An SVM-Based Anomaly Detection Method for Power System Security Analysis Using Particle Swarm Optimization and t-SNE for High-Dimensional Data Classification

Abstract

Research on the detection and identification of anomalies in electric power systems is crucial for ensuring their secure and stable operation. Anomaly detection models based on Support Vector Machines (SVMs) effectively process high-dimensional data while maintaining strong generalization capabilities. However, the performance of SVMs significantly depends on the choice of parameters, where improper parameter settings can lead to overfitting or underfitting, consequently decreasing the accuracy of anomaly detection. Furthermore, the dimensions of anomaly data in electric power systems are often unknown, making it difficult for existing methods to maintain a high precision in multidimensional data detection, and the segmentation of such data lacks intuitive display. In response, this article proposes an improved SVM model for electric power system anomaly detection, enhanced by parameter optimization algorithms, alongside a method for nonlinear dimension reduction and visualization using t-Distributed Stochastic Neighbor Embedding (t-SNE). Initially, traditional SVM parameters are optimized using the following four algorithms: Grid Search (GS), Particle Swarm Optimization (PSO), Genetic Algorithms (GAs), and Artificial Bee Colony (ABC) algorithms, in order to establish the optimized SVM model for electric power system anomaly detection. Finally, the effectiveness of the proposed method is verified through simulations. The simulation results indicate that, in the IEEE-14 node system case study, the accuracy for normal data reaches 97.58%, the accuracy for load step change detection reaches 99.52%, the accuracy for bad data detection reaches 99.92%, and the accuracy under fault conditions reaches 100%.

1. Introduction

As modern electrical power systems continue to expand in scale and level of intelligence, ensuring their secure and stable operation becomes increasingly critical [1]. However, during the operation of these power systems, issues such as equipment aging, environmental factors, and communication failures may lead to malfunctions. Additionally, inaccuracies in measurement equipment, delays in data transmission, and interference can introduce erroneous data, known as bad data. The presence of such anomalous data can jeopardize the secure operation of the power grid, potentially leading to misjudgments, which, in turn, can trigger significant economic losses and social impacts. Therefore, effectively detecting anomalous data within power systems and distinguishing between normal operation, faults, and bad data are of crucial importance for ensuring the security and reliability of such power systems [2,3].

Methods for detecting and identifying anomalies in electric power systems can generally be divided into statistical analysis methods and artificial intelligence techniques [4]. Statistical analysis methods primarily rely on the statistical properties of data, such as mean, residuals, skewness, and kurtosis. Methods based on residual analysis include the residual search method and the residual sensitivity method [5]. The residual search method can handle single faulty data points. However, when multiple uncorrelated faulty data points occur, this method requires iterative identification, which is slow and fails to recognize scenarios with multiple strongly correlated faulty data points. The residual sensitivity method, on the other hand, can handle multiple strongly correlated faulty data points in some cases. Reference [6] employs a set of linear recursive formulas, where state variables, residuals, and their variances are updated to the remaining dataset after suspicious data points are removed from the measurement set. This identification process does not require re-estimation or a residual sensitivity matrix, significantly enhancing computational speed. Reference [7] introduces a new method of state estimation aimed at handling bad data in PMU measurements. This approach establishes a state estimation model and provides a method for calculating errors in processed bad data. It employs a state estimation method based on the Cubature Kalman Filter, where Cubature points are generated using the spherical–radial rule. The state estimates are then transformed through equations applied to these Cubature points, yielding predicted state values. References [8,9] propose methods to detect and differentiate bad data and load spikes using skewness and residuals with fixed thresholds, respectively. However, the method based on skewness cannot accurately distinguish between the two types of data anomalies in large power networks, and the use of fixed thresholds lacks universal applicability.

In recent years, artificial intelligence technologies such as machine learning and deep learning have been widely applied in the detection of anomalies in power systems. Machine learning methods, such as SVM and Random Forests (RFs), are capable of learning the patterns of normal data from large datasets, thus effectively identifying anomalous data with a high accuracy and robustness [10]. Reference [11] introduces a detection method based on an Improved Local Outlier Factor (ILOF) and Kernel Support Vector Regression (KSVR) to evaluate the actual measurement errors of power metering equipment. Reference [12] proposes a method for detecting anomalies in power systems using deep learning techniques such as autoencoders and Generative Adversarial Networks (GANs). Their simulation results show that this method can automatically learn complex feature representations of data, demonstrating an enhanced capability to handle large-scale, high-dimensional data. In addition, recent advancements in machine learning have demonstrated promising solutions. Federated model-agnostic meta-learning frameworks, such as those applied to short-term load forecasting in distribution transformer supply zones [13], showcase the potential of distributed learning paradigms for preserving data privacy while handling spatially distributed grid data. However, these approaches prioritize forecasting accuracy over explicit anomaly detection mechanisms. Similarly, multi-task learning architectures with modal feature extraction, like those employed for multi-node load forecasting [14], effectively capture cross-node dependencies through shared latent representations. Yet, their computational complexity and reliance on aligned temporal resolutions limit real-time anomaly detection in high-dimensional state estimation scenarios.

A summary of existing techniques, their strengths, and their limitations is provided in Table 1.

In the field of anomaly detection in power systems, SVMs have demonstrated unique advantages. As a classification method based on statistical learning theory, SVMs effectively handle high-dimensional data and maintain a good generalization ability, even with small sample sizes [15]. However, the performance of SVMs heavily depends on the selection of their parameters. Inappropriate parameter choices can lead to model overfitting or underfitting, thus reducing the accuracy and reliability of anomaly detection [16]. Additionally, the dimensions of anomaly data in power systems are often unknown, making it challenging for existing research methods to maintain a high accuracy in multidimensional data detection. The segmentation of multidimensional data is not intuitively displayed, and training models with multidimensional data can be time-consuming.

To address these issues, this paper proposes an SVM-based method for electric power system anomaly detection using particle swarm optimization and t-SNE for high-dimensional data classification. The key innovations of this paper can be summarized as follows:

• (1). In this paper, GS, PSO, GA, and ABC are used for the optimization of traditional SVM hyperparameters, respectively, and an improved SVM model for power system anomaly detection is constructed. Through the comparison and optimization of multiple algorithms, the accuracy and generalization ability of anomaly detection can be effectively improved

• (2). In this paper, based on the establishment of a high-precision anomaly detection model, the t-SNE method is introduced to nonlinearly downscale and visualize high-dimensional data and map these complex high-dimensional data into two-dimensional or three-dimensional space. The method can provide an intuitive understanding and visual analysis of the distribution of power system anomaly detection data, which helps with the interpretation of anomaly detection results and model optimization.

While emerging metaheuristic algorithms such as Grey Wolf Optimizer (GWO) [17], Whale Optimization Algorithm (WOA) [18], and hybrid quantum-inspired methods [19] have demonstrated promising results in recent years, conventional approaches like PSO and ABC remain widely adopted in power system applications. This preference stems from their proven convergence properties in high-dimensional parameter spaces [20], computational efficiency for real-time implementation [21], and extensive validation in power system state estimation tasks. Our methodological selection aims to establish baseline performance metrics using well-understood algorithms before exploring more complex hybrid architectures in future work.

The remainder of this paper is organized as follows: Chapter 2 discusses traditional methods for detecting anomalous data in electric power systems. Chapter 3 elucidates the basic principles of SVMs. Chapter 4 describes the following four parameter optimization algorithms: the GS, PSO, GA, and ABC algorithms, and establishes an SVM parameter selection model. Chapter 5 presents a method for classifying high-dimensional data based on t-SNE. Chapter 6 provides a case study analysis. Finally, Chapter 7 concludes the paper. The specific workflow of this paper is shown in Figure 1.

2. Traditional Power System Data Anomaly Detection Methods

Power system forecasting-aided state estimation can synchronize different types of measurements through prediction and increase the redundancy of such measurements, which can enhance the accuracy of state estimation, thereby improving the reliability and safety of a system.

2.1. Power System Forecasting-Aided State-Space Model

The measurement equation in the power system is represented by the following formula:

(1)$z=h(x)+\delta$

where z is the measurement vector, x is the state vector consisting of the magnitudes and phase angles of the bus voltages, h(x) is the nonlinear function relating the measurements of the state variables x, and δ is the measurement error vector, assumed to have a mean of zero and a covariance matrix R of zero.

The power system in actual operation is a quasi-steady-state system, and the operating state of the power system changes in real time according to the dynamic characteristics of the load. Static state estimation cannot capture the dynamic changes in the power system, while forecasting-aided state estimation algorithms can establish the time-varying characteristics of the system between state variables and observation time. These algorithms are capable of estimating the current system state while also predicting the state values for the next time step in advance.

Forecasting-aided state estimation based on Kalman filtering includes the following two parts: prediction and correction (filtering). The prediction equation is as follows:

(2)$x_{k|k-1}=\mathrm{f}\left(x_{k-1|k-1}\right)+v_{k-1}$

(3)$P_{k|k-1}=F_k{P}_{k-1|k-1}{F}_k^T+Q_k$

The correction (filtering) equation is as follows:

(4)$K_k=P_{k|k-1}{H}_k^T{\left(H_k{P}_{k|k-1}{H}_k^T+R_k\right)}^{-1}$

(5)$x_{k|k}=x_{k|k-1}+K_k\left[z_k-h\left(x_{k|k-1}\right)\right]$

(6)$P_{k|k}=\left(\begin{array}{c}I-K_k{H}_k\end{array}\right)P_{k|k-1}$

where x_k and x_k−1 are the state variables at time steps k and k − 1, respectively, and η_k is the excitation noise, which follows a Gaussian distribution with covariance Q_k. P_k is the covariance matrix of x_k, F is the Jacobian matrix of f with respect to x, H is the Jacobian matrix of h with respect to x, K_k is the Kalman gain, z_k is the measurement vector, x_k is the final estimated state after correction, and I is the identity matrix.

Dynamic state estimation uses the Kalman filter to estimate the state from the measurement values and predicted measurements based on information from previous time steps. Since the power system is a nonlinear system, the Kalman filter must be linearized in order to be applied to dynamic state estimation. The difference from forecasting-aided state estimation lies in that Equation (2) is rewritten in the following form:

(7)${\dot{x}}_{k|k-1}=\mathrm{f}\left(x_{k-1|k-1}\right)+v_{k-1}$

In the traditional FASE algorithm, Holt’s double-parameter linear exponential smoothing technique is typically used to update the parameters of the prediction function. The state evolution model of the power system can be described in the following state-space form:

(8)$x_{k+1}=F_k{x}_k+g_k+q_k$

(9)$z_k=h\left(x_k,k\right)+r_k$

where x is the state vector; F and g are the transition matrix and input vector, respectively; z is the measurement vector; h is the nonlinear measurement function; q and r are the process and measurement noise error vectors, respectively, with covariance matrices Q and R; and the subscript k denotes the instant. Here, the matrix F and the vector g are estimated using Holt’s linear exponential smoothing technique, as follows:

(10)$F_k={\alpha }_k(1+{\beta }_k)I$

(11)$g_k=(1+{\beta }_k)(1-{\alpha }_k)x_k^{-}-{\beta }_k{a}_{k-1}+(1-{\beta }_k)b_{k-1}$

where I is the identity matrix; α_k and β_k are parameters within the range from 0 to 1; and $x_k^{-}$ is the state vector at time k. The parameter vectors a and b at time k can be estimated as follows:

(12)$a_k={\alpha }_k{x}_k+(1-{\alpha }_k)x_k^{-}$

(13)$b_k={\beta }_k(a_k-a_{k-1})+(1-{\beta }_k)b_{k-1}$

2.2. Existing Methods for Anomaly Detection in Power System Data

Traditional anomaly data detection and identification methods refer to techniques specifically designed to adapt to the anomaly detection and identification needs of various power systems based on their distribution characteristics. Depending on the stage at which anomaly detection and identification occur in state estimation, traditional methods can be categorized into the following three types: detection and identification before estimation, during estimation, and after estimation. Their advantages and disadvantages are as follows:

• Detection and identification before estimation: This method avoids the cumbersome calculations associated with repeated state estimation, offering a strong flexibility and broad applicability. However, its theoretical framework and method completeness need further improvement.

• Detection and identification during estimation: This method allows for the simultaneous detection of multiple anomalous data points, making it more efficient. However, its results may not be optimal.

• Detection and identification after estimation: This method provides better detection and identification results. However, it involves a large computational effort, and residuals may obscure detection, potentially reducing sensitivity.

• Reference [22] proposes an anomaly detection framework based on the residual innovation ratio, which can detect sudden topological changes caused by line faults.

• The residual innovation ratio algorithm is used to distinguish between bad data and state changes, and is defined as follows:

(14)$Inred=\underset{\epsilon \to \infty }{\lim }{\displaystyle \frac{\max \left(\left|z_{k,true}+\epsilon -h({\widehat{x}}_{k|k})\right|/\sqrt{P_{k|k}^{zz}}\right)}{\max \left(\left|z_{k,true}+\epsilon -h({\widehat{x}}_{k|k-1})\right|/\sqrt{P_{k|k-1}^{zz}}\right)}}$

where $z_{k,true}$ is the real measurement vector with large errors ε, and $P_{k|k}^{zz}$, $P_{k|k-1}^{zz}$ are the covariance matrices of $z_{k,true}+\epsilon -h({\widehat{x}}_{k|k})$ and $z_{k,true}+\epsilon -h({\widehat{x}}_{k|k-1})$, respectively.

Inred has the following attributes:

• Inred → 1: SCADA bad data

• Inred → (0, 1]: PMU bad data

• Inred → 0: Sudden load change

• Inred > 0: Fault state

Traditional methods fail to meet the requirements for mining massive amounts of data, while machine learning-based approaches have rapidly developed in recent years for big data applications. Machine learning techniques have been widely applied by researchers in the power system field. Machine learning-based models for electricity consumption anomaly analysis predict and make decisions by analyzing and learning from large amounts of existing or generated data. These models rely on historical power system data to train the model for state estimation, enabling the quick and accurate identification of abnormal electricity usage data. They can then further analyze the causes of anomalies. This approach is of significant research importance for improving the stability of daily maintenance in the power sector, helping electrical systems to operate more stably and efficiently and reducing economic losses caused by technical or non-technical losses. However, machine learning has not been able to detect all types of data. In practical applications, traditional simulation techniques based on deterministic methods still dominate in power grids. At the same time, the high performance of machine learning solutions in terms of accuracy, computational speed, and scalability has brought a new transformation to grid management and control. The shift from traditional power systems to smart grids has become a new challenge.

3. The Basic Concept of SVMs and Their Application Areas

SVMs were first proposed by Cortes and Vapnik in 1995 [23]. They demonstrate many unique advantages in solving small-sample, nonlinear, and high-dimensional pattern recognition problems and can be extended to other machine learning tasks such as function fitting.

The main idea of SVMs is to construct an optimal decision hyperplane that maximizes the margin between the closest samples of two classes on either side of the hyperplane, thereby providing a good generalization ability for classification problems.

3.1. Basic Principle of SVMs

The objective function and constraints of SVMs are represented as follows:

(15)$\begin{array}{l}\min {\displaystyle \frac{1}{2}}{‖\omega ‖}^2+C{\displaystyle \sum _{i=1}^l{\zeta }_i}\\ s.t. y_i(\omega x_i+b)\ge 1-{\zeta }_i\end{array}$

where $\omega ,x$ are the vectors, b is the constant, and l is the number of samples. y_i represents the attributes of the object. ξ_i are the non-negative slack variables, and C is the penalty coefficient.

The main idea of SVMs is to construct an optimal decision hyperplane that maximizes the distance between the two classes of samples closest to the hyperplane on either side, thereby providing a good generalization ability for classification problems (Figure 2).

SVMs have the following advantages:

• Small Sample Size: Compared to the complexity of the problem, the SVM algorithm requires relatively fewer samples.

• High Dimensionality: Even with a high number of dimensions, SVMs do not cause significant storage or computational problems.

• Minimum Structural Risk: SVMs minimize the sum of “empirical risk” and “confidence risk”, which is known as structural risk minimization (SRM). This helps in reducing overfitting and improving generalization.

• Nonlinear Classification: SVMs excel in situations where the sample data are not linearly separable, primarily achieved through the use of slack variables (also known as penalty variables) and kernel function techniques.

3.2. SVM Classification Boundary

SVMs essentially seek a classification hyperplane that separates positive and negative class samples under certain constraints [24]. In the sample space, the dividing hyperplane can be expressed as follows:

(16)${\omega }^{T}x+b=0$

where w determines the direction of the hyperplane and b is the displacement, which determines the distance between the hyperplane and the origin. Equation (15) represents the SVM model, which is a convex quadratic programming problem. The optimization problem is solved using the method of Lagrange multipliers, where ${\alpha }_i\ge 0,{\mu }_i\ge 0$ are the Lagrange multipliers, as follows:

(17)$L(\omega ,b,\alpha ,\xi ,\mu )={\displaystyle \frac{1}{2}}{‖\omega ‖}^2+C{\displaystyle \sum _{i=1}^l{\zeta }_i}+{\displaystyle \sum _{i=1}^m{\alpha }_i}(1-{\zeta }_i-y_i({\omega }^T{x}_i+b))-{\displaystyle \sum _{i=1}^m{\mu }_i{\xi }_i}$

Taking the partial derivative of $L(\omega ,b,\alpha ,\xi ,\mu )$ with respect to $\omega ,b,{\xi }_i$ and setting it to zero, we obtain the following:

(18)$\omega ={\displaystyle \sum _{i=1}^m{\alpha }_i{y}_i{x}_i}$

(19)$0={\displaystyle \sum _{i=1}^m{\alpha }_i{y}_i}$

(20)$C={\alpha }_i+{\mu }_i$

By substituting (18) and (19) into (17), we can obtain the dual problem of (15), as follows:

(21)$\underset{\alpha }{\max } {\displaystyle \sum _{i=1}^m{\alpha }_i}-{\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^m{\alpha }_i{\alpha }_j{y}_i{y}_j{x}_i^T}}{x}_j$

(22)$\begin{array}{l}s.t. {\displaystyle \sum _{i=1}^m{\alpha }_i{y}_i=0},\\ 0\le {\alpha }_i\le C, i,j=1,2,\dots ,m.\end{array}$

After solving for α, we can calculate w and b to obtain the following model:

(23)$f(x)={\omega }^T+b={\displaystyle \sum _{i=1}^m{\alpha }_i{y}_i{x}_i^T}+b$

In summary, for any training sample (x_i,y_i), there will always be either α_i = 0 or $y_{i}f(x_i)=1-{\xi }_i$.

3.3. Kernel Functions for SVM Classification

The basic idea for solving nonlinear separability issues is to transform the data into a higher-dimensional space, where they become linearly separable.

A kernel function is the inner product of the mapping function $\varphi (x)$, which itself is just a mapping relation and does not inherently increase dimensionality. By utilizing the properties of kernel functions, we can construct kernels that effectively increase dimensionality. Mapping from two dimensions to three or more dimensions makes it easier to separate data, which is why kernel functions are commonly used in clustering and classification tasks.

Several commonly used kernel functions are as follows:

• Linear Kernel: $\kappa (x,x_i)=(x\cdot x_i)$

• Polynomial kernel: $\kappa (x,x_i)={(s(x\cdot x_i)+c)}^d$

• Radical basis function: $\kappa (x,x_i)=\exp (-\gamma {\left|x-x_i\right|}^2)$

• Sigmoid tanh: $\kappa (x,x_i)=\tanh (s(x\cdot x_i)+c)$

Among the commonly used kernel functions, the Radial Basis Function (RBF) kernel is the most widely applied and has excellent learning capabilities. Regardless of whether the data are in low-dimensional or high-dimensional space and whether the sample size is small or large, the RBF kernel adapts well. It has a wide convergence domain, making it an ideal choice for classification tasks.

4. Principles of Parameter Optimization

SVMs can handle nonlinear high-dimensional data by using kernel functions. When dealing with nonlinear problems, SVMs first seek a nonlinear mapping function and then map the low-dimensional nonlinear samples to a high-dimensional feature space. Moreover, SVMs can replace the inner product operations in the high-dimensional space with kernel functions, which enhances their predictive performance and generalization capability. SVMs are known for their fast computation speed, high simulation accuracy, strong generalization ability, and ability to achieve a global optimum. These characteristics make SVMs suitable for solving practical problems such as nonlinearity, small sample sizes, high dimensionality, and local minima. Their application in prediction is of great value.

However, a drawback of the SVM algorithm is that it cannot determine whether data samples contain redundant information. The selection of the kernel function and its parameters is generally based on experience. To improve their prediction accuracy, SVMs can be combined with other algorithms to optimize parameter selection.

Model parameter optimization is achieved by minimizing the objective function to achieve the best fit between the model output and the actual observed data. Due to the inherent complexity of environmental models, conventional optimization algorithms often struggle to reach the global optimum in the parameter space.

The function estimation $f(x,a)=⟨w,\varphi (x)⟩+b$ attempts to minimize empirical risk, and the optimal value of ε ensures that this empirical risk is minimized. When ε is not sensitive to the empirical risk (ε ≤ 0), the feasible solution space for the parameters is formed by the constraints from structural risk minimization, and the feasible solution set S_n can be defined by the inequality $w\cdot w\le c_n$. This leads to a globally optimal solution for regression estimation.

The training data set $T={\{(x_i,y_i)\}}_{i=1}^n$ is a subset of $(X\times Y)$, where each $x_i\in X=R^n, y_i\in Y=R$. The feasible solution space for the parameters is constructed by the constraint that limits the sum of squared errors C to a value that does not exceed the empirical risk, thus ensuring an optimal solution for regression estimation, as follows:

(24)$\varphi (w)={\displaystyle \frac{1}{2}}‖w‖+C{\displaystyle \sum _{i=1}^{n}L(f(x_i),y_i)}$

where C is the penalty coefficient, which indicates the tolerance for errors. A higher C value implies a lower tolerance for errors, which can lead to overfitting. Conversely, a smaller C makes the model prone to underfitting. If C is too large or too small, the model’s generalization ability deteriorates. L represents the loss function. Non-linear data are mapped to a higher-dimensional space using kernel functions to perform linear regression in that space. The optimization in the form of quadratic programming can be transformed into the following:

(25)$\begin{array}{l}\min W(\alpha ,\alpha *)={\displaystyle \frac{1}{2}}{{\displaystyle \sum _{i=1,j=1}^n({\alpha }_i,{\alpha }_j^{*})}}^T\kappa (x_i,x_j)({\alpha }_j-{\alpha }_j^{*})\\ +{\displaystyle \sum _{i=1}^n({\alpha }_i+{\alpha }_i^{*})+}{\displaystyle \sum _{i=1}^n{y}_i({\alpha }_i-{\alpha }_i^{*})}\end{array}$

The function $\kappa (x_i,x_j)$ is compliant with Mercer’s theorem of kernel functions, commonly utilizing the RBF kernel, g. The RBF kernel is one of the most versatile, used to effectively handle the spatial relationships in data and enhance g, the complexity, and the flexibility of the model, which facilitates the capture of intricate patterns in the data. The application of this kernel helps in constructing more powerful and adaptable regression models.

From the analysis above, it is evident that, in regression-type SVMs, the regularization parameter C and the kernel parameter g both require careful selection. To ensure that SVMs have good applicational capabilities, it is necessary to use optimization algorithms to adjust these parameters.

4.1. Grid Search Optimization Method

The GS method searches for the global optimal values of C and g within a rectangular range according to a given step size [25]. It divides the search parameters within a certain spatial range into grids of an equal length and width according to a predetermined coordinate system. Each point in the coordinate system represents a set of parameters. The performance of the SVM system can then be validated by testing each point within the given range, identifying the point that yields the best overall system performance as the optimal parameter. The specific implementation steps are as follows:

• (1). Set the search range and step size for C and g in the GS method, and then construct a two-dimensional grid in the C and g coordinate system.

• (2). Divide the experimental data into n groups as the sample set. Select any one group as the training sample, and use the remaining data as the prediction sample. From the two-dimensional grid coordinate system, choose a set of parameters randomly, train on n − 1 groups of data, and then predict on one group of data (using cross-validation techniques). Record the prediction error.

• (3). Repeat step (2) until all C and g values have been trained once in the two-dimensional grid.

• (4). The mean squared error (MSE) serves as the criterion for prediction accuracy. For each combination of C and g, map the corresponding MSE onto a three-dimensional coordinate system, where the MSE values are represented as contour lines. Ultimately, the optimal parameter values for C and g are determined when the MSE is minimized.

The entire optimization process is a repetitive search iteration, continuing until the maximum number of iterations is reached or the minimum error condition is satisfied. The optimization process and results are illustrated in Figure 3 and Figure 4.

4.2. Particle Swarm Optimization Method

PSO is a parallel evolutionary computing technique inspired by the social behavior of flocks of birds searching for food. In the particle swarm algorithm, each member of the population, also known as a particle, searches for better regions within a multidimensional search space [26].

The PSO algorithm starts with a group of randomly initialized particles, and then iteratively searches for the optimal solution by updating generations. The velocity and position of each particle are updated according to the following formulas:

(26)$v_{id}^{t+1}=\omega v_d^t+c_1{r}_1(p_{BES{T}_{id}}^t-x_{id}^t)+c_2{r}_2(g_{BES{T}_d}^t-x_{id}^t)$

(27)$x_{id}^{t+1}=x_{id}^t+v_{id}^{t+1}$

The meanings of the main symbols here are described as follows: c₁ and c₂ are constants for acceleration coefficients, typically set as c₁ = c₂ = 2. r₁ and r₂ are two independent random numbers uniformly distributed within the range [0, 1].

In this context, $p_{BES{T}_{id}}^t$ represents the best prior position along the dth dimension of particle i in iteration t (as retained in each particle’s memory). $g_{BES{T}_d}^t$ represents the best prior position along the dth dimension among all particles in iteration t (stored in a common repository). w denotes the inertia weight.

The entire optimization process is a repetitive one, continuing until the maximum number of iterations is reached or the minimum error condition is met. The optimization process and results are shown in Figure 5.

4.3. Genetic Algorithm Optimization Method

The Genetic Algorithm (GA) utilizes the principles of biological genetics, combining the concepts of survival of the fittest and random information exchange [27]. Through mechanisms such as natural selection, crossover, and mutation, it achieves the evolution of a population. In the optimization process, the GA randomly generates multiple starting points within the solution space and begins the search simultaneously. The search direction is guided by the fitness function, allowing it to quickly seek a global optimal solution in a complex search space. The specific steps involved are as follows:

• (1). Initialization:

  • Collect data and perform normalization.

  • Randomly generate an initial population of N individuals, where each individual represents a potential solution encoded as a chromosome (e.g., binary or real-valued strings).

  • Set genetic parameters, including crossover probability p_c, mutation probability p_m, and maximum generations G_max.

• (2). Decoding and Evaluation:

  • Decode each chromosome into SVM parameters (kernel function type, penalty factor C, and kernel parameter g).

  • Train the SVM model using k-fold cross-validation and calculate fitness values. The fitness function is defined as follows:

    (28)$\mathrm{Fitness}={\displaystyle \frac{1}{1+\mathrm{MSE}}}$

    where MSE is the mean squared error of classification.

• (3). Selection:

  • Randomly select k individuals from the population and choose the one with the highest fitness. Repeat this process until a mating pool of size N is formed.

• (4). Crossover:

  • For two parent solutions p₁ and p₂, generate offspring o₁ and o₂ as follows:

    (29)$o_1=0.5\left[(1+\beta )p_1+(1-\beta )p_2\right]$

    (30)$o_2=0.5\left[(1-\beta )p_1+(1+\beta )p_2\right]$

    where β is a spread factor determined by a uniform random number.

• (5). Mutation:

  • For a gene value v, the mutated value v′ is as follows:

    (31)$v^{\prime }=v+\delta \times (v_{max}-v_{min})$

    where δ is a small perturbation sampled from a polynomial distribution, and v_min, v_max define the parameter search range.

• (6). Replacement:

  • Replace the worst 10% of the population with newly generated individuals to maintain diversity.

• (7). Termination Check:

  • If the maximum generation G_max is reached or the fitness plateaus, output the optimal parameters. Otherwise, return to Step (2).

The optimization process and results are shown in Figure 6.

4.4. The Artificial Bee Colony Optimization Method

The ABC algorithm is an optimization algorithm inspired by the behavior of bee swarms. This algorithm mimics the actions of individual bees searching for suitable food sources in the search space to find the global optimal solution. The ABC algorithm does not require knowledge of specific information about the problem, but instead compares the quality of solutions. Through the local search behaviors of each artificial bee individual, the algorithm ultimately highlights the global optimal solution within the population and has a relatively fast convergence speed.

By using the ABC algorithm to optimize the SVM, the parameters of the SVM are optimized. The range of parameter optimization corresponds to the range within which bees search for food sources. The bee swarm reaches the optimal food source, which corresponds to the optimal penalty factor C and kernel function parameters g of the SVM.

The specific steps of the ABC optimization for the SVM are as follows:

• (1). Collect data and perform data normalization and standardization to form the sample database.

• (2). Algorithm initialization—determine the value range for the penalty factor C and kernel function parameter g. Set the basic parameters of the ABC algorithm, including the maximum number of generations, the number of food sources (initial solutions) NP, and the initial solutions.

• (3). Observe the bees selecting nectar sources, record the global optimal solution and its corresponding fitness value, and update the positions of the nectar sources by comparing fitness values. The position with the best fitness value is the global optimal solution.

• (4). Check if the scout bee phase is reached. If the fitness value can no longer improve, the foraging bee at that position is converted to a scout bee, and a new solution is randomly generated to replace it. This process continues until the stopping criteria are met, at which point the optimal solution is output.

• (5). The optimal parameters found by the ABC algorithm are used as the optimal parameters for the SVM, and the training and testing sets are used to obtain the classification results.

5. High-Dimensional Data Classification Based on t-Distributed Stochastic Neighbor Embedding

The dimensions of measurement data are generally unknown, and most data have more than two dimensions. SVMs can still maintain a high accuracy in classifying high-dimensional data, but the partitioning of high-dimensional data cannot be intuitively visualized [28]. Additionally, training models on high-dimensional data requires a significant amount of time. However, with the help of t-SNE, data can be visualized, and dimensionality reduction will significantly reduce model training time. t-SNE is a nonlinear dimensionality reduction algorithm that is suitable for reducing high-dimensional data to 2D or 3D for visualization.

5.1. Basic Principles of Stochastic Neighbor Embedding

Stochastic Neighbor Embedding (SNE) maps data points to the corresponding probability distribution through affine transformations, and primarily involves the following two steps:

• (1). Construct a probability distribution P between data points in the high-dimensional space, such that similar data points have a higher probability of being selected, while dissimilar data points have a lower probability of being selected.

• (2). Then, reconstruct the probability distribution Q of these points in the low-dimensional space, aiming to make the two probability distributions as similar as possible.

Let the input space be $X\in R_n$ and the output space be $Y\in R_t (t<<n)$. Assume that it contains m sample data points $\{x^{(1)},x^{(2)},\dots ,x^{(m)}\}$, where $x^{(i)}\in X$ and the data after dimensionality reduction are $\{y^{(1)},y^{(2)},\dots ,y^{(m)}\}$, $y^{(i)}\in Y$.

SNE first converts Euclidean distances into conditional probabilities to express the similarity between points. Specifically, it starts by calculating the conditional probability $p_{j|i}$, which is proportional to the similarity between x⁽ⁱ⁾ and x^(j). The formula for calculating $p_{j|i}$ is as follows:

(32)$p_{j|i}={\displaystyle \frac{\exp (-\frac{{‖x^{(i)}-x^{(j)}‖}^2}{2{\sigma }_i^2})}{{\displaystyle \sum _{k\ne i}\exp (-\frac{{‖x^{(i)}-x^{(k)}‖}^2}{2{\sigma }_i^2})}}}$

where a parameter σ_i is introduced, which takes different values for different data points x⁽ⁱ⁾, since the focus is on the similarity between pairs of different data points, hence, p_i|i can be set to 0. For data points y⁽ⁱ⁾ in lower dimensions, the similarity between y⁽ⁱ⁾ and y^(j) is characterized by the conditional probability q_j|i. The formula for calculating q_j|i is as follows:

(33)$q_{j|i}={\displaystyle \frac{\exp (-{‖y^{(i)}-y^{(j)}‖}^2)}{{\displaystyle \sum _{k\ne i}\exp (-{‖y^{(i)}-y^{(k)}‖}^2)}}}$

Similarly, q_i|i can be set to 0.

If the dimensionality reduction is effective and local features are preserved completely, then p_i|j = q_i|j holds. Therefore, the loss function constructed by optimizing the KL divergence between the two distributions is as follows:

(34)$C(y^{(i)})={\displaystyle \sum _{i}KL(P_i||Q_i)}={\displaystyle \sum _i{\displaystyle \sum _j{p}_{j|i}\log \frac{p_{j|i}}{q_{j|i}}}}$

where P_i represents the conditional probability distribution of all other data points given a high-dimensional data point x⁽ⁱ⁾ and Q_i represents the conditional probability distribution of all other data points given a low-dimensional data point y⁽ⁱ⁾. From the loss function, it can be seen that, when p_j|i is large and q_j|i is small, the penalty is higher, and when p_j|i is small and q_j|i is large, the penalty is lower. That is, if two data points are close in the high-dimensional space but far apart after mapping to the low-dimensional space, a high penalty will be incurred; conversely, if two data points are far apart in the high-dimensional space but close after mapping to the low-dimensional space, a low penalty will be incurred. Thus, SNE’s loss function focuses more on local features while neglecting the global structure.

5.2. Basic Principles of t-Distributed Stochastic Neighbor Embedding

The improvement of t-SNE over symmetric SNE is that it initially converts distances into probability distributions using a Gaussian distribution in high-dimensional space. Then, in low-dimensional space, it uses a distribution that is more biased towards heavy tails to convert distances into probability distributions. This allows medium and low distances in the high-dimensional space to have a larger distance after mapping. t-SNE is a dimensionality reduction algorithm that is very suitable for visualizing high-dimensional data. The idea is to embed high-dimensional points into low-dimensional points in a way that respects the similarities between them. Points that are close in the high-dimensional space correspond to closely embedded points in the low-dimensional space, and points that are far apart in the high-dimensional space correspond to closely embedded points in the low-dimensional space.

The gradient update of t-SNE has the following two advantages:

• (1). For dissimilar data points in low-dimensional space, a smaller distance generates a larger gradient that pushes these data points apart.

• (2). This repulsion is not infinite, thus preventing the dissimilar data points from being too far apart.

6. Simulation Analysis

To validate the effectiveness of the proposed anomaly detection method for power system data based on an SVM, this chapter selects the IEEE 14, IEEE 33, and IEEE 118 bus test systems to analyze the performance of the employed SVM classifier [29]. All tests were conducted in a MATLAB (R2022a, MathWorks, Natick, MA, USA) environment using an Intel Core i7-9750 CPU (@2.6 Hz, Intel Corporation, Santa Clara, CA, USA) with 16 GB of memory.

To ensure the generality of the results, all results shown in this chapter are the average values obtained from 500 Monte Carlo simulation outcomes. The conventional measurements and PMU measurements for the IEEE 14 bus test system are as follows.

For Gaussian errors, the standard deviation is set to a certain percentage of the measured value, depending on the type of measurement, as follows:

• SCADA: P, Q measurements (0.02), Voltage (0.002)

• PMU: Amplitude (0.002), Phase angle (0.01)

This chapter selects the IEEE 14, IEEE 33, and IEEE 118 bus test systems to analyze the performance of the t-SNE and SVM methods employed. The faults considered are line-to-ground faults, categorized into single-phase, two-phase, and three-phase ground faults. Section 6.1 of this chapter applies different optimization methods to optimize the SVM parameters according to the optimization principles mentioned in Chapter 4 and records their accuracy.

Section 6.2 will discuss the classification accuracy under benchmark configurations by conducting extensive Monte Carlo experiments in the IEEE 14 bus system. Section 6.3 will introduce benchmark configuration simulations for different bus systems.

6.1. Simulation 1: Comparison of Parameter Optimization Results

Different penalty factors and kernel functions affect classification accuracy. Currently, four different optimization methods are employed to optimize the SVM parameters. The accuracy of these parameters is validated across different bus systems, with the accuracy of each optimization method shown in Table 2.

In Table 2, Gaussian noise is added to the measurement data. The variance of the measurement noise is chosen based on the assumed measurement accuracy and historical data about measurement errors, and not all measurement noise can be strictly represented by a Gaussian distribution. Based on this, Laplace noise is added to the data of the IEEE 14 bus system, and the accuracy of each optimization method is recalculated (Table 3).

From the comparison results above, it is evident that the PSO_SVM and ABC_SVM optimization algorithms have a higher accuracy. However, during experimental validation, the classification speed of PSO_SVM is faster than that of ABC_SVM. Therefore, in all subsequent simulations, the prediction data are provided by SVR, and classification is performed using the PSO_SVM classification method.

6.2. IEEE 14-Bus System Simulation Example

In the IEEE 14-bus system simulation, initial baseline configuration conditions are set where the bad data error rate is 100%, meaning that one data point is randomly selected from a dataset and set to zero; the load change rate is 40%, meaning that the load changes to 60% of the original load; Gaussian noise is added to all data; and the data include predicted data. Then, simulation examples are set up where one parameter of the baseline configuration conditions is changed arbitrarily, including different load change rates, different error rates based on the number of data points, the addition of non-Gaussian noise, and changes in accuracy when prediction data are missing.

6.2.1. Simulation 2: Baseline Configuration Simulation Example

For IEEE 14 distribution network measurements, each data dimension is visualized as follows: in the system, 90 sets of bad data are randomly added, with an error rate of 100% (i.e., one measurement data point is randomly set to zero). The classification effect is as follows. The data in the graph are determined by the PSO-SVM method to train the model and label each data point. Based on this, t-SNE is used to reduce the dimensionality to two-dimensional data, and a scatter plot is drawn, as shown in Figure 7. Then, the SVM maps the data back to the high-dimensional space for classification, and classification boundaries are drawn in the two-dimensional space. Based on the class of the corresponding new observations, colors are used in the drawing area to distinguish data categories.

The scatter plot has clear boundaries, and the SVM delineates these boundaries effectively, as shown in Figure 8.

Under the conditions described, the SVM classification performs well, with all data classes being independent, the clusters being closely packed, and similar data having short spatial distances between them. However, for t-SNE classification, there are still instances where clusters are not as compact and spatial distances are slightly larger. Nonetheless, t-SNE classification is primarily for visualizing data categorization and does not impact the final accuracy of the classification. When data clustering is not ideal, SVM classification remains effective. For data with less apparent spatial clustering, boundary plots are automatically drawn based on the data’s class relationships, as shown in Figure 9.

The SVM’s discriminative classification performance is evaluated using 2590 sets of data to categorize various types of data. Based on the simulation data mentioned above, the classification accuracy is shown in Figure 10.

The method proposed in this paper was compared with three other existing methods in terms of detection accuracy, as shown in Table 4. Method 1 uses normalized innovation and the skewness of normalized innovation to detect anomalies. Under SLC (sudden load change) conditions, its detection accuracy is poor, because the skewness of the normalized residuals becomes significant unless the load change is very small. Method 2 uses normalized innovation and normalized residuals to differentiate between SLC and bad data scenarios. However, the residuals under SLC conditions can be very large, making it difficult to distinguish between SLC and bad data.

As can be seen from Table 5, the method proposed in this paper was validated with the same data types as those in Method 3. Except for 0.48% of SLC and 0.08% of bad data not detected, the rest were correctly identified by the proposed method. By comparison, Methods 1 and 2 have too high false detection rates, while Method 3 has a lower false detection rate and the method proposed in this paper has the lowest false detection rate and does not require the consideration of threshold setting methods. For any data, this method will automatically set thresholds and continuously update the values as the system conditions change. Therefore, this method is feasible for online/real-time anomaly detection.

6.2.2. Simulation 3: Classification of Bad Data Under Different Error Rate Conditions

In the simulations above, results were obtained under baseline configuration conditions, that is, with a bad data error rate of 100%. However, in actual situations, the error rate is not certain. We introduce different sizes of errors and compare the state classification accuracy under the performance results of the measurement noise probability density function following a standard Gaussian distribution. We change the error rate to 20%, 40%, 60%, 80%, and 100% and conduct tests, respectively. When the bad data error rate is 20% (i.e., any data are reduced to 80% of the original data), the change is minor, making it easier to confuse with other data types. Since bad data represent a smaller proportion compared to other normal data, we use the classification accuracy of all data to test the results as follows.

Based on the analysis from Table 6, the following conclusions can be drawn. As the error rate increases, the classification accuracy becomes higher, reaching nearly 100% accuracy when the error rate is 100%. Even with lower error rates, the classification still maintains a high accuracy. When the error rate is low, the difference between anomalous data and normal data is relatively small, making it difficult for the model to distinguish these subtle variations. However, when the error rate is high, the characteristics of anomalous data become more pronounced, making it easier for the model to identify and classify them.

6.2.3. Simulation 4: State Classification with Non-Gaussian Noise

The accuracy rates of state classification mentioned above were all calculated with Gaussian noise added to the simulation data. To verify the general applicability of the proposed SVM classifier, Laplace noise was added to the data. Classification accuracy was validated using 2600 sets of data to categorize various types, with the classification results shown in Figure 11.

As can be seen from the Figure 11, the actual test set classification result and the predicted test set classification result are almost completely overlapped, which indicates that the actual classification result and the predicted classification result are almost consistent, proving the effectiveness of this method.

Based on the analysis from Table 7, the following conclusion can be drawn. When Laplace noise is added to the data, the SVM classification remains effective, but the accuracy decreases compared to when Gaussian noise is present.

6.2.4. Simulation 5: State Classification with Missing Prediction Data

All the accuracy rates mentioned above include predicted data. Regarding the simulation results under the baseline configuration conditions described earlier, the classifications were re-evaluated without the predicted values. The accuracy rates are shown in Table 8, indicating that the accuracy of data classification decreases when prediction values are missing.

6.3. Simulation 6: Comparison of Simulation Examples Across Different Bus Systems

To verify the general applicability of the proposed SVM classifier, it was tested on the IEEE 33 bus system. Classification decisions were made on 3200 sets of data, where the bad data error rate was 100% and the injected power change rate was 60%. According to Figure 12 and Table 9, the classification accuracy is quite good.

In the IEEE 118-bus system test, 3200 sets of data were used for classification to determine the data type, and the classification results are shown in Figure 13 and Table 10.

7. Conclusions

The detection and identification of anomalies are important for the security analysis of power system operation. To achieve accurate anomaly detection, this paper deployed an SVM for data classification to demonstrate the system’s actual operating state and utilized t-SNE for dimensionality reduction to achieve the purpose of visualizing classification. Based on extensive simulations set in different scenarios, the following conclusions were drawn.

First, the SVM used for data classification in power systems had a high accuracy. Second, models trained using the PSO-SVM method could determine each data label, and with t-SNE, reduce it to two-dimensional data to enable the visualization of classification. Third, different penalty factors and kernel functions affected the classification accuracy. This paper employed four different optimization methods to optimize the parameters in the SVM, with PSO_SVM and ABC_SVM showing a high accuracy and PSO_SVM having a faster classification speed. Lastly, the methods proposed in this paper were demonstrated in the IEEE 118 node test system to differentiate data classifications in the power system and achieve data visualization.

While the proposed method demonstrated promising results, several directions warrant further exploration. First, investigating hybrid optimization algorithms that combine evolutionary strategies (e.g., PSO and GA) with emerging metaheuristic techniques could enhance parameter tuning efficiency and robustness. Second, improving model interpretability remains critical, especially for high-dimensional power system data; integrating explainable AI (XAI) frameworks may bridge the gap between complex decision boundaries and actionable insights.

Author Contributions

Conceptualization, Y.T. and J.Y.; methodology, Y.T.; software, E.N.; validation, Y.T., J.Y. and S.Z.; formal analysis, Y.T. and P.Z.; investigation, P.Z.; resources, E.N.; data curation, J.Y.; writing—original draft preparation, Y.T.; writing—review and editing, Y.T.; visualization, Y.T.; supervision, S.Z. project administration, Y.T. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

Dataset available on request from the authors.

Conflicts of Interest

Y.T. and S.Z. are employed by the School of Electrical Engineering at Shandong University. Author J.Y. is employed by the Key Laboratory of Power System Intelli-gent Dispatch and Control of Ministry of Education at Shandong University. Author E.N. is employed by the Shandong Luruan Digital Technology Co., Ltd. Author P.Z. is employed by the Wenshan Power Supply Bureau of Yunnan Power Grid Co., Ltd. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as potential conflicts of interest.

Nomenclature

Abbreviations
ABC	Artificial Bee Colony Algorithm
FASE	Forecasting-Aided State Estimation
GA	Genetic Algorithm
GS	Grid Search
ILOF	Improved Local Outlier Factor
KSVR	Kernel Support Vector Regression
PMU	Phasor Measurement Unit
PSO	Particle Swarm Optimization
RBF	Radial Basis Function
SCADA	Supervisory Control and Data Acquisition
SVM	Support Vector Machine
t-SNE	t-Distributed Stochastic Neighbor Embedding
Parameters
w	Hyperplane normal vector (Direction vector)
b	Hyperplane displacement (Bias term)
C	SVM regularization parameter (Penalty coefficient)
Γ	RBF kernel width parameter
ξi	Slack variable for non-separable data
αi	Lagrange multiplier
xk	System state vector at time k
zk	Measurement vector at time k
Qk	Process noise covariance matrix
R	Measurement noise covariance matrix
Inred	Residual innovation ratio

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1. Overall process flow chart.

Figure 2. SVM classification schematic diagram.

Figure 3. SVM parameter selection result diagram (contour diagram).

Figure 4. SVM parameter selection result graph (3D view).

Figure 5. Parameter selection of support vector machine based on particle swarm optimization algorithm.

Figure 6. Parameter selection of support vector machine based on genetic algorithm.

Figure 7. Data scatter diagram of IEEE14 dimension reduction.

Figure 8. Demonstration of RBF classification results for power distribution network data based on high-dimensional data mapping.

Figure 9. Scatter diagram RBF classification during dispersion of six classifications of distribution network data.

Figure 10. IEEE14 node data classification accuracy.

Figure 11. Classification accuracy of IEEE14 node Laplacian noise data.

Figure 12. IEEE33 node data classification accuracy.

Figure 13. IEEE118 node data classification accuracy.

Table 1

Comparison of existing anomaly detection techniques in power systems.

Category	Method	Advantages	Limitations	Reference
Statistical	Residual Search	Simple for single bad data detection	Fails for correlated bad data; slow for iterative multi-fault identification	[5]
Statistical	Residual Sensitivity	Handles multiple correlated bad data	Suboptimal results; high computational complexity	[6]
Statistical	Skewness-Based Detection	Easy implementation with fixed thresholds	Poor distinction between bad data/load spikes; lacks adaptability	[8,9]
AI	SVM	Strong generalization; handles high-dimensional data	Performance sensitive to parameter selection (C, γ)	[10,15]
AI	Random Forest	Robust to overfitting; high accuracy	Computationally heavy; poor interpretability	[10]
AI	Deep Learning (Autoencoders)	Learns complex patterns; scalable for large datasets	Requires extensive labeled data; high training costs	[12]
Hybrid	ILOF + KSVR	Detects localized anomalies; combines regression for error assessment	Limited applicability to dynamic grid conditions	[11]

Table 2

Accuracy comparison of different optimization methods after introducing Gaussian noise.

Data Type	SVM (C = 2, g = 1)	GS-SVM	PSO-SVM	GA-SVM	ABC-SVM
IEEE14	97.51%	98.59%	98.67%	98.39%	98.56%
IEEE33	97.06%	98.71%	98.92%	98.51%	99.08%
IEEE118	97.66%	99.03%	99.19%	99.06%	99.11%

Table 3

Accuracy comparison of different optimization methods after introducing Laplace noise.

Noise Type	SVM (C = 2, g = 1)	GS-SVM	PSO-SVM	GA-SVM	ABC-SVM
Gaussian noise	96.23%	97.91%	99.27%	99.27%	99.22%
Laplace noise	96.14%	97.36%	97.54%	97.27%	97.53%

Table 4

Classification accuracy of IEEE14 node systems with different error rates.

Data Type	Normal	1 Phase Fault	2 Phase Fault	3 Phase Fault	Load Step Change	Bad Data
Accuracy	96.53%	99.72%	99.83%	99.91%	95.18%	99.87%

Table 5

Comparison between the detection accuracy of the method and other methods.

Detection Method	Normal	Load Step Change	Bad Data	Fault Condition
Method 1	70.5%	57.02%	17.75%	4.84%
Method 2	70.5%	11.94%	99.81%	4.84%
Method 3	95.5%	96.32%	99.57%	100%
Method in this paper	97.58%	99.52%	99.92%	100%

Table 6

Classification accuracy of different error rates.

Bad data error rate	20%	40%	60%	80%	100%
Accuracy	94.37%	95.79%	97.13%	99.06%	99.58%

Table 7

Laplacian noise classification accuracy.

Data Type	Normal	1 Phase Fault	2 Phase Fault	3 Phase Fault	Load Step Change	Bad Data
Accuracy	96.53%	99.72%	99.83%	99.91%	95.18%	99.87%

Table 8

Comparison of the accuracy of data anomaly classification with different error rates in the IEEE14 node system when missing predicted values.

Data Type		Normal	1 Phase Fault	2 Phase Fault	3 Phase Fault	Load Step Change	Bad Data
Accuracy	Including prediction	96.53%	99.72%	99.83%	99.91%	95.38%	99.87%
	Without prediction	94.05%	99.02%	99.13%	99.16%	94.92%	98.03%

Table 9

Classification accuracy of IEEE33 node systems with different error rates.

Data Type	Normal	1 Phase Fault	2 Phase Fault	3 Phase Fault	Load Step Change	Bad Data
Accuracy	95.13%	97.72%	98.83%	98.99%	94.21%	99.10%

Table 10

Classification accuracy of IEEE118 node systems with different error rates.

Data Type	Normal	1 Phase Fault	2 Phase Fault	3 Phase Fault	Load Step Change	Bad Data
Accuracy	92.46%	95.18%	96.87%	97.02%	92.23%	99.02%

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