Extreme Grid Operation Scenario Generation Framework Considering Discrete Failures and Continuous Output Variations

Abstract

In recent years, extreme weather events have occurred more frequently. The resulting equipment failure, renewable energy extreme output, and other extreme operation scenarios affect the smooth operation of power grids. The occurrence probability of extreme operation scenarios is small, and the occurrence frequency in historical operation data is low, which affects the modeling accuracy for scenario generation. Meanwhile, extreme operation scenarios in the form of discrete temporal data lack corresponding modeling methods. Therefore, this paper proposes a definition and generation framework for extreme power grid operation scenarios triggered by extreme weather events. Extreme operation scenario expansion is realized based on the sequential Monte Carlo sampling method and the distribution shifting algorithm. To generate equipment failure scenarios in discrete temporal data form and extreme output scenarios in continuous temporal data form for renewable energy, a Gumbel-Softmax variational autoencoder and an extreme conditional generative adversarial network are respectively proposed. Numerical examples show that the proposed models can effectively overcome limitations related to insufficient historical extreme data and discrete extreme scenario training. Additionally, they can generate improved-quality equipment failure scenarios and renewable energy extreme output scenarios and provide scenario support for power grid planning and operation.

1. Introduction

The ongoing construction and development of new power systems has led to an increase in power grid architectural complexity and a significant increase in the proportion of renewable energy sources connected to the power grid [1]. Concurrently, extreme weather conditions, including severe cold, typhoons, and droughts, have become increasingly prevalent [2]. These conditions have led to substantial fluctuations in renewable energy output and have affected the balance between the supply and demand of power sources and loads. Furthermore, these conditions pose a significant challenge for stable power grid equipment operation and have resulted in substantial power grid losses. That means not only the continuous renewable energy output scenarios, but also discrete equipment failure scenarios, should be incorporated into power grid extreme operation scenarios under extreme weather conditions. Therefore, the accurate probability feature characterization of low-probability and high-impact extreme operation scenarios, as well as the further generation of such scenarios for power grids, is essential for the optimization of decision-making processes related to power grid planning and operation.

Presently, the most effective extreme scenario generation methods can be broadly categorized into two distinct approaches: post-processing and direct generation [3]. Post-processing methods entail extreme scenario set determination from historical scenarios based on various indices or methodologies. In [4], a rapid search technique was employed to identify density peaks, thereby yielding extreme and typical scenarios. This paper assumes that the wind power obeys a normal distribution, but the actual wind power may have complex non-Gaussian characteristics. In [5], surface irradiance was derived in regions significantly impacted by the plum rain phenomenon, employing meteorological data as a foundation. But the reanalysis meteorological data may be less extreme than the actual weather data. In [6], an extreme scenario screening method was utilized by defining the minimum and maximum wind power output values. Their approach also incorporates four uncertainty sets of positive and negative extreme climbs, situated between the aforementioned maximum and minimum power output values, while the continuous temporal features of the whole scenarios, like average power output and volatility, are not taken into consideration. In [7], the self-organizing map clustering method was employed to screen extreme scenarios with high and low net load levels, while the definition of extreme days is limited to peak and trough net load values, overlooking compound extreme conditions. As posited in [8], the “unloading cost” and “highest cost” methods are proposed as extreme operation scenario identification methods. According to [9], the hour with the highest overall load loss is to be considered as the extreme scenario and should be screened out accordingly. Nevertheless, renewable generation and demand patterns often exhibit significant year-to-year variability due to weather fluctuations. A single year of data may not capture these long-term variations in both [8,9], leading to suboptimal investment decisions in generation and storage capacity. While the aforementioned methods can generate extreme scenario sets that align with historical probability characteristics, the occurrence probability of extreme events is low. Consequently, the extreme scenario data obtained thusly is inadequate for meeting power grid planning and operational demands.

Direct-generation methods refer to directly modeling the probability of extreme scenarios and obtaining extreme operation scenario sets via sampling. A taxonomy of these methods can be established through two distinct classifications: those grounded in statistical analysis and those founded on deep learning [10]. A variety of statistical methodologies have been extensively employed, including extreme value theory and the generalized Pareto distribution, among many others [11]. In [12,13], the probability distribution of extreme wind speed events is fitted using distribution equations, and an extreme wind speed event prediction methodology is proposed based on hierarchical classification regression. But the relationship between extreme wind speed and wind power output has not been further studied. In [14], a generalized additive model was employed in conjunction with quantile regression for probability prediction, utilizing the generalized Pareto distribution to forecast extreme scenarios pertaining to wind and solar power based on net load. Even though the conditional GPD improves the static assumption, it still needs to preset the distribution form, and the shape parameter of the generalized Pareto distribution was fixed. In [15], a generalized linear model was employed for power outage prediction in distribution networks caused by extreme weather, such as ice and snowstorms, while their accuracy is still limited by the data range and the model assumptions. From the above references, the use of statistical methods in modeling is predicated on assumptions regarding the distribution parameters. Consequently, the models invariably exhibit a certain degree of deviation, rendering them incapable of accurately depicting the intricate, high-dimensional, and nonlinear characteristics of extreme scenarios. This limitation compromises their practicality.

Deep-learning-based methods entail the construction of generative networks through condition or label parameter specification and the use of substantial datasets to generate extreme operation scenarios under predetermined conditions. In [16], a methodology for the concatenation of random noise and conditions as input was demonstrated, employing conditional generative adversarial networks (CGANs) for transformer failure sample data generation under specified conditions. But the model still focuses on the scenario generation with continuous data rather than the discrete temporal scenario generation of the equipment failure. Although reference [17] realized the scenario generation of specified types by embedding interpretable features in the latent space, this approach may not lead to satisfactory results under extreme weather conditions. In [18], extreme value theory was combined with GANs to model the probability of the distribution’s extreme tail and generate samples of specified extreme degrees, while they cannot directly specify the specific type or feature of the extreme scenarios. Another study accomplished controllable generation by establishing a relationship between latent vectors in the feature and manifold of the generated scenarios [19]. But the insufficient extreme data may also constrain the performance of extreme scenario generation. Although the use of deep-learning methodologies in modeling has certain advantages in terms of fitting the probability characteristics of scenarios, like reducing the reliance on distribution assumptions, the scarcity of historical extreme scenarios poses a significant challenge in facilitating the effective learning of extreme scenario distribution characteristics via generative networks. Concurrently, there is a lack of research on equipment failure scenarios in the form of discrete temporal data.

In summary, there still exist the following research gaps in extreme power grid scenario generation: (1) At present, the research on scenario generation using deep models mainly focuses on the prediction of renewable energy power generation and load, and there are relatively few works on the generation of both continuous extreme output scenarios and discrete grid fault scenarios. The existing literature lacks a framework to unify the generation of equipment failure scenarios and extreme renewable energy output scenarios caused by extreme weather. (2) The existing models put less attention on the equipment failure scenario generation with discrete temporal data and the corresponding structure design. And the historical extreme scenarios of power grid failure may not be enough for the training of deep-learning methods, which is also an issue in extreme renewable energy output scenarios.

In light of the aforementioned issues, this paper proposes an extreme operation scenario generation framework for power grids. The proposed framework considers discrete failures and continuous output variations, as well as the generation of equipment failure and extreme output scenarios for renewable energy. The main contributions and innovations of this study can be summarized as follows:

• In light of the escalating equipment failures observed in power grids, compounded by the extreme renewable energy outputs resulting from frequent extreme weather events, a comprehensive definition and generation framework for extreme power grid operation scenarios is proposed. This framework encompasses both equipment failure and extreme output.
• To address the lack of samples for extreme power grid operation scenarios, the equipment failure rate is calculated based on extreme weather data. Subsequently, the sequential Monte Carlo sampling method is employed to expand the discrete equipment failure scenario samples. During the extreme output scenario training process for renewable energy, a distribution shifting algorithm is introduced to gradually shift the training data distribution towards the tail distribution of the extreme dataset, thereby assisting the model in learning the extreme output characteristics of renewable energy.
• A Gumbel-Softmax variational autoencoder (Gumbel-Softmax VAE) suitable for modeling discrete temporal data is proposed for the power grid extreme operation scenario generation problem. A discrete latent space structure is designed to better fit the change characteristics of discrete temporal data. The proposed extreme CGAN (ExCGAN) model is designed for continuous temporal data analysis. The model incorporates four extreme metrics as scenario labels, a strategy that is intended to enhance model interpretability.

2. Power Grid Extreme Operation Scenario Definition and Generation Framework

As renewable energy increasingly penetrates the market, most power and energy imbalance problems in new power systems are influenced by extreme weather, either directly or indirectly [3]. The occurrence of extreme weather events has been demonstrated to exert a substantial influence on wind and solar power output. Furthermore, these events have been shown to result in power grid equipment aging and damage, thereby significantly increasing renewable energy non-consumption risks, imbalance between supply and demand, and unstable power system operation. In essence, equipment failure and extreme output scenarios for renewable energy have the potential to induce supply and demand risks, as well as safety concerns, during power system operation. Therefore, when studying power system operation and planning optimization, it is insufficient to consider only equipment failure or extreme output scenarios for renewable energy. To ensure optimal support for power grid planning and operation, it is imperative to comprehensively assess these two extreme scenarios. Accordingly, this paper delineates power grid extreme operation scenarios as time series scenarios, encompassing both discrete temporal data forms for equipment failure and continuous temporal data forms for renewable energy extreme outputs, triggered by extreme weather phenomena such as typhoons and cold waves. Among them, the equipment failure scenarios contain only 1 and 0 discrete data points, corresponding to equipment normal operation and failure, respectively. Similarly, the extreme output scenarios for renewable energy contain continuous output per-unit values between 0 and 1. Both scenarios have time series characteristics.

The equipment failure scenarios are characterized by the presence of discrete temporal data, while the extreme output scenarios for renewable energy are typified by continuous temporal data. Given the variety of data forms, it is challenging to attain high-precision scenario generation through a solitary generation model. Consequently, the design of independent generation models must be informed by their distinct characteristics. For the equipment failure scenario generation, due to challenges related to data sparsity, time series, and failure probability and duration modeling, the sequential Monte Carlo sampling method was used in this study to obtain time series of discrete equipment state scenarios based on historical data, failure rates, and repair rates, and to expand the training samples. Subsequently, the Gumbel-Softmax VAE model is proposed, and the discrete latent space structure is designed by combining the reparameterization technique to extract failure features and fit the complex relationship between random noise and failure data. Regarding extreme output scenario generation for renewable energy, such as wind and solar power, which exhibit low frequency and tail distribution characteristics, direct screening samples may not adequately meet the training requirements. Accordingly, the distribution shifting algorithm is introduced to expand extreme samples, and the ExCGAN model is proposed to model the renewable energy extreme output uncertainty and enhance model interpretability. By using the aforementioned methodologies, equipment failure scenarios due to extreme weather conditions and extreme output scenarios for renewable energy can be generated with greater precision. This, in turn, provides data support for power system operation and dispatch planning. The power grid extreme operation scenario generation framework is illustrated in Figure 1.

3. Extreme Operation Scenario Generation Model for Power Grids

3.1. Equipment Failure Scenario Generation Model

A VAE is typically comprised of an encoder and a decoder [20]. The former maps historical data $x$ to encoding variable $\mathit{z}$, while the latter decodes $\mathit{z}$ to obtain the generated data $x\prime$ with the same dimension as $x$. The VAE loss function is expressed as follows:

$$L_{\mathrm{VAE}}=E_{q_{\varphi }(\mathit{z}|x)}\left[\log p_{\theta }(x|\mathit{z})\right]-D_{\mathrm{KL}}(q_{\varphi }(\mathit{z}|x)|| p(\mathit{z})),$$

(1)

where $p_{\theta }(x|\mathit{z})$ and $q_{\varphi }(\mathit{z}|x)$ represent the probability distributions corresponding to the decoding and encoding networks, respectively, and $D_{\mathrm{KL}}$ is the KL divergence.

In conventional VAEs, the latent space prior distribution is typically implemented as a Gaussian distribution. However, the equipment failure scenario is represented by discrete temporal data containing only 0 and 1, making it difficult for the network to effectively fit the actual failure probability distribution. Conversely, the continuous form of latent space modeling is incompatible with the failure scenario’s discrete form. The use of one-hot encoding technology is imperative for latent space discretization. However, the network may encounter challenges in differentiating discrete temporal data, thereby failing to meet the criteria for gradient backpropagation. To address the aforementioned issues, this paper proposes the Gumbel-Softmax VAE, which models the discrete categorical latent space distribution and employs the reparameterization trick to enable gradient backpropagation. We choose Gumbel-Softmax for three primary reasons: (1) It provides valid gradient estimates aligned with variational inference principles, unlike the Straight-Through Estimator (STE) and Vector Quantized VAE (VQ-VAE), which rely on approximations. (2) The temperature annealing schedule ensures stable optimization, addressing the convergence issues associated with the STE and VQ-VAE. (3) Gumbel-Softmax supports arbitrary categorical distributions, whereas the VQ-VAE is constrained by fixed codebook sizes and the STE faces challenges in multi-class discretization. The network structure is illustrated in Figure 2.

The Gumbel-Softmax VAE encodes input data $\mathit{x}$ into characteristic variable $\mathit{z}$, which follows a categorical distribution with category probabilities $\mathit{\pi }=\left[{\pi }_1,{\pi }_2,\dots ,{\pi }_k\right]$, as follows:

$$\mathit{z}=\text{one\_hot}\left(\underset{k\in [1,2,\dots ,K]}{\mathrm{argmax}}[g_k+\log \left({\pi }_k\right)]\right),$$

(2)

where $\mathit{z}$ is a one-hot vector determined by the argmax function, $g_k$ is sampled from the Gumbel distribution, and K is the total number of latent vector categories. Each dimension of $\mathit{z}$ corresponds to a latent vector category. In this study, K was set to 2, representing the normal operation and failure states of the equipment.

The one-hot encoding for latent space discretization makes discrete temporal data difficult to differentiate, prohibiting gradient backpropagation because of the argmax function. As the argmax function is not differentiable, the continuous and differentiable Softmax function was used to approximate argmax and generate a $k$-dimensional vector $\left[{\mathrm{z}}_1,z_2,\dots ,z_k\right]$, known as the Gumbel-Softmax distribution, as shown in (3) and (4):

$$z_i={\displaystyle \frac{\exp \left(\left(\log {\pi }_i+g_i\right)/\tau \right)}{{\displaystyle \sum _{j=1}^k\exp }\left(\left(\log {\pi }_j+g_j\right)/\tau \right)}},\forall i=1,2,\dots ,k,$$

(3)

$$g_i=-\log \left(-\log \left(u_i\right)\right),$$

(4)

where $\tau$ is the temperature coefficient of the Softmax function, which is greater than zero; $u_i~U(0,1)$.

The Gumbel-Softmax reparameterization technique is a method that separates the sampling randomness from the gradient calculation. It uses the Gumbel-Softmax distribution to sample and approximate the one-hot encoding result. This ensures that gradient backpropagation occurs during training. By employing formula derivation, $D_{\mathrm{KL}}$ can be simplified to negative entropy form and is expressed as follows:

$$D_{\mathrm{KL}}(q_{\varphi }(\mathit{z}|\mathit{x})\left|\right| p(\mathit{z}))=-\mathit{\pi }\log (\mathit{\pi }).$$

(5)

In the context of the equipment failure scenario, the network is unable to acquire specialized knowledge regarding the failure state features due to the limited proportion of failure states in the total number of samples. Consequently, a focal loss was introduced to effectively address the class imbalance problem, as demonstrated in (6):

$$L_{FL}=-\alpha {\left(1-\mathit{\pi }\right)}^{\gamma }\log \left(\mathit{\pi }\right),$$

(6)

where $\alpha$ is the weight for balancing the ratio of normal operation samples to failure samples, and $\gamma$, often called focusing parameter, is the degree to which $\alpha$ is reduced, which can reduce the weight of easily classified samples and increase that of difficult samples, making the network focus more on learning difficult samples and improving model performance.

The Gumbel-Softmax VAE loss function is expressed as follows:

$$L=L_{\mathrm{VAE}}+L_{\mathrm{FL}}.$$

(7)

3.2. Extreme Scenario Generation Model for Renewable Energy

CGANs have the capacity to generate a specified set of samples based on the conditions that are provided [21]. The network first concatenates noise $\xi$ with condition $\mathit{c}$ as the generator input. After mapping transformation through the network layers, the network obtains the generated sample $x\prime$. Subsequently, the discriminator integrates $x\prime$, $\mathit{x}$, and $\mathit{c}$ to assess the approximation degree of the generated sample distribution, denoted by $p(x\prime )$, and the actual sample distribution, represented by $p(\mathit{x})$. If $p(x\prime )$ and $p(\mathit{x})$ do not overlap, the JS divergence between them is a constant, and the CGAN parameters will no longer be updated. To circumvent the CGAN mode collapse, the Wasserstein distance with a gradient penalty function was employed to delineate the disparity between the two distributions. Consequently, the objective function is derived as follows:

$$\begin{array}{cc}\hfill \underset{G}{\min } \underset{D}{\max }V\left(D,G\right)=& E_{\mathit{x}~p(\mathit{x})}[D(\mathit{x}|\mathit{c})]-\hfill \\ & E_{x\prime ~p\left(x\prime \right)}\left[\begin{array}{l}D(x\prime |\mathit{c})\\ -\lambda E[{(‖\nabla D(~)‖-1)}^2]\end{array}\right]\hfill \end{array},$$

(8)

where $E(·)$ represents the expected value, $D(\mathit{x}|\mathit{c})$ and $D(x\prime |\mathit{c})$ are the discriminator output values for the real and generated samples, respectively, and $\lambda$ is the penalty coefficient. $D(~)$ is represented as $\beta \mathit{x}+(1-\beta )G(\mathit{z})$, where $\beta ~U(0,1)$ and $G(\mathit{z})$ is the generated sample.

In this study, in addition to static extreme metrics such as maximum power and minimum power, temporal extreme metrics including average power and power volatility were also selected as conditional variables to better guide the model in generating corresponding types of extreme scenarios. The extreme metrics are presented in

Table 1. Extreme metrics.

Metric	Calculation Formula
Maximum power	$P_{\max }=\max \left\{P_{1:T}\right\}$
Minimum power	$P_{\min }=\min \left\{P_{1:T}\right\}$
Average power	$P_{\mathrm{ave}}={\displaystyle \frac{1}{T}}{\displaystyle \sum _{t=1}^T{P}_t}$
Volatility	$P_{\mathrm{flu} }={\displaystyle \frac{1}{T-1}}{\displaystyle \sum _{t=2}^T\left|P_t-P_{t-1}\right|}$

.

In Table 1, $P_{1:T}$ represents the renewable energy output scenarios, $P_t$ represents the renewable energy power value at time t, and $T$ is the total time sequence length of the scenarios.

In this paper, the CGAN model is used as a foundation, with the extreme metrics functioning as conditional variables. The loss function $L_{\mathrm{EX}}$ of the extreme metric value is introduced to assess the discrepancy between the extreme metric values of the generated and training samples [22]. The introduction of $L_{\mathrm{EX}}$ allows the ExCGAN to focus specifically on optimizing the generation of extreme scenarios, whereas the CGAN is not designed with this goal in mind, which may lead to suboptimal performance in extreme cases. The ExCGAN model is employed to generate the specified type of extreme scenarios. $L_{\mathrm{EX}}$ is expressed as follows:

$$L_{\mathrm{EX}}=E_{\xi ,\mathit{e}}\left[{\displaystyle \frac{\left|\mathit{e}-EX(G(\xi |e))\right|}{\mathit{e}}}\right],$$

(9)

where $\mathit{e}$ is the training sample’s extreme metric value, $\xi$ is the random noise, and $EX(G(\xi |e))$ is the generated sample’s extreme metric value. During training, the model minimizes the distance between $\mathit{e}$ and $EX(G(\xi |e))$ according to $L_{\mathrm{EX}}$, reducing the possibility of generating samples without the corresponding extreme conditions.

To further address the challenge of inadequate data from the initial extreme output scenarios, the distribution shifting algorithm is introduced [18], and extreme samples selected from the generated samples are used to substitute the ordinary samples in the training samples. The specific process is delineated in Algorithm 1. Given a dataset x of N samples, the ExCGAN is train firstly on the original dataset. Then, the dataset is sorted in descending order based on extreme value metrics $\mathit{e}$, and $\epsilon N$ relatively extreme samples are selected as the first-part scenario set $x_{\epsilon ,1}$. Then, the trained ExCGAN model generates $(N-\epsilon N)/\epsilon$ samples and selects $N-\epsilon N$ of the most extreme samples among them as the second-part scenario set $x_{\epsilon ,2}$. Finally, these two sets of extreme scenario samples are combined to form new training scenario set $x_{\epsilon }$ for the next iteration. This ensures that there is always sufficient data to train the ExGAN in a stable manner, while the generated distribution also gets closer to the tail end of the extreme distribution. Additionally, retaining the generator parameters obtained after each cycle as the initial values for the generator model in the next iteration can help speed up the whole training process. The shift factor and training round settings are based on reference [18].

Algorithm 1. Extreme output scenario generation for renewable energy

Input: Original scenario set x, number of scenarios N, extreme metric $e$, shift factor $\epsilon$, training rounds k

Sort all scenarios in the original scenario set $x$ in descending order according to the extreme metric $e$

Initialize the training set $x_{\epsilon }\leftarrow x$

for $i=1,\dots ,k-1,k$ do

Train the ExCGAN generator and discriminator on $x_{\epsilon }$

Select $\epsilon N$ samples that are relatively extreme from $x_{\epsilon }$ as the first-part scenario set $x_{\epsilon ,1}$

Generate ${\displaystyle \frac{N-\epsilon N}{\epsilon }}$ samples using the ExCGAN

Select $N-\epsilon N$ samples as the second-part scenario set $x_{\epsilon ,2}$ Incorporate $x_{\epsilon ,1}$ and $x_{\epsilon ,2}$ together to form the new training scenario set $x_{\epsilon }$

end

Output: Extreme scenario set $x_{\epsilon }$

The ExCGAN structure is illustrated in Figure 3, where the solid lines represent forward computation and the dashed lines denote gradient backpropagation.

3.3. Extreme Scenario Generation Quality Evaluation Indices

3.3.1. Evaluation Indices for Equipment Failure Scenarios

In this study, three evaluation indices, namely, the system equipment failure probability, equipment failure frequency, and equipment failure duration, were selected for comparing and analyzing the equipment failure scenario generated samples against the real samples [23]. This is expressed as follows:

$$P_f={\displaystyle \frac{{\displaystyle \sum _{i=1}^I{T}_i^{\mathrm{d}}}}{{\displaystyle \sum _{i=1}^I{T}_i^{\mathrm{d}}}+{\displaystyle \sum _{j=1}^J{T}_j^{\mathrm{u}}}}},$$

(10)

$$F_f={\displaystyle \frac{I}{{\displaystyle \sum _{i=1}^I{T}_i^{\mathrm{d}}}+{\displaystyle \sum _{j=1}^J{T}_j^{\mathrm{u}}}}},$$

(11)

$$D_f={\displaystyle \frac{{\displaystyle \sum _{i=1}^I{T}_i^{\mathrm{d}}}}{I}},$$

(12)

where $P_f$ is the equipment failure probability, I and J represent the number of equipment failures and normal operation states within a specified time period, respectively, $T_i^{\mathrm{d}}$ and $T_j^{\mathrm{u}}$ represent the durations of the i-th equipment failure state and the j-th normal operation state, respectively, $F_f$ is the equipment failure frequency, and $D_f$ is the average equipment failure duration.

To draw parallels between the deviation of the generated and real data, as well as to verify the scenario generation accuracy, the percentage errors of equipment failure probability, equipment failure frequency, and average equipment failure duration, represented by ${\delta }_{P_f}$, ${\delta }_{F_f}$, and ${\delta }_{D_f}$, respectively, are introduced to represent the deviation degree of the generated samples relative to the real samples under the three evaluation indices. The relative error measures are expressed as follows:

$${\delta }_{P_f}={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n\frac{|P_{f,i}-{\widehat{P}}_{f,i}|}{P_{f,i}}},$$

(13)

$${\delta }_{F_f}={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n\frac{|F_{f,i}-{\widehat{F}}_{f,i}|}{F_{f,i}}},$$

(14)

$${\delta }_{D_f}={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n\frac{|D_{f,i}-{\widehat{D}}_{f,i}|}{D_{f,i}}},$$

(15)

where n is the number of real failure samples, $P_{f,i}$ is the equipment failure probability of the i-th real failure sample, ${\widehat{P}}_{f,i}$ is the equipment failure probability of the i-th generated failure sample, $F_{f,i}$ is the equipment failure frequency of the i-th real failure sample, ${\widehat{F}}_{f,i}$ is the equipment failure frequency of the i-th generated failure sample, $D_{f,i}$ is the average equipment failure duration of the i-th real failure sample, and ${\widehat{D}}_{f,i}$ is the average equipment failure duration of the i-th generated failure sample.

3.3.2. Extreme Output Scenario Evaluation Indices for Renewable Energy

In this study, two evaluation indices, namely, the reconstruction loss error $I_{\mathrm{RCE}}$ (RCE) and mean absolute percentage error (MAPE) $I_{\mathrm{MAPE}}$, were selected to measure the generation effect of extreme output scenarios for renewable energy [24]. $I_{\mathrm{MAPE}}$ is expressed as in (9), while $I_{\mathrm{RCE}}$ is expressed as follows:

$$I_{\mathrm{RCE}}={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{‖G(\xi |EX({\mathit{x}}_i))-{\mathit{x}}_i‖}_2^2},$$

(16)

where n is the number of scenarios, ${\mathit{x}}_i$ is the i-th extreme real scenario, and $G(\xi |EX({\mathit{x}}_i))$ is the i-th extreme generated scenario. The smaller the values of $I_{\mathrm{RCE}}$ and $I_{\mathrm{MAPE}}$, the higher the scenario generation accuracy.

4. Case Study

4.1. Case Study Setup

In this study, typhoon weather was used as a case study example, formulating two case studies: line failure scenario generation and extreme renewable energy output scenario generation. These case studies were designed to verify the effectiveness of the proposed model. Three distinct datasets were employed to validate the proposed methods:

(1)

For the first Gumbel-Softmax VAE case study, the IEEE 39-bus system (46 lines) was chosen to simulate line failure scenarios. In the context of the line probability failure model in the NaFIRS database, the failure rate was established in a 0.06–0.1 range under typhoon weather conditions, while the repair rate was set to 0.8. Through sequential Monte Carlo sampling [25,26], 682 days of failure scenarios with a time granularity of 1 h were obtained, which were partitioned into 582 training days and 100 test days.

(2)

The second case study utilized real-world operational status data of 1222 lines in a northern Chinese city, covering the first quarter of 2022 (January–March), with a time granularity of 5 min. The dataset spans 90 days, divided into 70 training days and 20 test days, preserving temporal continuity for validation.

(3)

The ExCGAN case study employs two-year (2021–2022) normalized power generation data with a time granularity of 15 min from 11 wind farms in the same region. The 730-day dataset was divided into 600 training days and 130 test days, ensuring adequate representation of seasonal variations while maintaining sufficient test samples for performance evaluation.

The PyTorch (version 1.7.1) deep-learning framework was adopted, and the computer environment consisted of an Intel Core i9-7920X 2.90-GHz CPU, 128 GB of memory, and an NVIDIA GeForce RTX 2080Ti. The model hyperparameters and settings are listed in

Table 2. Model hyperparameter settings.

Parameter	Value
Training epoch	500
Gumbel-Softmax VAE learning rate	0.001
LeakyReLU slope	0.2
ExCGAN generator learning rate	0.0002
ExCGAN discriminator learning rate	0.0001
Shift factor $\epsilon$	0.75
Training round k	10
Weight for balancing the ratio of normal operation samples to failure samples $\alpha$	0.3
Focusing parameter $\gamma$	0.5
Annealing range	1→0.01

.

4.2. Equipment Failure Scenario Generation

4.2.1. IEEE 39-Bus System Example

The Gumbel-Softmax VAE was first trained on the failure scenarios generated from the IEEE 39-bus system. Subsequent to the training phase, the decoder portion of the Gumbel-Softmax VAE was extracted and evaluated on the test set, thereby generating 300 equipment failure scenario sets. A statistical analysis of the failure probabilities of the generated and sampling failure scenarios was performed, as illustrated in Figure A1. The discrepancy in line failure probabilities between the generated and sampling failure scenarios remains within ±0.004, indicating that the proposed model can effectively learn the probability distribution of equipment failure scenarios and generate equipment failure scenarios with relatively low error.

To illustrate the line failure scenarios, two scenarios were selected, as displayed in Figure 4. In the figure, “0” and “1” denote the failure and normal operation states, respectively. The orange lines represent the generated failure scenarios. The blue lines with hollow circles represent the sampling failure scenarios generated by sequential Monte Carlo sampling. As illustrated in Figure 4a, a single line failure occurred within one day, while multiple line failures occurred within the same time span, as illustrated in Figure 4b. It is evident that the generated scenarios, subsequent to rounding, exhibit a high degree of compatibility with the sampling failure scenarios, thereby demonstrating the model’s ability to effectively learn the characteristics of discrete time series failure scenarios.

To examine the advantages of the focal loss function for failure feature extraction and evaluate the Gumbel-Softmax VAE model stability, the case study was reconfigured with and without the focal loss function introduction and taking the number of training set samples as 500, 300, and 200 days. To further enhance the credibility of the results, multiple experiments were conducted. As presented in

Table 3. Error indices under different training samples.

Number of Training Samples	With Focal Loss			Without Focal Loss
	${\delta }_{P_f}$/%	${\delta }_{F_f}$/%	${\delta }_{D_f}$/%	${\delta }_{P_f}$/%	${\delta }_{F_f}$/%	${\delta }_{D_f}$/%
500-day training sample	4.07	1.07	9.58	6.08	10.52	17.12
300-day training sample	9.48	1.35	11.50	22.16	14.91	18.40
200-day training sample	10.83	2.83	15.33	22.87	17.59	18.77

, with the same number of training samples, all generated scenario error metrics undergo a substantial decrease following the focal loss function introduction. This finding suggests that the focal loss function can effectively enhance the model performance. The experimental results of different training sample numbers demonstrate a clear degradation in model performance as the number of training samples decreases, as evidenced by the progressively higher error indices across all error indices in Table 3. This finding substantiates that employing sequential Monte Carlo sampling to generate extensive failure scenarios can effectively enhance the learning capability of the Gumbel-Softmax VAE. Furthermore, a GAN was selected as the comparison method, and the error metric comparison between the models is presented in

Table A1. Error index comparison between the different models.

Method	Error Index
	${\delta }_{P_f}$/%	${\delta }_{F_f}$/%	${\delta }_{D_f}$/%
GAN	22.6	8.46	12.9
Gumbel-Softmax VAE	4.07	1.07	9.58

. The error metrics for the GAN were all higher than those of the Gumbel-Softmax VAE, thereby demonstrating the superiority of the proposed method.

4.2.2. Actual Power Grid Test Case

To analyze the impact of the latent space dimension on the model and highlight the model’s advantages, two methods, namely, conventional and Gumbel-Softmax VAEs, were used to generate failure scenarios. The results for the three indices are presented in

Table 4. Fault scenario generation accuracy comparison between different models.

Method	Latent Space Dimension 144			Latent Space Dimension 288
	${\delta }_{P_f}$/%	${\delta }_{F_f}$/%	${\delta }_{D_f}$/%	${\delta }_{P_f}$/%	${\delta }_{F_f}$/%	${\delta }_{D_f}$/%
Conventional VAE	42	51	55	32	54	49
Gumbel-Softmax VAE	13	22	19	5.4	7.4	3.1

. As can be seen, when the latent space dimension is set to 288, the decoder’s generation effect is enhanced compared with the same model with the latent space dimension of 144. This is attributed to the increased dimension, which leads to improved model performance and enhanced network mapping capability. A comparison between the conventional and Gumbel-Softmax VAEs reveals a substantial enhancement in terms of the relative error indices of equipment failure probability, equipment failure frequency, and average equipment failure duration, suggesting that the proposed reparameterization method is capable of effectively addressing the distribution issue of discrete temporal data and generating scenarios with higher precision.

4.3. Extreme Output Scenario Generation for Renewable Energy

The confidence intervals of the generated scenarios are plotted in Figure 5, and the blue line represents the mean of all generated scenarios. The different degree of grey area represents the different degree of confidence. The width of each confidence interval is relatively narrow, which indicates that the proposed model has good performance on learning the probability distribution of extreme output scenarios. As shown in Figure 5a, for the high-volatility extreme scenario, the observed wind speed at the selected site exceeded the cut-out wind speed of the wind turbine, a consequence of the impact of a typhoon. This extreme event led to the shutdown of a significant number of wind turbines, resulting in a substantial decline in wind power output (from 0.8 to approximately 0.1), accompanied by considerable fluctuations. As shown in Figure 5b, for the low-power extreme scenario, the typhoon persisted for an extended period, resulting in the majority of wind turbines experiencing operational challenges. Consequently, the wind power output remained within 0–0.1 throughout the day.

The accuracy rates of the extreme metrics for the aforementioned generated scenarios are presented in

Table 5. Accuracy of different extreme metrics.

Extreme Metrics	High-Power Accuracy Rate/%	Low-Power Accuracy Rate/%	Average-Power Accuracy Rate/%	Volatility Accuracy Rate/%
[0.8, 0.1, 0.3, 0.7]	98.2	90.1	99.7	92.6
[0.1, 0, 0.05, 0.1]	91.2	98.7	96	90.4

. The accuracy rates of the four extreme metrics were all above 90%. The frequency distribution histograms of the extreme metrics for the aforementioned generated scenarios are shown in Figure A2. The red line represents the kernel density estimation curve, and the purple bar represents the frequency of corresponding extreme metric values. In this figure, the x-axis represents the extreme metric values, while the y-axis represents the corresponding frequencies. It is evident that the power distribution of the generated scenarios is predominantly concentrated at the corresponding extreme metrics, thereby meeting the specified condition characteristics. This observation indicates that the proposed model is capable of accurately generating specific extreme scenarios.

For the case study under consideration, three models, the CGAN, CGAN + distribution shift, and ExCGAN, were compared. The evaluation indices are presented in

Table 6. Scenario generation index comparison between different models.

Method	Generation Index
	$I_{\mathbf{RCE}}$	$I_{\mathbf{MAPE}}$
CGAN	0.0375	0.0320
CGAN + distribution shift	0.0266	0.0159
ExCGAN	0.0236	0.0132

. The introduction of distribution shift and extreme metric loss resulted in the two error metrics for the proposed model being lower than those for the other two models.

5. Conclusions

In this paper, a power grid extreme operation scenario generation framework is proposed, considering discrete failures and continuous output variations. Based on simulation results, the main conclusions of this study can be summarized as follows:

• The proposed Gumbel-Softmax VAE model, which is based on the Gumbel-Softmax reparameterization technique and focused loss function introduction, has the ability to effectively generate equipment failure scenarios in discrete data form. This provides a new approach for failure scenario modeling.
• Four extreme metrics are proposed. When employed in conjunction with the distribution shifting algorithm, this approach effectively addresses the challenge posed by the lack of historical extreme output scenario data. Compared with the conventional CGAN, the proposed ExCGAN model demonstrated its efficiency in terms of generating particular types of extreme output scenarios for renewable energy. This capability offers significant advantages and provides data support for power system extreme situation analysis.

However, some assumptions were made during this research that may limit the generalizability of the findings. For instance, we assumed that the failure rates and repair rates of the sequential Monte Carlo sampling method are within a constant range, which could affect the applicability and generalizability of our methods in different extreme contexts. And the spatial correlation between different lines or stations was not considered. So, incorporating the extreme weather conditions and the spatial topologies into the generation of power grid extreme operation scenarios might help improve the generalizability.

In future work, we will focus on integrating spatial dependencies into extreme scenario generation by coupling grid topology with extreme weather patterns. This will provide theoretical and practical support for a safe and stable power grid operation.

6. Patents

A national invention patent has been submitted under Application No. 202410987153.3.