## 3 Load Curve Clustering under DPVS Influence
### Gaussian Mixture Model Cluster (GMM)  
The GMM (Gaussian Mixture Model) is a probabilistic clustering method [36]. The GMM assumes that the input samples are generated from k Gaussian distributions with unknown parameters, and it clusters samples that follow the same distribution.  
GMM clustering divides members based on probabilities and has more flexible cluster shapes. The result of GMM clustering is a series of probability values, where each individual in the sample corresponds to a specific category with its corresponding probability. The category with the highest probability is selected as the basis for classification.  
GMM clustering uses the Expectation-Maximization (EM) algorithm to estimate the parameters of individual Gaussian distribution functions in the GMM. The EM algorithm consists of an expectation step and a maximization step.  
Expected Step: Based on the given initial value or the previous step's maximized step, three sets of parameters are calculated to determine the probability \(\gamma_{j}(x_{i})\) of sample \(x_{i}\) being generated by the \(j\)-th category. The three sets of parameters are represented by Equation (3), and the calculation formula is given by Equation (4).  
\[\left\{\begin{array}{l}\alpha=\left[\alpha_{1},\alpha_{2},\ldots,\alpha_{j},\ldots,\alpha_{k}\right]^{\mathrm{T}}\\ \mu=\left[\mu_{1},\mu_{2},\ldots,\mu_{j},\ldots,\mu_{k}\right]^{\mathrm{T}}\\ \mathbf{\Sigma}=\left[\mathbf{\Sigma}_{1},\mathbf{\Sigma}_{2},\ldots,\mathbf{\Sigma}_{j}, \ldots,\mathbf{\Sigma}_{k}\right]^{\mathrm{T}}\end{array}\right. \tag{3}\]  
\[\gamma_{j}(x_{i})=\frac{\frac{\alpha_{j}}{\left(2\pi\right)^{\frac{\alpha_{j}}{ 2}}\left|\mathbf{\Sigma}_{j}\right|^{\frac{1}{2}}\mathrm{e}^{-\frac{1}{2}\left(x_{ i}-\mu_{j}\right)^{\mathrm{T}}\mathbf{\Sigma}_{j}^{-1}(x_{i}-\mu_{j})}}{\sum_{j=1}^{D} \frac{\alpha_{j}}{\left(2\pi\right)^{\frac{\alpha_{j}}{2}}\left|\mathbf{\Sigma}_{ j}\right|^{\frac{1}{2}}\mathrm{e}^{-\frac{1}{2}\left(x_{i}-\mu_{j}\right)^{\mathrm{T}} \mathbf{\Sigma}_{j}^{-1}(x_{i}-\mu_{j})}}} \tag{4}\]  
where \(\alpha_{j}\), \(\mu_{j}\), and \(\mathbf{\Sigma}_{j}\) are the weight, mean value, and covariance of the \(j\)-th Gaussian distribution, respectively; \(k\) is the total number of Gaussian distributions; \(x_{i}\) is the input data of the daily load scenario division; \(n\) is the dimension of \(x_{i}\); \(D\) Indicates the total number of days in the daily load database.  
Maximization step: Update the parameters \(\alpha_{j}\), \(\mu_{j}\), and \(\mathbf{\Sigma}_{j}\) in the above three groups according to the maximum likelihood estimation, as shown in Equation (5).  
\[\left\{\begin{array}{l}\alpha_{j}^{\prime}=\frac{1}{D}\sum\limits_{j=1}^{D} \gamma_{j}(x_{i})\\ \mu_{j}^{\prime}=\frac{\sum_{i=1}^{D}\gamma_{j}(x_{i})x_{i}}{\sum_{i=1}^{D} \gamma_{j}(x_{i})}\\ \mathbf{\Sigma}_{j}^{\prime}=\frac{\sum_{i=1}^{D}\gamma_{j}(x_{i}-\mu_{j}^{\prime} )\left(x_{i}-\mu_{j}^{\prime}\right)^{\mathrm{T}}}{\sum_{i=1}^{D}\gamma_{j}(x _{i})}\end{array}\right. \tag{5}\]  
Perform the expectation-maximization (EM) algorithm iteratively until the parameters converge or the likelihood function converges, to obtain the hyperparameters (optimal number of clusters) of the Gaussian mixture model (GMM) clustering for the daily load scenarios. The Bayesian information criterion (BIC) is commonly used to determine the optimal number of clusters, where the clustering quantity corresponding to the lowest BIC value is selected as the optimal value [37].  
Figure 2: Variation in the load curve under DPVS influence.