CVT is a special transformer used to convert high-voltage primary measurement signals into low-voltage secondary outputs suitable for measurement. Its structure, shown in Figure 1 , consists of two parts: a capacitive voltage divider and an intermediate transformer. The capacitive voltage divider achieves a voltage reduction through capacitive division to obtain a low-voltage output suitable for measurement. In this setup, the division capacitors $C_M$ and $C_H$ are connected in series, and different voltage division ratios can be obtained by adjusting their sizes. Figure 1 , $U_p$ represents the primary input voltage, $U_s$ represents the secondary output voltage measurement, L denotes the compensating inductor, and D denotes residual windings. The intermediate transformer further reduces the low voltage signal from the capacitive divider to a suitable range for measurement, protection, and communication CVTs are manufactured by different manufacturers, and their parts can be affected by the surrounding conditions. This leads to differences between the voltage they actually produce and what they should ideally produce, which is referred to as a measurement error. Official rules state that these errors can affect how well the voltage matches in terms (ratio errors) or timing (phase errors). The primary focus is on the first type: the proximity of the actual voltage level to the expected value. In voltage measurement,

this discrepancy is referred to as the “ratio difference,” which can be described using an equation, as shown in Equation (1).

$\epsilon \left(t\right)=\frac{k_r{U}_s\left(t\right)-U_p\left(t\right)}{U_p\left(t\right)}\times 100\text{\%}$(1)

where $\epsilon \left(t\right)$ represents the ratio error of the CVT at time $t$, $k_r$ is the nominal transformation ratio of the CVT; and $U_p\left(t\right)$ and $U_s\left(t\right)$ represent the primary and secondary voltages of the CVT at time $t$, respectively.

The measurement error level of CVT can be described by its accuracy class. According to national standards, within the specified range of primary voltage and secondary load variations, when the load power factor is at its rated value, the maximum value of the voltage error represents the accuracy class. This is typically expressed as a percentage, such as 0.2, 0.5, 1, 3, etc., as shown in Table 1 .

The operation mode of the power system and the electrical connection relationship between CVTs impose restrictions on the layout of CVTs, exhibiting the characteristics of symmetrical distribution and electrical correlation. In high-voltage substations, the configuration of CVTs is affected by the primary wiring method. Typically, multiple groups of CVTs are used to measure the voltage on the same bus to ensure accuracy. In double-bus or double-bus segmented wiring and 3/2 wiring, the primary side of all the CVTs is connected to the same bus; therefore, the input voltage is equal.

$U_{p1}\left(t\right)=U_{p2}\left(t\right)=\cdots =U_{pn}\left(t\right)$(2)

where $U_p\left(t\right)$ represents the amplitude of the busbar voltage and $U_{pn}\left(t\right)$ represents the amplitude of the primary measurement voltage of CVT-n.

CVTs are a special type of transformer that converts high-voltage signals from the busbar into low-voltage signals. According to Equation (3), the secondary output of the CVT can be derived as follows.

$\begin{array}{c}{U}_s=U_s+\text{Δ}U\\ =\left(\begin{array}{cccc}\frac{1+{\epsilon }_1\left(t\right)}{k_r}& 0& \cdots & 0\\ 0& \frac{1+{\epsilon }_2\left(t\right)}{k_r}& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & \frac{1+{\epsilon }_n\left(t\right)}{k_r}\end{array}\right)\times \left(\begin{array}{c}{U}_{p1}\left(t\right)\\ U_{p2}\left(t\right)\\ ⋮\\ U_{pn}\left(t\right)\end{array}\right)\pm \left(\begin{array}{c}\text{Δ}{U}_1\left(t\right)\\ \text{Δ}{U}_2\left(t\right)\\ ⋮\\ \text{Δ}{U}_n\left(t\right)\end{array}\right)\end{array}$(3)

where ${\epsilon }_n\left(t\right)$ represents the ratio error of the nth CVT at time $t$, $U_{sn}\left(t\right)$ and $U_{pn}\left(t\right)$ represent the primary and secondary voltages of the nth CVT at time $t$ respectively, and $\text{Δ}{U}_n\left(t\right)$ represents the magnitude of the voltage fluctuations caused by the three-phase load asymmetry.

Owing to the absence of load asymmetry in the correlation relationship, $\text{Δ}{U}_n\left(t\right)$ is 0 in Equation (3), which is simplified to Equation (4).

$\left(\begin{array}{c}\begin{array}{c}{U}_{s1}\left(t\right)\\ U_{s2}\left(t\right)\\ ⋮\\ U_{sn}\left(t\right)\end{array}\end{array}\right)=\left(\begin{array}{cccc}\frac{1+{\epsilon }_1\left(t\right)}{k_r}& 0& \cdots & 0\\ 0& \frac{1+{\epsilon }_2\left(t\right)}{k_r}& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & \frac{1+{\epsilon }_n\left(t\right)}{k_r}\end{array}\right)\times \left(\begin{array}{c}\begin{array}{c}{U}_{p1}\left(t\right)\\ U_{p2}\left(t\right)\\ ⋮\\ U_{pn}\left(t\right)\end{array}\end{array}\right)$(4)

According to the research in Section B regarding the connection methods of CVTs, the primary-side input voltage of the same-phase CVTs on the same bus is equal. Therefore, the secondary voltage amplitudes of single-phase CVTs operating at the same voltage level exhibit the following relationship.

$\begin{array}{l}\frac{u_{s1}\left(t\right)}{u_{sn}\left(t\right)}=\frac{1+{\epsilon }_1\left(t\right)}{1+{\epsilon }_n\left(t\right)}\hfill \end{array}$(5)

$u_{s1}\left(t\right)$ and $u_{sn}\left(t\right)$ represent the initial and subsequent secondary-side output voltages, respectively. ${\epsilon }_1\left(t\right)$ is defined as the first ratio error at time $t$, and correspondingly, ${\epsilon }_n\left(t\right)$ represents the n-th ratio error at the same time $t$.

CVTs with at least three channels of the same phase were examined. Based on Equations (4) and (5), which represent the invariance of the ratio relationship between same-phase channels, the co-proportional matrix $X\left(t\right)\in {ℝ}^{n\times n}$ at sampling point $t$ is defined as shown in Equation (6). This matrix captures the ratio relationships between the measurement errors of all the channels in the CVTs at time $t$.

$\begin{array}{c}X\left(t\right)=\left(\begin{array}{c}{u}_{s1}\left(t\right)\\ ⋮\\ u_{sn}\left(t\right)\end{array}\right)\times {\left(\begin{array}{c}{u}_{s1}^{-1}\left(t\right)\\ ⋮\\ u_{sn}^{-1}\left(t\right)\end{array}\right)}^{\text{T}}=\left(\begin{array}{ccc}1& \cdots & \frac{u_{s1}\left(t\right)}{u_{sn}\left(t\right)}\\ ⋮& \ddots & ⋮\\ \frac{u_{sn}\left(t\right)}{u_{s1}\left(t\right)}& \cdots & 1\end{array}\right)\\ =\left(\begin{array}{ccc}1& \cdots & \frac{1+{\epsilon }_1\left(t\right)}{1+{\epsilon }_n\left(t\right)}\\ ⋮& \ddots & ⋮\\ \frac{1+{\epsilon }_n\left(t\right)}{1+{\epsilon }_1\left(t\right)}& \cdots & 1\end{array}\right)\end{array}$(6)

When a CVT undergoes error variation at time $t$, the co-proportional matrix $X\left(t\right)$ exhibits significant fluctuations compared to $X\left(t-1\right)$. Therefore, matrix $H\left(t\right)\in {ℝ}^{i\times j}$ is defined to represent the relationship between the co-proportional matrices at times $t$ and $t-1$, and its elements are calculated according to Equation (7).

$\begin{array}{l}{H}_{i,j}\left(t\right)=\frac{X_{i,j}\left(t-1\right)}{X_{i,j}\left(t\right)}=\frac{1+{\epsilon }_i\left(t-1\right)}{1+{\epsilon }_j\left(t-1\right)}\cdot \frac{1+{\epsilon }_j\left(t\right)}{1+{\epsilon }_i\left(t\right)}\hfill \end{array}$(7)

In the equation, $H_{i,j}\left(t\right)$ and $X_{i,j}\left(t\right)$ represent the elements in the ith row and jth column of matrices $H\left(t\right)$ and $X\left(t\right)$ at time $t$. Additionally, $X_{i,j}\left(t\right)$ represents the ratio of the secondary voltage measurements between CVT-i and CVT-j. Furthermore, $H_{i,j}\left(t\right)$ is calculated based on the amplitude of the secondary output voltage.

Combining Equation (6), matrix $H\left(t\right)$ can be represented as Equation (8).

$H\left(t\right)=\left(\begin{array}{ccc}1& \cdots & \frac{1+{\epsilon }_1\left(t-1\right)}{1+{\epsilon }_j\left(t-1\right)}\cdot \frac{1+{\epsilon }_i\left(t\right)}{1+{\epsilon }_1\left(t\right)}\\ ⋮& \ddots & ⋮\\ \frac{1+{\epsilon }_i\left(t-1\right)}{1+{\epsilon }_1\left(t-1\right)}\cdot \frac{1+{\epsilon }_1\left(t\right)}{1+{\epsilon }_j\left(t\right)}& \cdots & 1\end{array}\right)$(8)

According to Equation (7), the measurement error ${\epsilon }_j$ of the jth CVT at time $t$ can be derived using the calculation detailed in Equation (9).

${\epsilon }_j\left(t\right)=\left(1+{\epsilon }_j\left(t-1\right)\right)\cdot \frac{1+{\epsilon }_i\left(t\right)}{1+{\epsilon }_i\left(t-1\right)}\cdot H_{i,j}\left(t\right)-1$ (9)

At time $t$, in Equation (9), CVT-i and CVT-j at time $t-1$ are known values. If this is the first evaluation point of the sample, it represents the initial error of CVT.

When there is no change in the measurement error at times $t$ and $t-1$, that is, ${\epsilon }_i\left(t\right)={\epsilon }_i\left(t-1\right)$, the magnitude of the measurement error of CVT-j is denoted as ${{\epsilon }_j\left(t\right)|}_i$, as shown in Equation (10).

${{\epsilon }_j\left(t\right)|}_i=\left(1+{\epsilon }_j\left(t-1\right)\right)\cdot H_{i,j}\left(t\right)-1$(10)

where ${{\epsilon }_j\left(t\right)|}_i$ represents the measurement error of the jth CVT when the ith CVT remains constant.

By combining Equation (8) and Equation (10), a matrix composed of errors can be obtained, referred to as the co-prediction matrix ${\epsilon }^{\text{pre}}\left(t\right)\in {ℝ}^{i\times j}$, where ${\epsilon }^{\text{pre}}$ represents the predicted error. This is detailed in Equation (11).

${\epsilon }^{\text{pre}}\left(t\right)=\left(\begin{array}{c}{\epsilon }_1^{\text{pre}}\left(t\right)\\ {\epsilon }_2^{\text{pre}}\left(t\right)\\ ⋮\\ {\epsilon }_i^{\text{pre}}\left(t\right)\end{array}\right)=\left(\begin{array}{cccc}{{\epsilon }_1\left(t-1\right)|}_1& {{\epsilon }_2\left(t\right)|}_1& \cdots & {{\epsilon }_j\left(t\right)|}_1\\ {{\epsilon }_1\left(t\right)|}_2& {{\epsilon }_2\left(t-1\right)|}_2& \cdots & {{\epsilon }_j\left(t\right)|}_2\\ ⋮& ⋮& \ddots & ⋮\\ {{\epsilon }_1\left(t\right)|}_i& {{\epsilon }_2\left(t\right)|}_i& \cdots & {{\epsilon }_j\left(t-1\right)|}_i\end{array}\right)$(11)

The physical meaning of this equation is as follows: at time t, compared with time t-1, if the i-th CVT does not experience an error drift, it indicates the measurement error magnitude of the other CVTs. ${\epsilon }_i^{\text{pre}}\left(t\right)$ represents the measurement error row vector at time t in the i-th row. The establishment of the co-prediction matrix enables quantitative calculation and real-time tracking of the monitored CVT measurement errors.

In the co-prediction matrix, the elements related to the CVT with error changes correspondingly change, whereas those unaffected will remain relatively stable. Based on this characteristic, these changes can be identified by comparing the predicted error values over the time series, and an error localization matrix $\text{Δ}\epsilon \left(t\right)$ can be defined. The Specific Calculation Methods for this matrix are shown in Equation (12).

$\text{Δ}\epsilon \left(t\right)=\left(\begin{array}{c}{\epsilon }_1^{\text{pre}}\left(t\right)\\ {\epsilon }_2^{\text{pre}}\left(t\right)\\ ⋮\\ {\epsilon }_i^{\text{pre}}\left(t\right)\end{array}\right)-\left(\begin{array}{c}{\epsilon }^{ret}\\ {\epsilon }^{ret}\\ ⋮\\ {\epsilon }^{ret}\end{array}\right)$(12)

where ${\epsilon }^{ret}$ represents the newly updated error vector.

Owing to the random fluctuations present in CVTs, even if they are co-related, the elements in the co-proportion matrix may not be exactly equal over the time series when no error changes occur in the CVTs. However, they may exhibit only minor fluctuations. These fluctuations can be filtered by setting a threshold, calculated using Equation (13).

$\begin{array}{l}-\alpha <\text{Δ}{\epsilon }_{i,j}\left(t\right)<\alpha ,\left(i\ne j\right)\hfill \end{array}$(13)

where $\text{Δ}{\epsilon }_{i,j}\left(t\right)$ and $\alpha \in \left[-0.0001,0.0001\right]$ represent the error in the i-th row and j-th column of $\text{Δ}\epsilon \left(t\right)$ and the set threshold, respectively.

In the event of a power outage, if the system deviation prior to the outage remains within the accuracy standard of 0.2, the latest error vector of the system post-restoration can be estimated by averaging the most recent 100 error vectors ${\epsilon }^{ret}$. This approach ensures data continuity and maintains system stability.

The experiment was conducted on CVTs operating in a 330 kV substation in Northwest China, which employs a 3/2 wiring configuration where outgoing lines are connected to CVTs labeled A, B, and C. All CVTs, model TYD300/3-0.005 with an accuracy class of 0.2, were manufactured by Xi’an Xirong Electric Power Capacitor Co., Ltd. To ensure experimental accuracy, the initial errors of all the CVTs were set to zero. To precisely capture the secondary voltage signals, a high-precision acquisition device with an accuracy level of 0.01 was selected (see Figure 3 and schematic in Figure 4 ). This device features a 24-bit A/D converter at the sampling rate of 10 kHz and outputs data every three minutes. According to the acceptance test for substation, the ratio errors of three CVTs of the A phase are recorded from the test reports, as shown in Table 2 .

Over a continuous period of 15 days, comprehensive data acquisition was conducted on nine CVTs, with each CVT collecting 7200 data points. To verify the high accuracy and reliability of the collected data, the measured voltages ( $U_s$) were converted to primary side voltages ( $U_p$) using the nominal transformation ratio, and the ratio errors of the three CVTs were calculated using Equation (1). The ratio error curves and voltage waveforms are shown in Fig.5. By comparing the ratio error curves in Figure 5 with the numerical values listed in Table 2 , it was observed that the ratio errors of the CVTs showed no significant deviation from the acceptance test results and complied with the 0.2 level precision requirement specified in Table 1 .

1) Based on the basic conditions and data collection of the substation, the initial dataset was constructed from the first one thousand sampling points of three phase a CVTs, $X_i^{in}=\left[x_1^{in},x_2^{in},x_3^{in}\right]\in {ℝ}^{1000\times 3}$. Subsequently, sampling points 1001 to 7200 were designated as the monitoring dataset without added set error, $X_i^{Ev}=\left[x_1^{Ev},x_2^{Ev},x_3^{Ev}\right]\in {ℝ}^{6200\times 3}$.

2) Different true error datasets were constructed for the CVTs with varying settings across different periods, denoted as ${\epsilon }^{Tru}=\left[{\epsilon }_1^{Tru},{\epsilon }_2^{Tru},\cdots ,{\epsilon }_{5200}^{Tru}\right]$. This dataset was added to the last 5200 data points of the monitoring dataset $X_i^{Ev}=\left[x_1^{Ev},x_2^{Ev},x_3^{Ev}\right]\in {ℝ}^{6200\times 3}$ for CVT1, whereas the other two groups of CVTs remained unchanged. After processing, the first set of experimental data was obtained as $X_i^{Ev}=\left[x_1^{Ev}\times \left(1+{\epsilon }^{Tru}\right),x_2^{Ev},x_3^{Ev}\right]\in {ℝ}^{6200\times 3}$, from which the measurement error value and its variation for the CVT were calculated.

3) Step 2 was repeated for the remaining two CVTs. The predicted error value and variations in the predicted errors for the CVT were recorded separately for subsequent analysis and discussion.

In the experiment, we conducted a detailed analysis of the predicted error values, with the key metric being the predicted bias ${\epsilon }^{\text{bia}}$, which is defined as the difference between the predicted error value ${\epsilon }^{\text{pre}}$ obtained through the co-prediction matrix and the true error value ${\epsilon }^{\text{tru}}$. The calculation formula is ${\epsilon }^{\text{bia}}={\epsilon }^{\text{pre}}-{\epsilon }^{\text{tru}}$. To evaluate the performance under different conditions, various magnitudes of true errors were introduced into the CVT monitoring dataset at different times (see Table 3 ): for CVT1, with errors of 0.002, 0.0016, and 0.0015 for CVT2 and CVT3, errors of 0.001 and 0.0012 were set, respectively. Fig.6 shows the changes in each CVT over time or sample sequence, with the x-axis representing the time or sample number and the y-axis representing the predicted error value ${\epsilon }^{\text{pre}}$.

As shown in Figure 6 , CVT1 did not have a true error added in the sample interval from 0 to 1000, resulting in a predicted error that is essentially zero. In the sample interval from 1000 to 2500, the predicted error aligns well with the set true error of 0.002. Subsequently, in the sample interval from 2500 to 4000, the predicted error is approximately 0.0015, which was consistent with the added true error. Finally, in the sample interval from 4000 to 6200, the predicted error remained at approximately 0.0016, which matched the set true error. Similar error tracking results were obtained for CVT2 and CVT3, where the predicted errors at different sample intervals matched the set true errors. These results clearly demonstrate the effectiveness of using a co-prediction matrix for error tracking.

Combining the data from Table 4 and Figure 7 further illustrates the maximum absolute predicted bias values for each CVT. CVT1 performed exceptionally well,

with a maximum predicted bias of only 0.0016%, whereas the maximum predicted biases for the other two CVTs were controlled within 0.0037%. Overall, the measurement error accuracy obtained by this method is less than 0.01%, which is significantly better than the requirements of the traditional 0.5-class transformer calibration standard. This demonstrates that the proposed method is not only effective but also highly practical, showing significant potential for improving the monitoring accuracy of Capacitive Voltage Transformers in power systems.

To evaluate the effectiveness of different monitoring methods in handling measurement errors in Capacitive Voltage Transformers, true errors of varying magnitudes were introduced into CVT1-A at different time points in the dataset. Specifically, error values of 0.002, 0.0015, and 0.0016 were added to the three sample segments, as shown in Table 5 . Subsequently, the performance of the proposed method was compared with that of ICA based on three-phase balance [13] , PCA based on three-phase balance [14] .

Figure 8 shows the results of three-phase balanced independent component analysis (TP-ICA). In Figure 8 , the statistical value in the sample interval from 0 to 1000 should not exceed the threshold, whereas in the sample interval from 1000 to 6200, the statistical value should not be lower than the threshold. However, in this interval, only 47.19% of the Q value exceeded the set threshold, resulting in failure of effective error tracking. Figure 9 shows the results of the three-phase balanced principal component analysis (TP-PCA). Similar to the results of TP-ICA, in Figure 9 , the threshold value was exceeded in the sample interval from 0 to 1000, whereas the statistical value was below the threshold value in the sample interval from 1000 to 6200. Similarly, in this interval, only 32.88% of the Q values exceeded the set threshold, which made it impossible to carry out effective error tracking.

Figure 10 shows the results obtained using the co-prediction matrix method. This method does not produce error changes in the 0 - 1000 sample interval, but the predicted error value in the 1000 - 2500 sample interval is consistent with 0.002 in Table 5 . The predicted error value in the 2500 - 4000 sample interval and 4000 - 6200 sample interval were also consistent with the data in Table 5 . This method can monitor and estimate the error amplitude in real time, which is consistent with the actual situation, and shows higher accuracy and response speed.

In summary, although TP-ICA and TP-PCA could detect anomalies to some extent, their detection rates were relatively low. The proposed method not only has high sensitivity but also effectively tracks errors in real-time, making it particularly suitable for CVT online monitoring scenarios. As shown in Table 6 , the method developed in this study demonstrates significant advantages in handling measurement errors, efficiently identifying abnormal states without requiring complex statistical computations, providing immediate dynamic feedback on errors, and enhancing data responsiveness and overall efficiency in rapidly changing environments.

1) This method is suitable for recently commissioned substations. If a

long-operating substation has recently undergone periodic on-site calibration, as long as the recent calibration error is considered as the initial error, this method can also be applied.

2) A single-phase CVT group in a station must consist of at least three or more CVTs, and the ratio error must not be changed for the same set of samples. When the group contains only two CVTs and the error of one CVT changes, Equation (13) does not hold. Therefore, it is impossible to identify faulty CVT accurately.