBU blocks the protection relay. After blocking the protection relay, the PSBU enters into the process of PSD. The flow chart of the PSD function is shown in Figure 5.

As Figure 5 shows, after blocking the protection relay, the PSBU continues calculating P_UI and

Δ P

by using the three-phase voltage and current data. If (21) is not satisfied, the value of P_UI at this moment is saved for the next calculation. If (21) is satisfied, the PSBU deblocks the protection relay. Otherwise, the value of P_UI at this moment is saved for the next calculation.

4. Simulation Tests

4.1. Test Results Based on Two-End System

For testing the performance of the proposed method, a two-end system is built in PSCAD/EMTDC. The sketch of the system is shown in Figure 6. In this figure,

{\dot{E}}_{m}

and

{\dot{E}}_{n}

are the equivalent power sources at two ends, where

{\dot{E}}_{m} = 500 ∠ 33 ° kV

and

{\dot{E}}_{n} = 500 ∠ 0 ° kV

Z_{m}

and

Z_{n}

are the equivalent impedances of the systems on the two sides, where

Z_{m} = Z_{n} = 28.07 ∠ 86 ° Ω

. M is the measuring point. The length of the transmission line is 300 km, and the detail of its parameters are given in Table 1.

During the simulation, the power frequency is 50 Hz and the sampling frequency is 4 kHz. The value of the threshold

C_{set}

is set to avoid the fault occurring at the end of the line, while considering a certain margin upwards. Since the value of

C_{avr}

after a three-phase shorting fault occurs at the end of the line is 61.57,

C_{set}

in (19) is set to 70. In addition, the value of the threshold

P_{set}

is set based on the corresponding calculation result of

Δ P

under the fault occurring at the end of the line, while considering a certain margin downwards. Since the value of

Δ P

after a three-phase shorting fault occurs at the end of the line during a power swing is 204.89,

P_{set}

is set to 100. In addition, I_set in (22) is set to 1.1 times the peak value of the current during normal operation. Based on the two-end system shown above, power swings of different oscillation frequencies and different three-phase shorting fault situations are simulated to test the performance of the proposed method.

The frequency of the system on the right side is gradually changed from 50 Hz to 48 Hz, resulting in a power swing with an oscillation frequency of 2 Hz. Taking the distance protection with directional circle characteristics as an example, the setting value of the distance protection I is set to 80% of the line length. The impedance circle is shown in Figure 7. The blue line in the figure indicates the trajectory of the measured impedance after the power swing occurs, with the direction of movement being from right to left. From Figure 7, it can be seen that the measured impedance enters the operation zone of Zone I of the distance protection, and remains there for 138.3 ms after the power swing, resulting in a mal-operation of the distance protection. Therefore, there is a necessity to identify oscillations to prevent mal-operation of the distance protection.

Figure 8 shows the waveform of the three-phase voltage under the power swing of 2 Hz. The corresponding value of C is shown in Figure 9. The power swing occurs at 1.2 s; 6 ms later (1.206 s, “

t_{1}

” in Figure 9), the instantaneous current value exceeds the set threshold, and PSB is activated. After a delay of 10 milliseconds (1.216 s, “

t_{2}

” in Figure 9), based on the three-phase voltage data of three sampling points after

t_{2}

and matrix P shown in (14), the horizontal and vertical coordinates

x_{U}

and

y_{U}

in the new two-dimensional coordinate system, corresponding to the three-phase voltage, can be obtained. After obtaining

x_{U}

and

y_{U}

, they can be substituted into (18) to obtain index C as 95.85, 95.77, and 95.70. According to the definition of

C_{avr}

in (19), the value of

C_{avr}

is 95.77. In such a case, (19) is satisfied, and the PSBU will block the protection relay. More test results are shown in Table 2, Table 3 and Table 4.

Table 2 shows the performance of the proposed method under different power swing situations, where “

f_{ps}

” represents the oscillation frequency, “

t_{0}

” represents the starting moment of the power swing, and “PS” indicates the test result of the power swing. It is observed in Table 2 that the proposed method can identify power swings reliably, and it is not affected by the starting moment.

Table 3 shows the performance of the proposed method under different three-phase shorting fault situations, where “Distance” represents the distance from the fault point to the measuring point, “

t_{0}

” represents the occurring moment of the fault, and “F” indicates the test result of the fault. It is observed in Table 3 that the proposed method can also identify faults reliably, and is not affected by the starting moment of the fault.

For testing the effectiveness of the PSD function of the proposed method, several scenarios were simulated. During the simulation, a power swing with an oscillation frequency of 5 Hz occurred at 1.2 s. Different fault distances and power angles between the two ends during the occurrence of the fault were also considered in the simulation. The test results are presented in Table 4, where

δ

represents the power angle between the two ends. According to the results of No. 1–5 shown in the table, the proposed method can reliably detect faults during power swings. Moreover, since

Z_{m}

is equal to

Z_{n}

in the simulation system shown in Figure 6, the fault point 150 km away from the measuring point is actually the oscillation center. From the results of No. 6–12, it is observed that the proposed method can also reliably detect a fault occurring at the oscillation center during a power swing.

4.2. Test Results Based on Nine-Bus System

Some simulation tests were also conducted based on a 400 kV system with three alternators and nine buses (WSCC system), as shown in Figure 10. The parameters of different components of this system are presented in Table A1 in Appendix A [24].

For testing the performance of the proposed method under different fault conditions, some three-phase shorting faults on L78 are simulated. Figure 11 shows the waveform of the three-phase voltage under a fault, and the corresponding value of C under a fault is shown in Figure 12. The fault distance is 125 km. The fault occurs at 1.2 s, and the PSB scheme is enabled at 1.201 s (“

t_{1}

” in the figure). In such a case, the value of

C_{avr}

is 31.68, and the PSBU will not operate. More obtained results are presented in Table 5.

For testing the performance of the proposed method under different power swing conditions, the power swing should be observed. Therefore, during the simulation process, transmission line L57 is tripped because of a fault at 1.2 s, and a power swing phenomenon is observed on transmission line L78. The waveforms of three-phase voltage and current are shown in Figure 13. In such a case, the value of

C_{avr}

becomes 102.18, and the PSB unit blocks the protection relay.

The above test results also indicate the effectiveness of the proposed method in discriminating between faults and power swings.

For testing the PSD function, L57 is still tripped because of a fault at 1.2 s. Several faults on L78 were simulated after observing the power swing on it. Table 6 shows the test results, from which it can again be seen that the proposed method can detect faults during power swings.

4.3. Comparison and Discussion

Some comparisons are made here. In terms of comparison with traditional methods, the method introduced in [5] is chosen as a representative traditional method for comparison with the proposed method. For comparisons with novel methods, those found in references like [8,9], which demand high sampling frequencies and encounter practical challenges, are not selected for comparison. Moreover, methods akin to reference [28] that rely on phasor data are also excluded from the comparison, as the distinct features between these methods and the proposed methods have already been highlighted when contrasted with traditional methods. The method introduced in reference [27], which utilizes instantaneous values and has minimal frequency requirements, resembles the proposed method. Hence, it is selected as representative of the novel methods for comparison with the proposed method.

In addition to the judgment results of these three methods, the operating speed and the effect of the measuring noise on them are also discussed here. Different disturbance scenarios were simulated based on the two-terminal system shown in Figure 6. Table 7, Table 8, Table 9, Table 10 and Table 11 show the comparison results. In these tables, “SNR” represents the signal–noise ratio, “R” represents the test result, “DI” is the index used in the method in [27], for which the corresponding setting value is set to 0.01 here, and “k” is the index used in the traditional method in [5], for which the corresponding setting value is set to 0.3 here. For the method in [27], if the DI is less than 0.01, the disturbance is judged to be a power swing. Otherwise, it is judged to be a fault. For the traditional method in [5], if k is greater than 0.3, the PSB unit will deblock the protection relay. The symbol “T” in Table 7 and Table 8 represents the time required from the start of the method to obtain the corresponding value of the judgment index. “T” in Table 9, Table 10 and Table 11 represents the time from the moment at which the fault occurs to the moment at which the corresponding value of the judgment index is obtained.

Since the traditional method is applied for detecting faults during power swings in [5], Table 7 and Table 8 only show the performance of the proposed method and the method introduced in [27] for discriminating between power swings and faults. Table 7 shows the test results under different power swing situations and Table 8 shows the test results under different fault situations. From Table 7 and Table 8, it can be seen that both methods can identify power swings of different oscillation frequencies and faults with different fault distances in about 10 ms after the power swing or the fault occurs. In addition, it can be seen that the proposed method is less affected by measurement noise. Considering the effect of measurement noise, the method in [27] can more easily misjudge the power swing as a fault.

Table 9, Table 10 and Table 11 show the performance of the three methods in detecting three-phase shorting faults during power swings. Comparing Table 9 and Table 10, it can be seen that both the proposed method and the traditional method can detect faults reliably, no matter where the fault occurs on the line, and are less affected by measurement noise. At the same time, the proposed method has an advantage in terms of response speed. Comparing Table 9 and Table 11, it can be seen that both the proposed method and the method in [27] can detect faults quickly. However, the reliability of the method in [27] is not sufficient. It cannot detect some faults occurring in the oscillation center, and it is easily affected by measurement noise, while the proposed method performs well in such situations.

The effect of sampling frequency is also discussed here. Since the criterion for PSB depends on the value of

C_{avr}

to discriminate between power swings and faults, the sampling frequency has little effect on it. Table 12 shows the calculation results of

C_{avr}

under different sampling frequencies for the same power swing and fault situations. In the table,

f_{s}

represents the sampling frequency.

On the other hand, the effectiveness of the criterion of the PSD function will be affected by the sampling frequency. This is because the criterion depends on the value of

Δ P

to detect the faults under power swing. Figure 14 and Figure 15 show the value of

Δ P

under different sampling frequencies for the same power swing and fault situations. As seen in Figure 14 and Figure 15, the maximum value of

Δ P

decreases significantly as the sampling frequency increases under a power swing, while the value of

Δ P

is less affected by the sampling frequency when a fault occurs. Therefore, the higher the sampling frequency, the easier it is for the criterion of PSD to detect a fault during a power swing. According to Figure 15a, a criterion of PSD with a sampling frequency of 1 kHz is actually able to detect a faults during a power swing. Considering the simulation results regarding the effect of measurement noise on the criterion of PSD, a sampling frequency of 4 kHz is used here, which is not difficult to achieve in actual projects.

5. Conclusions

A method for a PSBU is proposed in this paper. This method uses the characteristics of the measured voltage to discriminate between power swings and faults. To make the characteristics more intuitive, PCA is applied. Moreover, an index P_UI is proposed, and its change rate is observed to detect faults during power swings. Since the coordinate basis that is needed in the calculation process of the method can be obtained in advance, the proposed method only involves simple multiplication and addition operations. Therefore, the proposed method requires few computational resources, and can be easily implemented. Moreover, it only needs the instantaneous values of the voltage and current when detecting faults during power swings, so the proposed method can deblock the protection relay quickly after a fault occurs during a power swing. Certainly, the proposed method has its limitations, particularly in terms of its resistance to transition resistance. However, this is not a significant concern, since the method is primarily used to differentiate between three-phase symmetrical short-circuit faults and power swings. The transition resistance in three-phase symmetrical short-circuit faults is typically very low. Consequently, the impact of transition resistance on the proposed method can be disregarded.

To verify the effectiveness of the proposed method, many tests have been performed. The results show that this method can not only reliably block the protection relay after a power swing occurs, but also quickly deblock the protection relay after detecting the fault during the power swing. In addition, the proposed method was finally compared with a traditional method and a novel method in [27]. The comparison results indicate that the proposed method has an advantage in terms of response speed and is less affected by measurement noise.

Author Contributions

Conceptualization, H.W.; data curation, W.Z.; methodology, H.W.; resources, Q.Y.; software, Q.Y; validation, X.L.; writing—original draft, H.W.; writing—review and editing, X.L. All authors have read and agreed to the published version of the manuscript.

Funding

This work was supported in part by the Science and Technology Project of the State Grid Sichuan Electric Power Company, under Grant 521997230003.

Data Availability Statement

The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author/s.

Conflicts of Interest

Authors Hao Wang, Xiaopeng Li, and Wenyue Zhou were employed by the State Grid Sichuan Electrical Power Research Institute. Author Qi Yang was employed by the State Grid Sichuan Electrical Power Company. All the authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Appendix A

Table A1. Parameters of nine-bus system.

Components	Specification
G1, G2, and G3	Capacity: $588 MVA$ Voltage rating: $V_{1} = 21 ∠ 0 ° kV, V_{2} = 21 ∠ 30 ° k V, V_{3} = 21 ∠ 0 ° kV$ Leakage reactance: $X_{L} = 0.147 Ω$ Direct axis reactance: $X_{d} = 2.31 Ω$ Quadrature axis reactance: $X_{q} = 2.19 Ω$ Transient reactance: ${X^{'}}_{d} = 0.253 Ω, {X^{'}}_{q} = 0.665 Ω$ Sub-transient reactance: ${X^{″}}_{d} = 0.191 Ω, {X^{″}}_{q} = 0.233 Ω$ Transient time constant: ${T^{'}}_{d 0} = 9.14 s, {T^{'}}_{q 0} = 2.5 s$ Sub-transient time constant: ${T^{″}}_{d 0} = 0.04 s, {T^{″}}_{q 0} = 0.2 s$ Inertia constant: $T_{J} = 3.117 s$
T1, T2, and T3	$21 kV / 400 kV, 600 MVA$
Transmission line	Positive sequence: $R_{1} = 0.0308 Ω / km, L_{1} = 0.998 mH / km, C_{1} = 11.2 nF / km$ Zero sequence: $R_{0} = 0.1788 Ω / km, L_{1} = 2.565 mH / km, C_{1} = 7.3 nF / km$ Length: $L 45 = 150 km, L 46 = 180 km, L 57 = 220 km$ $L 69 = 200 km, L 78 = 250 km, L 89 = 250 km$
Load	Load1: 500 MVA, p.f. 0.90 (lagging) Load2: 350 MVA, p.f. 0.85 (lagging) Load3: 400 MVA, p.f. 0.92 (lagging)

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Figure 1. Schematic diagram of PCA processing results.

Figure 2. Projection trajectories under different disturbances. (a) Three-phase shorting fault; (b) power swing.

Figure 3. The value of C.

Figure 4. Flow chart of blocking protection relay.

Figure 5. Flow chart of deblocking the protection relay.

Figure 6. Sketch of simulation system.

Figure 7. Trajectory of measured impedance during power swing.

Figure 8. The waveform of the three-phase voltage under a power swing of 2 Hz.

Figure 9. The value of C under a power swing of 2 Hz.

Figure 10. Sketch of 400 kV system.

Figure 11. The waveform of the three-phase voltage under a fault.

Figure 12. The value of C under a fault.

Figure 13. Waveforms of voltage and current.

Figure 14. ΔP under different sampling frequencies of power swings.

Figure 15. ΔP under different sampling frequencies for faults.

Table 1. Parameters of transmission line.

Parameter	Positive Sequence	Zero Sequence
R (Ω/km)	0.0196	0.1828
L (mH/km)	0.8917	2.739
C (μF/km)	135	92

Table 2. Test results under different power swing situations.

No.	f_ps (Hz)	t₀ (s)	C_avr	Result
1	7.00	1.200	85.24	PS
2	5.00	1.200	88.06	PS
3	2.50	1.200	94.36	PS
4	2.00	1.200	95.55	PS
5	1.25	1.200	96.77	PS
6	1.00	1.200	97.44	PS
7	0.50	1.200	98.30	PS
8	0.40	1.200	98.24	PS
9	0.20	1.200	98.50	PS
10	7.00	1.201	83.51	PS
11	7.00	1.202	84.19	PS
12	7.00	1.203	84.89	PS
13	7.00	1.204	83.17	PS
14	7.00	1.205	83.85	PS
15	7.00	1.206	84.89	PS
16	7.00	1.207	85.24	PS
17	7.00	1.208	83.85	PS
18	7.00	1.209	84.53	PS

Table 3. Test results under different three-phase shorting fault situations.

No.	Distance (km)	t₀ (s)	C_avr	Result
1	10	1.200	1.57	F
2	30	1.200	5.28	F
3	70	1.200	16.90	F
4	110	1.200	27.37	F
5	150	1.200	36.10	F
6	190	1.200	42.96	F
7	230	1.200	49.32	F
8	270	1.200	55.39	F
9	290	1.200	58.05	F
10	290	1.201	58.05	F
11	290	1.202	58.05	F
12	290	1.203	58.05	F
13	290	1.204	58.05	F
14	290	1.205	58.05	F
15	290	1.206	58.05	F
16	290	1.207	58.05	F
17	290	1.208	58.05	F
18	290	1.209	58.05	F

Table 4. Test results of PSD for two-end system.

No.	Distance (km)	$δ$	ΔP	Result
1	10	50°	1605.52	F
2	80	50°	895.34	F
3	150	50°	1162.82	F
4	220	50°	1325.18	F
5	290	50°	1441.59	F
6	150	80°	770.14	F
7	150	120°	125.90	F
8	150	150°	276.45	F
9	150	180°	698.84	F
10	150	220°	1076.44	F
11	150	260°	1325.72	F
12	150	330°	593.41	F

Table 5. Test results under different fault situations for nine-bus system.

No.	Distance (km)	t₀ (s)	C_avr	Result
1	10	1.200	2.11	F
2	25	1.200	4.87	F
3	55	1.200	13.62	F
4	85	1.200	22.12	F
5	125	1.200	31.68	F
6	155	1.200	36.67	F
7	185	1.200	44.05	F
8	225	1.200	56.94	F
9	240	1.200	54.69	F
10	240	1.201	54.76	F
11	240	1.202	54.70	F
12	240	1.203	54.69	F
13	240	1.204	54.74	F
14	240	1.205	54.69	F
15	240	1.206	54.70	F
16	240	1.207	54.72	F
17	240	1.208	54.75	F
18	240	1.209	54.72	F

Table 6. Test results of PSD for nine-bus system.

No.	Distance (km)	$δ$	ΔP	Result
1	10	50°	1279.13	F
2	70	50°	1068.65	F
3	125	50°	873.70	F
4	180	50°	683.19	F
5	240	50°	533.45	F
6	240	80°	676.34	F
7	240	120°	769.19	F
8	240	150°	711.32	F
9	240	180°	560.03	F
10	240	220°	413.78	F
11	240	260°	388.03	F
12	240	330°	374.51	F

Table 7. Test results of PSD of proposed method and method in [27].

f_ps (Hz)	SNR (dB)	Proposed Method			Method in [27]
f_ps (Hz)	SNR (dB)	C_avr	R	T (ms)	DI	R	T (ms)
7.00	-	85.24	√	10.07		√	10.09
5.00	-	88.06	√	10.09	$2.28 \times 1 0^{- 5}$	√	10.08
2.00	-	95.55	√	10.09	$1.65 \times 1 0^{- 5}$	√	10.10
1.00	-	97.44	√	10.07	$1.50 \times 1 0^{- 5}$	√	10.09
0.40	-	98.24	√	10.08	$1.49 \times 1 0^{- 5}$	√	10.10
7.00	50	85.29	√	10.07	0.03	×	10.07
7.00	40	85.83	√	10.08	0.25	×	10.08
7.00	30	83.06	√	10.08	3.82	×	10.07

Table 8. Test results under different fault situations of proposed method and method in [27].

Distance (Hz)	SNR (dB)	Proposed Method			Method in [27]
Distance (Hz)	SNR (dB)	C_avr	R	T (ms)	DI	R	T (ms)
10	-	1.57	√	10.09	0.01	√	10.09
70	-	16.90	√	10.09	0.76	√	10.08
150	-	36.10	√	10.08	0.31	√	10.10
230	-	49.32	√	10.09	0.42	√	10.09
290	-	58.05	√	10.09	0.47	√	10.10
290	50	58.12	√	10.09	0.63	×	10.07
290	40	58.41	√	10.09	1.63	×	10.08
290	30	57.46	√	10.08	14.26	×	10.07

Table 9. Test results for PSD of proposed method.

f_ps (Hz)	Distance (km)	$δ$	SNR (dB)	$Δ P$	R	T (ms)
7.00	10	50°	-	85.24	√	0.5
	150	50°	-	88.06	√	0.75
	290	50°	--	94.36	√	1.25
	150	120°	-	95.55	√	0.75
	150	180°	-	96.77	√	0.75
	150	220°	-	97.44	√	0.75
	150	120°	50	98.30	√	0.75
	150	120°	40	98.24	√	0.75
	150	120°	30	98.50	√	0.50
0.40	10	50°	-	83.51	√	0.50
	150	50°	-	84.19	√	0.75
	290	50°	--	84.89	√	1.00
	150	120°	-	83.17	√	0.75
	150	180°	-	83.85	√	1.50
	150	220°	-	84/89	√	0.75
	150	120°	50	85.24	√	0.75
	150	120°	40	83.85	√	0.75
	150	120°	30	84.53	√	0.75

Table 10. Test results for PSD of traditional method in [5].

f_ps (Hz)	Distance (km)	$δ$	SNR (dB)	k	R	T (ms)
7.00	10	50°	-	0.29	√	29.50
	150	50°	-	0.27	√	31.75
	290	50°	--	0.29	√	32.00
	150	120°	-	0.29	√	23.50
	150	180°	-	0.12	√	12.75
	150	220°	-	0.21	√	33.00
	150	120°	50	0.29	√	23.50
	150	120°	40	0.29	√	23.50
	150	120°	30	0.29	√	23.50
0.40	10	50°	-	0.28	√	30.25
	150	50°	-	0.27	√	32.50
	290	50°	--	0.29	√	32.50
	150	120°	-	0.28	√	36.00
	150	180°	-	0.27	√	8.50
	150	220°	-	0.29	√	35.75
	150	120°	50	0.28	√	35.50
	150	120°	40	0.24	√	35.00
	150	120°	30	0.27	√	36.25

Table 11. Test results for PSD of traditional method in [27].

f_ps (Hz)	Distance (km)	$δ$	SNR (dB)	DI	R	T (ms)
7.00	10	50°	-	0.92	√	0.50
	150	50°	-	0.12	√	1.25
	290	50°	--	0.33	√	1.25
	150	120°	-	-	×	-
	150	180°	-	0.04	√	0.75
	150	220°	-	0.09	√	0.75
	150	120°	50	-	×	-
	150	120°	40	-	×	-
	150	120°	30	-	×	-
0.40	10	50°	-	1.33	√	0.75
	150	50°	-	0.19	√	1.00
	290	50°	--	0.49	√	1.50
	150	120°	-	0.33	√	1.00
	150	180°	-	-	×	-
	150	220°	-	0.02	√	1.00
	150	120°	50	-	×	-
	150	120°	40	-	×	-
	150	120°	30	-	×	-

Table 12. Effect of sampling frequency on

C_{avr}

Table 12. Effect of sampling frequency on

C_{avr}

Disturbance	Fs (kHz)	$C_{avr}$
Power swing	1	88.51
	2	89.42
	4	88.06
	8	88.11
Fault	1	34.25
	2	35.28
	4	36.10
	8	33.52

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Wang, H.; Yang, Q.; Li, X.; Zhou, W. A Method of Discriminating Between Power Swings and Faults Based on Principal Component Analysis. Appl. Sci. 2025, 15, 2867. https://doi.org/10.3390/app15052867

AMA Style

Wang H, Yang Q, Li X, Zhou W. A Method of Discriminating Between Power Swings and Faults Based on Principal Component Analysis. Applied Sciences. 2025; 15(5):2867. https://doi.org/10.3390/app15052867

Chicago/Turabian Style

Wang, Hao, Qi Yang, Xiaopeng Li, and Wenyue Zhou. 2025. "A Method of Discriminating Between Power Swings and Faults Based on Principal Component Analysis" Applied Sciences 15, no. 5: 2867. https://doi.org/10.3390/app15052867

APA Style

Wang, H., Yang, Q., Li, X., & Zhou, W. (2025). A Method of Discriminating Between Power Swings and Faults Based on Principal Component Analysis. Applied Sciences, 15(5), 2867. https://doi.org/10.3390/app15052867

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A Method of Discriminating Between Power Swings and Faults Based on Principal Component Analysis

Abstract

1. Introduction

2. Principal Component Analysis

3. Power Swing Blocking Method Using PCA

3.1. Basic Principle

3.2. Criterion for PSB and PSD Function

3.3. Flow Chart of PSB and PSD Function

4. Simulation Tests

4.1. Test Results Based on Two-End System

4.2. Test Results Based on Nine-Bus System

4.3. Comparison and Discussion

5. Conclusions

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

Appendix A

References

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