Global Sensitivity Analysis of Tie-Line Power on Voltage Stability Margin in Renewable Energy-Integrated System

Abstract

With the increasing load and renewable energy capacity in interconnected power grids, the system voltage stability faces significant challenges. Tie-line transmission power is a critical factor influencing the voltage stability margin. To address this, this paper proposes a fully data-driven global sensitivity calculation method for the tie-line power-voltage stability margin, aiming to quantify the impact of tie-line power on the voltage stability margin. The method first constructs an online estimation model of the voltage stability margin based on system measurement data under ambient excitation. To adapt to changes in system operating conditions, an online updating strategy for the parameters of the margin estimation model is further proposed, drawing on incremental learning principles. Subsequently, considering the source–load uncertainty of the system, a global sensitivity calculation method based on analysis of variance (ANOVA) is proposed, utilizing online acquired voltage stability margin and tie-line power data, to accurately quantify the impact of tie-lines on the voltage stability margin. The accuracy of the proposed method is verified through the Nordic test system and the China Electric Power Research Institute (CEPRI) standard test case; the results show that the error of the proposed method is less than 0.3%, and the computation time is within 1 s.

1. Introduction

With the continuous advancement of energy transition, the “dual-high” characteristics of high proportions of renewable energy and power electronic devices in new power systems are increasingly prominent, leading to a sustained increase in system operating pressure [1]. The system operation is gradually approaching its stability limit, and the continuous growth of load can easily trigger voltage instability or even collapse. Typical accident cases include the 2019 UK blackout, the 2021 Xi’an blackout, and the 2023 Brazilian 8.15 incident [2,3,4]. Due to the tight coupling of interconnected power grids and the increasingly frequent power exchange between regional grids, tie-line transmission power has a significant impact on system voltage stability.

With the large-scale grid connection of intermittent and uncertain renewable energy sources, the power fluctuations transmitted by tie-lines become more frequent. As a weak link in the system [5], heavily loaded tie-lines have a more complex impact on the system voltage stability. Therefore, timely assessment of the system voltage stability margin and accurate quantification of the impact of tie-lines on the system voltage stability margin are of great significance for formulating voltage stability margin optimization and enhancement strategies.

Sensitivity analysis enables quantitative analysis of the system input on the output results [6,7,8,9]. Therefore, existing studies quantify the impact of key links on system stability by calculating the sensitivity of control variables to the power system state. Among them, reference [10] calculates the sensitivity of control variables to the voltage stability margin based on eigenvalue analysis; reference [11] establishes the sensitivity of generator active-power output to the voltage stability margin through the Jacobian matrix; in order to further improve the response speed of voltage control strategies, reference [12] uses the sensitivity of the generator active-power output to the load margin to quickly screen the generator set for participation, thereby effectively reducing the subsequent optimization search range.

However, the calculation of the sensitivity of control variables and the voltage stability margin in the above studies generally relies on the power system power flow model. In the current new power system with large-scale renewable energy grid connection, the system operating conditions are becoming more complex and variable, and the randomness and uncertainty are increasing. It is difficult to obtain sensitivity results during system operation based on traditional modeling methods, which is not conducive to the online implementation of voltage stability control means.

To this end, this paper proposes a fully data-driven tie-line power-voltage stability margin global sensitivity calculation method for large-scale renewable energy grid-connected systems with high source–load uncertainty. The main contributions are as follows:

(1)

An online estimation model of the voltage stability margin is proposed based on measurement data such as the system node voltage and phase angle under ambient excitation to obtain the system voltage stability margin online.

(2)

To update the parameters of the proposed margin estimation model, the incremental learning algorithm is adopted to realize the online updating of the system voltage stability margin.

(3)

Through the obtained voltage stability margin estimation results and system tie-line transmission power operation data, an online calculation method of voltage stability margin global sensitivity based on analysis of variance (ANOVA) is proposed to accurately quantify the impact of tie-line power on the voltage stability margin.

(4)

The accuracy and adaptability of the proposed global sensitivity calculation method are verified by the Nordic test system simulation and the China Electric Power Research Institute standard test case.

2. Data-Driven Online Extraction of Voltage Stability Margin Based on Incremental Learning

2.1. Voltage Stability Margin Online Estimation Model Construction

Current methods for establishing voltage stability margin estimation models based on machine learning have reached a relatively mature stage [13]. This section first constructs a training sample set of system operating variables and voltage stability margins. To determine the mapping relationship between the input operating variables and the output voltage stability margin, a voltage stability margin estimation model is constructed based on the Gradient Boosting Decision Tree (GBDT). Furthermore, incorporating the concept of incremental learning, an online updating method for model parameters is proposed to adapt to changes in system operating conditions, thereby achieving data-driven online extraction of the voltage stability margin.

Considering system operating variables that are strongly correlated with the voltage stability margin: node voltage magnitudes, phase angles, and tie-line power, are used as input variables for the voltage stability margin estimation model training data. The voltage stability margin calculated by Continuation Power Flow (CPF) is used as the output variable of the training data to construct the following training sample set:

$$\begin{array}{l}(\mathit{X},\mathit{Y})=\left\{\begin{array}{c}{\mathit{x}}^1,{\mathit{y}}^1\\ ⋮\\ {\mathit{x}}^j,{\mathit{y}}^j\end{array}\right\}\\ =\left[\begin{array}{c}{v}_1^1,v_2^1,\dots ,v_p^1,{\delta }_1^1,{\delta }_2^1,\dots ,{\delta }_p^1,P_1^1,P_2^1,\dots ,P_q^1,y^1\\ ⋮\\ v_1^j,v_2^j,\dots ,v_p^j,{\delta }_1^j,{\delta }_2^j,\dots ,{\delta }_p^j,P_1^j,P_2^j,\dots ,P_q^j,y^j\end{array}\right]\end{array}$$

(1)

where p and q are the number of load buses and tie-lines in the system, respectively. j is the sampling time, i.e., the number of samples. $v, \delta , P,y$ represent the node voltage magnitudes, phase angles, tie-line active power, and voltage stability margin, respectively.

GBDT is an integrated decision tree model that updates the decision tree by iteratively calculating the negative gradient of the samples, achieving rapid model training and prediction. Based on the above training sample set (X, Y), a GBDT base prediction model is established [14]:

$$f_0(\mathit{x})=\underset{c}{\underbrace{\arg \min }}{\displaystyle \sum _{j=1}^{J}L({\mathit{y}}^j,c)}$$

(2)

where $L(\mathit{y},f(x))$ is the loss function generated after predicting the sample x. c is the initial estimate, which can generally be initialized as the mean of all sample output values.

To ensure that the loss function is reduced in each iteration process, GBDT calculates and updates the negative gradient of the samples for the number of iterations k (k = 1, 2, …, K), that is, the residual value:

$${\mathit{g}}_{kj}={\left[{\displaystyle \frac{\partial L({\mathit{y}}^j,f({\mathit{x}}^j))}{\partial f({\mathit{x}}^j)}}\right]}_{f(x)=f_{(k-1)}(x)}$$

(3)

When all samples in the training sample set traverse f_k(x) to each leaf node in the k-th iteration, calculate and update the residual value. Use each updated residual value as the new sample input value for the next iteration.

For the leaf node region R_km, where m = 1, 2, …, M leaf nodes, calculate the best fitting value:

$$c_{km}=\arg {\min }_c{\displaystyle \sum _{x^j\in R_{km}}L}\left(y^j,f_{k-1}\left(x^j\right)+c\right)$$

(4)

where $c_{km}$ is the smallest square loss of the leaf node R_km.

Update the base prediction model:

$$f_k(x)=f_{k-1}(x)+{\displaystyle \sum _{m=1}^M{c}_{km}}I\left(x\in R_{km}\right)$$

(5)

where I is the indicator function, if (condition), then I = 1; otherwise, I = 0.

Furthermore, the final GBDT prediction model can be trained to obtain:

$$\widehat{f}(x)=f_K(x)=f_0(x)+{\displaystyle \sum _{k=1}^K{\displaystyle \sum _{m=1}^M{c}_{km}}}I\left(x\in R_{km}\right)$$

(6)

After the above training process, a voltage stability margin estimation model based on GBDT is formed. During online application, it is only necessary to input the system operating variables at a certain moment as the model input to output the voltage stability margin at the current moment.

However, due to the widespread access of new energy and power electronic equipment, system operating conditions are becoming more complex and variable. The adaptability of models trained based on system operating data obtained offline under limited operating conditions will decrease, resulting in a decrease in the accuracy of the margin estimation results. Therefore, in order to ensure the accuracy and adaptability of the proposed margin estimation model, the second part of this section incorporates the idea of incremental learning and proposes an online updating method for the margin estimation model parameters to realize the online extraction of the voltage stability margin under multiple operating conditions.

2.2. Online Extraction of Voltage Stability Margin Based on Incremental Learning

To achieve online updating of the margin estimation model parameters and ensure that the proposed margin estimation model has good adaptive performance under changing system operating conditions, let f_t−1 be the margin estimation model before parameter updating. Based on its original training sample set D^t−1(x^t−1, y^t−1), a sample distribution function (weight parameter) is defined as:

$${\mathit{w}}_{D_{t-1}}^{t-1}=\{{\mathit{w}}_1^{t-1},{\mathit{w}}_2^{t-1},\dots ,{\mathit{w}}_j^{t-1}\}$$

(7)

where j is the number of samples.

When the system operation mode changes, a new update training sample set D^t(x^t, y^t) is obtained. Its initial weight is calculated as:

$${\mathit{w}}_{D_t}^{t-1}=\{{\mathit{w}}_1^{t-1},{\mathit{w}}_2^{t-1},\dots ,{\mathit{w}}_n^{t-1}\}$$

(8)

where n is the number of samples collected by the system under the new operating mode. Generally, the initial weight is calculated as: $w_j^{t-1}={\displaystyle \frac{1}{n-m-1}}$.

Subsequently, ${\widehat{f}}_{t-1}(x)$ is used to estimate the voltage stability margin of the update training sample set, and the estimation error is calculated [15]:

$${\mathit{T}}_t=\underset{n-j}{\underbrace{\arg \max }}\left|{\mathit{y}}_n-{\widehat{f}}_{t-1}({\mathit{x}}_n)\right|$$

(9)

$${\mathit{e}}_n^t=\left|{\mathit{y}}_n-{\widehat{f}}_{t-1}({\mathit{x}}_n)\right|/{\mathit{T}}_t$$

(10)

Based on the estimation error obtained from Equation (10), the adjustment error required for the parameter update is further calculated:

$${\mathit{\epsilon }}_t={\displaystyle \sum _{i=1}^n({\mathit{w}}_j^{t-1}\cdot {\mathit{e}}_n^t}/{\displaystyle \sum _{i=1}^n{\mathit{w}}_j^{t-1}})$$

(11)

The margin estimation model weight parameters are then updated:

$${\mathit{w}}_n^t={\mathit{w}}_n^{t-1}{\mathit{\beta }}_t^{e_n^t}/z$$

(12)

Based on the above parameter updating process, the proposed margin estimation model can adapt to the changes in the system operating conditions by updating the set weight parameters, realizing online extraction of the voltage stability margin under the large-scale renewable energy grid connection, as shown in Figure 1.

Remark 1.

In practical applications, the input to the proposed method is system measurement data collected from PMUs, specifically, the response data of power, voltage, and phase angle under ambient excitations, such as renewable energy and load fluctuations.

3. Global Sensitivity Calculation of Voltage Stability Margin Based on Analysis of Variance

3.1. The Proposed Method

Traditional methods for determining the sensitivity of the voltage stability margin, which rely on system modeling analysis and the existing Monte Carlo sampling methods, face challenges in application within new power systems due to incompatibility with the dynamic operating conditions of the system and the inability to be applied online. Therefore, this section, based on the acquired voltage stability margin and tie-line power measurement data from system operation, proposes an online calculation method for the global sensitivity of the voltage stability margin based on Analysis of Variance (ANOVA), grounded in Sobol’ global sensitivity theory. This method extracts the voltage stability margin sensitivity online, thereby quantifying the impact of tie-line power on the voltage stability margin, providing a basis for subsequent control measures.

Figure 2 shows 1000 sets of calculation results obtained through the proposed method using online measured voltage magnitudes, phase angles, and tie-line power for online extraction of the voltage stability margin in a certain actual power system.

As can be seen from Figure 2, the voltage stability margin obtained through the proposed method exhibits a high degree of randomness, and the online measured tie-line power also demonstrates random characteristics.

Accordingly, taking the Sobol’ method’s global sensitivity as the foundation, the analytical function is expanded in the form of an Analysis of Variance (ANOVA) kernel. And the interpolation function is further rewritten into the following form of the kernel function:

$$m(\mathit{X})=\mathit{k}{(\mathit{X})}^t{K}^{-1}\mathit{F}$$

(13)

where $\mathit{F}=f(\mathit{X})$ is the output random data column vector. K represents the Gram matrix, and $K_{i,j}=K(x_i,x_j)$, where $\mathit{X}=(x_1,\dots ,x_i,\dots ,x_j,\dots ,x_d)$ is the input random data, and $\mathit{k}(\mathbf{·})$ is a column vector of the function $K{(x_i,.)}_{(1\le i\le d)}$.

Expand the above equation using the ANOVA kernel:

$$\mathit{k}(\mathit{X})=1+{\displaystyle \sum _{I\subset 1,\dots ,d}\underset{i\in I}{\odot }}{\mathit{k}}^i\left(x_i\right)$$

(14)

where 1 is the bias term, i.e., a column vector of all 1; ⊙ represents the element-wise multiplication of matrices and has ${\left(\underset{i\in I}{\odot }{\mathit{k}}^i\left(x_i\right)\right)}_j={\displaystyle \prod _{i\in I}{k}^i}(x_i,x_j)$.

Therefore, the interpolation function can be further rewritten as:

$$m(\mathit{X})=1^t{K}^{-1}\mathit{F}+{\displaystyle \sum _{I\subset 1,\dots ,d}(\underset{i\in I}{\odot }}{\mathit{k}}^i\left(x_i\right){)}^t{K}^{-1}\mathit{F}$$

(15)

Since $m_0=1^t{K}^{-1}\mathit{F}$ and $m_I(\mathit{X})={(\underset{i\in I}{\odot }{\mathit{k}}^i\left(x_i\right))}^t{K}^{-1}\mathit{F}$, Equation (15) can be written as:

$$m(\mathit{X})=m_0+{\displaystyle \sum _{i=1}^d{m}_i}(x_i)+{\displaystyle \sum _{1\le i<j\le d}{m}_{i,j}(x_i, x_j)}+\cdots +m_{1, \dots , d}(\mathit{X})$$

(16)

Further obtaining the Sobol’ global sensitivity expression based on the ANOVA kernel [16]:

$$S_I={\displaystyle \frac{\mathrm{var}(m_I({\mathit{X}}_I))}{\mathrm{var}(m(\mathit{X}))}}$$

(17)

where $\begin{array}{l}\begin{array}{c}\mathrm{var}\left(m_I\left({\mathit{X}}_I\right)\right)=\mathrm{var}\left(\left(\underset{i\in I}{{\displaystyle \odot }}{\mathit{k}}_0^i{\left(X_i\right)}^t\right)K^{-1}\mathit{F}\right)\\ ={\mathit{F}}^t{K}^{-1}\mathrm{cov}\left(\underset{i\in I}{{\displaystyle \odot }}{\mathit{k}}_0^i\left(X_i\right)\right)K^{-1}\mathit{F}\\ ={\mathit{F}}^t{K}^{-1}\underset{i\in I}{{\displaystyle \odot }}\mathrm{cov}\left({\mathit{k}}_0^i\left(X_i\right)\right)K^{-1}\mathit{F}\end{array}\\ ={\mathit{F}}^t{K}^{-1}\underset{i\in I}{{\displaystyle \odot }}\left({\displaystyle {\int }_{D_i}{\mathit{k}}_0^i}\left(x_i\right){\mathit{k}}_0^i{\left(x_i\right)}^{t}d{\mu }_i\left(x_i\right)\right)K^{-1}\mathit{F}\end{array}$.

Therefore, the proposed Sobol’ global sensitivity calculation method based on ANOVA can be further organized as:

$$S_I={\displaystyle \frac{\mathrm{var}\left(m_I\left({\mathit{X}}_I\right)\right)}{\mathrm{var}(m(\mathit{X}))}}={\displaystyle \frac{{\mathit{F}}^t{K}^{-1}\left(\underset{i\in I}{{\displaystyle \odot }}{\Gamma }_i\right)K^{-1}\mathit{F}}{{\mathit{F}}^t{K}^{-1}\left({{\displaystyle \odot }}_{i=1}^d\left(1_{n\times n}+{\Gamma }_i\right)-1_{n\times n}\right)K^{-1}\mathit{F}}}$$

(18)

In summary, the proposed ANOVA-based Sobol’ global sensitivity calculation method expands the interpolation function m(X) through a kernel function, obtaining the global sensitivity expression of input variables to the system model f(X) in an analytical manner. The random data (X, Y) of the input and output can be substituted to obtain the global sensitivity. Therefore, the proposed method can achieve online calculation and analysis of the global sensitivity.

3.2. Implementation Procedure

The implementation procedure of the proposed global sensitivity calculation method consists of three main stages.

(1)

Offline Stage Training. To simulate the uncertainty and stochastic of power systems with renewable energies, time-domain simulation is first carried out under different conditions with ambient excitation to obtain input features (voltage amplitude, phase angle of the load buses, and active power of the tie-lines), which are then used to train the GBDT model.

(2)

Online Voltage Stability Margin Estimation. The ambient response of the system (consisting of voltage magnitudes, phase angles, and power) is acquired online via PMUs. Then, the GBDT model is updated based on incremental learning online, yielding an estimate of the power system’s voltage stability margin.

(3)

Calculation of the Global Sensitivity. Using the online-derived voltage stability margin and online-measured tie-line power as inputs, the proposed ANOVA-based method calculates the global sensitivity of the system’s voltage stability margin with respect to tie-line power. This sensitivity is then transmitted to the control center.

4. Case Studies

4.1. Nordic 32 Test System

The Nordic 32 test system is used in this Section, as shown in Figure 3, comprising 74 nodes, 20 generators, and 22 load nodes, and is divided into four main regions: EQUIV, NORTH, CENTRAL, and SOUTH. The CENTRAL region is heavily loaded, with electrical power primarily transferred from the NORTH region through five 400 kV transmission lines. This system is extensively utilized in current voltage stability research. To verify the effectiveness of the proposed strategy in the current power system with large-scale renewable energy integration, wind farms are connected to buses 1012, 1013, 1014, 4042, and 4047 in the Nordic system through multi-stage step-up transformers, replacing the synchronous generators in the original system. Specific parameters of the connected wind farms can be found in reference [17].

To verify the accuracy of the proposed method, the following error E_s is defined as the verification index of the proposed global sensitivity calculation method:

$$E_s={\displaystyle \frac{\left|{\widehat{S}}_i-S_i\right|}{S_i}}\times 100\%$$

(19)

where ${\widehat{S}}_i$ is the calculated global sensitivity value, and $S_i$ is the global sensitivity value obtained based on the model method.

Similarly, the proposed voltage stability margin estimation error E_m is:

$$E_m={\displaystyle \frac{\left|{\widehat{\mathit{y}}}_i-{\mathit{y}}_i\right|}{{\mathit{y}}_i}}\times 100\%$$

(20)

where ${\widehat{\mathit{y}}}_i$ is the margin estimation value, and ${\mathit{y}}_i$ is the actual value based on the voltage stability margin.

The application of the proposed method was implemented under the MATLAB^® version 2016a environment on a computer equipped with a 3.7 GHz processor and 16 GB of memory.

First, consider generating a training sample set for the voltage stability margin estimation model under different grid structures and load growth methods: Under two conditions, considering and not considering the N−1 grid structure of the test system, the continuation power flow analysis is performed with different load growth methods. The voltage magnitudes and phase angles of each node and the active power of the five tie-lines are selected as input variables, and the voltage stability margin calculated based on CPF is used as the output variable to train the proposed voltage stability margin estimation model.

For the test system, two operating scenarios are set up: S1 (0–5 s) normal operation and S2 (5 s–10 s) tie-line 4032–4044 disconnected. Based on the online measurement data, the online extracted voltage margin evaluation results are shown in Figure 4. According to Figure 4, the voltage stability margin of the system fluctuates around 12% when the system is running normally. When one tie-line is disconnected in the system, the voltage stability margin of the system drops directly to around 5%. The above calculation results can show the influence relationship between the system inter-area tie-lines and the voltage stability margin. At the same time, according to Figure 4, the error between the estimated margin and the actual value in the two operating conditions S1 and S2 is less than 0.7%, indicating that the proposed method can accurately extract the voltage stability margin, and the proposed parameter updating method based on incremental learning further ensures the adaptability of the voltage stability margin estimation model when the system operating conditions change.

Based on the online extraction results of the voltage stability margin and the online measurement data of the acquired tie-line transmission power, the proposed method is used to calculate the global sensitivity, S_AN, of the two. The traditional sensitivity calculation method S_pf based on the system power flow model is used as the benchmark, and the calculation results of the Sobol’ method S_mc based on Monte Carlo are compared.

Among them, the Monte Carlo sampling is set to 2000 times, and the window length of the proposed method is set to 1 s (N = 100). The comparison results of the three global sensitivities under the two operating conditions are shown in

Table 1. The comparison results of the three global sensitivities.

Method	Scenario
	S1		S2
	Es/%	CPU/s	Es/%	CPU/s
Spf	/	135.60	/	124.15
Smc	1.33	7.45	1.45	7.12
SAN	0.15	0.36	0.14	0.33

. Further analysis of Table 1 shows that when the system operating mode changes, compared with the Sobol’ method using Monte Carlo sampling, the error between the proposed method and the benchmark value is smaller, indicating that the global sensitivity calculation results obtained by the proposed method are more adaptable to the system operating mode changes and can accurately quantify the relationship between the interconnected system tie-line power and the system voltage margin.

In addition, analyzing Table 1, it can be found that compared with the Monte Carlo-based calculation method, the proposed method only needs a small amount of measurement data (1 s) to complete the calculation, and the actual application takes less time, only 0.3 s–0.4 s, which effectively improves the global sensitivity acquisition speed and realizes the online calculation and analysis of global sensitivity.

In order to prove that the proposed global sensitivity calculation method has the advantage of realizing the online application of voltage stability margin sensitivity, the online extraction results of the system voltage stability margin and the online measurement data of tie-line active power are used to calculate online the voltage stability margin global sensitivity and compare it with the GRNN [18] and the robust estimation method [19] to compare the online calculation performance of voltage stability margin global sensitivity.

The time-domain results of online calculation of the voltage stability margin global sensitivity under the two operating conditions S1 and S2 are shown in Figure 5. According to Figure 5, in actual operation, due to the system source–load uncertainty, the voltage stability margin global sensitivity has time-varying characteristics, and when the system topology changes (one tie-line is disconnected), the voltage stability global sensitivity has a large change, where the global sensitivity of the tie-line 4032–4044 becomes 0 directly under the S2 operating condition. Among the three comparison methods, GRNN and the robust estimation method are both sensitivity calculation methods based on system linearization, which cannot capture the dynamic changes in sensitivity well, resulting in a large gap between the calculation results and the benchmark value and low accuracy.

The proposed method can quickly calculate the system’s voltage stability margin global sensitivity based on only 1 s of online measurement data while ensuring the calculation accuracy. The calculation time is short, the calculation speed is fast, and the adaptability to the changes in system operating conditions is stronger. Therefore, the proposed method can realize the online tracking calculation of the voltage stability margin global sensitivity, so as to ensure the effectiveness of the implementation of the voltage stability optimization and enhancement strategy.

4.2. CSEE-VS Standard Case System

In this section, the CSEE-VS standard case system for the voltage stability of the Electric Power Research Institute is used to further verify the effectiveness and applicability of the proposed online calculation method for the global sensitivity of the voltage stability margin. See references [20,21] for detailed parameters of CSEE-VS. The system consists of 97 nodes, with 45 nodes belonging to the 500 kV main grid. The installed capacity ratio of renewable energy sources to conventional power sources in the system is 2.4 GW to 6.3 GW (1:2.62). The system includes a single HVDC line that receives and transmits 800 MW of power. Due to the receiving-end system being a heavily loaded area with large-scale renewable energy integration, its transient voltage stability problem is prominent and can reflect typical characteristics of voltage instability following system faults.

The original single DC is changed to four inter-area tie-lines; the historical operating data of the voltage magnitude, phase angle, and tie-line active power of each node of the receiving end system are collected; and the continuous power flow analysis is performed on the system to obtain the voltage stability margin [22], construct the training sample set, and establish the voltage stability margin estimation model. In online application, the voltage stability margin is obtained through the proposed online extraction method of the voltage stability margin. The voltage stability margin estimation results are shown in Figure 6.

Combined with the active-power operating data of the four tie-lines, the proposed method is used to calculate the global sensitivity of the voltage stability margin; the comparison results are shown in Figure 7. And the online calculation results of the global sensitivity through 10 s in the time domain are shown in Figure 8. The statistical calculated results are listed in

Table 2. Sensitivity Comparison of CSEE-VS Standard Case System Connecting Lines.

Tie-Line	Tie-Line 1	Tie-Line 2	Tie-Line 3	Tie-Line 4
Es/%	0.19	0.16	0.19	0.18
CPU/s	0.42	0.39	0.38	0.40

.

The calculation results of the proposed method are close to the benchmark value with E_s less than 0.20%, indicating the accuracy of the proposed method. It is also clear in Figure 7 that the proposed method maintains good performance when used in an online manner.

4.3. A Real-World Power System

To validate the effectiveness of the proposed global sensitivity calculation method in a larger power system, the Northeast China Power Grid (NECPG) is used in this section. NECPG consists of two closely connected small grids (west power grid, WPG, and east power grid, EPG). The WPG delivers power to the EPG through three 500 kV transmission lines—tie-lines 1, 2, and 3—whose active power is recorded in Figure 9.

By applying the proposed method to the voltage magnitudes, phase angles, and tie-line power of key buses within the receiving-end EPG, we first perform online voltage stability margin estimation. The results are then combined with tie-line measurement data to compute the global sensitivity of the voltage stability margin to the tie-line power. The accuracy and computation time results are detailed in

Table 3. Sensitivity Comparison of Real-World Power System Connecting Lines.

Tie-Line	Tie-Line 1	Tie-Line 2	Tie-Line 3	Average
Es/%	0.20	0.19	0.18	0.19
CPU/s	0.69	0.70	0.68	0.69

.

The findings presented in Table 3 reveal that the computational error remains small; additionally, the time required to compute sensitivities for three tie-lines is roughly 1.75 s, therefore showing that the proposed method is both rapid and accurate in producing global sensitivity calculation results.

Moreover, recognizing the presence of measurement noise in practical applications, this section investigates the effect of varying noise intensities on the measurement data. The measurement errors of PMUs are added to the dataset. In particular, the errors are simulated by white noise following the Gaussian distribution N(0, ρ). The resulting global sensitivity calculation errors are presented in Figure 10. The results displayed in Figure 10 demonstrate that the proposed method exhibits good noise resilience, confirming its robustness for real-world system deployment.

5. Conclusions

This paper considers the source–load uncertainty of large-scale renewable energy grid-connected systems, establishes a voltage stability margin estimation model based on incremental learning, and proposes an online calculation method for the global sensitivity of the voltage stability margin based on the online extracted voltage stability margin and tie-line power online measurement data. The main conclusions are as follows:

(1)

Considering the multi-change characteristics of the operating conditions of renewable energy grid-connected systems, a voltage stability margin estimation method based on incremental learning is established. By online updating the parameters of the margin estimation model, the adaptability of the voltage stability margin extraction method to the changes in the operating conditions is improved, and the online extraction of the voltage stability margin is realized.

(2)

Based on the online acquired voltage stability margin and the online measurement data of the tie-line power, a fully data-driven online calculation method for the global sensitivity of the voltage stability margin is proposed to improve the calculation speed of the global sensitivity of the voltage stability margin and the adaptability to the multi-change operating conditions of the renewable energy grid-connected system.

(3)

The accuracy of the proposed online calculation method for the global sensitivity of the voltage stability margin is verified by the Nordic test system and the CSEE-VS standard case. The verification results show that the proposed method can realize the online calculation of the global sensitivity of the voltage stability margin of a large-scale renewable energy grid-connected system.

Future research will focus on developing voltage stability margin enhancement strategies based on the calculated sensitivity results.