Research on Time Delay Estimation Method of Partial Discharges Signal with Improved Weighted Function

Abstract

Positioning systems based on the Time Difference of Arrival (TDOA) often struggle to accurately estimate time delays for partial discharge signals in complex electromagnetic environments. To address this, we introduce an improved joint-weighted generalized cross-correlation (GCC) time delay estimation algorithm. Traditional GCC time delay estimations, when reliant on conventional weighted functions, significantly underperform in situations characterized by low signal–noise ratios (SNRs). Our proposed improved joint weighted function combines the advantages of both phase transform (PHAT) and correlation transform (SCOT) weightings, resulting in a composite function. Compared to the conventional weighted GCC, the improved joint weighted GCC displays a more distinct peak in cross-correlation functions and ensures more robust delay estimations, especially under low SNR conditions. Experimental results indicate that the improved joint-weighted GCC algorithm achieves an error margin below 1 ns in challenging outdoor electromagnetic scenarios, demonstrating its aptitude for detecting and localizing partial discharge signals in practical engineering applications.

1. Introduction

The partial discharge phenomenon generated by electrical equipment has been an important causative factor for the insulation deterioration of power equipment [1]. Partial discharges contain a large amount of characteristic information about potential power system failures. The accurate positioning of partial discharge sources achieved by the detection, processing, and analysis of the partial discharge sources has drawn significant attention, which can contribute to the early warning and immediate maintenance of electrical equipment faults.

Based on the input feature quantities of spatial localization algorithms, partial discharge localization methods can be classified into Time of Arrival (TOA), Time Difference of Arrival (TDOA), Angle of Arrival (AOA), Direction of Arrival (DOA), and Received Signal Strength (RSS) [2]. Among these localization algorithms, the TDOA-based partial discharge source localization algorithm has become the most widely used and researched because of its high localization accuracy and low computational power requirement, which can quickly meet the actual engineering requirements. The accuracy of TDOA-based partial discharge source localization is greatly associated with time delay differences, station errors, and array deployment methods [3]. The time delay estimation error has gained much attention because of the crucial importance of the accuracy of TDOA-based partial discharge source estimation.

Currently, there are various methods for time delay estimation, including the thresholding method, signal feature point estimation, energy profile estimation, and correlation detection method [4]. Different time delay estimation algorithms are selected according to different experimental environments and measurement needs. Knapp and Carter initially proposed the GCC method in 1976, which is still one of the most popular algorithms for time delay estimation due to its computational simplicity and superior noise immunity [5,6]. According to the different scenarios and signal characteristics, the GCC method can use different weighted functions, such as PHAT, ROTH, HB, SCOT, etc. [7]. The generalized cross-correlation with phase transform (PHAT–GCC) is the most commonly used algorithm for generalized weighted cross-correlation time delay estimation in practical scenarios affected by noise and multipath. However, the performance of time delay estimation based on the generalized weighted cross-correlation algorithm deteriorates severely in the presence of multipath signals and low SNR [8]. Actually, multipath effects and noise are unavoidable in natural scenes. Therefore, improvements to the traditional generalized weighted cross-correlation algorithms are urgently needed. At present, the time delay estimation algorithm based on generalized cross-correlation is mainly used to raise the accuracy of time delay estimation under low SNR and multipath effects by enhancing the weighted function or preprocessing the partial discharge signal [9,10]. However, it is difficult to obtain accurate time delay estimation results in complex electromagnetic environments.

Traditional algorithms are not ideal for the detection and processing of weak signals in complex electromagnetic environments. Here, to address the problem of multiple spurious peaks in the cross-power spectrum of time delay estimation under low SNR conditions, which leads to poor delay estimation accuracy, we propose a GCC algorithm with improved weighted functions. The improved weighted function incorporates and makes better use of the advantages of the PHAT and SCOT weighted functions. The cross-power spectrum is filtered using the improved weighted function, which optimizes time delay estimation accuracy in complex electromagnetic environments.

2. Theories and Methods

2.1. Generalized Cross-Correlation Time Delay Estimation for Joint Weighted Function

In this manuscript, the weighted function is revised based on the PHAT–GCC time delay estimation algorithm. The joint phase transform and smooth correlation transform (PHAT–SCOT) weighted function is used instead of the PHAT weighted function to improve the problem that the PHAT–GCC algorithm cannot obtain the effective cross-power spectrum at low SNR, which leads to poor time delay estimation accuracy.

2.1.1. Generalized Cross-Correlation Time Delay Estimation

Cross-correlation is an algorithm representing the degree of similarity between two signals in the time domain [11]. The time delay results of the two received signals can be obtained by using the cross-correlation algorithm, and the received signal model is constructed as follows:

$$\{\begin{array}{l}{x}_1(t)=s(t)+n_1(t)\\ x_2(t)=s(t-d)+n_2(t)\end{array}$$

(1)

where x₁(t) and x₂(t) are two received partial discharge signals, S(t) is a partial discharge power signal; n₁(t) and n₂(t) are noise, S(t) and n(t) are independent, and d is the relative time delay of signals x₁(t) and x₂(t).

The cross-correlation function of the two received partial discharge signals is expressed as:

$$\begin{array}{l}{R}_{12}(\tau )=E[x_1(t)x_2(t+\tau )]\\ =E\{[s(t)+n_1(t)][s(t+\tau -d)+n_2(t+\tau ))]\}\\ =R_{ss}(\tau -d)+R_{s{n}_2}(\tau )+R_{n_{1}s}(\tau -d)+R_{n_1{n}_2}(\tau )\end{array}$$

(2)

Assumptions n₁(t) and n₂(t) are independent of each other, as well as of the noise and the signal. Where R_sn2(τ), R_sn1(τ − d), and R_n1n2(τ) are all zero. i.e.,

$$R_{12}(\tau )=R_{ss}(\tau -d)$$

(3)

where R_SS(τ − d) is the autocorrelation function of the partial discharge signal when τ = d when the correlation function peak reaches the maximum. At this time, the correlation between the partial discharge signals received by the two antennas is at its maximum, and the maximum peak position of the correlation function is located where τ represents the time delay value of the signals.

GCC can overcome the disadvantage of the significant error in the basic cross-correlation time delay estimation algorithm at a low SNR [12]. According to the Wiener–Khintchine theorem, it can be known that the generalized cross-correlation time delay estimation method for the essential cross-correlation function in the form of a Fourier transform of the correlation function Equation (3) is Fourier transformed, that is:

$$R_{12}(\tau )={\displaystyle {\int }_{-\infty }^{+\infty }\phi (f)G_{12}(f)e^{j2\pi ft}}df$$

(4)

where τ denotes the time delay, $\phi (f)$ represents the introduced power function, and G₁₂(f) is the cross-power spectrum of the signals received by the two antennas.

In applying a generalized weighted cross-correlation function to solve the delay estimation problem, different weighted functions should be selected for different application scenarios and interferences to suppress the noise and improve the accuracy of delay estimation.

2.1.2. Generalized Weighted Cross-Correlation Algorithm

The expression of the PHAT weighted function is

$${\phi }_{PHAT}(f)=\frac{1}{\left|G_{12}(f)\right|}$$

(5)

where G₁₂(f) is the cross-power spectrum of the partial discharge signal received by the two antennas. The PHAT weighted function is the most commonly used weighted method to effectively suppress noise interference and sharpen its peaks [13]. Because of the accuracy of cross-correlation analysis, even for phase inaccuracies, PHAT–GCC can solve problems such as noise, interference, and nonlinear distortions in real-world environments. Moreover, PHAT–GCC can quickly compute the cross-correlation results due to its low computational complexity, making it suitable for factual signal processing and control applications. On the other hand, it discards the amplitude information of the signal and applies the cross-power spectrum phase for time delay estimation, significantly improving time delay estimation accuracy. However, the numerator of its weighted function is always constant at low SNR, and multiple spurious peaks appear when the valuable signal power is insufficient, leading to impaired time delay estimation accuracy [14].

The SCOT weighted function expression is:

$${\phi }_{SCOT}(f)=\frac{1}{\sqrt{G_{11}(f)G_{22}(f)}}$$

(6)

where G₁₁(f) and G₂₂(f) are the auto-power spectra of the two partial discharge signals, respectively. The SCOT weighted function considers the correlation at different points in time in the series, which captures the transformation of the temporal correlation in the series and deals well with non-smooth and nonlinear time series [15]. The SCOT weighted function also considers the two partial discharge signals to reduce the influence of the interfering signals on the cross-power spectrum. However, when the power spectral densities of the two partial discharge signals are similar, the peaks of their correlation functions broaden, resulting in the inability to calculate the actual delay value accurately.

2.1.3. Principle Analysis of Improved Weighted Function

The above two traditional GCC algorithms for weighted functions are verified by simulation and experiment. The results display that the performance of the algorithms decreases sharply and deteriorates severely under low SNR conditions. To solve the problem that the traditional weighted function cannot obtain accurate time delay results in complex electromagnetic environments, the PHAT–SCOT joint weighted algorithm is proposed. The functional expression of PHAT–SCOT is:

$${\phi }_{PHAT-SCOT}(f)=\left|\frac{\sqrt{G_{11}(f)G_{22}(f)}-\left|G_{12}(f)\right|}{{\left|G_{12}(f)\right|}^{\alpha }}\right|$$

(7)

where G₁₁(f) and G₂₂(f) are the auto-power spectra of the two partial discharge signals, respectively. G₁₂(f) is the cross-power spectrum of partial discharge signals x₁(t) and x₁(t), and α is the exponential adjustment factor. According to the characteristics of the partial discharge signal, α takes the value range of [0,1] to receive the accurate time delay value. Collating Equation (7):

$$\begin{array}{ll}{\phi }_{P-S}(f)& =\left|\frac{\sqrt{G_{11}(f)G_{22}(f)}-\left|G_{12}(f)\right|}{{\left|G_{12}(f)\right|}^{\alpha }}\right|\\ & =\left|\frac{1}{{\left|G_{12}(f)\right|}^{\alpha }\frac{1}{\sqrt{G_{11}(f)G_{22}(f)}}}-{\left|G_{12}(f)\right|}^{1-\alpha }\right|\\ & =\left|\frac{1}{\frac{1}{{\phi }_{PHAT-\alpha }}{\phi }_{SCOT}}-{\left|G_{12}(f)\right|}^{1-\alpha }\right|\\ & =\left|\frac{{\phi }_{PHAT-\alpha }}{{\phi }_{SCOT}}-{\left|G_{12}(f)\right|}^{1-\alpha }\right|\end{array}$$

(8)

The improved joint weighted function suppresses the noise band through 1/G₁₂(f) in the PHAT weighted function. The PHAT–SCOT overcomes the effect of spurious peaks while suppressing noise through the auto-power spectra G₁₁(f) and G₂₂(f) of the two partial discharge signals. In actual measurements, the noise is not strictly uncorrelated with each other, i.e., R_sn2(τ), R_sn1(τ − d), and R_n1n2(τ) are not absolutely zero, so accuracy can be affected by noise. The introduction of the exponential adjustment factor α in the denominator can effectively improve the situation that the denominator of the PHAT weighted function tends to zero when the SNR is low. The α suppresses the noise signal to varying degrees, thus enabling the regulation of the cross-power spectrum density. While the partial discharge signal is submerged in noise, the power spectrum density of the partial discharge signal is less than the interfering noise. To ensure the stability of the weighted function value, the auto-power spectrum density function and cross-power spectrum density function of the two partial discharge signals are introduced into the numerator of the PHAT–SCOT. Results in both the denominator and numerator of the PHAT–SCOT decrease as the power spectral density of the partial discharge signal decreases. Therefore, the error of the improved joint weighted function can be diminished, leading to an optimized accuracy of the time delay estimation.

This PHAT–SCOT weighted function enhances the deficiencies of the PHAT-weighted and SCOT-weighted functions as well as improved delay estimation accuracy. The PHAT–SCOT generalized cross-correlation (PHAT–SCOT–GCC) algorithm raised the stability and generalizability of the delay estimation results of partial discharge signals in complex electromagnetic environments.

2.1.4. Algorithm Flow Analysis

The PHAT–SCOT–GCC time delay estimation algorithm (Figure 1) can effectively raise the time delay estimation accuracy of partial discharge signals under periodic narrowband interference and Gaussian white noise. Moreover, improved joint weighting was applied to delay estimation in real-world scenarios.

Step 1: The two partial discharge signals are processed by adding Hann window filtering, which effectively reduces the signal tail affected by the multipath effect and prevents frequency leakage after the Fourier transform is performed.

Step 2: Perform the Fourier transform and generalized cross-correlation operation on the two partial discharge signals.

Step 3: Introduce a modified joint weighted function ${\phi }_{P-S}(f)$ for the cross-correlation function to obtain the cross-power spectrum function.

Step 4: Convert the cross-power spectrum by Fourier inverse transform to the time domain for peak detection to obtain the delay value.

3. Simulation Analysis

3.1. Partial Discharge Signal Simulation

A double exponential decay oscillatory pulse model is used to simulate the partial discharge signal [16], which is expressed as follows:

$$S(t)=A(e^{\frac{-1.3t}{{\tau }_1}}-e^{\frac{-2.2t}{{\tau }_2}})\sin (2\pi f_{c}t)$$

(9)

We set the sampling frequency Fs = 100 Hz, the oscillation frequency f_c = 3 MHz, and the attenuation coefficients 1 and 2 to 0.5 and 0.1, respectively. The simulated signals S₁(t), S₂(t), and S₃(t) amplitude parameters and delay points are shown in

Table 1. Parameters for partial discharge signal simulation.

	S1 (t)	S2 (t)	S3 (t)
Amplitude A/v	3	0.6	0.1
Delay Points	0	100	200

.

To simulate the aliasing produced by the tail of the actual partial discharge signal due to the multipath effect, the signals S₁(t), S₂(t), and S₃(t) are accumulated to obtain x₁(t), i.e.,

$$x_1(t)=S_1(t)+S_2(t)+S_3(t)$$

(10)

The first partial discharge signal x₁(t) is shown in Figure 2:

3.2. Effect of External Noise on Time Delay Estimation

In practical engineering, partial discharge signals usually occur in complex electromagnetic environments, resulting in a low SNR of the collected partial discharge signals, which affects the accuracy of time delay estimation. Simulation of Gaussian white noise and narrow bandwidth interference signals to analyze the effect of noise interference on the accuracy of time delay estimation of the PHAT–SCOT–GCC algorithm [17].

The broad spectrum and random generation of Gaussian white noise are utilized to simulate the bottom noise generated by the oscilloscope and the background noise in the environment [18]. The signal x₁(t) adds Gaussian white noise with an SNR of −5 dB, as shown in Figure 3.

According to Equation (9), each parameter in Equation (9) is the same as x₁(t), and comparing x₁(t) delay τ is 200 points to obtain x₂(t), and the absolute value of the delay of the two signals is 2 μs. In the following section, time delay estimation of simulated partial discharge signals will be performed using the generalized cross-correlation algorithm with an improved weighted function proposed. The results of time delay estimation using three algorithms (PHAT weighted function, SCOT weighted function, and improved generalized cross-correlation algorithm with joint weighted function) for the SNR of −5 dB are shown in Figure 4:

Figure 4a–c presents the time delay estimation results obtained using PHAT–GCC, SCOT–GCC, and PHAT–SCOT–GCC. Comparing the simulation results of the three algorithms, it can be seen that the PHAT–GCC and SCOT–GCC algorithms have poor immunity to interference noise. The principal peak value of the correlation function is almost submerged in the noise interference, which produces several false peaks, and the error of their delay results is more significant than 0.5 μs. The PHAT–SCOT–GCC algorithm peak sharpening is obviously without the interference of false peaks, with a delay value of −2.01 us and an error of 0.01 us, which indicates that the delay accuracy is still higher than the PHAT–GCC and SCOT–GCC algorithms. Therefore, the PHAT–SCOT–GCC algorithm improves the delay estimation accuracy at low SNR and is more resistant to noise interference.

For far-field measurement of weak signals in complex electromagnetic environments, the SNR is set to decrease linearly from 0 dB to −10 dB in 5 dB steps, and 1000 Monte Carlo experiments are carried out to experimentally estimate the three kinds of weighted functions for each fixed SNR. Meanwhile, the distribution of delay results for the three weighting functions with an SNR of −10 dB is shown in Figure 5.

Upon examining the distribution of delay results in Figure 5a–c, the distribution of delay results of PHAT–GCC and SCOT–GCC is more irregular, mainly concentrated around −2 μs and 0 μs, with occurrence counts of −2 μs being 34 and 28 times, respectively. The delay results for the PHAT–SCOT–GCC algorithm mainly center around −2 μs, with 146 occurrences at this value. Therefore, the PHAT–SCOT–GCC algorithm demonstrates more accuracy than the other two algorithms. We consider that the results of each delay estimation will be contingent due to the introduction of random noise. Therefore, based on the characteristics of the distribution of delay results for each algorithm, the absolute value (1 μs) of the median value of −2 μs and 0 μs in the centralized distribution area is chosen as the basis for determining whether the results are correct. We set an interval from −3 μs to −1 μs around the theoretical value of −2 μs, and results outside the interval are considered estimation failures. As a result, an analysis of correctness and root mean square error (RMSE) is necessary. The correctness of each algorithm at different SNRs is shown in

Table 2. Correctness rate of three algorithms under Gaussian noise.

SNR/dB	PHAT–GCC/μs	SCOT–GCC/μs	PHAT–SCOT–GCC/μs
0	93%	87%	100%
−5	71%	65%	100%
−10	15%	17%	76%

.

To further investigate the stability of the PHAT–SCOT–GCC algorithm under Gaussian white noise conditions with an SNR of −10 dB to 0 dB, the SNR is set to decrease linearly from 0 dB to −10 dB in 1 dB steps, and 1000 experiments are carried out to experimentally validate three kinds of weighted functions at each fixed SNR. The RMSE of their time delay estimation is shown in Figure 6.

As shown in Table 2 and Figure 6, when the SNR is large (SNR = 0 dB), the PHAT–GCC and SCOT–GCC time delay estimation algorithms are more than 80% correct, and the RMSE is below 0.05 μs. However, the performance of the algorithms of the PHAT–GCC and SCOT–GCC approaches decreases dramatically as the SNR decreases, and their correctness reduces by 20 percentage points at an SNR of −5 dB. While the SNR is further reduced, the performance deterioration is even more severe. The stability of the PHAT–SCOT–GCC algorithm proposed is significantly improved at low SNR. Typically, the correct rate of this algorithm reaches 76% (SNR = −10 dB), which is much higher than that of the PHAT–GCC and SCOT–GCC algorithms. Additionally, the RMSE is much lower than that of the PHAT–GCC and SCOT–GCC algorithms at low SNR. The results reveal that the proposed method effectively improves the stability of the algorithm and the accuracy of time delay estimation.

To simulate the impact of broadcasting, communication, and other periodic narrowband interference on the performance of the PHAT–SCOT–GCC algorithm proposed, we conducted simulations. We take the superposition of three sinusoids of different frequencies to simulate the periodic narrowband interference, represented as follows:

$$n_1(t)=A_1\sin (2\pi f_{1}t)+A_2\sin (2\pi f_{2}t)\sin (2\pi f_{3}t)$$

(11)

where A_i is the signal amplitude; i = 1, 2. F_i represents the oscillation frequency, i = 1, 2, 3. The values of the parameters in Equation (10) are shown in

Table 3. Periodic narrowband interference simulation parameters.

	i = 1	i = 2	i = 3
Signal amplitude Ai/v	1	1.5
Signal frequency fi/MHz	3	2.5	1.5

.

As shown in Figure 2, a mixed interference signal of Gaussian white noise and periodic narrowband noise is superimposed on the original partial discharge signal.

$$X(t)=x_1(t)+n(t)$$

(12)

where x₁(t) is the simulated first partial discharge signal shown in Figure 2, n(t) is the mixed superimposed interference signal, and X(t) is the measured signal affected by the simulation.

The SNR of the simulated analog actual signal can be expressed as:

$$SNR=20\lg \frac{V_{x-\max }}{V_{n-\max }}$$

(13)

With an SNR = −5 dB, the first simulation mimics the measured partial discharge signal, as shown in Figure 7.

As mentioned above, the PHAT–SCOT–GCC will be utilized in the following section for time delay estimation of simulated partial discharge signals. The results of time delay estimation using three algorithms (PHAT weighted function, SCOT weighted function, and improved generalized cross-correlation algorithm with joint weighted function) for an SNR of −5 dB are shown in Figure 8:

As shown in Figure 8a–c, the PHAT–GCC and SCOT–GCC algorithms cannot accurately calculate the actual delay value when the mixed superposition noise SNR = −5 dB, and the error of their delay results in more than 2 μs. The delay error of the PHAT–SCOT–GCC algorithm is 0.01 μs, which can accurately calculate the delay results. The PHAT–GCC and SCOT–GCC methods increase the false peaks of the cross-correlation function in low SNR, seriously affecting the performance of the algorithm. However, the PHAT–SCOT–GCC, with its mountains of the cross-correlation function, is free from the influence of false peaks, which can be observed intuitively due to the time delay. Therefore, the PHAT–SCOT–GCC algorithm further verifies the accuracy of the time delay estimation results in complex noise environments and is more resistant to noise interference than the PHAT–GCC and SCOT–GCC algorithms.

Again, this is consistent with the conclusions of the above analysis of Figure 5a–c. The SNR is set to decrease linearly from 0 dB to −10 dB in 5 dB steps, and 1000 experiments are conducted at each fixed SNR for time delay estimation of the three weighted functions. The correct rate (within 1 μs error) of time delay estimation is shown in

Table 4. Correctness rate of three algorithms under mixed noise.

SNR/dB	PHAT–GCC/μs	SCOT–GCC/μs	PHAT–SCOT–GCC/μs
0	93%	92%	100%
−5	25%	30%	97%
−10	15%	10%	66%

.

To further understand the performance of the PHAT–SCOT–GCC algorithm under mixed noise conditions with an SNR of −10 to 0 dB, the SNR is set to decrease linearly from 0 dB to −10 dB in the 1 dB step. The experimental validation of the three weighted functions is carried out 1000 times for each fixed SNR. The RMSE of time delay estimation is shown in Figure 9.

From Table 4 and Figure 9, it can be seen that the algorithms of the PHAT–GCC and SCOT–GCC perform excellently when the SNR is large and the correct rate is above 90%. As the SNR decreases, the performance of all three algorithms is affected. The performance of the PHAT–GCC and SCOT–GCC algorithms reduces dramatically, with the correctness rate below 20% while the SNR = −10 dB. However, the PHAT–SCOT–GCC algorithm has significantly improved its stability at low SNR; the correct rate of this algorithm reaches 66% when SNR = −10 dB, much higher than the PHAT–GCC and SCOT–GCC. Meanwhile, the RMSE of the PHAT–SCOT–GCC algorithm is lower than the PHAT–GCC and SCOT–GCC algorithms under the SNR from −10 dB to 0 dB.

In summary, under different types of noise interference, the PHAT–SCOT–GCC algorithm has superior anti-interference performance compared with the PHAT–GCC and SCOT–GCC algorithms. The reduction in SNR, the time delay estimation accuracy, and the stability of the PHAT–SCOT–GCC are significantly improved compared with the traditional algorithm.

4. Experimental Verification

To verify the performance of the time delay estimation algorithm, we build a partial discharge TDOA localization system under an outdoor complex electromagnetic environment and two station antennas for the time delay estimation test.

Partial discharge signals belong to broadband signals, with the selection of two broadband receiving antennas as the base station. Partial discharge signals are generated by the high-voltage partial discharge simulation device. The oscilloscope selection of the maximum sampling rate of the 5 GHz RIGOL oscilloscope is shown in the experimental setup shown in Figure 10. The two antennas and the discharge source are linearly arranged, and the height is in the same horizontal plane. To verify the influence of the SNR on the algorithm of the PHAT–SCOT–GCC in the actual scene, the distance L of the discharge source from antenna A is set to 1 m, 5 m, and 10 m, respectively. The SNR of the received signal is changed by changing the distance L. The partial discharge signals received by the two antennas are shown in Figure 11.

As seen in Figure 11, x₁(t) and x₂(t) are weak signals with amplitudes around 0.3 v, subject to more severe noise interference. The distance difference between the two station antennas and the partial discharge source is 3 m, so the theoretical delay is 10 ns, the oscilloscope sampling frequency is 5 GHz, and the distance of the discharge source from the antenna A is 1 m. The three algorithms (PHAT weighted function, SCOT weighted function, and the improved generalized mutual correlation algorithm with the joint weighted function) are used to carry out 30 sets of time delay estimation experiments. The cross-correlation is shown in Figure 12, and the error of the delay results is shown in Figure 13.

As presented in Figure 12a–c and Figure 13, in the real partial discharge time delay estimation scenario, the PHAT–GCC and SCOT–GCC algorithms have ten sets of anomalies showing poor stability. The PHAT–SCOT–GCC algorithm has an accuracy of 0.5 ns or less with no monsters, which exhibits improved time delay estimation accuracy and stability. At the same time, the peak of the cross-correlation function of the PHAT–GCC and SCOT–GCC algorithms is easily submerged in the noise, resulting in multiple false peaks, which seriously interferes with the accuracy of time delay estimation. The PHAT–SCOT–GCC algorithm has a significant peak value of the cross-correlation function and no false peak interference, which improves the accuracy and anti-interference ability of time delay estimation.

To further verify the application effect of the algorithm in practical engineering, we conducted 30 groups of experiments in the same testing environment. These experiments involved setting the discharge power source from the antenna A at a distance of 5 m and 10 m. We compared the mean value and RMSE of different algorithms under different SNRs to verify the performance of the algorithms, as shown in Table 4 and

Table 5. Comparison of the mean values of the three algorithms with different SNRs.

Spacing	PHAT–GCC/ns	SCOT–GCC/ns	PHAT–SCOT–GCC/ns
1 m	10.8	10.8	9.6
5 m	6.9	6.7	10.7
10 m	4.8	5.4	10.9

.

As shown in Table 5 and

Table 6. Comparison of RMSE for three algorithms with different SNRs.

Spacing	PHAT–GCC/ns	SCOT–GCC/ns	PHAT–SCOT–GCC/ns
1 m	0.93	0.93	0.24
5 m	8.9	9.5	0.7
10 m	11.4	12.3	1.14

, for the time delay estimation of partial discharge signals in practical engineering, the mean error of the PHAT–GCC and SCOT–GCC algorithms is higher than 3 ns while the distance L is 5 m–10 m. In contrast, the PHAT–SCOT–GCC algorithm can obtain accurate delay results with an error of less than 1 ns under the condition of L = 1 m–10 m, which reflects a more noise-resistant performance. Meanwhile, the RMSE of the PHAT–SCOT–GCC algorithm is lower than the PHAT–GCC and SCOT–GCC algorithms. When L is 10 m, the RMSE of the PHAT–SCOT–GCC algorithm is 1.14, which is far less than 11.4 and 12.3 of the PHAT–GCC and SCOT–GCC algorithms, respectively. So, the PHAT–SCOT–GCC algorithm has higher stability and effectiveness.

5. Conclusions

This paper proposes a GCC algorithm with an improved PHAT–SCOT weighting function. The experimental scene is a far-field test in a relatively complex outdoor electromagnetic environment, and the measured signal is weak, resulting in a low SNR. In order to simulate the noise interference in a real environment, the effects of Gaussian white noise and mixed noise interference on the estimation of the delay of partial discharge signals were investigated at low SNRs (SNR = −10–0 dB). It is verified that the delay accuracy of the –SCOT–GCC algorithm is significantly improved compared with the traditional algorithm.

• Simulation and analysis of partial discharge signals with the addition of Gaussian white noise. The PHAT–SCOT–GCC algorithm has significant cross-correlation peaks without the interference of spurious peaks, so the delay results can be obtained accurately. Through 1000 experiments at a lower SNR (SNR = −10 dB), the PHAT–SCOT–GCC algorithm is 76% correct. The RMSEs of PHAT–SCOT–GCC are all less than 0.15 μs when the SNR is below 0 dB, and the stability is higher than that of the PHAT–GCC and SCOT–GCC algorithms. Thus, the superior noise immunity of PHAT–SCOT–GCC is verified.
• To deeply investigate the noise immunity performance of PHAT–SCOT–GCC, the cross-correlation peaks remain obvious when mixed noise interference is introduced. At low SNR (SNR = −10 dB), the PHAT–SCOT–GCC algorithm’s correctness is improved by at least 50 percentage points. The RMSE of PHAT–SCOT–GCC is less than 0.3 μs when the SNR is less than 0 dB, which further validates the noise immunity performance of PHAT–SCOT–GCC.
• Compare the mean and RMSE of the algorithms at different SNRs in time delay estimation experiments for partial discharge signal detection in real scenarios. The algorithm proposed in this paper has an error of less than 1 ns, and the RMSE is lower than the traditional algorithm, which is more stable and effective. The PHAT–SCOT–GCC can better meet the accuracy requirements of practical engineering in complex electromagnetic environments.