Computationally efficient tail distribution-aware large-scale power system overloading risk assessment

Domain: physics
Published: 2026-01-05
Editorial outcome: minor_revision

Abstract
The uncertainties and intermittency associated with renewable generation sources, such as solar and wind, can pose significant overloading risks to power systems under N−k contingencies, potentially leading to cascading outages. Accurately quantifying these risks in independent system operator scale power systems, which may include tens of thousands of buses, remains a grand challenge. This paper proposes a computationally efficient, tail-distribution-aware approach for accurate overloading risk quantification in large-scale power systems. Specifically, a deep-kernel sparse vector-valued Gaussian process is developed and serves as a surrogate model. This model incorporates generation dispatch, predefined contingencies, and uncertain inputs, such as photovoltaic power and load demand, to predict their impacts on branch power flows, which are treated as the model outputs. To improve the fidelity of overloading risk assessment, we introduce an adaptive resampling mechanism based on power flow solver, which corrects biases in surrogate model predictions near the overloading threshold. Extensive results obtained on the realistic 21k+ bus New England power system demonstrate that the proposed method accelerates the risk assessment process by 22 times compared to the benchmark Monte Carlo sampling method, while maintaining high accuracy. Additionally, we validate the robustness of the approach across a wide range of distribution types and correlation scenarios between renewable generation and load demands.

Introduction
The increased penetrations of intermittent renewable generations, the uncertainties that propagate from sources such as renewable generation and load to branch flow. Numerous research efforts have been conducted on PPF solution strategies that can be generally categorized as Monte Carlo sampling (MCS) methods, statistical methods and surrogate model-based methods. MCS is widely used in PPF due to its simplicity and accuracy. MCS repeatedly samples from the power production distribution of the uncertain sources and obtains the power flow solution for each power production sample. Once the quantity of interest (QoI), such as branch flow in the repeated power flow converges, MCS can accurately estimate the distribution of the QoI2. This estimation is usually utilized as the baseline3. However, the MCS is a time-consuming process as it involves thousands or even more power flow calculations. To reduce the computational burden, several variant sampling strategies, such as adaptive importance sampling4, Latin hypercube sampling5 and Hammersley sampling6 have been proposed. These strategies, however, continue to suffer from heavy computational burden as they still require a large number of samples in the high-dimensional sample space, especially when modeling large-scale power systems. To further accelerate computation, DC power flow is used to calculate branch power7. Nevertheless, it introduces non-negligible errors since it is only an approximation of AC power flow, and it cannot provide assessment results for bus voltage violation risks.

To avoid the requirement for numerous power flow calculations, some statistical methods have been developed, i.e., analytical modeling8,9 and approximation modeling. In analytical modeling, a linearized power flow model is used to describe the explicit relationship between the cumulants of uncertain sources and those of the power flow results7–9. These derived cumulants are further used to construct the probability density functions (PDF) of power flow results via expansion series, such as the Gram-Charlier expansion10. However, analytical results contain estimation errors due to model linearization.

In approximation modeling, the moments of outputs are approximated using some selected representative points, which requires significantly less computation. The well-known approximation modeling methods include the 2m point estimation method (PEM)11,12 and the 2m+1 PEM13,14. Meanwhile, the approximation errors and computational burden of these methods will increase with the number of uncertain inputs.

To reduce the errors due to linearization or approximation, the surrogate model-based method is proposed. Under the MCS framework, the key of the surrogate model-based method is to use a computationally efficient surrogate model to replace the time-consuming power flow solver. Consequently, many efforts have been dedicated to developing different surrogate models for PPF, such as low rank approximation (LRA)15,16, polynomial chaos expansion (PCE)17–19, Gaussian process emulator (GPE)20–23, and deep neural network (DNN)24–27. LRA and PCE were early surrogate models used in PPF when the initial research focused on addressing the curse of dimensionality15–17 or correlations among inputs18,19. However, these models impose restrictions on input distributions. To overcome this limitation, the GPE, which does not require specific input distributions, was introduced to PPF20–22. Although the computational efficiency of the GPE is significantly improved compared to MCS, it is still limited by the curse of dimensionality and the accuracy is relatively low when the training data is a relatively small dataset. To further enhance interpretability, ref. 23 utilizes hyperparameters in the GPE to assess the sensitivity between input and output. However, the GPE has a limited capacity in extracting deep representations from data, which makes it susceptible to topology changes. Consequently, various DNN methods have been developed. Among them, ref. 24 utilizes voltage variations before and after branch outages to encode different topology changes although it imposes additional computational burdens due to the need for power flow calculations under varying topologies; ref. 27 proposes a physics-informed graph neural network that directly incorporates the admittance matrix as an input and improves its ability to represent topology changes. However, this approach has only been validated for N−k, k≤2 topology changes while the system operators frequently encounter higher-order branch outages in extreme weather events. Additionally, the existing work often overlooks the tail distribution that is critical for risk quantification. The surrogate model-based methods commonly struggle with accuracy in the tail distribution as these regions are underrepresented in the training data.

This paper proposes a computationally efficient tail distribution-aware PPF assessment approach for transmission overloading risk assessment. Specifically, a deep-kernel sparse vector-valued Gaussian process (DKS-VVGP) is developed as the surrogate model to replace a time-consuming nonlinear power flow solver. In this model, the generation dispatch, predefined contingencies, and uncertain sources, such as PV and load power, are embedded as inputs, while the outputs are the branch power of interest. This enables rapid scanning of branch overloading risks under N−k contingencies and various dispatch scenarios. To mitigate the DKS-VVGP model error at the overloading boundary, a tail distribution-aware resampling mechanism is proposed. This mechanism adaptively resamples the overloading boundary by re-evaluating the DKS-VVGP assessment results using a power flow solver.

Validation of this approach is performed on the realistic ISO New England (NE) power system using historical contingency records. Results reveal that this approach achieves accurate overloading risk estimation, matching the accuracy of the benchmark, i.e., MCS, while significantly accelerating the assessment process. Furthermore, through correlation analysis, further insights are obtained about the relationship between uncertainties and ISONE power system overloading risks.

Results
Surrogate model-based power system overloading risk formulation
In this study, we explore a power network characterized by a set of nodes N = {1,...,n_b} and lines L = {1,...,l_g}. Within this network, a selection of nodes, denoted as G ⊂ N, is equipped with generators, whereas another subset, L_D ⊂ N, represents the load demands. For an AC power flow with a constant network configuration, it is generally expressed using the following formula:

P_G - P_D = V_i \sum_{j \in N} V_j (G_{ij} \cos(\theta_{ij}) + B_{ij} \sin(\theta_{ij}))
Q_G - Q_D = V_i \sum_{j \in N} V_j (G_{ij} \sin(\theta_{ij}) - B_{ij} \cos(\theta_{ij})) (1)

where P_G and Q_G denote the active and reactive power generated, and P_D and Q_D represent the respective demands; V_i is the bus voltage; G_{ij} and B_{ij} stand for the conductance and susceptance between nodes i and j, with \theta_{ij} = \theta_i - \theta_j signifying the angle difference. Once the AC power flow in equation (1) is solved, the active power of a branch can be calculated as:

P_{B,ij} = V_i^2 G_{ij} - V_i V_j (G_{ij} \cos \theta_{ij} + B_{ij} \sin \theta_{ij}) (2)

where P_B denotes branch power.

From equations (1) and (2), it is evident that branch power is influenced by the generation, load, and system topology. Topology changes, often resulting from predefined contingencies, must be thoroughly assessed to ensure system security before the scheduled operating conditions are implemented. Consequently, the relationship between branch power and generation, load, and system topology can be implicitly expressed as:

y = M(x, \tau) (3)

where y \in R^{n_1 \times 1} denotes n_1 branch power quantities of interests, x = [P_G^T, P_{PV}^T, P_D^T]^T \in R^{n_b \times 1} represents the active power injections across n_b nodes (including contributions from synchronous generators, photovoltaics (PV) systems, and loads), and \tau denotes the predefined contingency (topology change), mathematically represented by the absolute values of the diagonal elements of the admittance matrix. The function M denotes the power flow computation process.

In this paper, we aim to quantify the risk of overloading considering generation dispatch, predefined contingencies, and stochastic PV and load variations. More specifically, the overloading risk is influenced by the following factors:
• Synchronous Generation Dispatch: Dispatch is typically determined based on the load and renewable energy variations, meaning that variations in load and renewable distribution and magnitude at different times can lead to changes in dispatch. Therefore, dispatch plays a critical role in defining the operating conditions, including branch power flows.
• Predefined Contingencies: We consider a set of predefined contingencies, \tau = {\tau_1, \tau_2, ..., \tau_{n_{topo}}}, as all considered scenarios, with most being N−k contingencies. Here, n_{topo} denotes the total number of predefined contingencies. If using power flow solver, each contingency must be evaluated individually under the scheduled operating conditions for security assessment.
• Random PV and Loads: Both PV generation and loads follow specific probability distributions, introducing stochastic variations that can impact branch power flows.

To summarize, branch power flows follow a distribution under generation dispatch and predefined contingencies, as uncertainties from PV generation and loads propagate through the system. A widely used approach to obtain the distribution of branch power flows is MCS, as shown in Supplementary Fig. 1. This method repeatedly samples the distributions of PV generation and loads for each predefined contingency under the current generation dispatch, with power flow calculations performed for each sample. Consequently, the branch flow distribution can be expressed as:

Y = M(X, T) (4)

where Y denotes the branch power flow samples with size of N_s \times l. As a result, the overloading of k-th branch can be approximated using MCS as follows:

p^{(k)} \approx p_{MCS}^{(k)} = \frac{1}{n_{topo} N_s} \sum_{\tau \in T} \sum_{i=1}^{N_s} I(|y_i^{(k)}| \ge P_{Lim}^{(k)}) (5)

where I(condition) is an indicator function that equals 1 if the condition is true, and 0 otherwise. P_{Lim}^{(k)} is the power limit of the k-th branch.

However, although MCS is sufficiently accurate, it is time-consuming due to the computational burden of the power flow solver, often requiring hundreds of hours, far exceeding the time scale of power system dispatch (i.e., 15 min to half an hour). To accelerate MCS, we propose using a computationally efficient surrogate model, DKS-VVGP, to replace the time-consuming power flow solver denoted by equation (3). Specifically, DKS-VVGP is formulated as:

\hat{y} = M_f(x, \tau | \theta) (7)

In equation (7), \hat{y} denotes the branch power predictions. DKS-VVGP is essentially a machine learning model based on GPE. Unlike traditional GPE, which relies on simple nonlinear functions, such as the radial basis function, to model the nonlinearity of the learning objective and is limited to single-output support, GPE also suffers from computational complexity that grows polynomially with the size of the training data. Consequently, these characteristics restrict its applicability in nonlinear and high-dimensional power systems. In contrast, DKS-VVGP addresses these limitations and achieves better generalization in approximating equation (3) by embedding a universal approximator neural network into the kernel matrix of a multi-output Gaussian process. Additionally, it employs a stochastic variational sparse approximation to enable efficient learning and training on large datasets.

While DKS-VVGP can effectively model the overall distribution of branch power, significant errors may still arise when it is used for overloading risk assessment. This is primarily due to the limited number of generated samples in the tail of the distribution, where overloading events are most likely to occur. As a result, the predictions of the DKS-VVGP in the tail region may be inaccurate, leading to substantial errors in the risk assessment. To address this issue, this paper proposes a adaptive tail-resampling mechanism.

In this paper, F^{(k)} = {y^{(k)} : |y^{(k)}| \ge P_{Lim}^{(k)}} is defined as the overloading distribution for k-th branch, which resides in the tail of the overall distribution. Due to insufficient training in this region, the DKS-VVGP has limited ability to accurately distinguish samples in the tail. As a result, it may struggle to classify samples near the boundary, i.e., |y^{(k)}| = P_{Lim}^{(k)}. This can lead to misclassifications, where samples with |y^{(k)}| \ge P_{Lim}^{(k)} are identified as |y^{(k)}| < P_{Lim}^{(k)}, and vice versa. Such misclassifications represent a significant source of error in overloading risk assessment when using the DKS-VVGP as a surrogate model. The resampling procedure can be summarized in the following steps:
• Step One: Assume \hat{\Phi}^{(k)} = {\hat{y}_i^{(k)} : |\hat{y}_i^{(k)}| \ge P_{Lim}^{(k)}}, i = 1, ..., N_s represents the PPF results of the k-th branch predicted by the DKS-VVGP, considering uncertain power injections and predefined contingencies. The iteration number j is initialized to 1.
• Step Two: To mitigate errors in overloading risk assessment, the resampling process begins at the risky boundary. Specifically, \hat{\Phi}^{(k)} is used to identify the samples closest to the boundary |\hat{y}_i^{(k)}| \approx P_{Lim}^{(k)}.
• Step Three: The overloading risk of the k-th branch at the j-th iteration, denoted p^{(k)}, is calculated using equation (5) and \hat{\Phi}^{(k)}.
• Step Four: Tail Resampling. The \lambda predicted samples \hat{y}_i closest to P_{Lim}^{(k)} are selected, where the domain where the uncertainty is high is defined as |\hat{y}_i^{(k)}| \in [P_{Lim}^{(k)} - \delta, P_{Lim}^{(k)} + \delta]. The power flow solver is then used to calculate the true branch power y_i^{(k)} for these \lambda samples. The surrogate model is then updated using these new samples, and the iteration number is updated as j = j + 1.
• Step Five: If the difference between the overloading risk at the current iteration and the previous iteration is less than a threshold \epsilon, the resampling will be terminated. Otherwise, return to Step Three.

From the above procedure, it can be observed that the proposed adaptive tail-resampling mechanism is purely surrogate-error-driven. In other words, the physics is incorporated solely through the training data of the surrogate.

Transfer learning for quick model updating
Although the proposed DKS-VVGP and adaptive tail-resampling mechanism can significantly reduce the computational burden of MCS, the training of the DKS-VVGP still requires a large volume of data due to the numerous parameters that need to be optimized. This means that once new generation dispatches or contingencies are introduced, the previously trained DKS-VVGP becomes invalid. Training a new DKS-VVGP from scratch can be time-consuming, as it requires generating substantial new simulation data and retraining the model.

To address this issue, transfer learning is employed to fine-tune the previously trained DKS-VVGP M_f(x, \tau | \theta). Since the historically trained DKS-VVGP provides good initial parameters for model training, only a small number of samples from the new, unseen scenario are needed to fine-tune the model and adapt it to the new conditions. Consequently, the fine-tuned DKS-VVGP for the new scenarios, denoted as M_f(x, \tau | \theta^0), can be obtained, where \theta^0 represents the fine-tuned parameters for the DKS-VVGP.

Proposed overloading risk assessment framework
As shown in Supplementary Fig. 3, the proposed overloading risk assessment framework consists of four essential steps: data preparation, surrogate model training, overloading risk assessment, and surrogate model fine-tuning. These steps are detailed as follows:
• Step One: Data Preparation. The initial step involves collecting new operational data based on the inferred distribution of PVs and loads. When the marginal distributions and their correlations for loads and PV generations are unknown, they can be estimated from collected data by using Copula theory, facilitating the data generation based on identified Copula structures, such as Gaussian Copulas, t Copulas, or Vine Copulas. It is worth mentioning that no uncertainty modeling is applied to the contingencies (including rare events such as the simultaneous outage of up to 8 lines) in this paper; instead, each contingency is treated as a deterministic event that is scanned and quantified individually. Therefore, the inclusion of contingencies in this context does not introduce new challenges to the uncertainty modeling. As a result, the active power of PVs and loads is sampled from their respective inferred distributions and combined with predefined contingencies as inputs for power flow calculations. This process generates the training dataset, denoted as D = {X, Y} = {x_i, y_i}_{i=1}^{N_s}.
• Step Two: Surrogate Model Training. Once the training dataset D is generated, it can be used to train the DKS-VVGP. The trained DKS-VVGP is denoted as M_f(x, \tau | \theta).
• Step Three: Overloading Risk Assessment. Using the trained DKS-VVGP, sufficient data can be sampled from the inferred distribution of PVs and loads under each predefined contingency. These samples are then fed into the inputs of the trained DKS-VVGP, and the adaptive tail-resampling mechanism is employed to correct the predictions near the overloading boundary, enabling accurate quantification of the overloading risk.
• Step Four: Surrogate Model Updating via Fine-Tuning. If unseen scenarios arise, such as unseen generation dispatch or predefined contingencies, a small number of training samples, denoted as N_{transfer}, will be generated following step one. Steps Two and Three will then be repeated to transfer the previously trained surrogate model M_f(x, \tau | \theta) to a new model M_f(x, \tau | \theta^0), enabling overloading risk assessment for the new scenario.

Remark 1: In this paper, the generation dispatches are obtained from the historical data of the ISONE power system. Specifically, the generation dispatches determine the outputs of synchronous generators, which are calculated by the system operator using an economic dispatch optimization model based on load forecasts and renewable power forecasts.

Remark 2: Compared with the instant on approach in ref. 7, the proposed method differs in the following aspects:
• Objective: The objective of the proposed method is to evaluate the overloading risk in a statistical sense, whereas the instant on approach aims to identify the most serious operating instance of wind generation that causes line overloading.
• Inclusion of Contingencies: The proposed method considers the combined impact of critical contingencies and uncertain sources, including loads and PV generation, on the overloading risk. In contrast, the instant on approach only accounts for fluctuations in wind generation.
• Different Solvers: The proposed method relies on an AC power flow solver, which provides higher accuracy in risk assessment, while the instant on approach adopts a DC power flow solver to improve computational efficiency at the cost of reduced accuracy.

ISONE power system
In this section, we conducted experiments on the ISONE power system, which is a large-scale system comprising 21754 nodes, 24280 branches, 1959 loads, and 804 generators, among which 291 generators were replaced with PV generations, as shown in Fig. 1 and Supplementary Fig. 4. Since modeling the uncertainty of load and solar generation is not the main focus of this paper and we do not have access to the actual historical load or solar data of the ISONE power system, the distributions and their parameters are selected based on representative references to ensure practicality. For the purpose of validation, a Gaussian or Beta distribution is commonly used to characterize solar power uncertainty, while a Gaussian distribution is widely adopted for load uncertainty29–31. This dataset includes 1959+291=2250 uncertain sources and 24280 potential outputs. The branch power flows of n_1 = 10 branches (named as L1, L2, L3, L4, L5, L6, L7, L8, L9, L10) that are prone to power flow violations were used as outputs for overloading risk assessment.

To assess the performance of our proposed method, we compared it to the benchmark results obtained through MCS with 2 \times 10^4 samples, where n_{topo} = 40 contingencies (CTGCs) are involved. These 40 critical CTGCs represent simultaneous outages of multiple important transmission lines (up to 8 lines). In this paper, the 40 contingencies are selected by the ISONE system operator as they have historically caused line overloading and are therefore considered more severe critical contingencies compared with others. If these critical contingencies do not lead to line overloading risk, then other contingencies are unlikely to cause such risks. It is worth mentioning that the contingencies are deterministic and involve no randomness. The mean absolute percentage error (MAPE) was utilized as a metric to measure the overall model error:

MAPE = \frac{1}{N} \sum |\frac{y - \hat{y}}{y}| \times 100\% (8)

where N is the number of assessment samples, and y and \hat{y} represent the true and estimated power flow, respectively. We calculate the risk and expected excess using the same methodology as given in equation (5), allowing for a direct comparison between DKS-VVGP and our previous state-of-the-art method, i.e., partial parallel Gaussian process (PPGP)32. Note that when the branch power is negative, its absolute value is used in equation (5) to calculate the overloading risk. In PPGP, the Laplacian matrix L corresponding to the power system topology is decomposed as L = U_e \Sigma_e W_f^T, and W_f is used to transform the load and solar power vectors P_D and P_{PV} into the input

P_{W_f} = [P_D, P_{PV}] W_f (9)

The PPGP model for the power flow on the k-th line is then constructed as

y^{(k)}(P_{W_f}) = M_{PPGP,k}(P_{W_f}) + \epsilon_{PPGP,k} (10)

where
• M_{PPGP,k}(P_{W_f}) \sim GP(\mu_{PPGP,k}, \sigma^2_{PPGP,k} K(P_{W_f}, P_{W_f}^T)) is a Gaussian process with mean \mu_{PPGP,k} and kernel covariance function K(\cdot, \cdot),
• \epsilon_{PPGP,k} \sim N(0, \sigma^2_{PPGP,k}) represents independent observation noise,
• y^{(k)}(P_W) is the predicted power flow on the k-th line,
• U_e and \Sigma_e are the left singular vectors and singular values from the SVD of L.

Handling different contingencies without transfer learning
In this experiment, we assumed 1959 uncertain loads and 291 PV sources, modeled as Gaussian distributions with means equal to their nominal values and fixed standard deviations set to 0.12 times the means. The correlation r among all uncertain sources was set to 0.2. Figure 2 illustrates the probabilistic power flow analysis results for the proposed method across ten branches under all CTGCs. Due to the substantial assessment error of PPGP, Fig. 3 only presents its assessment results under CTGC 1 defined in Supplementary Table 1. It can be observed that the proposed method (DKS-VVGP with resampling) achieves PDF accuracy comparable to that of MCS. This performance is attributed to the embedded neural network in DKS-VVGP, which effectively extracts topological feature information under different contingencies. In contrast, the state-of-the-art PPGP method fails to accurately quantify branch power uncertainties under varying contingencies, as it relies solely on kernel functions for topological information capture, which is inadequate for managing the high-dimensional and nonlinear characteristics of large-scale power grids. Table 1 further demonstrates that, although resampling increases the evaluation time of the proposed method, it enables overloading risk assessment accuracy consistent with MCS while requiring only approximately one-fortieth of MCS’s evaluation time. Therefore, the slight computational burden introduced by resampling is worthwhile to ensure exact overloading risk assessment. By comparison, PPGP underperforms significantly in terms of MAPE, p_F, and assessment time, highlighting the effectiveness and practicality of the proposed method for overloading risk assessment in real-world power systems.

Figures 4 and 5 present the probabilistic power flow results for the branches L1, L3, and L4 under different CTGCs. Once again, the performance of the proposed approach generally outperforms that of PPGP in these scenarios. It is essential to note that a single branch, under different CTGCs, exhibits varying PDFs of power flow. Consequently, the potential risk associated with these branches differs significantly. For example, from the (a) and (b) subplots of Figs. 4 and 5, it can be observed that CTGC 1 induces greater branch power fluctuations than CTGC 6 defined in Supplementary Table 1, demonstrating that CTGC 1 poses a higher overloading risk to the power system than CTGC 6. This observation further validates the importance of considering different CTGCs in probabilistic analysis, as it accurately accounts for the diverse operational conditions that can impact the power system’s reliability.

In the last part of this section, we discuss the impact of hyperparameters on the performance of the proposed method. The most influential hyperparameter affecting overloading risk assessment accuracy is the resampling step size, \lambda, which determines the number of samples resampled at each step. As shown in Table 2, increasing \lambda allows the resampling mechanism to achieve more accurate risk assessment. This is because only a limited number of samples near the overloading boundary are misclassified by DKS-VVGP. If the resampling size is too small, the proposed method may converge prematurely, failing to correct these misclassified samples, thus leading to assessment bias. However, larger values of \lambda result in increased computational cost due to a greater number of resampled points. To balance evaluation time and assessment accuracy, we select \lambda = 150 as the optimal resampling step size for this study.

Performance under different distribution correlations
To investigate the impact of distribution correlations among uncertain sources on the proposed method, this study continued to assume that all uncertain sources follow Gaussian distributions with means equal to their nominal values and fixed standard deviations set to 0.05 times the means. However, the correlations among these sources were varied across three scenarios: r=0.3, r=0.4, and r=0.5. During simulations, we observed that certain CTGCs failed to converge as the distribution correlation among uncertain sources increased. Consequently, the analysis was limited to the 30 CTGCs that converged successfully, resulting in an MCS sample size of 1.5 \times 10^4. Similar to Handling Different Contingencies without Transfer Learning Section, 6000 samples were used to train the DKS-VVGP model. Since it was found that the overloading risk for branch L4 was zero in all three scenarios, branches L1 and L3 were selected for performance analysis of the proposed method. Therefore, in this scenario, branches L1 and L3 are the top contributors to the overloading risk. Figure 6 shows the PDFs produced by the proposed method under different distribution correlations. Thanks to the strong feature extraction capability of DKS-VVGP, it can be observed that the proposed method accurately quantifies the uncertainty of branch power in all scenarios. In other words, the proposed method is not affected by variations in uncertain source distribution correlation.

Table 3 presents the overloading risk, MAPE, and assessment time of the proposed method under different distribution correlations. It can be observed that, due to the proposed resampling mechanism, the method can accurately calculate the overloading risk, which matches the benchmark results obtained from MCS. However, the proposed approach accelerates the assessment process by up to 22 times compared to MCS. An interesting phenomenon can be observed from this table: as the correlation coefficient increases, the system’s overloading risk decreases, but the number of required resamplings increases. This may be due to the reduction in samples near the overloading boundary as the overloading risk decreases. Consequently, DKS-VVGP may not learn sufficient representations at the overloading boundary, resulting in a higher number of misclassified samples and the need for more resamplings.

Performance under Various Distributions
In Handling Different Contingencies without Transfer Learning and Performance under Different Distribution Correlation Sections, numerical results demonstrate that the proposed method achieves overloading risk assessment results as accurate as those of MCS under Gaussian distributions and can compute branch power distributions that closely match MCS. To verify the effectiveness of the proposed method under different distributions of uncertain sources, this section tests its performance when uncertain sources follow a Beta distribution, with a correlation coefficient of 0.2. As tests show that the overloading risk for branch L4 was zero, we only analyze the results for branches L1 and L3. Additionally, since all 40 CTGCs are analyzed and yielded convergent power flow results, the number of MCS samples is set to 2 \times 10^4.

Table 4 presents the overloading risk assessment results of the proposed method under a Beta distribution. It can be observed that the proposed method continues to achieve overloading risk assessment results equivalent to those of MCS, while improving assessment speed by approximately 16 times. This demonstrates the effectiveness of the proposed method for different types of uncertain distributions.

Predicting overloading risk with model updating
In this section, the transfer performance of the proposed method is validated under different dispatch scenarios. As described in Step Four of Proposed Overloading Risk Assessment Framework Section, the surrogate model for the current dispatch is fine-tuned using a small number of new samples based on the surrogate model obtained in the previous operating condition. Assuming the operating conditions at the previous timestep correspond to the scenario with a correlation coefficient r=0.3, we recursively conduct transfer learning along the timeline and evaluate the risk for branch L3 under three different dispatch scenarios, namely, Disp. 1, Disp. 2, and Disp. 3. These three dispatch scenarios correspond to system status under heavy loads at three different time periods, which lead to a high line overloading risk. They are selected to test the accuracy and computational efficiency of the proposed method in quantifying this risk when the system operating conditions change significantly, in contrast to the previous sections where the loads were relatively less heavy. In each dispatch, 30 CTGCs out of the aforementioned 40 CTGCs are considered, leading to an MCS sample size of 1.5 \times 10^4. This is because the unselected 10 contingencies would lead to system collapse, which would certainly require corrective actions to avoid such failures. By contrast, the selected 30 contingencies allow the power flow to converge but may cause line overloading risks, which need to be quantified using the proposed method. The branch power limits (P_{Lim}) for these dispatches are set to 210 MW, 230 MW, and 215 MW, respectively.

Figure 7 illustrates the number of resampled samples required under different training sample sizes for transfer learning in Disp. 1. It can be observed that as the number of training samples used for transfer learning increases, the required number of resampled samples does not change significantly. However, when the training sample size is too small, the required resampled samples exceed the training data, indicating poor accuracy of the surrogate model due to insufficient training data. Based on this observation, we set the training sample size for transfer learning to N_{transfer} = 1000, as it is the first point where the number of resampled samples decreases below the training data, indicating that the surrogate model accuracy is acceptable.

Table 5 presents the evaluation results of the proposed method for the three dispatch scenarios. Due to the transfer learning mechanism, the training time is reduced to approximately one-sixth of the time required for training from scratch as shown in Handling Different Contingencies without Transfer Learning Section, significantly enhancing model training efficiency. Even with a 15-min dispatch interval, the proposed method can rapidly evaluate future dispatch scenarios using a dual-core CPU. Furthermore, the results demonstrate that the proposed method achieves zero error in overloading risk assessment under transfer learning conditions, validating its adaptability to various power system dispatch scenarios. This adaptability, especially in rapidly evolving operational environments, makes the proposed method a viable solution for real-time power system analysis and decision-making.

Methods
Formulation of deep-kernel sparse vector-valued Gaussian process
Traditional GPE have strong uncertainty quantification capabilities, but they rely on simple nonlinear functions, such as the radial basis function (RBF), to model the nonlinearity of the learning objective and are limited to single-output support. Consequently, these characteristics restrict their applicability in highly nonlinear and high-dimensional power systems. To address these limitations and achieve better generalization in approximating equation (3), this paper proposes DKS-VVGP. Neural networks are employed to construct the mean and kernel functions of the VVGP, referred to as the neural mean \phi(\cdot) and neural kernel \phi^0(\cdot), respectively. The inputs to both \phi(\cdot) and \phi^0(\cdot) are the same, namely s. Consequently, the joint information of the topology, which indicates predefined contingencies and the power injection can be extracted, enabling accurate learning of equation (3).

In the DKS-VVGP, n_l latent functions F(s) = [f^{(1)}(s), ..., f^{(n_Q)}(s)]^T are employed to encode the l-dimension output, namely y. Specifically, for j-th latent function f^{(q)}(s), it is assumed that its realizations follow a Gaussian prior N(\phi(S), K_{S,S}^{(q)}), which can be formulated as:

F^{(q)}(s) = [f^{(q)}(s_1), ..., f^{(q)}(s_{N_t})]^T \sim N(\phi(S), K_{S,S}^{(q)}) (11)

where K_{S,S}^{(q)} is j-th covariance matrix representing the model uncertainty. The covariance matrix is defined as:

K_{S,S}^{(q)} = \begin{bmatrix} \kappa^{(q)}(s_1, s_1) & \dots & \kappa^{(q)}(s_1, s_{N_t}) \\ \vdots & \ddots & \vdots \\ \kappa^{(q)}(s_{N_t}, s_1) & \dots & \kappa^{(q)}(s_{N_t}, s_{N_t}) \end{bmatrix} (12)

In equation (12), \kappa^{(q)}(\cdot, \cdot) is the kernel function, i.e., RBF kernel \kappa^{(q)}(s, s^0) = \exp(-\frac{1}{2} ||\phi(s) - \phi(s)^0||^2 / (\ell^{(q)})^2) and \ell^{(q)} is the corresponding hyperparameter.

The set Y = {y_1, y_2, ..., y_{N_t}} represents observations from the power system. On one hand, it is assumed that Y represents the system observations, i.e., y_i = A f(s_i) + \epsilon, where \epsilon \sim N(0, \sigma^2 I). On the other hand, the DKS-VVGP employs neural network functions to encode the output through the linear model of coregionalization33. Consequently, the output can be expressed as:

y = A f(s) + \epsilon (13)

where f(s) = [f^{(1)}(s), ..., f^{(n_Q)}(s)]^T are latent functions; A \in R^{n_l \times n_Q} is the coregionalization matrix that encodes the relationship between the latent functions and the outputs. The outputs can be computed as:

p(Y|F) = \prod_{j=1}^{n_l} \prod_{i=1}^{N_t} N(y_i^{(j)} | \sum_{q=1}^{n_Q} A_{jq} f^{(q)}(s_i), \sigma_j^2) (14)

where y_i^{(j)} denotes the j-th dimension of the i-th sample of y. To maximize the likelihood p(Y) based on equation (14), it requires inverting the covariance matrix n_Q times, leading to a computational complexity of O(n_Q N_t^3). This could result in significant memory usage, potentially causing storage overflow issues. To address this, a stochastic variational sparse approximation is employed to enable scalable learning for the DKS-VVGP.

Stochastic variational sparse approximation for deep-kernel sparse vector-valued Gaussian process
To improve the training efficiency, M inducing points Z = {s_i}_{i=1}^M can be used to approximately reconstruct the covariance matrix instead of using all N_t training data points. Note that these M inducing points are a subset of the original training dataset. If M \ll N_t, the computational burden can be significantly reduced.

Defining the q-th latent function over Z as U^{(q)}, the joint probability distribution of U^{(q)} can be formulated as:

U^{(q)} = [u^{(q)}(s_1), ..., u^{(q)}(s_M)]^T \sim N(\phi(Z), K_{Z,Z}^{(q)}) (15)

where K_{Z,Z}^{(q)} is the covariance matrix associated with the inducing points for the q-th latent function. Consequently, the conditional distribution of F^{(q)} can be derived as:

p(F^{(q)} | U^{(q)}, S, Z) = N(F^{(q)} | \mu^{(q)}, \Sigma^{(q)}) (16)

where \mu^{(q)} and \Sigma^{(q)} represent the mean and covariance, respectively. These can be expressed as:

\mu^{(q)} = \phi(S) + K_{S,Z}^{(q)} (K_{Z,Z}^{(q)})^{-1} (U^{(q)} - \phi(Z)) (17)
\Sigma^{(q)} = K_{S,S}^{(q)} - K_{S,Z}^{(q)} (K_{Z,Z}^{(q)})^{-1} K_{Z,S}^{(q)}

To conduct the stochastic variational inference for the DKS-VVGP, we define a variational distribution p_e(U^{(q)}) = N(U^{(q)} | m^{(q)}, W^{(q)}) for the q-th latent function, which approximates p(U^{(q)}). Consequently, the log marginal likelihood, i.e., log P(Y), can be expressed by factoring the evidence lower bound:

L = E_{p_e(F)} [log p(Y|F)] - KL[p_e(U) || p(U)] (18)

= \sum_{j=1}^{n_l} \sum_{i=1}^{N_t} log N(y_{ij} | \sum_{q=1}^{n_Q} A_{jq} \mu_{iq}, \sigma_j^2) - \frac{1}{2\sigma_j^2} \sum_{q=1}^{n_Q} (A_{jq})^2 \nu_{iq} - \sum_{q=1}^{n_Q} KL[p_e(U^{(q)}) || p(U^{(q)})] (19)

where \mu_{iq} and \nu_{iq} are the mean and variance of the q-th latent function at the i-th sample, respectively. The Kullback-Leibler (KL) divergence is formulated as:

KL[p_e(U^{(q)}) || p(U^{(q)})] = \frac{1}{2} [log \frac{|K_{Z,Z}^{(q)}|}{|W^{(q)}|} - M + tr((K_{Z,Z}^{(q)})^{-1} W^{(q)}) + (m^{(q)})^T (K_{Z,Z}^{(q)})^{-1} m^{(q)}] (20)

By maximizing equation (18), the DKS-VVGP can be trained using \theta = {\phi, \phi^0, m, W, \sigma, A} as hyperparameters. Each element of the coregionalization matrix A is randomly initialized according to a normal distribution with a mean of 0 and a standard deviation of 1, without any regularization. Notably, minibatch training is employed in this paper to improve training efficiency. Once training is completed, the new inputs s^* can be predicted as:

\hat{y} = A \mu^* (21)

where q-th column of \mu^* is computed as \phi(s^*) + K_{s^*, Z}^{(q)} (K_{Z,Z}^{(q)})^{-1} (m^{(q)} - \phi(Z)). In equation (21) can be expressed as \hat{y} = M_f(x, \tau | \theta). Note that the proposed surrogate model is developed based on Gaussian processes. However, this does not mean that the proposed Gaussian process-based surrogate model is unsuitable for data sampled from non-Gaussian distributions. According to the definition in ref. 34: A Gaussian process is a collection of random variables, any finite number of which have a joint Gaussian distribution. A basic assumption behind this definition is that a finite number of realizations can be covered by the Gaussian process. Consequently, the proposed Gaussian process-based surrogate model is not limited to data sampled from a Gaussian distribution.

Simulations settings
For the proposed MNV-VVGP, the structures of the neural mean and kernel are respectively 24517-50-20-1 and 24517-50-20-10, with training epochs set to 100, a minibatch size of 64, M=300, and n_Q set to 10. The input dimension of 24517 is determined due to the inclusion of uncertain PV generation (291 dimensions), loads (1959 dimensions), generators (510 dimensions), and \tau, which represents the absolute values of the admittance diagonal elements (21754 dimensions). The stopping tolerance \epsilon is set as 0.01%.

All simulations were performed on a computer equipped with a 3.2-GHz Intel Core i9-12900KF CPU, an NVIDIA GeForce RTX 3090 GPU, and 128GB of RAM. Power flow calculations were conducted using PowerWorld, and the DKS-VVGP was implemented using the Python library PyTorch35. All PDFs were generated through kernel density estimation36.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "BriefSummaryofthePaper: Themanuscriptdevelopsatwo–stageworkflowforestimatingtheprobability\nthatanindividualtransmissionline/branchexceedsitsemergency(thermal)ratingunderafixedlibraryof\nN–kcontingencies:\n1. A vector–valued, deep–kernel, sparse Gaussian–process (GP) surrogate maps injections x and a\ncontingencytagτ tothebranch–flowvectory˜.\n2. An adaptive “tail–resampling” loop repeatedly re–runs the full AC power–flow on Monte–Carlo\nsampleswhosepredictedflowslieclosesttothethermallimituntiltheoverload-probabilityestimate\np(k)converges.\nF\nOna21,000–busISO–NEmodelwith40contingenciesthemethodachievesapproximately22× speed–up\novercrudeMonte–Carlowhilematchingitsaccuracy.\nMajor Strengths\nTheReviewer’sComment#1\nMajorStrengths:\n• Scalabilitydemonstration: ISO-scale,fullACflows,thousandsofuncertaininjections.\n• Vectorsurrogate: predictsalllinessimultaneously.\n• Simplecorrectiveloop: easytoembedinexistingreal–timecontingency-analysispipelines.\n• Clearengineeringvalue: per–lineprobabilitiesspeakthelanguageofNERCcompliance.\nMajor Weaknesses & Required Actions\nTheReviewer’sComment#2\nMajorWeaknesses&RequiredActions#1:\n• Issue: “Importancesampling”mis–labelling.\n• Whyitmatters: ClassicalISprovidesvarianceguarantees;theproposedmethoddoesnot.\n• Howtofix: Recastasadaptivetail–resampling;citetheadaptiveimportancesampling(EJS)\nandinstantonpapers;discussdifferences(youuseACPFvsDCPFinthepapers).\nTheReviewer’sComment#3\nMajorWeaknesses&RequiredActions#2:\n• Issue: Physicsguidanceisindirectonly.\n• Whyitmatters: Readersmayassumephysics–informedsamplingwhenitispurelysurro-\ngate–error–driven.\n• Howtofix: Stateexplicitlythatphysicsenterssolelythroughthesurrogate’strainingdata.\nTheReviewer’sComment#4\nMajorWeaknesses&RequiredActions#3:\n• Issue: Line–by–linefocusignoresthesystem-widerisk.\n• Whyitmatters: Plannersandsomeoperatorsneedunionprobabilities;connectstoinstanton\nliterature.\n• How to fix: Use existing samples to identify top contributing lines and report respective\noverloadprobabilities;comparetotheaforementionedinstantonapproach.\nTheReviewer’sComment#5\nMajorWeaknesses&RequiredActions#4:\n• Issue: Noablations: neuralkernelandλstepnotjustified.\n• Whyitmatters: Neededtoestablishmethodologicalnovelty.\n• Howtofix: Reportresultswith(i)plainRBFGP;(ii)variedλ.\nTheReviewer’sComment#6\nMajorWeaknesses&RequiredActions#5:\n• Issue: Reproducibility. Hyper-parameters,inducing-pointplacement,andcodeavailabil-\nityaremissing..\n• Whyitmatters: NatureCommunicationsreadersexpectreusablecode.\n• Howtofix: Releasescriptsandfullsettingsinapublicrepositoryorassupplementarymaterial.\nMinor Comments and Clarifications\nTheReviewer’sComment#7\nMinorCommentsandClarifications#1:\nReplace“Neural–Mean–VarianceGP”withthestandardterm“deep–kernelsparsevector–valuedGP”\ntoavoidconfusion.\nTheReviewer’sComment#8\nMinorCommentsandClarifications#2:\nStatehowthecoregionalisationmatrixAisinitialisedandregularised.\nTheReviewer’sComment#9\nMinorCommentsandClarifications#3:\nProvidepseudo–codefortheλ–resamplingloopandspecifythestoppingtoleranceϵ.\nTheReviewer’sComment#10\nMinorCommentsandClarifications#4:\nClarifywhetherunseencontingencytypes(notonlydispatchchanges)canbeaccommodatedwith\n1,000newACsimulations.\nSignificance for a Broad Audience\nTheReviewer’sComment11\nThestudyistechnicallysolidandclearlyusefulforpower–systempractitioners.However,twoaspects\nlimititsbroaderappeal:\n• Thestatisticalmachinery(deep–kernelGP+activeresampling)isstandardoutsidepowergrids;\nnoveltyliesmainlyinscaleandintegration.\n• Results are tightly coupled to power–system operational rules; immediate transfer to other\nfields(aerospace,climate,finance)wouldrequirenon–trivialadaptation.\nIthereforecautiontheeditorsthatthepaper’simpactislikelystrongestwithintheenergy–systems\ncommunity.", "reviewer_2": "The proposed manuscript studies the use of a neural mean-variance vector-valued Gaussian process (NMV-VVGP) to\nestimate the overload probability for transmission lines under different system operating conditions, with these being a result\nuncertain solar PV generation and loads, as well as generator dispatch and contingencies. Overall, the manuscript is well\nwritten and addresses an existing literature gap, making it suitable for publication in this journal. Nevertheless, several\naspects need to be first improved, which are mainly related to providing more details on the modeling approach and\ncomputer simulations. Please refer to the following comments for more details:\n1) Please include a detailed explanation of the steps presented in Figure 2, specially the ‘Tail resampling’ step and\ntermination condition (IF YES/NO block).\n2) In page 10, line 204, it is not clear where generation dispatch is included. Only solar PV generation, loads, and\ncontingencies are mentioned; but nothing about how generation dispatch is mentioned. Is it also modeled from a probability\ndistribution? Is it obtained from an AC power flow? Please explain.\n3) Throughout the manuscript, the set of contingencies \\tau is used, but the type of contingency is not made explicit (e.g., line\nor generator loss). Based on what is indicated in the Simulation Settings section, I can infer these are line contingencies, but\nthis should be made explicit way earlier in the manuscript.\n4) Throughout the manuscript, uncertainties are described through pre-defined probability distributions (e.g., Gaussian or\nBeta) with pre-defined parameters. However, the selection of these specific distributions and parameters is not justified. How\ncan it be demonstrated that these are realistic? Please include a reference to backup the selection of such\ndistributions/parameters.\n5) In the Case Study Section (page 12), 10 line power flows are selected to be evaluated, which are chosen as they are\nmore prone to overloading. Have you tested the proposed algorithm with other lines? Namely, those with lower probability of\noverloading. It would be interesting to see the performance of the proposed NMV-VVGP for these lines, especially given that\nthe system has a total of 24280 lines, and only 10 of those are being evaluated right now.\n6) In the Case Study section (page 12), a total of 40 contingencies are selected. Please indicate the criteria for selecting\nsuch contingencies. Why use those 40 contingencies in particular and not others? Furthermore, please include a brief\ndescription of such contingencies (e.g., the selected contingencies represent the failure of main transmission corridors).\n7) In Figure 5, it is not clear if the PDFs are result of only uncertainty from solar/loads, or if contingencies are also included. I\nraise this question, as no probability of contingencies is mentioned in the Case Study section in page 12, but the description\nof the Figure mentions that it is for “all CTCGs”.\n8) Throughout the simulation results, specific contingencies (e.g., CTGG1 and CTCG6) are studied, for which it would be\nrelevant to know what such contingency entails. Please be explicit about the characteristics of CTCG1 and CTCG6 (e.g.,\nthey imply the failures of lines A and B, respectively).\n9) Please provide more details on the PPGP method used as a benchmark, including its inputs and mathematical model.\nThis is particularly relevant, as in page 14, line 246 it is mentioned that PPGP fails to adapt to contingencies due to the\nexclusive use of kernel functions.\n10) Considering that the proposed manuscript focusing on modeling tail distributions, the PDFs presented in Figures 5, 8\nand 9 should also include a “zoom” on the tails.\n11) In page 17, line 272, it is mentioned that there is tradeoff between model accuracy and computational cost. Please\nindicate explicitly what are the associated computational costs (i.e., times) associated to each step \\lambda. This will give an\nidea is such additional computational cost is worth it (or not) to incur.\n12) In page 17, line 278, 30 (instead of 40) contingencies are selected to due converge issues. Please provide a brief\ndescription of these 30 contingencies, or mention how these differentiate from the original 40 (e.g., these no longer consider\nthe contingencies related to the north area of the system).\n13) In page 21, lines 320, three dispatch scenarios are presented. Please provide a brief description of the characteristics of\nsuch dispatch scenarios. Why are they chosen? What operating conditions do they describe? (e.g., significant solar PV\nparticipation).\n14) I would suggest adding a future work section, as this manuscript focuses exclusively on predicting active line flows. For\nexample, future work on assessing reactive/apparent line flows or bus voltages should be mentioned.\n15) There are missing acronym definitions, such as Low Rank Approximation (LRA) around line 64 page 3.", "reviewer_3": "This study presents a neural mean-variance vector-valued Gaussian process (NMV-VVGP)-based surrogate model that\nintegrates generation dispatch, contingencies, and uncertainty modeling. The key advantage of the proposed approach lies\nin its computational efficiency compared to the widely used Monte Carlo Simulation (MCS) methods. While the model\nrepresents an advancement over the current state of the art by incorporating AI/ML techniques, several significant concerns\nwere raised by the reviewer:\n1. Uncertainty Modeling – Particularly for Solar Generation\nThe treatment of uncertainty, especially with respect to solar generation, lacks clarity. In the case study, solar generation\nreplaces certain traditional generators in the NE-ISO system, but the origin and nature of the input data remain unexplained.\nIs the data region specific? The use of varying correlation coefficients for $r$ is noted, but it is unclear how these correlations\nare integrated into the modeling process. Renewable generation often exhibits strong local but weak global correlations; a\ndetailed explanation of how such spatial relationships are captured in the model and data generation is necessary. While\nNE-ISO may not currently experience high levels of renewable penetration, leveraging real-world NE-ISO data would\nenhance the study’s credibility.\n2. Tail Distribution-Aware Resampling\nThe proposed tail-aware resampling mechanism appears conceptually simple but depends heavily on accurate tail\ndistribution modeling. However, it is unclear how the distribution tails are represented, especially given the rarity of such\nextreme data points. This issue ties back to the broader challenge of uncertainty modeling. Moreover, since both renewable\ngeneration and load are influenced by weather, they are not independent. The types of conditions that lead to rare events\nand how those scenarios are generated or simulated should be discussed in detail.\n3. Power Flow Calculation and Surrogate Model Accuracy\nAs shown in Figure 3, the surrogate model is used to perform power flow analysis and subsequently estimate probabilistic\npower flow violation metrics. While results for certain buses are shown and appear accurate, it would be more compelling to\ncompute and report mean absolute percentage errors (MAPEs) for all buses, thereby demonstrating the model’s accuracy\nacross the entire system.\n4. Assumption of Beta Distribution for Uncertainty Sources\nThe case study appears to rely on Beta distributions, presumably to model solar generation uncertainty. However, other\nuncertainty sources may follow different distributions. Line 285 mentions the model’s “strong feature extraction capability,”\nyet the underlying model is based on Gaussian processes, as detailed in the Methods section. It is unclear how the Beta\ndistribution assumption influences the outcomes and whether the model is sensitive to the choice of distribution. Additional\nexperiments using alternative distributions should be conducted to evaluate robustness and generalizability.\n5. Inconsistencies in Figure 7 and Table 2\nIn Figure 7, the active power values for branches L1 and L3 are shown as negative, while Table 2 lists positive Plim values\nfor the same branches. Clarification is needed on the definitions and sign conventions of these variables to ensure\nconsistency and interpretability.\n6. Missing Section References\nThe manuscript refers to Sections IV-A, IV-B, etc., but these sections do not exist in the current draft. The text should be\nrevised to ensure all section references are accurate and properly formatted.\n7. Terminology Consistency – CTGC vs. CTCG\nBoth “CTGC” and “CTCG” appear in the manuscript, but only one term is presumably correct. This inconsistency should be\naddressed to maintain clarity and avoid confusion.\n8. Lack of Contingency Description\nAlthough the manuscript mentions different contingencies, it does not specify their nature or classification (e.g., line outages,\ngenerator failures). It remains unclear whether these contingencies have a material impact on the results or if they are\nillustrative. A clearer explanation of the contingency selection and their implications for the analysis is warranted.\n(Remarks on code availability)\nIt is unclear which code in the github page is for the submitted work."}, "author_response": "RESPONSE TO REVIEWERS' COMMENTS\nReviewer #1\nBriefSummaryofthePaper: Themanuscriptdevelopsatwo–stageworkflowforestimatingtheprobability\nthatanindividualtransmissionline/branchexceedsitsemergency(thermal)ratingunderafixedlibraryof\nN–kcontingencies:\n1. A vector–valued, deep–kernel, sparse Gaussian–process (GP) surrogate maps injections x and a\ncontingencytagτ tothebranch–flowvectory˜.\n2. An adaptive “tail–resampling” loop repeatedly re–runs the full AC power–flow on Monte–Carlo\nsampleswhosepredictedflowslieclosesttothethermallimituntiltheoverload-probabilityestimate\np(k)converges.\nF\nOna21,000–busISO–NEmodelwith40contingenciesthemethodachievesapproximately22×speed–up\novercrudeMonte–Carlowhilematchingitsaccuracy.\nMajor Strengths\nTheReviewer’sComment#1\nMajorStrengths:\n• Scalabilitydemonstration: ISO-scale,fullACflows,thousandsofuncertaininjections.\n• Vectorsurrogate: predictsalllinessimultaneously.\n• Simplecorrectiveloop: easytoembedinexistingreal–timecontingency-analysispipelines.\n• Clearengineeringvalue: per–lineprobabilitiesspeakthelanguageofNERCcompliance.\nResponse:\nWesincerelythankthereviewerforhighlightingthesestrengths. Wearegladthatvaluesofourworkare\nrecognized.\nMajor Weaknesses & Required Actions\nTheReviewer’sComment#2\nMajorWeaknesses&RequiredActions#1:\n• Issue: “Importancesampling”mis–labelling.\n• Whyitmatters: ClassicalISprovidesvarianceguarantees;theproposedmethoddoesnot.\n• Howtofix: Recastasadaptivetail–resampling;citetheadaptiveimportancesampling(EJS)\nandinstantonpapers;discussdifferences(youuseACPFvsDCPFinthepapers).\nResponse:\nAdditionally,wehavecitedtheadaptiveimportancesamplingpaper(EJS[4])andtheinstantonpaper([7])to\nimprovetheliteraturereview. Thedifferencesbetweenthispaperandtheworksin[4,7]aresummarized\nasfollows:\nInthispaper,ACpowerflowisusedtoevaluatesystemoverloadingrisk,whereasDCpowerflowisadopted\nin[4,7]. SinceDCpowerflowisanapproximationofACpowerflow,ACpowerflowprovidesmoreaccurate\nsystemoverloadingriskassessment.Moreover,ACpowerflowenablestheevaluationofbusvoltageviolations,\nwhichDCpowerflowcannotcapture.\nReferences:\n[4] A. B. Owen, Y. Maximov, and M. Chertkov, “Importance sampling the union of rare events with an\napplicationtopowersystemsanalysis,”Electron. J.Statist.,vol. 13,no. 1,pp. 231-254,2019.\n[7]M.Chertkov,M.Stepanov,F.PanandR.Baldick,\"Exactandefficientalgorithmtodiscoverextreme\nstochasticeventsinwindgenerationovertransmissionPowerGrids,\"201150thIEEEConferenceonDecision\nandControlandEuropeanControlConference,Orlando,FL,USA,2011,pp. 2174-2180.\nChangesinthemanuscript:\nTherelatedworkhasbeenaddedtoenhancetheliteraturereview. Themanuscriptisrevisedasfollows:\n\"··· Tofurtheracceleratecomputation,DCpowerflowisusedtocalculatebranchpower[7]. Nevertheless,it\nintroducesnon-negligibleerrorssinceitisonlyanapproximationofACpowerflow,anditcannotprovide\nassessmentresultsforbusvoltageviolationrisks. ···\"\nTheReviewer’sComment#3\nMajorWeaknesses&RequiredActions#2:\n• Issue: Physicsguidanceisindirectonly.\n• Whyitmatters: Readersmayassumephysics–informedsamplingwhenitispurelysurro-\ngate–error–driven.\n• Howtofix: Stateexplicitlythatphysicsenterssolelythroughthesurrogate’strainingdata.\nResponse:\nThanksforyoursuggestions! Wehaveexplicitlystatedthatphysicsenterssolelythroughthetrainingdataof\nthesurrogate.\nChangesinthemanuscript:\nIntheSection\"Risk-AwareAdaptiveTail–ResamplingMechanism\",theclarificationofphysicsusedinthe\nproposedmethodhasbeenadded. Themanuscriptisrevisedasfollows:\n\"··· Fromtheaboveprocedure,itcanbeobservedthattheproposedadaptivetail–resamplingmechanismis\npurelysurrogate–error–driven. Inotherwords,thephysicsisincorporatedsolelythroughthetrainingdataof\nthesurrogate. ···\"\nMajorWeaknesses&RequiredActions#3:\n• Issue: Line–by–linefocusignoresthesystem-widerisk.\n• Whyitmatters: Plannersandsomeoperatorsneedunionprobabilities;connectstoinstanton\nliterature.\n• How to fix: Use existing samples to identify top contributing lines and report respective\noverloadprobabilities;comparetotheaforementionedinstantonapproach.\nResponse: Thanksforyoursuggestion! Actionsarelistedbelow:\n(1)Useexistingsamplestoidentifytopcontributinglinesandreportrespectiveoverloadprobabilities.\nTakingthecasewhereallloadsandPVgenerationfollowaGaussiandistributionwithmeansequaltotheir\nnominalvaluesandfixedstandarddeviationsof0.05timesthemeans,whilethecorrelationsamongthese\nsourcesaresettor =0.5.\nFromtheresultsinTablesR1,acrossthesystem,onlybranchesL1andL3haveoverloadingrisk,whilethe\noverloadingriskofotherbranchesisnegligible. Therefore,inthisscenario,branchesL1andL3arethetop\ncontributorstotheoverloadingrisk,andtheirrespectiveoverloadprobabilitiesareshowninTablesR1.\nTableR1: RiskAssessmentResultsunderAllCTGCs(λ=150,r =0.5)\nMCS DKS-VVGPwithresampling\np ×100% MAPE p ×100% MAPE\nF F\nBranchname:L1\n2.800 - 2.800 0.278\nP =180MW\nLim\nBranchname:L3\n3.144 - 3.144 0.264\nP =330MW\nLim\nResampledsamples\n-(8.08h) 560(0.3020h)\n(Assessmenttime)\n(2)Comparetotheaforementionedinstantonapproach.\nComparedwiththeinstantonapproachin[7],theproposedmethoddiffersinthefollowingaspects,asshown\ninTableR2:\n• Objective: Theobjectiveoftheproposedmethodistoevaluatetheoverloadingriskinastatistical\nsense,whereastheinstantonapproachaimstoidentifythemostseriousoperatinginstanceofwind\ngenerationthatcauseslineoverloading.\n3\n• Inclusionofcontingencies: Theproposedmethodconsidersthecombinedimpactofcriticalcontin-\ngenciesanduncertainsources,includingloadsandPVgeneration,ontheoverloadingrisk. Incontrast,\ntheinstantonapproachonlyaccountsforfluctuationsinwindgeneration.\n• DifferentSolvers: TheproposedmethodreliesonanACpowerflowsolver,whichprovideshigher\naccuracyinriskassessment,whiletheinstantonapproachadoptsaDCpowerflowsolvertoimprove\ncomputationalefficiencyatthecostofreducedaccuracy.\nTableR2: Comparisonbetwe: ComparisonbetweentheProposedMethodandtheInstantonApproach\nAspect ProposedMethod InstantonApproach\nIdentifiesthemostcriticaloperating\nEvaluatestheoverloadingriskina\nObjective instanceofwindgenerationcausingline\nstatisticalsense.\noverloading.\nConsidersthecombinedimpactofserious\nInclusionof Onlyaccountsforfluctuationsinwind\ncontingencies,loads,andPVgenerationon\nContingencies generation.\noverloadingrisk.\nUsesaDCpowerflowsolvertoimprove\nReliesonanACpowerflowsolverfor\nDifferentSolvers computationalefficiencyatthecostof\nhigheraccuracyinriskassessment.\nreducedaccuracy.\nReferences:\n[7]M.Chertkov,M.Stepanov,F.PanandR.Baldick,\"Exactandefficientalgorithmtodiscoverextreme\nstochasticeventsinwindgenerationovertransmissionPowerGrids,\"201150thIEEEConferenceonDecision\nandControlandEuropeanControlConference,Orlando,FL,USA,2011,pp. 2174-2180.\nChangesinthemanuscript:\nIntheSection\"PerformanceunderDifferentDistributionCorrelations\",thecalrificationofopcontributing\nlineshasbeenadded. Themanuscriptisrevisedasfollows:\n\"··· Therefore,inthisscenario,branchesL1andL3arethetopcontributorstotheoverloadingrisk. ···\"\nIntheSection\"TransferLearningforQuickModelUpdating\",thecomparisonbetwentheproposedmethod\nandtheinstantonapproachin[7]hasbeenadded. Themanuscriptisrevisedasfollows:\n\"··· Comparedwiththeinstantonapproachin[7],theproposedmethoddiffersinthefollowingaspects:\n• Objective: Theobjectiveoftheproposedmethodistoevaluatetheoverloadingriskinastatistical\nsense,whereastheinstantonapproachaimstoidentifythemostseriousoperatinginstanceofwind\ngenerationthatcauseslineoverloading.\n4\n• Inclusionofcontingencies: Theproposedmethodconsidersthecombinedimpactofcriticalcontin-\ngenciesanduncertainsources,includingloadsandPVgeneration,ontheoverloadingrisk. Incontrast,\ntheinstantonapproachonlyaccountsforfluctuationsinwindgeneration.\n• DifferentSolvers: TheproposedmethodreliesonanACpowerflowsolver,whichprovideshigher\naccuracyinriskassessment,whiletheinstantonapproachadoptsaDCpowerflowsolvertoimprove\ncomputationalefficiencyatthecostofreducedaccuracy.\n···\"\nTheReviewer’sComment#5\nMajorWeaknesses&RequiredActions#4:\n• Issue: Noablations: neuralkernelandλstepnotjustified.\n• Whyitmatters: Neededtoestablishmethodologicalnovelty.\n• Howtofix: Reportresultswith(i)plainRBFGP;(ii)variedλ.\nResponse: Thanksforyoursuggestion! Actionsarelistedbelow:\n(1)ReportresultswithplainRBFGP.\nWehavereportedtheresultsusingtheplainRBFGP,referredtoasVVGPinthepaper. FromTableR3,\nitcanbeobservedthattheproposedmethod(DKS-VVGPwithresampling)achieveshigheraccuracyin\ntermsofMAPEthanthevariantswithouttheneuralnetworkcomponent(i.e.,plainRBFGPorVVGPwith\nresampling). Thisdemonstratesthattheneuralnetworkplaysanimportantroleinenhancingtheperformance\noftheproposedmethod.\n(2)Reportresultswithvariedλ\nAsshowninTableR4,increasingλallowstheresamplingmechanismtoachievemoreaccurateriskassessment.\nThisisbecauseonlyalimitednumberofsamplesneartheoverloadingboundaryaremisclassifiedbythe\nsurrogatemodel,namelyDKS-VVGP.Iftheresamplingsizeistoosmall,theproposedmethodmayconverge\nprematurely,failingtocorrectthesemisclassifiedsamples,thusleadingtoassessmentbias. However,larger\nvaluesofλresultinincreasedcomputationalcostduetoagreaternumberofresampledpoints. Tobalance\nevaluationtimeandassessmentaccuracy,weselectλ=150astheoptimalresamplingstepsizeforthisstudy.\nDKS-VVGP\nDKS-VVGPwith VVGPwith\nMCS without PPGP\nresampling resampling\nresampling\npF×\nMAPE\npF×\nMAPE\npF×\nMAPE\npF×\nMAPE\npF×\nMAPE\n100% 100% 100% 100% 100%\nBranchname:L1\n2.229 - 2.229 0.289 2.136 0.310 2.229 0.323 2.557 2.901\nPLim =180MW\nBranchname:L3\n2.557 - 2.557 0.280 2.471 0.314 2.557 0.253 2.557 0.748\nPLim =330MW\nBranchname:L4\n0.25 - 0.25 1.136 0.2 1.200 0.25 1.30 0.310 5.276\nPLim =200MW\nTrainingtime - 3.34h 3.34h 3.34h 4.75h\nAssessmenttime 10.78h 0.340h ≈0h(0.44s) 0.340h 0.0168h\nTableR4: RiskAssessmentResultsunderAllCTGCswithVariousλ(r =0.2)\nλ=50 λ=100 λ=150 λ=200 λ=250\nBranchname:L1\n2.236 2.236 2.229 2.229 2.229\np ×100%(P =180MW)\nF Lim\nBranchname:L3\n2.557 2.557 2.557 2.557 2.557\np ×100%(P =330MW)\nF Lim\nBranchname:L4\n0.207 0.25 0.25 0.25 0.25\np ×100%(P =200MW)\nF Lim\nResampledsamples 191 553 630 831 1114\nTheReviewer’sComment#6\nMajorWeaknesses&RequiredActions#5:\n• Issue: Reproducibility. Hyper-parameters,inducing-pointplacement,andcodeavailabil-\nityaremissing..\n• Whyitmatters: NatureCommunicationsreadersexpectreusablecode.\n• Howtofix: Releasescriptsandfullsettingsinapublicrepositoryorassupplementarymaterial.\nResponse:\n18GB,wehaveprovidedaGoogleDrivelinktofacilitatereadersintestingthecodeassociatedwiththispaper:\n[REDACTED]\nMinor Comments and Clarifications\nTheReviewer’sComment#7\nMinorCommentsandClarifications#1:\nReplace“Neural–Mean–VarianceGP”withthestandardterm“deep–kernelsparsevector–valuedGP”\ntoavoidconfusion.\nResponse:\nThanksforyoursuggestion! Wehaverenamedtheproposed\"Neural-Mean-VarVector-ValuedGaussian\nProcess(NMV-VVGP)\"as\"Deep–KernelSparseVector-ValuedGaussianProcess(DKS-VVGP)\".\nTheReviewer’sComment#8\nMinorCommentsandClarifications#2:\nStatehowthecoregionalisationmatrixAisinitialisedandregularised.\nResponse:\nIntheproposedmulti-outputsurrogatemodel,eachelementofthecoregionalizationmatrixAisrandomly\ninitializedaccordingtoanormaldistributionwithameanof0andastandarddeviationof1,withoutany\nregularization.\nChangesinthemanuscript:\nIntheSection\"Methods\",theinitializationandregularizationofthecoregionalizationmatrixAhavebeen\nadded. Themanuscriptisrevisedasfollows:\n\"··· EachelementofthecoregionalizationmatrixAisrandomlyinitializedaccordingtoanormaldistribution\nwithameanof0andastandarddeviationof1,withoutanyregularization. ···\"\nTheReviewer’sComment#9\nMinorCommentsandClarifications#3:\nProvidepseudo–codefortheλ–resamplingloopandspecifythestoppingtoleranceϵ.\nResponse: Thanksforyoursuggestion! Inthispaper,thestoppingtoleranceϵissetas0.01%. Thepseudo\ncodesforλ–resamplingaredetailedinthefollowingsteps,asalsoshowninFig. R1.\n• Step One: Assume Φ(k) =\n(cid:110)\n(x ,τ ),||y˜(k)|−P(k) |,y˜(k)\n(cid:111)Ns\nrepresents the PPF results of the\ni i i Lim i\ni=1\nk-th branch predicted by the DKS-VVGP, considering uncertain power injections and predefined\ncontingencies. Theiterationnumberj isinitializedto1.\n7\n• StepTwo: Tomitigateerrorsinoverloadingriskassessment,theresamplingprocessbeginsattherisky\nboundary. Specifically,Φ(k)issortedinascendingorderaccordingto||y˜(k)|−P(k) |.\ni Lim\n• StepThree: Theestimatedoverloadingriskofthek-thbranchatthej-thiteration,denotedp(k),\nF,j\ncalculatedusingequation(5)andΦ(k).\n• StepFour: TailResamplingTheλpredictedsamplesy˜ closesttotheriskyboundaryattail(i.e.,\ni\ntheminimumvalueof||y˜ |−P(k) |)arerecalculatedusingthepowerflowsolver. Asaresult,Φ(k)\ni Lim\nisupdatedbyexcludingλpredictedsamples{y˜ }jλ andincludingtherecalculatedsamples\ni i=(j−1)λ+1\n{y }jλ . Then,setj =j+1andp(k),iscalculatedusingequation(5)andupdatedΦ(k).\ni i=(j−1)λ+1 F,j\n• StepFive: Ifthedifferencebetweentheestimatedp(k)atthecurrentiterationandthepreviousoneis\nF\nsmallerthanthepredefinedthresholdϵ,theresamplingwillbeterminated. Otherwise,returntoStep\nThree.\nFigureR1: Risk-awaretaildistributionresamplingmechanismfork-thbranch.\nChangesinthemanuscript:\nIntheResultsandDiscussionSection, thedescriptionoftheresamplingprocedurehasbeenadded. The\nmanuscriptisrevisedasfollows:\n8\n\"···\n• Step One: Assume Φ(k) =\n(cid:110)\n(x ,τ ),||y˜(k)|−P(k) |,y˜(k)\n(cid:111)Ns\nrepresents the PPF results of the\ni i i Lim i\ni=1\nk-th branch predicted by the DKS-VVGP, considering uncertain power injections and predefined\ncontingencies. Theiterationnumberj isinitializedto1.\n• StepTwo: Tomitigateerrorsinoverloadingriskassessment,theresamplingprocessbeginsattherisky\nboundary. Specifically,Φ(k)issortedinascendingorderaccordingto||y˜(k)|−P(k) |.\ni Lim\n• StepThree: Theestimatedoverloadingriskofthek-thbranchatthej-thiteration,denotedp(k),\nF,j\ncalculatedusingequation(5)andΦ(k).\n• StepFour: TailResamplingTheλpredictedsamplesy˜ closesttotheriskyboundaryattail(i.e.,\ni\ntheminimumvalueof||y˜ |−P(k) |)arerecalculatedusingthepowerflowsolver. Asaresult,Φ(k)\ni Lim\nisupdatedbyexcludingλpredictedsamples{y˜ }jλ andincludingtherecalculatedsamples\ni i=(j−1)λ+1\n{y }jλ . Then,setj =j+1andp(k),iscalculatedusingequation(5)andupdatedΦ(k).\ni i=(j−1)λ+1 F,j\n• StepFive: Ifthedifferencebetweentheestimatedp(k)atthecurrentiterationandthepreviousoneis\nF\nsmallerthanthepredefinedthresholdϵ,theresamplingwillbeterminated. Otherwise,returntoStep\nThree.\n···\"\nChangesinthemanuscript:\nIntheSection\"SimulationSettings\",thesettingofthestoppingtoleranceϵhasbeenadded. Themanuscript\nisrevisedasfollows:\n\"··· Thestoppingtoleranceϵissetas0.01%. ···\"\nTheReviewer’sComment#10\nMinorCommentsandClarifications#4:\nClarifywhetherunseencontingencytypes(notonlydispatchchanges)canbeaccommodatedwith\n1,000newACsimulations.\nResponse: Basedonourexperiments,1000newACsimulations(samples)areusuallysufficient. However,\nitisdifficulttoconcludethat1000newACsimulations(samples)aresufficienttoupdatetheproposedmodel\nforallunseencontingencies. Thisisbecausetherequiredamountofdataforupdatingdependsonseveral\nfactors,suchasthesimilaritybetweentheunseencontingenciesandtheexistingones,aswellasthesampling\nstrategy. Therefore,aninterestingdirectionforfutureworkistoinvestigatehowtotheoreticallyquantifythe\namountofdatarequiredtoupdatetheproposedmodel.\nChangesinthemanuscript:\nIntheSection\"Discussion\",thefutureworkofhandlingunseencontingencieshasbeenadded.Themanuscript\nisrevisedasfollows:\n9\n\"··· Whenunseencontingenciesarise,itisdifficulttodeterminehowmanynewACsimulations(samples)\naresufficienttoupdatetheproposedmodel. Thisisbecausetherequiredamountofdatadependsonseveral\nfactors,suchasthesimilaritybetweentheunseencontingenciesandtheexistingones,aswellasthesampling\nstrategy. Therefore,aninterestingdirectionforfutureworkistoinvestigatehowtotheoreticallyquantifythe\namountofdataneededtoupdatetheproposedmodel. ···\"\nSignificance for a Broad Audience\nTheReviewer’sComment11\nThestudyistechnicallysolidandclearlyusefulforpower–systempractitioners.However,twoaspects\nlimititsbroaderappeal:\n• Thestatisticalmachinery(deep–kernelGP+activeresampling)isstandardoutsidepowergrids;\nnoveltyliesmainlyinscaleandintegration.\n• Results are tightly coupled to power–system operational rules; immediate transfer to other\nfields(aerospace,climate,finance)wouldrequirenon–trivialadaptation.\nIthereforecautiontheeditorsthatthepaper’simpactislikelystrongestwithintheenergy–systems\ncommunity.\nResponse:\nWe thank the reviewer for the positive assessment and insightful comments. We agree that the novelty\nprimarilyliesinthescaleandintegrationofthemethodwithinpower–systemapplications,andourgoalis\ntocontributetoimprovingtheefficiencyandaccuracyofoverloadingriskassessmentinpracticalpower\nsystems.\nTheproposedmanuscriptstudiestheuseofaneuralmean-variancevector-valuedGaussianprocess(NMV-\nVVGP)toestimatetheoverloadprobabilityfortransmissionlinesunderdifferentsystemoperatingconditions,\nwith these being a result uncertain solar PV generation and loads, as well as generator dispatch and\ncontingencies. Overall,themanuscriptiswellwrittenandaddressesanexistingliteraturegap,makingit\nsuitableforpublicationinthisjournal. Nevertheless,severalaspectsneedtobefirstimproved,whichare\nmainlyrelatedtoprovidingmoredetailsonthemodelingapproachandcomputersimulations. Pleasereferto\nthefollowingcommentsformoredetails:\nTheReviewer’sComment#1\nPleaseincludeadetailedexplanationofthestepspresentedinFigure2,speciallythe‘Tailresampling’\nstepandterminationcondition(IFYES/NOblock).\nResponse:\nThanksforyoursuggestion! Theresamplingprocedurecanbedetailedinthefollowingsteps,asalsoshown\ninFig. R2.\n• Step One: Assume Φ(k) =\n(cid:110)\n(x ,τ ),||y˜(k)|−P(k) |,y˜(k)\n(cid:111)Ns\nrepresents the PPF results of the\ni i i Lim i\ni=1\nk-th branch predicted by the NMV-VVGP, considering uncertain power injections and predefined\ncontingencies. Theiterationnumberj isinitializedto1.\n• StepTwo: Tomitigateerrorsinoverloadingriskassessment,theresamplingprocessbeginsattherisky\nboundary. Specifically,Φ(k)issortedinascendingorderaccordingto||y˜(k)|−P(k) |.\ni Lim\n• StepThree: Theestimatedoverloadingriskofthek-thbranchatthej-thiteration,denotedp(k),\nF,j\ncalculatedusingequation(5)andΦ(k).\n• StepFour: TailResamplingTheλpredictedsamplesy˜ closesttotheriskyboundaryattail(i.e.,\ni\ntheminimumvalueof||y˜ |−P(k) |)arerecalculatedusingthepowerflowsolver. Asaresult,Φ(k)\ni Lim\nisupdatedbyexcludingλpredictedsamples{y˜ }jλ andincludingtherecalculatedsamples\ni i=(j−1)λ+1\n{y }jλ . Then,setj =j+1andp(k),iscalculatedusingequation(5)andupdatedΦ(k).\ni i=(j−1)λ+1 F,j\n• StepFive: Ifthedifferencebetweentheestimatedp(k)atthecurrentiterationandthepreviousoneis\nF\nsmallerthanthepredefinedthresholdϵ,theresamplingwillbeterminated. Otherwise,returntoStep\nThree.\nChangesinthemanuscript:\nIntheResultsandDiscussionSection, thedescriptionoftheresamplingprocedurehasbeenadded. The\nmanuscriptisrevisedasfollows:\n\"···\n12\n• Step One: Assume Φ(k) =\n(cid:110)\n(x ,τ ),||y˜(k)|−P(k) |,y˜(k)\n(cid:111)Ns\nrepresents the PPF results of the\ni i i Lim i\ni=1\nk-th branch predicted by the DKS-VVGP, considering uncertain power injections and predefined\ncontingencies. Theiterationnumberj isinitializedto1.\n• StepTwo: Tomitigateerrorsinoverloadingriskassessment,theresamplingprocessbeginsattherisky\nboundary. Specifically,Φ(k)issortedinascendingorderaccordingto||y˜(k)|−P(k) |.\ni Lim\n• StepThree: Theestimatedoverloadingriskofthek-thbranchatthej-thiteration,denotedp(k),\nF,j\ncalculatedusingequation(5)andΦ(k).\n• StepFour: TailResamplingTheλpredictedsamplesy˜ closesttotheriskyboundaryattail(i.e.,\ni\ntheminimumvalueof||y˜ |−P(k) |)arerecalculatedusingthepowerflowsolver. Asaresult,Φ(k)\ni Lim\nisupdatedbyexcludingλpredictedsamples{y˜ }jλ andincludingtherecalculatedsamples\ni i=(j−1)λ+1\n{y }jλ . Then,setj =j+1andp(k),iscalculatedusingequation(5)andupdatedΦ(k).\ni i=(j−1)λ+1 F,j\n• StepFive: Ifthedifferencebetweentheestimatedp(k)atthecurrentiterationandthepreviousoneis\nF\nsmallerthanthepredefinedthresholdϵ,theresamplingwillbeterminated. Otherwise,returntoStep\nThree.\n···\"\nTheReviewer’sComment#2\nInpage10,line204,itisnotclearwheregenerationdispatchisincluded. OnlysolarPVgeneration,\nloads,andcontingenciesarementioned;butnothingabouthowgenerationdispatchismentioned. Isit\nalsomodeledfromaprobabilitydistribution? IsitobtainedfromanACpowerflow? Pleaseexplain.\nResponse:\nThanksforyoursuggestion!Inthispaper,thegenerationdispatchesareobtainedfromthehistoricaldataofthe\nISONewEnglandpowersystem.Specifically,thegenerationdispatchesdeterminetheoutputsofsynchronous\ngenerators,whicharecalculatedbythesystemoperatorusinganeconomicdispatchoptimizationmodelbased\nonloadforecastsandrenewablepowerforecasts. Inotherwords,differentgenerationdispatchescorrespond\ntodifferentoperatingconditions,includingdifferentgenerationoutputsandloadlevels. Therefore,theyare\nneithermodeledasprobabilitydistributionsnorderivedfromACpowerflow,butareobtainedthrough\ntheeconomicdispatchoptimizationmodel. Notethat,weusetheactualsystemandfielddataforvalidation\nandthusgenerationdispatchhasalreadybeenincludedinthetestdata.\nChangesinthemanuscript:\nIntheSection\"TransferLearningforQuickModelUpdating\",thedefinitionsofgenerationdispatchhave\nbeenadded. Themanuscriptisrevisedasfollows:\n\"··· Remark1: Inthispaper,thegenerationdispatchesareobtainedfromthehistoricaldataoftheISO\nNewEnglandpowersystem. Specifically,thegenerationdispatchesdeterminetheoutputsofsynchronous\ngenerators, whicharecalculatedbythesystemoperatorusinganeconomicdispatchoptimizationmodel\nbasedonloadforecastsandrenewablepowerforecasts. ···\"\nThroughoutthemanuscript,thesetofcontingenciesτ isused,butthetypeofcontingencyisnotmade\nexplicit(e.g.,lineorgeneratorloss). BasedonwhatisindicatedintheSimulationSettingssection,I\ncaninferthesearelinecontingencies,butthisshouldbemadeexplicitwayearlierinthemanuscript.\nResponse:\nThanks for your suggestion! In our paper, contingencies refer to line outages, such as the loss of one or\nseveraltransmissionlines(upto8lines). Wehaveclarifiedthisinformationintherevisedmanuscript.\nChangesinthemanuscript:\nIntheSection\"CaseStudyontheLarge-ScaleISONEPowerSystem\",thedefinitionsofcontingencieshave\nbeenadded. Themanuscriptisrevisedasfollows:\n\"··· These40criticalCTGCsrepresentsimultaneousoutagesofmultipleimportanttransmissionlines(upto\n8lines). ···\"\nTheReviewer’sComment#4\nThroughoutthemanuscript,uncertaintiesaredescribedthroughpre-definedprobabilitydistributions\n(e.g., Gaussian or Beta) with pre-defined parameters. However, the selection of these specific\ndistributions and parameters is not justified. How can it be demonstrated that these are realistic?\nPleaseincludeareferencetobackuptheselectionofsuchdistributions/parameters.\nResponse:\nSincemodelingtheuncertaintyofloadandsolargenerationisnotthemainfocusofthispaper,\nthedistributionsandtheirparametersareselectedbasedonrepresentativereferencestoensurepracticality.\nForthepurposeofvalidation,aGaussianorBetadistributioniscommonlyusedtocharacterizesolarpower\nuncertainty,whileaGaussiandistributioniswidelyadoptedforloaduncertainty[29,30,31]. Detailsonthe\nselectionofdistributionsandparameterscanbefoundin[29]. Inreal-worldapplications,historicaldatacan\nbeanalyzedusingmaximumlikelihoodestimationtoinfertheappropriatedistributionsandtheirparameters\nforloadsandsolargeneration,therebyensuringpracticality.\nReferences:\n[29]U.H.Ramadhani,etal.,“Reviewofprobabilisticloadflowapproachesforpowerdistributionsystems\nwith photovoltaic generation and electric vehicle charging,” International Journal of Electrical Power &\nEnergySystems,vol.120,pp. 106003,2020.\n[30]V.Singh,etal.,“Uncertaintyhandlingtechniquesinpowersystems: Acriticalreview,”ElectricPower\nSystemsResearch,vol. 203,pp. 107633,2020.\n[31]K.Ye,J.Zhao,N.DuanandY.Zhang,\"Physics-InformedSparseGaussianProcessforProbabilistic\nStabilityAnalysisofLarge-ScalePowerSystemWithDynamicPVsandLoads,\"IEEETransactionsonPower\nSystems,vol. 38,no. 3,pp. 2868-2879,May2023\nIn the Section \"Case Study on the Large-Scale ISO NE Power System\", the clarification of uncertainty\nmodelingforloadsandsolargenerationhasbeenadded. Themanuscriptisrevisedasfollows:\n\"··· Sincemodelingtheuncertaintyofloadandsolargenerationisnotthemainfocusofthispaperand\nwedonothaveaccesstotheactualhistoricalloadorsolardataoftheISONewEnglandpowersystem,the\ndistributionsandtheirparametersareselectedbasedonrepresentativereferencestoensurepracticality. For\nthe purpose of validation, a Gaussian or Beta distribution is commonly used to characterize solar power\nuncertainty,whileaGaussiandistributioniswidelyadoptedforloaduncertainty[27,28,29]. ···\"\nTheReviewer’sComment#5\nIntheCaseStudySection(page12), 10linepowerflowsareselectedtobeevaluated, whichare\nchosenastheyaremorepronetooverloading. Haveyoutestedtheproposedalgorithmwithother\nlines? Namely, those with lower probability of overloading. It would be interesting to see the\nperformanceoftheproposedNMV-VVGPfortheselines,especiallygiventhatthesystemhasatotal\nof24280lines,andonly10ofthosearebeingevaluatedrightnow.\nResponse:\nWealsotestedtheproposedalgorithmonothertransmissionlines,butnooverloadingwasobserved\nonthoselines. Therefore,inthispaper,wefocusouranalysisonthe10linesthataremorelikelyto\nexperienceoverloading. Specifically,basedontheexperienceofsystemoperatorsintheISONewEngland\npowersystem,theselectedlinesarecriticalmonitoringlinesthatarepronetooverloading. Ourcomputational\nresultsfurtherconfirmthattheremaininglines,beyondthese10,arenotonlyfreefromoverloadingbutalso\nfarfromtheoverloadingmargin. Hence,the10linesthatareclosetooractuallyexperienceoverloading\nwerechosenforanalysistovalidatetheaccuracyoftheproposedmethod. Thischoicehastwoadvantages:\nfirst,byfocusingontheoperator-identifiedcriticalmonitoringlines,wecanclearlyanalyzehowoverloading\noccursandwhatcontrolmeasurescanbetaken;second,itsignificantlyimprovescomputationalefficiency,\nsincemonitoringalllineswouldresultinmorethan20koutputdimensions,requiringatleast2000times\nmorekernelmatrixmultiplicationsduringmodeltrainingcomparedtoanalyzingonlythe10keylines,which\nrequiresfarfewermatrixmultiplications.\nTheReviewer’sComment#6\nIn the Case Study section (page 12), a total of 40 contingencies are selected. Please indicate the\ncriteriaforselectingsuchcontingencies. Whyusethose40contingenciesinparticularandnotothers?\nFurthermore,pleaseincludeabriefdescriptionofsuchcontingencies(e.g.,theselectedcontingencies\nrepresentthefailureofmaintransmissioncorridors).\nResponse:\nThanksforyoursuggestions! These40criticalcontingenciesrepresentsimultaneousoutagesofmultiple\nimportanttransmissionlines(upto8lines). Inthispaper,the40contingenciesarethoserecommendedbythe\nsystemoperatorsoftheISONewEnglandpowersystemforanalysis,astheyhavehistoricallycausedline\noverloadingandarethereforeconsideredmoreseverecriticalcontingenciescomparedwithothers. Ifthese\ncriticalcontingenciesdonotleadtolineoverloadingrisk,thenothercontingenciesareunlikelytocausesuch\n15\nrisks.\nChangesinthemanuscript:\nIntheSection\"CaseStudyontheLarge-ScaleISONEPowerSystem\",thedescriptionsofcontingencies\nhavebeenadded. Themanuscriptisrevisedasfollows:\n\"··· These40criticalCTGCsrepresentsimultaneousoutagesofmultipleimportanttransmissionlines(up\nto8lines). Inthispaper,the40contingenciesarethoserecommendedbythesystemoperatorsoftheISO\nNewEnglandpowersystemforanalysis,astheyhavehistoricallycausedlineoverloadingandaretherefore\nconsideredmoreseverecriticalcontingenciescomparedwithothers. Ifthesecriticalcontingenciesdonot\nleadtolineoverloadingrisk,thenothercontingenciesareunlikelytocausesuchrisks. ···\"\nTheReviewer’sComment#7\nInFigure5,itisnotclearifthePDFsareresultofonlyuncertaintyfromsolar/loads,orifcontingencies\narealsoincluded. Iraisethisquestion,asnoprobabilityofcontingenciesismentionedintheCase\nStudysectioninpage12,butthedescriptionoftheFigurementionsthatitisfor“allCTCGs”.\nResponse:\nGiventhecontingencies,thispaperfocusesonanalyzingtheimpactofstochasticfluctuationsinloadsand\nsolargenerationonthelineoverloadingriskoftheISONewEnglandpowersystem. Asaresult,theline\noverloadingriskcanbeobtainedforeachcontingency. Inotherwords,thecontingenciesaredeterministic\nandinvolvenorandomness,whereastheloadandsolargenerationarestochastic.\nChangesinthemanuscript:\nIntheSection“CaseStudyontheLarge-ScaleISONEPowerSystem,”descriptionsregardingthestochastic\nnatureofcontingencieshavebeenadded. Therevisedmanuscriptisasfollows:\n\"··· Itisworthmentioningthatthecontingenciesaredeterministicandinvolvenorandomness. ···\"\nTheReviewer’sComment#8\nThroughoutthesimulationresults,specificcontingencies(e.g.,CTGG1andCTCG6)arestudied,\nforwhichitwouldberelevanttoknowwhatsuchcontingencyentails. Pleasebeexplicitaboutthe\ncharacteristicsofCTCG1andCTCG6(e.g.,theyimplythefailuresoflinesAandB,respectively).\nResponse:\nThanksforyoursuggestion! CTGC1andCTGC6representspecificlineoutagecontingencies. Inparticular:\n• CTGC1correspondstothesimultaneousoutagesofthefollowinglines: (13287→13285),(13285→\n13284),(17707→17709),(17710→17708),(17723→17722),and(17721→17720).\n• CTGC6 corresponds to the outages of the lines: (3477 → 3476), (3472 → 3477), and (3486 →\n3480).\nInthesection“CaseStudyontheLarge-ScaleISONEPowerSystem,”atabledescribingCTGC1andCTGC6\nhasbeenadded. Therevisedmanuscriptisasfollows:\nTableR5: LineoutagesetsofCTGC1andCTGC6\nContingencyName LineOutages(StartBus→EndBus)\n(13287→13285),(13285→13284),(17707→17709),(17710→\nCTGC1\n17708),(17723→17722),(17721→17720)\nCTGC6 (3477→3476),(3472→3477),(3486→3480)\nTheReviewer’sComment#9\nPlease provide more details on the PPGP method used as a benchmark, including its inputs and\nmathematicalmodel. Thisisparticularlyrelevant,asinpage14,line246itismentionedthatPPGP\nfailstoadapttocontingenciesduetotheexclusiveuseofkernelfunctions.\nResponse:\nInPPGP,theLaplacianmatrixLcorrespondingtothepowersystemtopologyisdecomposedas\nL=U(cid:101)Σ(cid:101)W(cid:102),\nandW(cid:102) isusedtotransformtheloadandsolarpowervectorsP\nD\nandP\nPV\nintotheinput\n \nP\nD\nPW(cid:102) = W(cid:102)\nP\nPV\nThePPGPmodelforthepowerflowonthek-thlineisthenconstructedas\ny(k)(PW(cid:102))=M\nPPGP,k\n(PW(cid:102))+ϵ\nPPGP,k\n,\nwhere\n⊤\n• M\nPPGP,k\n(PW(cid:102))∼GP(µ\nPPGP,k\n,σ\nP\nK(PW(cid:102),PW(cid:102) ))isaGaussianprocesswithmeanµ\nPPGP,k\nandkenelcovariancefunctionK(·,·),\n• ϵ ∼N(0,σ2 )representsindependentobservationnoise,\nPPGP,k PPGP,k\n• y(k)(PW)isthepredictedpowerflowonthek-thline,\n• U(cid:101) andΣ(cid:101) aretheleftsingularvectorsandsingularvaluesfromthesingularvaluedecompositionofL.\nFrom the PPGP model, it can be observed that performing a linear transformation on the system\ntopology may result in the loss of nonlinear information inherent in the topology. In contrast, our\n17\nproposedmodeldirectlyleveragesaneuralnetworktoextractfeaturesfromthenodaladmittance\nmatrixrepresentingthepowersystemtopology,therebypreservingthenonlinearcharacteristicsofthe\nfeatures. Consequently,theproposedmodelachievessuperiorperformancethanPPGPtoadapttotopology\nchangescausedbycontingencies.\nChangesinthemanuscript:\nIntheSection“CaseStudyontheLarge-ScaleISONEPowerSystem,”descriptionsregardingdetailedmodel\nofPPGPhavebeenadded. Therevisedmanuscriptisasfollows:\n\"··· In PPGP, the Laplacian matrix L corresponding to the power system topology is decomposed as\nL=U(cid:101)Σ(cid:101)W(cid:102),andW(cid:102) isusedtotransformtheloadandsolarpowervectorsP\nD\nandP\nPV\nintotheinput\n \nP\nD\nPW(cid:102) = W(cid:102)\nP\nPV\nThePPGPmodelforthepowerflowonthek-thlineisthenconstructedas\ny(k)(PW(cid:102))=M\nPPGP,k\n(PW(cid:102))+ϵ\nPPGP,k\n,\nwhere\n⊤\n• M\nPPGP,k\n(PW(cid:102))∼GP(µ\nPPGP,k\n,σ\nP\nK(PW(cid:102),PW(cid:102) ))isaGaussianprocesswithmeanµ\nPPGP,k\nandkenelcovariancefunctionK(·,·),\n• ϵ ∼N(0,σ2 )representsindependentobservationnoise,\nPPGP,k PPGP,k\n• y(k)(PW)isthepredictedpowerflowonthek-thline,\n• U(cid:101) andΣ(cid:101) aretheleftsingularvectorsandsingularvaluesfromthesingularvaluedecompositionofL.\n···\"\nTheReviewer’sComment#10\nConsideringthattheproposedmanuscriptfocusingonmodelingtaildistributions,thePDFspresented\ninFigures5,8and9shouldalsoincludea“zoom”onthetails.\nResponse:\nWehaveaddeda“zoom”viewforFigure8(indexedasFigureR3inthisletter)andFigure9(indexedas\nFigure R4 in this letter). However, we did not include a “zoom” view for Figure 5, since it contains 10\nsubfiguresandaddingadditionalzoomswouldmakethefigureoverlycluttered.\n18\n0\n-170\n(a)\n0\n-280\n(b)\n0\n180 200\n(c)\nFigureR3: PDFsofactivepowerofdifferentbranchesunderCTGC6whenthecorrelationamong\nall uncertain sources was set to 0.2. a results of branch L1. b results of branch L3. c results of\nbranchL4.\n19\n0\n-190 -180\n(a)\n0\n-190 -180\n(b)\n0\n-190 -180\n(c)\nFigureR4: PDFsofactivepowerofbranchL1underallCTGCswhenthecorrelationamongall\nuncertain sources is different. a correlation is set as r = 0.3; b correlation is set as r = 0.4; c\ncorrelationissetasr =0.5.\nTheReviewer’sComment#11\nInpage17,line272,itismentionedthatthereistradeoffbetweenmodelaccuracyandcomputational\ncost. Pleaseindicateexplicitlywhataretheassociatedcomputationalcosts(i.e.,times)associatedto\neachstepλ. Thiswillgiveanideaissuchadditionalcomputationalcostisworthit(ornot)toincur.\nResponse:\nThanksforyoursuggestion! Thecomputationalcosts(i.e.,runtime)associatedwitheachstepsizeλarise\nfromtheresamplingprocess. Inotherwords,duringeachiterationofresampling,theλsurrogatepredictions\nclosesttotheriskyboundaryarerecalculatedusingthepowerflowsolver,whichincreasesthecomputational\nburden. AsshowninTableR6,theassessmenttimeincreasesfromnearlyzeroto0.34hoursafterapplying\n20\nresampling.However,comparedwiththe10.78hoursrequiredbyMonteCarlosampling(MCS)basedentirely\nonthepowerflowsolver,theproposedresamplingapproachachievesveryaccurateriskassessment,whereas\nomittingresamplingwouldresultincertainriskassessmenterrors. Therefore,theslightcomputational\nburdenintroducedbyresamplingisworthwhiletoensureexactoverloadingriskassessment.\nTableR6: RiskAssessmentResultsunderAllCTGCs(λ=150,r =0.2)\nMNV-VVGPwith MNV-VVGP\nMCS resampling withoutresampling PPGP\np ×100% MAPE p ×100% MAPE p ×100% MAPE p ×100% MAPE\nF F F F\nBranchname:L1\n2.229 - 2.229 0.289 2.136 0.310 2.557 2.901\nP =180MW\nLim\nBranchname:L3\n2.557 - 2.557 0.280 2.471 0.314 2.557 0.748\nP =330MW\nLim\nBranchname:L4\n0.25 - 0.25 1.136 0.2 1.200 0 5.276\nP =200MW\nLim\nTrainingtime - 3.34h 3.34h 4.75h\nAssessmenttime 10.78h 0.340h ≈0h(0.44s) 0.0168h\nChangesinthemanuscript:\nIntheSection“CaseStudyontheLarge-ScaleISONEPowerSystem,”descriptionsregardingtheclarification\noftheneedforresamplinghavebeenadded. Therevisedmanuscriptisasfollows:\n\"··· Therefore, the slight computational burden introduced by resampling is worthwhile to ensure exact\noverloadingriskassessment. ···\"\nTheReviewer’sComment#12\nInpage17,line278,30(insteadof40)contingenciesareselectedtodueconvergeissues. Please\nprovideabriefdescriptionofthese30contingencies,ormentionhowthesedifferentiatefromthe\noriginal40(e.g.,thesenolongerconsiderthecontingenciesrelatedtothenorthareaofthesystem).\nResponse:\nThanksforyoursuggestions! Tovalidatetheproposedmethodunderunseengenerationdispatches,only\n30contingenciesareselectedfromthe40historicalcontingencies. Thisisbecausetheunseengeneration\ndispatchescorrespondtooperatingconditionswithheavyloads,underwhichtheremaining10contingencies\nwould cause power flow divergence. In other words, these unselected 10 contingencies would lead to\nsystemcollapse,whichwouldcertainlyrequirecorrectiveactionstoavoidsuchfailures. Bycontrast,\ntheselected30contingenciesallowthepowerflowtoconvergebutmaycauselineoverloadingrisks,\nwhichneedtobequantifiedusingtheproposedmethod.\nIntheSection“PredictingOverloadingRiskwithModelUpdating,”abriefdescriptionofthese30contingen-\ncieshasbeenadded. Therevisedmanuscriptisasfollows:\n\"··· Ineachdispatch,30CTGCsoutoftheaforementioned40CTGCsareconsidered,leadingtoanMCS\nsamplesizeof1.5×104.Thisisbecausetheunselected10contingencieswouldleadtosystemcollapse,which\nwouldcertainlyrequirecorrectiveactionstoavoidsuchfailures. Bycontrast,theselected30contingencies\nallowthepowerflowtoconvergebutmaycauselineoverloadingrisks,whichneedtobequantifiedusingthe\nproposedmethod. ···\"\nTheReviewer’sComment#13\nInpage21,lines320,threedispatchscenariosarepresented. Pleaseprovideabriefdescriptionofthe\ncharacteristicsofsuchdispatchscenarios. Whyaretheychosen? Whatoperatingconditionsdothey\ndescribe? (e.g.,significantsolarPVparticipation).\nResponse:\nThesethreedispatchscenarioscorrespondtosystemstatusunderheavyloadsatthreedifferenttimeperiods,\nwhichleadtoahighlineoverloadingrisk. Theyareselectedtotesttheaccuracyandcomputationalefficiency\noftheproposedmethodinquantifyingthisriskwhenthesystemoperatingconditionschangesignificantly,in\ncontrasttotheprevioussectionswheretheloadswererelativelylessheavy.\nChangesinthemanuscript:\nIntheSection“PredictingOverloadingRiskwithModelUpdating,”abriefdescriptionofthesethreedispatch\nscenarioshasbeenadded. Therevisedmanuscriptisasfollows:\n\"··· Thesethreedispatchscenarioscorrespondtosystemstatusunderheavyloadsatthreedifferenttime\nperiods,whichleadtoahighlineoverloadingrisk. Theyareselectedtotesttheaccuracyandcomputational\nefficiency of the proposed method in quantifying this risk when the system operating conditions change\nsignificantly,incontrasttotheprevioussectionswheretheloadswererelativelylessheavy. ···\"\nTheReviewer’sComment#14\nIwouldsuggestaddingafutureworksection,asthismanuscriptfocusesexclusivelyonpredicting\nactivelineflows. Forexample,futureworkonassessingreactive/apparentlineflowsorbusvoltages\nshouldbementioned.\nResponse:\nThankyouforyoursuggestions!Wehaveaddedthefutureworksection,whichincludesthefollowingaspects:\nInthefuture,theproposedmethodwillbeadvancedalongthefollowingdirections:\n• ValidationandExtension: Theproposedmethodwillbevalidatedusingreal-timetime-seriesdata\n22\nfromtheISONewEnglandpowersystemandextendedtoquantifytheriskofreactiveandapparent\nlineflows,aswellasbusvoltages,inadditiontotheactivelineflowaddressedinthispaper.\n• ModelInterpretabilityandIntegration: TheinterpretabilityoftheDKS-VVGPmodelwillalsobe\nfurtherexplored,includingdeterminingthesensitivityofeachuncertainsourcetooverloadingrisk\nandunderstandinghowthesesourcesinfluencetheriskofoverloading. Additionally,embeddingthe\nassessedoverloadingriskintotheenergymanagementframeworkiscrucial. Theseadvancementswill\nhelpoperatorsoptimizetheplanningandallocationofrenewableenergy.\n• UncertaintyModeling: Itisalsointerestingtoinvestigatetheuncertaintymodelingamongcontin-\ngencies,renewablegenerations,andloads. Thisapproachcouldavoidconsideringthemostextreme\ncontingencies(whichwouldrequiresignificantcoststomitigatesystemrisk)andinsteadadoptmore\neconomicalmethodstoreducesystemrisk. OnepossibleapproachisutilizingCopulatheory. Specifi-\ncally,foreachline,theoutageprobabilitycanbeestimatedfromhistoricaldata,andthelineoutage\ncanbemodeledusingaBernoullidistribution,whichallowssimulatingrareeventswhenmultipleline\noutagesoccursimultaneously. Copulatheorycanthenbeusedtofurtherinferthecorrelationbetween\nthelineoutagedistributionandthedistributionsofrenewablegenerationandloads. OncetheCopula\nstructureisidentified,datacanbegeneratedbasedonthisCopula.\n• DataRequirementforUnseenContingencies: Whenunseencontingenciesarise,itisdifficultto\ndeterminehowmanynewACsimulations(samples)aresufficienttoupdatetheproposedmodel. This\nisbecausetherequiredamountofdatadependsonseveralfactors,suchasthesimilaritybetweenthe\nunseencontingenciesandtheexistingones,aswellasthesamplingstrategy. Therefore,aninteresting\ndirectionforfutureworkistoinvestigatehowtotheoreticallyquantifytheamountofdataneededtoupdatetheproposedmodel.\nChangesinthemanuscript:\nIntheDiscussionSection,thefutureworkhasbeenadded. Themanuscriptisrevisedasfollows:\n\"··· Inthefuture,theproposedmethodwillbeadvancedalongthefollowingdirections:\n• Theproposedmethodwillbevalidatedusingreal-timetime-seriesdatafromtheISONewEngland\npower system and extended to quantify the risk of reactive and apparent line flows, as well as bus\nvoltages,inadditiontotheactivelineflowaddressedinthispaper.\n• TheinterpretabilityoftheDKS-VVGPmodelwillalsobefurtherexplored, includingdetermining\nthe sensitivity of each uncertain source to overloading risk and understanding how these sources\ninfluencetheriskofoverloading. Additionally,embeddingtheassessedoverloadingriskintotheenergy\nmanagementframeworkiscrucial. Theseadvancementswillhelpoperatorsoptimizetheplanningand\nallocationofrenewableenergy.\n• Itisalsointerestingtoinvestigatetheuncertaintymodelingamongcontingencies,renewablegenerations,\nandloads.Thisapproachcouldavoidconsideringthemostextremecontingencies(whichwouldrequire\nsignificant costs to mitigate system risk) and instead adopt more economical methods to reduce\nsystemrisk. OnepossibleapproachisutilizingCopulatheory. SSpecifically,foreachline,theoutage\nprobabilitycanbeestimatedfromhistoricaldata,andthelineoutagecanbemodeledusingaBernoulli\ndistribution, whichallowssimulatingrareeventswhenmultiplelineoutagesoccursimultaneously.\nCopulatheorycanthenbeusedtofurtherinferthecorrelationbetweenthelineoutagedistributionand\nthedistributionsofrenewablegenerationandloads. OncetheCopulastructureisidentified,datacanbe\ngeneratedbasedonthisCopula.\n23\n• Whenunseencontingenciesarise,itisdifficulttodeterminehowmanynewACsimulations(samples)\naresufficienttoupdatetheproposedmodel. Thisisbecausetherequiredamountofdatadependson\nseveralfactors,suchasthesimilaritybetweentheunseencontingenciesandtheexistingones,aswell\nasthesamplingstrategy. Therefore,aninterestingdirectionforfutureworkistoinvestigatehowto\ntheoreticallyquantifytheamountofdataneededtoupdatetheproposedmodel.\n···\"\nTheReviewer’sComment#15\nTherearemissingacronymdefinitions,suchasLowRankApproximation(LRA)aroundline64page\n3.\nResponse:\nWehavecheckedandaddedthedefinitionsofallacronyms.\nChangesinthemanuscript:\nIntheIntroductionSection,thedefinitionsofallacronymshavebeenadded. Themanuscriptisrevisedas\nfollows:\n\"··· Sinceitwasfirstproposed,manyeffortshavebeendedicatedtodevelopingdifferentsurrogatemodelsfor\nPPF,suchaslowrankapproximation(LRA)[15,16],polynomialchaosexpansion(PCE)[17–19],Gaussian\nprocess emulator (GPE) [18–21], and deep neural network (DNN) [24–27]. LRA and PCE were early\nsurrogatemodelsusedinPPFwhentheinitialresearchfocusedonaddressingthecurseofdimensionality\n[15–17]orcorrelationsamonginputs[18,19]. ···\"\nThisstudypresentsaneuralmean-variancevector-valuedGaussianprocess(NMV-VVGP)-basedsurrogate\nmodelthatintegratesgenerationdispatch,contingencies,anduncertaintymodeling. Thekeyadvantageofthe\nproposedapproachliesinitscomputationalefficiencycomparedtothewidelyusedMonteCarloSimulation\n(MCS)methods. Whilethemodelrepresentsanadvancementoverthecurrentstateoftheartbyincorporating\nAI/MLtechniques,severalsignificantconcernswereraisedbythereviewer:\nRemarks to the Author:\nTheReviewer’sComment#1\nUncertaintyModeling–ParticularlyforSolarGeneration\nThetreatmentofuncertainty,especiallywithrespecttosolargeneration,lacksclarity. Inthecase\nstudy,solargenerationreplacescertaintraditionalgeneratorsintheNE-ISOsystem,buttheorigin\nand nature of the input data remain unexplained. Is the data region specific? The use of varying\ncorrelationcoefficientsforrisnoted,butitisunclearhowthesecorrelationsareintegratedintothe\nmodelingprocess. Renewablegenerationoftenexhibitsstronglocalbutweakglobalcorrelations;a\ndetailedexplanationofhowsuchspatialrelationshipsarecapturedinthemodelanddatageneration\nis necessary. While NE-ISO may not currently experience high levels of renewable penetration,\nleveragingreal-worldNE-ISOdatawouldenhancethestudy’scredibility.\nResponse:\nToansweryourquestion,explanationsarelistedbelow:\n(1)Inthecasestudy,solargenerationreplacescertaintraditionalgeneratorsintheNE-ISOsystem,but\ntheoriginandnatureoftheinputdataremainunexplained. Isthedataregionspecific?\nIntheNE-ISOsystem,PVgeneratorsareinstalledonBuseswithactivepowerinjectionslargerthan20MW.\nTheseareactualPVgeneratorsinreality. Therefore,thePVgenerationdataisnotregion-specific. However,\nitdoesn’taffectthevalidationoftheproposedmethod.\n(2)Theuseofvaryingcorrelationcoefficientsforrisnoted,butitisunclearhowthesecorrelationsare\nintegratedintothemodelingprocess. Renewablegenerationoftenexhibitsstronglocalbutweakglobal\ncorrelations;adetailedexplanationofhowsuchspatialrelationshipsarecapturedinthemodeland\ndatagenerationisnecessary.\nThespatialcorrelationmodelingforuncertainsources,includingloadsandPVgeneration,isachieved\nbyusingCopula[26]inthispaper. Specifically,Copulasprovideaframeworkformodelingthedependency\nstructurebetweenrandomvariables,decoupledfromtheirmarginaldistributions. ForaM-dimensionaljoint\ncumulativedistributionfunction(CDF)Λ ,itcanbedecomposedintoitsmarginalCDFsandaCopulathat\nΨ\nencapsulatestheirdependencyasfollows[28]:\nΛ =Copula(Λ (ψ ),Λ (ψ ),...,Λ (ψ )).\nΨ Ψ1 1 Ψ2 2 ΨM M\nHere,Λ (ψ )representsthemarginalCDFforthei-thvariableψ(e.g.,PVgenerationorloadpower),and\nΨi i\nCopula(·)describesthedependencyamongthesevariables. Whenthemarginaldistributionsandtheir\ncorrelationsforloadsandPVgenerationsareunknown,theycanbeestimatedfromcollecteddataby\nusingCopulatheory,facilitatingthedatagenerationbasedonidentifiedCopulastructures,suchas\nInthispaper,assumingknownmarginaldistributionsandcorrelationsforloadsandPVgeneration,we\nemploytheGaussianCopulatomodelthedependenciesbyspecifyingthemarginaldistributionsand\ncorrelations. ThesamplingprocessfromaGaussianCopulastructureforloadpowerandPVgenerationis\ndescribedasfollows:\n1. Definer ⊆Rasthecorrelationmatrixforthevariablesψ,includingloadpowerandPVgeneration.\n2. ComputetheCholeskydecomposition,L,ofRsothatR=L⊤L.\n3. GenerateW∼N(0,I ),whereN(·,·)denotesthenormaldistributionandI isanidentitymatrix.\nM M\n4. SetE=L⊤W.\n5. ObtainH=(Φ(E ),··· ,Φ(E )),whereΦisthestandardnormalCDF.\n6. Returnsamplesforthei-thdimensionofψ: ψ =Λ−1(H ).\ni Ψi i\n(3) While NE-ISO may not currently experience high levels of renewable penetration,\nleveragingreal-worldNE-ISOdatawouldenhancethestudy’scredibility.\nInthepaper,wehavealreadyusedthehistoricalactualISONewEnglandsystemanddatafortestingand\nvalidations. Thegenerationdispatchdataarealsofromhistoricaldata. Fortheuncertaintiesofloadand\nrenewablegenerationdata,wehaveusedthehistoricaldatatoinferrepresentativedistributionsforobtaining\nuncertaintiesassociatedthem. Inthefuturework,theproposedmethodwillbevalidatedonthereal-world\nNE-ISOoperationaldata. Pleasenotethatoperationaldataaredifferentfromthehistoricaldataweusedin\nthiswork,astheyaremoreproprietaryandhardtoget.\nReferences\n[28]R.B.Nelsen,AnIntroductiontoCopulas. Berlin,Germany: Springer,2007.\nChangesinthemanuscript:\nIn the Section \"Proposed Overloading Risk Assessment Framework\", the descriptions of the correlation\nmodelinghavebeenadded. Themanuscriptisrevisedasfollows:\n\"··· When the marginal distributions and their correlations for loads and PV generations are unknown,\ntheycanbeestimatedfromcollecteddatabyusingCopulatheory,facilitatingthedatagenerationbasedon\nidentifiedCopulastructures,suchasGaussianCopulas,tCopulas,orVineCopulas. ···\"\nChangesinthemanuscript:\nIntheDiscussionSection,thefutureworkhasbeenadded. Themanuscriptisrevisedasfollows:\n\"··· Inthefuture,theproposedmethodwillbevalidatedusingreal-timetime-seriesdatafromtheISONew\nEnglandpowersystem ···\"\nTailDistribution-AwareResampling\nTheproposedtail-awareresamplingmechanismappearsconceptuallysimplebutdependsheavilyon\naccuratetaildistributionmodeling. However,itisunclearhowthedistributiontailsarerepresented,\nespeciallygiventherarityofsuchextremedatapoints. Thisissuetiesbacktothebroaderchallengeof\nuncertaintymodeling. Moreover,sincebothrenewablegenerationandloadareinfluencedbyweather,\ntheyarenotindependent. Thetypesofconditionsthatleadtorareeventsandhowthosescenariosare\ngeneratedorsimulatedshouldbediscussedindetail.\nResponse:\nProbability\ndensity\nTail\nRisky boundary\nBranch power\nFigureR5: Exampleofbranchpowerdistribution\nToansweryourquestion,explanationsarelistedbelow:\n(1)However,itisunclearhowthedistributiontailsarerepresented,especiallygiventherarityofsuch\nextremedatapoints. Thisissuetiesbacktothebroaderchallengeofuncertaintymodeling.\nInthispaper,thetaildistributionrepresentstheprobabilitydensitydistributionofbranchpowerexceeding\nthepowerlimit,asshowninFig. R5,whichcorrespondstotheriskprobabilitybeyondtheriskyboundary.\nRegardlessofwhetherthesearerareevents,thispaperdefinesthetaildistributioninthismanner. Itisworth\nmentioningthatnouncertaintymodelingisappliedtothecontingencies(includingrareeventssuchasthe\nsimultaneousoutageofupto8lines)inthispaper;instead,eachcontingencyistreatedasadeterministic\neventthatisscannedandquantifiedindividually. Therefore,theinclusionofcontingenciesinthiscontext\ndoesnotintroducenewchallengestotheuncertaintymodeling.\n(2) Moreover, since both renewable generation and load are influenced by weather, they are not\nindependent. Thetypesofconditionsthatleadtorareeventsandhowthosescenariosaregeneratedor\nsimulatedshouldbediscussedindetail.\nAsmentionedinthepreviousexplanation,inthesettingofthispaper,nouncertaintymodelingisapplied\ntothecontingencies(includingrareeventssuchasthesimultaneousoutageofupto8lines);instead,each\ncontingencyistreatedasadeterministiceventthatisscannedandquantifiedindividually. Therefore,the\ninclusionofcontingenciesinthiscontextdoesnotintroducenewchallengestotheuncertaintymodeling.\nInotherwords,itisassumedthatthecontingenciesareindependentoftherenewablegenerationsand\nloadsandareanalyzedasextremecases.\nandloads. Thisapproachcouldavoidconsideringthemostextremecontingencies(whichwouldrequire\nsignificantcoststomitigatesystemrisk)andinsteadadoptmoreeconomicalmethodstoreducesystemrisk.\nOnepossibleapproachisutilizingCopulatheory. Specifically,foreachline,theoutageprobabilitycanbe\nestimatedfromhistoricaldata,andthelineoutagecanbemodeledusingaBernoullidistribution,which\nallowssimulatingrareeventswhenmultiplelineoutagesoccursimultaneously. Copulatheorycanthen\nbeusedtofurtherinferthecorrelationbetweenthelineoutagedistributionandthedistributionsof\nrenewablegenerationandloads. OncetheCopulastructureisidentified,datacanbegeneratedbased\nonthisCopula.\nChangesinthemanuscript:\nIntheSection\"ProposedOverloadingRiskAssessmentFramework\",theclarificationofuncertaintymodeling\nofcontingencieshasbeenadded. Themanuscriptisrevisedasfollows:\n\"··· Itisworthmentioningthatnouncertaintymodelingisappliedtothecontingencies(includingrareevents\nsuch as the simultaneous outage of up to 8 lines) in this paper; instead, each contingency is treated as a\ndeterministiceventthatisscannedandquantifiedindividually. Therefore,theinclusionofcontingenciesin\nthiscontextdoesnotintroducenewchallengestotheuncertaintymodeling. ···\"\nChangesinthemanuscript:\nIntheDiscussionSection,thefutureworkhasbeenadded. Themanuscriptisrevisedasfollows:\n\"··· Inthefuture,theproposedmethodwillbevalidatedusingreal-timetime-seriesdatafromtheISONew\nEnglandpowersystem ···\"\nIntheDiscussionSection,thefutureworkofuncertaintymodelinghasbeenadded. Themanuscriptisrevised\nasfollows:\n\"··· Itisalsointerestingtoinvestigatetheuncertaintymodelingamongcontingencies,renewablegenerations,\nandloads. Thisapproachcouldavoidconsideringthemostextremecontingencies(whichwouldrequire\nsignificantcoststomitigatesystemrisk)andinsteadadoptmoreeconomicalmethodstoreducesystemrisk.\nOnepossibleapproachisutilizingCopulatheory. Specifically,foreachline,theoutageprobabilitycanbe\nestimated fromhistorical data, andthe line outagecan bemodeled using aBernoulli distribution, which\nallowssimulatingrareeventswhenmultiplelineoutagesoccursimultaneously. Copulatheorycanthenbe\nusedtofurtherinferthecorrelationbetweenthelineoutagedistributionandthedistributionsofrenewable\ngenerationandloads. OncetheCopulastructureisidentified,datacanbegeneratedbasedonthisCopula.\n···\"\nPowerFlowCalculationandSurrogateModelAccuracy\nAsshowninFigure3,thesurrogatemodelisusedtoperformpowerflowanalysisandsubsequently\nestimateprobabilisticpowerflowviolationmetrics. Whileresultsforcertainbusesareshownand\nappearaccurate,itwouldbemorecompellingtocomputeandreportmeanabsolutepercentageerrors\n(MAPEs)forallbuses,therebydemonstratingthemodel’saccuracyacrosstheentiresystem.\nResponse:\nThanks for your suggestions! However, the proposed method has been tested on 10 lines that are\nlocatedindifferentpositionsandhavedifferentlinecapacities,whichissufficienttodemonstrateits\neffectiveness. Fromapracticalperspective,itisnotnecessarytoperformriskassessmentforalllinesand\nbusesinthesystem. Thereasonsareasfollows:\n• BasedontheexperienceofsystemoperatorsintheISONewEnglandpowersystem,onlytheselected\n10linesarecriticalmonitoringlinesthatarepronetooverloading. Ourcomputationalresultsfurther\nconfirmthattheremaininglinesbeyondthese10arenotonlyfreefromoverloadingbutalsofarfrom\ntheoverloadingmargin.\n• Byfocusingontheoperator-identifiedcriticalmonitoringlines,wecanclearlyanalyzehowoverloading\noccursandwhatcontrolmeasurescanbetaken.\nTheReviewer’sComment#4\nAssumptionofBetaDistributionforUncertaintySources\nThecasestudyappearstorelyonBetadistributions,presumablytomodelsolargenerationuncertainty.\nHowever,otheruncertaintysourcesmayfollowdifferentdistributions. Line285mentionsthemodel’s\n“strongfeatureextractioncapability,”yettheunderlyingmodelisbasedonGaussianprocesses,as\ndetailedintheMethodssection. ItisunclearhowtheBetadistributionassumptioninfluencesthe\noutcomesandwhetherthemodelissensitivetothechoiceofdistribution. Additionalexperiments\nusingalternativedistributionsshouldbeconductedtoevaluaterobustnessandgeneralizability.\nResponse\nThanksforyoursuggestions! Toaddressyourquestion,wewillexplainwhyourproposedGaussianprocess-\nbasedsurrogatemodelcanbeappliedtodifferentinputdistributionsandwhyithasstrongfeatureextraction\ncapabilities.\nTheproposedsurrogatemodelisdevelopedbasedonGaussianprocesses.\n1. However,thisdoesnotmeanthattheproposedGaussianprocess-basedsurrogatemodelisunsuitablefor\ndatasampledfromnon-Gaussiandistributions. Accordingtothedefinitionin[34]: AGaussianprocess\nisacollectionofrandomvariables,anyfinitenumberofwhichhaveajointGaussiandistribution. A\nbasicassumptionbehindthisdefinitionisthatafinitenumberofrealizationscanbecoveredbythe\nGaussianprocess. Consequently,theproposedGaussianprocess-basedsurrogatemodelisnotlimited\ntodatasampledfromaGaussiandistribution.\n2. Moreover,itpossessesstrongfeatureextractioncapabilitybecauseitusesaneuralnetworkasakernel\nfunctiontoextractinformationfromtheinputs,enablingittocapturenonlinearfeatureseffectively.\nsourcesfollowGaussianandBetadistributionsrespectively,asshowninTablesR7andR8. Ascanbeseen,\ntheproposedmethodachievesthesameoverloadingriskassessmentresultsasthebenchmarkMonteCarlo\nsampling(MCS)undertheseinputdistributions,demonstratingitsrobustnessandgeneralizabilityacross\ndifferentinputdistributions.\nTableR7: RiskAssessmentResultsunderAllCTGCsandGaussianDistribution((λ=150,r =0.2)\nMNV-VVGPwith MNV-VVGP\nMCS resampling withoutresampling PPGP\np ×100% MAPE p ×100% MAPE p ×100% MAPE p ×100% MAPE\nF F F F\nBranchname:L1\n2.229 - 2.229 0.289 2.136 0.310 2.557 2.901\nP =180MW\nLim\nBranchname:L3\n2.557 - 2.557 0.280 2.471 0.314 2.557 0.748\nP =330MW\nLim\nBranchname:L4\n0.25 - 0.25 1.136 0.2 1.200 0 5.276\nP =200MW\nLim\nTrainingtime - 3.34h 3.34h 4.75h\nAssessmenttime 10.78h 0.340h ≈0h(0.44s) 0.0168h\nTableR8: RiskAssessmentResultsunderAllCTGCsandBetaDistribution(λ=150,r =0.2)\nMCS MNV-VVGPwithresampling\np ×100% MAPE p ×100% MAPE\nF F\nBranchname:L1\n2.021 - 2.021 0.181\nP =180MW\nLim\nBranchname:L3\n2.171 - 2.171 0.163\nP =330MW\nLim\nResampledsamples\n-(10.78h) 782(0.423h)\n(Assessmenttime)\n[34]C.E.Rasmussen,etal.,GaussianProcessesforMachineLearning. Cambridge,MA,USA:MITPress,\n2006.\nChangesinthemanuscript:\nInSection“Methods”aclarificationregardingtheimpactsofinputdistributiontypeontheproposedmethod\n30\nhasbeenadded. Therevisedmanuscriptisasfollows:\n\"··· NotethattheproposedsurrogatemodelisdevelopedbasedonGaussianprocesses. However,thisdoes\nnotmeanthattheproposedGaussianprocess-basedsurrogatemodelisunsuitablefordatasampledfrom\nnon-Gaussiandistributions. Accordingtothedefinitionin[34]: AGaussianprocessisacollectionofrandom\nvariables,anyfinitenumberofwhichhaveajointGaussiandistribution. Abasicassumptionbehindthis\ndefinitionisthatafinitenumberofrealizationscanbecoveredbytheGaussianprocess. Consequently,the\nproposedGaussianprocess-basedsurrogatemodelisnotlimitedtodatasampledfromaGaussiandistribution.\n···\"\nTheReviewer’sComment#5\nInconsistenciesinFigure7andTable2\nInFigure7,theactivepowervaluesforbranchesL1andL3areshownasnegative,whileTable2\nlistspositivePlimvaluesforthesamebranches. Clarificationisneededonthedefinitionsandsign\nconventionsofthesevariablestoensureconsistencyandinterpretability.\nResponse\nIn Section “Risk-Aware adaptive tail–resampling Mechanism,” F(k) = {y(k) : |y(k)| ≥ P(k)} has been\nLim\ndefined as the overloading risk for the k-th branch, which resides in the tail of the overall distribution.\nHere,y(k)denotesthepowerflowonthek-thbranch,andP(k) isthetransmissionlimitofthek-thbranch.\nLim\nTherefore, thesignofthebranchpowerdoesnotaffectthecalculationoftheoverloadingrisk. Toavoid\nconfusioninthesubsequentsectionsregardingcaseswherebranchpowermaybenegative,wereiteratein\nSection“CaseStudyontheLarge-ScaleISONEPowerSystem”themethodusedinthispapertocalculate\nlineoverloadingrisk.\nChangesinthemanuscript:\nInSection“CaseStudyontheLarge-ScaleISONEPowerSystem,”aclarificationregardingthecalculation\nofriskwhenbranchpowerisnegativehasbeenadded. Therevisedmanuscriptisasfollows:\n\"··· Notethatwhenthebranchpowerisnegative,itsabsolutevalueisusedin(5)tocalculatetheoverloading\nrisk. ···\"\nTheReviewer’sComment#6\nMissingSectionReferences\nThemanuscriptreferstoSectionsIV-A,IV-B,etc.,butthesesectionsdonotexistinthecurrentdraft.\nThetextshouldberevisedtoensureallsectionreferencesareaccurateandproperlyformatted.\nResponse\nThankyouforpointingouttheincorrectsectionreferences. ThereferencestoSectionsIV-AandIV-Bhave\nbeencorrectedto\"HandlingDifferentContingencieswithoutTransferLearning\"and\"Performanceunder\nDifferentDistributionCorrelations\".\nTerminologyConsistency–CTGCvs. CTCG\nBoth“CTGC”and“CTCG”appearinthemanuscript,butonlyonetermispresumablycorrect. This\ninconsistencyshouldbeaddressedtomaintainclarityandavoidconfusion.\nResponse\nThankyouforpointingouttheincorrectabbreviation. Wehavecorrectedallmisusedoccurrencesof“CTCG”\nto“CTGC,”ascontingencyisabbreviatedasCTGC.\nTheReviewer’sComment#8\nLackofContingencyDescription\nAlthoughthemanuscriptmentionsdifferentcontingencies,itdoesnotspecifytheirnatureorclassifi-\ncation(e.g.,lineoutages,generatorfailures). Itremainsunclearwhetherthesecontingencieshave\namaterialimpactontheresultsoriftheyareillustrative. Aclearerexplanationofthecontingency\nselectionandtheirimplicationsfortheanalysisiswarranted.\nResponse\nThanksforyoursuggestions! These40criticalcontingenciesrepresentsimultaneousoutagesofmultiple\nimportanttransmissionlines(upto8lines). Inthispaper,the40contingenciesarethoserecommended\nbythesystemoperatorsoftheISONewEnglandpowersystemforanalysis,astheyhavehistorically\ncaused line overloading and are therefore considered more severe critical contingencies compared\nwithothers. Ifthesecriticalcontingenciesdonotleadtolineoverloadingrisk,thenothercontingenciesare\nunlikelytocausesuchrisks. Forexample,2contingencies,namelyCTGC1andCTGC6,outofthese40\ncontingenciesaredefinedasfollows:\n• CTGC1correspondstothesimultaneousoutagesofthefollowinglines: (13287→13285),(13285→\n13284),(17707→17709),(17710→17708),(17723→17722),and(17721→17720).\n• CTGC6 corresponds to the outages of the lines: (3477 → 3476), (3472 → 3477), and (3486 →\n3480).\nChangesinthemanuscript:\nIntheSection\"CaseStudyontheLarge-ScaleISONEPowerSystem\",thedescriptionsofcontingencies\nhavebeenadded. Themanuscriptisrevisedasfollows:\n\"··· These40criticalCTGCsrepresentsimultaneousoutagesofmultipleimportanttransmissionlines(up\nto8lines). Inthispaper,the40contingenciesarethoserecommendedbythesystemoperatorsoftheISO\nNewEnglandpowersystemforanalysis,astheyhavehistoricallycausedlineoverloadingandaretherefore\nconsideredmoreseverecriticalcontingenciescomparedwithothers. Ifthesecriticalcontingenciesdonot\nleadtolineoverloadingrisk,thenothercontingenciesareunlikelytocausesuchrisks. ···\"\nInthesection“CaseStudyontheLarge-ScaleISONEPowerSystem,”atabledescribingCTGC1andCTGC6\nhasbeenadded. Therevisedmanuscriptisasfollows:\nContingencyName LineOutages(StartBus→EndBus)\n(13287→13285),(13285→13284),(17707→17709),(17710→\nCTGC1\n17708),(17723→17722),(17721→17720)\nCTGC6 (3477→3476),(3472→3477),(3486→3480)\nRemarks on code availability:\nTheReviewer’sComment#1\nItisunclearwhichcodeinthegithubpageisforthesubmittedwork.\nResponse\nSinceGitHubPagesdoesnotsupportuploadingverylargefiles,andourcodealongwiththedataexceeds\n18GB,wehaveprovidedaGoogleDrivelinktofacilitatereadersintestingthecodeassociatedwiththispaper:\n[REDACTED]\nThe manuscript develops a two–stage workflow for estimating the probability that an individual\ntransmission line/branch exceeds its emergency (thermal) rating under a fixed library of N–k con-\ntingencies:\n1. A vector–valued, deep–kernel, sparse Gaussian–process (GP) surrogate maps in-\njections x and a contingency tag τ to the branch–flow vector y˜.\n2. Anadaptive“tail–resampling”looprepeatedlyre–runsthefullACpower–flowonMonte–\nCarlo samples whose predicted flows lie closest to the thermal limit until the overload-\n(k)\nprobability estimate p converges.\nF\nOn a 21000–bus ISO–NE model with 40 contingencies the method achieves ≈ 22× speed – up\nover crude Monte – Carlo while matching its accuracy.\n• Scalability demonstration: ISO-scale, full AC flows, thousands of uncertain injections.\n• Vector surrogate: predicts all lines simultaneously.\n• Simple corrective loop: easytoembedinexistingreal–timecontingency-analysispipelines.\n• Clear engineering value: per–line probabilities speak the language of NERC compliance.\n1\n# Issue Why it matters How to fix\nRecast as adaptive\n“Importance tail–resampling; cite the\nsampling” Classical IS provides adaptive importance\nmis–labelling. The variance guarantees; the sampling EJS) and\n1\nloop never changes proposed method does instanton papers; discuss\nthe proposal nor not. differences (you use AC\napplies IS weights. PF vs DC PF in the\npapers).\nReaders may assume\nState explicitly that\nphysics–informed\nPhysics guidance physics enters solely\n2 sampling when it is\nis indirect only. through the surrogate’s\npurely\ntraining data.\nsurrogate–error–driven.\nUse existing samples to\nidentify top contributing\nPlanners and some\nLine–by–line focus lines and report\noperators need union\n3 ignores the respective overload\nprobabilities; connects to\nsystem-wide risk. probabilities; compare to\ninstanton literature.\nthe aforementioned\ninstanton approach\nNo ablations:\nReport results with (i)\nneural kernel and Needed to establish\n4 plain RBF GP; (ii)\nλ step not methodological novelty.\nvaried λ.\njustified.\nReproducibility.\nHyper-parameters, Release scripts and full\nNature Communications\ninducing-point settings in a public\n5 readers expect reusable\nplacement, and repository or as\ncode.\ncode availability supplementary material.\nare missing.\n1. Replace “Neural–Mean–Variance GP” with the standard term “deep–kernel sparse vector–\nvalued GP” to avoid confusion.\n2. State how the coregionalisation matrix A is initialised and regularised.\n3. Provide pseudo–code for the λ–resampling loop and specify the stopping tolerance ε.\n4. Clarify whether unseen contingency types (not only dispatch changes) can be accommodated\nwith 1000 new AC simulations.\n2\nThe study is technically solid and clearly useful for power–system practitioners. However,\ntwo aspects limit its broader appeal:\n• The statistical machinery (deep–kernel GP + active resampling) is standard outside power\ngrids; novelty lies mainly in scale and integration.\n• Results are tightly coupled to power–system operational rules; immediate transfer to other\nfields (aerospace, climate, finance) would require non–trivial adaptation.\nI therefore caution the editors that the paper’s impact is likely strongest within the energy–\nsystems community.\nAccept after major revision addressing points 1–5 above and adding a discussion contrasting\nthe proposed method with importance–sampling approaches and related (two papers mentioned\nabove).\n3"}, {"round": 2, "reviewer_comments": {"reviewer_1": "I command the authors for detailed response. I am happy with the changes made. Happy to recommend for publications\nasis.", "reviewer_2": "The authors have successfully addressed all my previous comments. I have no further observations.", "reviewer_3": "The revised manuscript addressed all of my comments and concerns."}, "author_response": ""}]

Decision letter