Multi-Level Evaluation for Flexible Load Regulation Potential in Distribution Network Based on Ensemble Clustering

Abstract

With the rapid increase in the renewable energy penetration rate in distribution networks, the volatility and uncertainty on the power supply side have become prominent; thus, it is urgent to fully utilize the regulation potential of the flexible load on the user side to maintain the dynamic balance of power. A multi-level evaluation method for flexible load regulation potential based on ensemble clustering is proposed in the paper. First, a data-driven approach based on ensemble clustering is adopted to quantify the user-level regulation potential of flexible load. Second, the bus-level regulation potential of the flexible load is obtained by aggregation calculation. Finally, a quantitative evaluation of the system-level regulation potential of flexible load in the distribution network is realized by constructing three optimization models with different objectives. Case studies show that the proposed method can effectively evaluate the regulation potential of flexible load in the distribution network from multiple levels, i.e., user level, bus level, and system level.

1. Introduction

In the context of the global energy transition, the continuous increase in the proportion of renewable energy sources has posed challenges of volatility and uncontrollability to the power system [1]. To ensure the safe and stable operation of distribution networks, there is an urgent need to tap into the regulation potential of various controllable resources such as battery energy storage devices, flexible load, and soft open points. Among these resources, the flexible load on the user side demonstrates unique advantages by virtue of its flexible response characteristics [2,3]. Therefore, the scientific assessment of the regulation potential of flexible load serves as a key technical foundation for giving full play to its regulation potential and supporting the stable operation of the system [4,5,6].

Scientific evaluation of the regulation potential of flexible load often requires collecting multi-dimensional data such as equipment parameters, user behavior characteristics, and environmental parameters of various types of flexible load, as well as conducting refined modeling of flexible load. On this basis, the evaluation of regulation potential is realized through methods such as the superposition method [7], state sequence model [8], and conditional value-at-risk (CVaR) model [9]. Gao et al. proposed a method combining a hierarchical federated perception algorithm and a hierarchical architecture, which enables the decomposition of flexible load at the substation level without exposing the privacy of individual users [10]. Wang et al. provided a predictive data foundation for assessing the dispatchable capacity and adjustable range of large-scale electric vehicle clusters by accurately characterizing the spatiotemporal aggregation characteristics of electric vehicle load [11]. Some studies further adopt the method of reduced-order modeling to lower the difficulty of modeling. For instance, Ge et al. describe the aggregated regulation potential of large-scale flexible load groups using a highly simplified low-order mathematical model, thereby reducing the difficulty of modeling flexible load groups [12]. Other studies have introduced advanced risk and optimization theories to address the uncertainties brought about by a high proportion of renewable energy: Lu et al. proposed a stochastic unit commitment model based on Conditional Value-at-Risk (CVaR). By quantifying the operational risks of isolated systems under high-proportion renewable energy penetration through CVaR, this model enables accurate evaluation of the risk-averse regulation potential of various flexible resources in the system [13]. Cao et al. utilized the theory of distributionally robust optimization to provide strict mathematical guarantees for addressing the uncertainties encountered in assessing the regulation potential of flexible load, ensuring that the assessed regulation potential remains reliable and usable under various extreme scenarios [14]. Xu et al. established an uncertainty model for renewable energy output using multi-scenario generation technology, and further proposed a comprehensive reliability assessment method for active distribution networks [15].

Although the aforementioned methods based on refined modeling of flexible load theoretically offer advantages in precision, in practical scenarios where large-scale heterogeneous flexible loads are connected, the difficulties in multi-dimensional data collection and high computational complexity lead to a sharp increase in engineering implementation costs and significantly limited operability. Therefore, some scholars at home and abroad have adopted data-driven approaches to bypass complex modeling and tap into the regulation potential of flexible load from their operational characteristics [16,17]. Baoyi et al. proposed a residential electricity load pattern recognition method based on the combination of K-Means clustering and a deep belief network. This method provides reliable distribution network maintenance for residential electricity load based on classification and recognition [18]. Li et al. assessed the spatiotemporal distribution and regulation potential of aggregated charging load through cluster analysis based on real-world electric vehicle travel data [19]. Wang et al. used a density-based clustering algorithm to classify electricity users into different groups, modeled the dynamic characteristics of each user’s electricity consumption using a Markov model, and further determined the demand response potential of users in each category [20]. Chen et al. proposed a two-stage model that integrates non-intrusive load disaggregation and advanced sequence prediction. In this model, the accurate separation of flexible loads is first achieved through a CNN-BiLSTM network, which effectively extracts their dynamic features; subsequently, the informer deep learning model is used for prediction [21]. Wang et al. first applied the singular value decomposition method to reduce the dimensionality of flexible load data, and then adopted a combined approach of K-Means clustering and convolutional neural network deep learning to improve the prediction accuracy of flexible loads [22]. Sun et al. proposed a two-layer clustering model that integrates K-Means, SOM, and BP neural networks, and conducted a comprehensive analysis of users’ load characteristics and regulation potential based on large-scale power load data [23].

However, existing studies still have some limitations. First, most works only focus on a single dimension, such as the user-level or bus-level, failing to construct a complete evaluation chain of “user–bus–system” and making it difficult to quantify the comprehensive value of flexible load from the global perspective of the system. Second, current data-driven evaluation methods mostly rely on a single clustering algorithm or the combination of a single clustering algorithm with deep learning technologies. The former has poor clustering stability, resulting in insufficient credibility of evaluation results; the latter, which combines with deep learning, usually relies on a large amount of data for model training and has low interpretability, which limits its practical application. It is worth noting that some scholars have applied ensemble clustering algorithms to the fields of load pattern identification and feature extraction [24,25]; for example, Sarmas et al. verified the superiority of ensemble clustering algorithms in load pattern identification by comparing multiple clustering methods [26]. Inspired by this, the paper proposes a multi-level evaluation method for flexible load regulation potential based on ensemble clustering. The main contributions of this paper are as follows:

(1)

A “user–bus–system” evaluation framework is established, thereby enabling a comprehensive assessment of flexible load regulation potential across multiple levels.

(2)

The stability and robustness of load clustering are enhanced by the ensemble clustering algorithm, thus providing a reliable foundation for bus-level assessment.

(3)

The system-level regulation effects of flexible loads are quantified by solving multiple system-level optimization models, which address minimizing the average fluctuation of system load, maximizing renewable energy consumption, and reducing the peak-valley difference in system load, respectively, and by a set of system contribution indexes.

The paper is organized as follows. Section 2 introduces the processes of the ensemble clustering algorithm and the user-level regulation potential evaluation of flexible load; Section 3 introduces the processes of the bus-level and system-level regulation potential evaluation of flexible load; and Section 4 presents the case studies for validating the effectiveness of the proposed evaluation method. Finally, Section 5 provides a summary of the paper.

2. Quantitative Evaluation of User-Level Flexible Load Regulation Potential

2.1. Quantitative Evaluation Framework for Multi-Level Flexible Load Regulation Potential

Figure 1 illustrates the “user–bus–system” three-level evaluation framework for the regulation potential of flexible load on the user side of the distribution network. First, at the user level, an ensemble clustering algorithm is utilized to analyze users’ historical load data, to characterize the baseline load of each user and their bidirectional regulation potential. Then, at the bus level, based on the topology of distribution networks, the load regulation potential of all users connected to the same bus is aggregated to generate the regulation potential of each bus in the distribution network. Finally, at the system level, three optimization models with different objective functions, i.e., minimizing the average fluctuation of system load, maximizing renewable energy consumption, and minimizing the system load peak-valley difference, respectively, are constructed. By solving the three optimization models, the value and efficiency of flexible load at the system level are comprehensively evaluated.

2.2. Ensemble Clustering Algorithm

The ensemble clustering algorithm typically involves two important steps: the first step is to generate a set of base clusters, and the second step is to construct a consensus matrix from the generated set of base clusters and ensemble base clusters to obtain the final clustering result [27]. Figure 2 illustrates the basic workflow of the ensemble clustering algorithm.

The specific steps for clustering the original dataset ${\mathrm{X}}_N$ using the ensemble clustering algorithm constructed in this paper are detailed below:

(1)

Perform B times of Bootstrap Resampling on the original dataset ${\mathrm{X}}_N$ to generate B training sample sets $\mathrm{X}=\{{\mathrm{X}}_N^1,{\mathrm{X}}_N^2,\cdots ,{\mathrm{X}}_N^B\}$. The original dataset ${\mathrm{X}}_N=\{x_1,x_2,\cdots ,x_i,{\cdots ,x}_N\}$ contains N historical load curves, where $x_i$ represents the i-th load curve in the dataset ${\mathrm{X}}_N$.

The primary purpose of resampling is to enhance the overall stability of the algorithm by performing clustering on multiple sample sets, thereby mitigating the effects of randomness in initial center selection and interference from data outliers. Furthermore, it is established in the literature that a bootstrap size of 10 (B = 10) is typically sufficient to virtually encompass all distinct data points from the original dataset [24].

(2)

Perform K-Means clustering on each training sample set to obtain B base cluster results $\mathsf{\Pi }=\{{\mathsf{\pi }}^1,{\mathsf{\pi }}^2,\cdots ,{\mathsf{\pi }}^b,\cdots ,{\mathsf{\pi }}^B\}$, where ${\mathsf{\pi }}^b=\{{\mathrm{C}}_1^b,{\mathrm{C}}_2^b,\cdots ,{\mathrm{C}}_i^b\}$ represents the b-th base cluster, and ${\mathrm{C}}_i^b$ denotes the i-th cluster in the b-th base cluster.

(3)

Based on these base clusters, a consensus matrix $A$ is constructed. The consensus matrix $A$ records the frequency at which any two load curves in the original dataset ${\mathrm{X}}_N$ are assigned to the same cluster across all base clusters. It is an $N\times N$ matrix, and the calculation formula for its element $a_{ij}$ (located at the i-th row and j-th column) is as follows:

$$\begin{array}{c}{a}_{ij}={\displaystyle \frac{1}{B}}{\displaystyle \sum _{b=1}^B}\mathsf{\delta }\left({\mathsf{\pi }}^b\left(x_i,x_j\right)\right)\end{array}$$

(1)

$$\begin{array}{c}\delta \left({\mathsf{\pi }}^b\left(x_i,x_j\right)\right)=\left\{\begin{array}{c}1, x_i \text{and} x_j \text{belong to the same cluster in the base cluster }{\pi }^b\hfill \\ 0, \text{other cases}\end{array}\right.\end{array}$$

(2)

where $x_i$ and $x_j$, respectively, represent the i-th and j-th load curves in the original dataset ${\mathrm{X}}_N$.

(4)

Based on the consensus matrix, a hierarchical clustering algorithm is used to perform the final clustering division on the original dataset ${\mathrm{X}}_N$. Each load curve in the original dataset ${\mathrm{X}}_N$ is regarded as an independent cluster, and then the two clusters with the highest similarity are merged iteratively until the number of clusters reaches the optimal number of clusters $\mathrm{K}$, at which point the final clustering result is obtained. The calculation formula for the inter-cluster similarity $S\left(C_p,C_q\right)$ is as follows:

$$\begin{array}{c}S\left(C_p,C_q\right)={\displaystyle \frac{1}{n_p\cdot n_q}}{\displaystyle \sum _{i\in S_{C_p}}}{\displaystyle \sum _{j\in S_{C_q}}}{a}_{ij}\end{array}$$

(3)

where $n_p$ and $n_q$ represent the number of samples in the cluster $C_p$ and $C_q$, and $S_{C_p}$ and $S_{C_q}$ denote the index sets of samples in the cluster $C_p$ and $C_q$, respectively.

The DBI (Davies–Bouldin Index) is used to determine the optimal number of clusters $\mathrm{K}$ for the original dataset ${\mathrm{X}}_N$ in this paper. The DBI, shown as (4), measures the compactness of intra-cluster samples and the separation of inter-cluster samples and a smaller value indicates a better clustering effect [24].

$$\begin{array}{c}{I}_{\mathrm{DBI}}={\displaystyle \frac{1}{k}}{\displaystyle \sum _{i=1}^k}\left[\underset{i\ne j}{\max }\left({\displaystyle \frac{d\left(C_i\right)+d\left(C_j\right)}{d\left(C_i,C_j\right)}}\right)\right]\end{array}$$

(4)

where $I_{\mathrm{DBI}}$ denotes the value of DBI; $d\left(C_i\right)$ and $d\left(C_j\right)$ represent the intra-cluster average distances of clusters $C_i$ and $C_j$, respectively; $d\left(C_i,C_j\right)$ is the centroid distance between $C_i$ and $C_j$; and $k$ is the number of clusters.

When specifically determining the optimal number of clusters $\mathrm{K}$, it is necessary to iterate through the range of $k$ values, perform K-Means clustering on the original dataset ${\mathrm{X}}_N$ sequentially, and calculate the DBI. Finally, the value of $k$ corresponding to the minimum DBI is selected as the optimal number of clusters, denoted as $\mathrm{K}$. It should be noted that the value range for the number of clusters, k, is typically determined based on the data characteristics and research objectives. Considering the characteristics of the power load curve data used in this paper and the goal of cluster analysis, the value range for k is set from 2 to 10 [28].

Through the above steps, the original dataset ${\mathrm{X}}_N$ can be divided into $\mathrm{K}$ clusters using the ensemble clustering algorithm.

2.3. Quantitative Evaluation of User-Level Regulation Potential

The evaluation process of user-level regulation potential is as follows:

(1)

A full year of historical load data prior to the regulation date is first acquired for each user. This period is chosen as it is considered to virtually encompass the entire spectrum of the user’s electricity consumption characteristics [15].

(2)

By calculating the average power of the user’s load during the $N$ days before regulation, the user’s baseline load curve $P_{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{e}}\left(t\right)$ is constructed:

$$\begin{array}{c}{P}_{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{e}}\left(t\right)={\displaystyle \frac{{\sum }_{i=1}^N{P}_i\left(t\right)}{N}}, t=\mathrm{1,2},\dots ,T\end{array}$$

(5)

where $P_i$ represents the load curve of the i-th day before the regulation day and $T$ denotes the total number of time intervals. If the dataset adopts a 15 min time resolution, then $T=96$.

(3)

The baseline load curve $P_{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{e}}\left(t\right)$ is classified into the cluster $C_M$ with which it has the maximum membership degree. The calculation formula for the membership degree ${\xi }_i$ between the baseline load curve $P_{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{e}}\left(t\right)$ and cluster $C_i$ is as follows:

$$\begin{array}{c}{\xi }_i={\displaystyle \frac{1}{\sum _{j=1}^{\mathrm{K}}{\left(\frac{{‖P_{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{e}}(t)-{\overline{P}}_{C_i}(t)‖}^2}{{‖P_{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{e}}(t)-{\overline{P}}_{C_j}(t)‖}^2}\right)}^2}}, i=\mathrm{1,2},\dots ,\mathrm{K}\end{array}$$

(6)

where ${\overline{P}}_{C_i}(t)$ and ${\overline{P}}_{C_j}(t)$ represent the sample centroids of cluster $C_i$ and $C_j$, respectively.

(4)

After determining the cluster $C_M$ to which the user’s baseline load curve $P_{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{e}}\left(t\right)$ belongs, load power values are extracted at each time $t$ from the historical load data in the cluster $C_M$. The 0.95th and 0.05th percentiles of these values are then calculated, which are defined as the user’s upper power limit $P_{\mathrm{m}\mathrm{a}\mathrm{x}}\left(t\right)$ and lower power limit $P_{\mathrm{m}\mathrm{i}\mathrm{n}}\left(t\right)$ at time $t$, respectively. The interval ${[P}_{\mathrm{m}\mathrm{i}\mathrm{n}}\left(t\right),P_{\mathrm{m}\mathrm{a}\mathrm{x}}\left(t\right)]$ represents the typical power fluctuation range of the user at time t.

(5)

By comparing the baseline load $P_{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{e}}\left(t\right)$ on the regulation day with the fluctuation interval ${[P}_{\mathrm{m}\mathrm{i}\mathrm{n}}\left(t\right),P_{\mathrm{m}\mathrm{a}\mathrm{x}}\left(t\right)]$, the user-level regulation potential is quantified as follows:

$$\begin{array}{c}{∆P}_{\mathrm{u}\mathrm{p}}(t)=\max (0,P_{\mathrm{m}\mathrm{a}\mathrm{x}}\left(t\right)-P_{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{e}}(t))\end{array}$$

(7)

$$\begin{array}{c}{∆P}_{\mathrm{d}\mathrm{o}\mathrm{w}\mathrm{n}}\left(t\right)=\max (0,P_{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{e}}(t)-P_{\mathrm{m}\mathrm{i}\mathrm{n}}\left(t\right))\end{array}$$

(8)

where ${∆P}_{\mathrm{u}\mathrm{p}}(t)$ and ${∆P}_{\mathrm{d}\mathrm{o}\mathrm{w}\mathrm{n}}\left(t\right)$ represent the user’s upward regulation potential and downward regulation potential, respectively.

Finally, the upward and downward regulation potential curves of the user are formed by the sets $\left\{{∆P}_{\mathrm{u}\mathrm{p}}\left(t\right)\right|t=\mathrm{1,2},\cdots ,\mathrm{T}\}$ and $\left\{{-∆P}_{\mathrm{d}\mathrm{o}\mathrm{w}\mathrm{n}}\right(t\left)\right|t=\mathrm{1,2},\cdots ,\mathrm{T}\}$, respectively, thereby completing the quantification of the user-level regulation potential.

3. Quantitative Evaluation of Flexible Load Regulation Potential at Bus-Level and System-Level

3.1. Quantitative Evaluation of Bus-Level Regulation Potential

The quantification of bus-level regulation potential plays a connecting role in the entire evaluation framework: it is not only the spatial aggregation of user-level evaluation results, but also the data foundation for system-level optimization analysis.

The bus-level regulation potential is obtained by aggregating the potential curves of all users connected to the same bus. Specifically, for a given bus $m$, its bus-level baseline load $P_{\mathrm{b}\mathrm{u}\mathrm{s},m}^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{e}}(t)$ is obtained by adding the baseline load values of the users connected to bus m at time $t$, shown as follows:

$$\begin{array}{c}{P}_{\mathrm{b}\mathrm{u}\mathrm{s},m}^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{e}}(t)={\displaystyle \sum _{i=1}^{N_m}}{P}_{m_i}^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{e}}\left(t\right)\end{array}$$

(9)

where $P_{m_i}^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{e}}\left(t\right)$ represents the baseline load of the i-th user connected to bus $m$ at time $t$.

Similarly, the curves of the total upward regulation potential ${∆P}_{\mathrm{b}\mathrm{u}\mathrm{s},m}^{\mathrm{u}\mathrm{p}}\left(t\right)$ and total downward regulation potential ${∆P}_{\mathrm{b}\mathrm{u}\mathrm{s},m}^{\mathrm{d}\mathrm{o}\mathrm{w}\mathrm{n}}\left(t\right)$ of bus $m$ are generated by adding the values of user-level potential curves at the same time $t$:

$$\begin{array}{c}{∆P}_{\mathrm{b}\mathrm{u}\mathrm{s},m}^{\mathrm{u}\mathrm{p}}\left(t\right)={\displaystyle \sum _{i=1}^{N_m}}{∆P}_{m_i}^{\mathrm{u}\mathrm{p}}\left(t\right)\end{array}$$

(10)

$$\begin{array}{c}{∆P}_{\mathrm{b}\mathrm{u}\mathrm{s},m}^{\mathrm{d}\mathrm{o}\mathrm{w}\mathrm{n}}\left(t\right)={\displaystyle \sum _{i=1}^{N_m}}{∆P}_{m_i}^{\mathrm{d}\mathrm{o}\mathrm{w}\mathrm{n}}\left(t\right)\end{array}$$

(11)

where ${∆P}_{m_i}^{\mathrm{u}\mathrm{p}}\left(t\right)$ and ${∆P}_{m_i}^{\mathrm{d}\mathrm{o}\mathrm{w}\mathrm{n}}\left(t\right)$ represent the upward regulation potential and downward regulation potential of the i-th user connected to bus $m$ at time $t$, respectively.

Through the above calculations, the spatial aggregation of regulation potential from the user-level to the bus-level can be completed.

3.2. Quantitative Evaluation of System-Level Regulation Potential

Through the aforementioned method, the baseline load, upward regulation potential, and downward regulation potential of flexible load at each bus in the distribution network can be obtained. On this basis, according to the actual operational requirements of the distribution network, three objective functions are established: minimizing the average fluctuation of the system load, maximizing the renewable energy consumption, and minimizing the peak-valley difference in the system load. Comprehensive consideration is given to distribution network operational constraints, flexible load regulation constraints, and other factors, thereby constructing a system-level evaluation model for flexible load regulation potential. Furtherly, the contribution values of flexible load to the distribution network are assessed.

3.2.1. Optimization Objectives

(1)

With the objective of minimizing the average fluctuation of system load,

$$\begin{array}{c}\mathrm{min }{F}_{\mathrm{a}}={\displaystyle \frac{1}{N_{\mathrm{t}}-1}}{\displaystyle \sum _{i=1}^{N_{\mathrm{b}\mathrm{u}\mathrm{s}}}}{\displaystyle \sum _{t=1}^{N_{\mathrm{t}}-1}}\left|P_{\mathrm{b}\mathrm{u}\mathrm{s},i}\left(t+1\right)-P_{\mathrm{b}\mathrm{u}\mathrm{s},i}\left(t\right)\right|\end{array}$$

(12)

where $F_{\mathrm{a}}$ is the average fluctuation of system load; $N_{\mathrm{b}\mathrm{u}\mathrm{s}}$ is the number of buses in the distribution network; $N_{\mathrm{t}}$ is the number of sampling time points of the daily load; and $P_{\mathrm{b}\mathrm{u}\mathrm{s},i}\left(t\right)$ represents the load value of bus $i$ at time $t$.

(2)

With the objective of maximizing the consumption of renewable energy,

$$\begin{array}{c}\mathrm{max }{F}_{\mathrm{c}}={\displaystyle \sum _{i=1}^{N_{\mathrm{b}\mathrm{u}\mathrm{s}}}}{\displaystyle \sum _{t=1}^{N_{\mathrm{t}}}}{P}_{\mathrm{b}\mathrm{u}\mathrm{s},i}^{\mathrm{n}\mathrm{e}\mathrm{w}}(t)\end{array}$$

(13)

where $F_{\mathrm{c}}$ is the consumption of renewable energy $\text{and}{ P}_{\mathrm{b}\mathrm{u}\mathrm{s},i}^{\mathrm{n}\mathrm{e}\mathrm{w}}(t)$ represents the renewable energy consumption of bus $i$ at time $t$.

(3)

With the objective of minimizing the peak-valley difference in system load,

$$\begin{array}{c}\min F_{\mathrm{p}}=\underset{t\in \{\mathrm{1,2},\cdots ,N_{\mathrm{t}}\}}{\max }\left({\displaystyle \sum _{i=1}^{N_{\mathrm{b}\mathrm{u}\mathrm{s}}}}{P}_{\mathrm{b}\mathrm{u}\mathrm{s},i}\left(t\right)\right)-\underset{t\in \{\mathrm{1,2},\cdots ,N_{\mathrm{t}}\}}{\min }\left({\displaystyle \sum _{i=1}^{N_{\mathrm{b}\mathrm{u}\mathrm{s}}}}{P}_{\mathrm{b}\mathrm{u}\mathrm{s},i}\left(t\right)\right)\end{array}$$

(14)

where $F_{\mathrm{p}}$ is the peak-valley difference in system load.

3.2.2. Constraints

The constraint conditions include power flow balance constraints, bus voltage and branch current constraints, regulation potential constraints, etc. The details are as follows:

(1)

Power flow balance constraints. These constraints ensure the real-time balance of active power and reactive power in the distribution network:

$$\begin{array}{c}{\displaystyle \sum _{k\in w\left(j\right)}}{P}_{jk}\left(t\right)-{\displaystyle \sum _{i\in u\left(j\right)}}\left(P_{ij}\left(t\right)-R_{ij}{I}_{ij}^2\left(t\right)\right)=P_{\mathrm{b}\mathrm{u}\mathrm{s},j}^{\mathrm{n}\mathrm{e}\mathrm{w}}{\left(t\right)-P}_{\mathrm{b}\mathrm{u}\mathrm{s},j}^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{e}}\left(t\right)-{∆P}_{\mathrm{b}\mathrm{u}\mathrm{s},j}\left(t\right)\end{array}$$

(15)

$$\begin{array}{c}{\displaystyle \sum _{k\in w\left(j\right)}}{Q}_{jk}\left(t\right)-{\displaystyle \sum _{i\in u\left(j\right)}}\left(Q_{ij}\left(t\right)-X_{ij}{I}_{ij}^2\left(t\right)\right)={-(P}_{\mathrm{b}\mathrm{u}\mathrm{s},j}^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{e}}\left(t\right)+{∆P}_{\mathrm{b}\mathrm{u}\mathrm{s},j}\left(t\right))·\tan {\phi }_j\left(t\right)\end{array}$$

(16)

where $u\left(j\right)$ denotes the set of starting bus of branches with $j$ as the ending bus; $w\left(j\right)$ denotes the set of ending bus of branches with $j$ as the starting bus; $P_{ij}\left(t\right)$ and $Q_{ij}\left(t\right)$ represent the active power and reactive power flowing through branch $ij$ at time $t$, respectively; $P_{jk}\left(t\right)$ and $Q_{jk}\left(t\right)$ represent the active power and reactive power flowing through branch $jk$ at time $t$, respectively; ${\phi }_j\left(t\right)$ is the power factor angle of the load at bus $j$ at time $t$; and ${∆P}_{\mathrm{b}\mathrm{u}\mathrm{s},j}\left(t\right)$ is the load adjustment amount of bus $j$ at time $t$.

(2)

Bus voltage and branch current constraints. These constraints ensure that the voltage of all buses is within the allowable range and that the branch current does not exceed the thermal limit, to prevent equipment overload and ensure power supply safety:

$$\begin{array}{c}{V}_j^2\left(t\right)=V_i^2\left(t\right)-2\left(R_{ij}{P}_{ij}\left(t\right)+X_{ij}{Q}_{ij}\left(t\right)\right)+Z_{ij}^2{I}_{ij}^2\left(t\right)\end{array}$$

(17)

$$\begin{array}{c}{V}_{i,\mathrm{m}\mathrm{i}\mathrm{n}}\le V_i\left(t\right)\le V_{i,\mathrm{m}\mathrm{a}\mathrm{x}}\end{array}$$

(18)

$$\begin{array}{c}{I}_{ij}^2\left(t\right)\le I_{ij,\mathrm{m}\mathrm{a}\mathrm{x}}^2\end{array}$$

(19)

where $V_i\left(t\right)$ denotes the voltage amplitude of bus $i$ at time $t$; $R_{ij}$ represents the resistance of branch $ij$; $X_{ij}$ stands for the reactance of branch $ij$; $Z_{ij}$ indicates the impedance of branch $ij$; $V_{i,\mathrm{m}\mathrm{a}\mathrm{x}}$ and $V_{i,\mathrm{m}\mathrm{i}\mathrm{n}}$ represent the upper limit and lower limit of voltage of bus $i$, respectively; $I_{ij}\left(t\right)$ denotes the current-carrying capacity of branch $ij$ at time $t$; and $I_{ij,\mathrm{m}\mathrm{a}\mathrm{x}}$ represents the safe current of branch $ij$.

(3)

Regulation potential constraints. These constraints ensure that the actual load adjustment amount of each bus does not exceed its upper and lower limits, and that the output of renewable energy does not exceed the predicted value:

$$\begin{array}{c}-{∆P}_{\mathrm{b}\mathrm{u}\mathrm{s},j}^{\mathrm{d}\mathrm{o}\mathrm{w}\mathrm{n}}\left(t\right)\le {∆P}_{\mathrm{b}\mathrm{u}\mathrm{s},j}\left(t\right)\le {∆P}_{\mathrm{b}\mathrm{u}\mathrm{s},j}^{\mathrm{u}\mathrm{p}}\left(t\right)\end{array}$$

(20)

$$\begin{array}{c}0\le P_{\mathrm{b}\mathrm{u}\mathrm{s},j}^{\mathrm{n}\mathrm{e}\mathrm{w}}\left(t\right)\le P_{\mathrm{b}\mathrm{u}\mathrm{s},j}^{\mathrm{n}\mathrm{e}\mathrm{w},\mathrm{p}\mathrm{r}\mathrm{e}}\left(t\right)\end{array}$$

(21)

where ${∆P}_{\mathrm{b}\mathrm{u}\mathrm{s},j}\left(t\right)$ is the load adjustment amount of bus $j$ at time $t$; $P_{\mathrm{b}\mathrm{u}\mathrm{s},j}^{\mathrm{n}\mathrm{e}\mathrm{w}}\left(t\right)$ is the active power output of renewable energy of bus $j$ at time $t$; and $P_{\mathrm{b}\mathrm{u}\mathrm{s},j}^{\mathrm{n}\mathrm{e}\mathrm{w},\mathrm{p}\mathrm{r}\mathrm{e}}\left(t\right)$ is the predicted value of renewable energy generation of bus $j$ at time $t$.

3.2.3. Convex Relaxation of the Optimization Model

To improve the solution efficiency and numerical stability, the original non-convex optimization problem is transformed into a convex programming form for solution using convex relaxation techniques [9], as follows:

First, variables $l_{ij}\left(t\right)=I_{ij}^2\left(t\right)$ and $v_i\left(t\right)=V_i^2\left(t\right)$ are introduced to eliminate the quadratic terms of current and voltage. The active power and reactive power balance constraints are converted into the following forms:

$$\begin{array}{c}{\displaystyle \sum _{k\in w\left(j\right)}}{P}_{jk}\left(t\right)-{\displaystyle \sum _{i\in u\left(j\right)}}\left(P_{ij}\left(t\right)-R_{ij}{l}_{ij}\left(t\right)\right)=P_{\mathrm{b}\mathrm{u}\mathrm{s},j}^{\mathrm{n}\mathrm{e}\mathrm{w}}{\left(t\right)-P}_{\mathrm{b}\mathrm{u}\mathrm{s},j}^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{e}}\left(t\right)-{∆P}_{\mathrm{b}\mathrm{u}\mathrm{s},j}\left(t\right)\end{array}$$

(22)

$$\begin{array}{c}{\displaystyle \sum _{k\in w\left(j\right)}}{Q}_{jk}\left(t\right)-{\displaystyle \sum _{i\in u\left(j\right)}}\left(Q_{ij}\left(t\right)-X_{ij}{l}_{ij}\left(t\right)\right)={-(P}_{\mathrm{b}\mathrm{u}\mathrm{s},j}^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{e}}\left(t\right)+{∆P}_{\mathrm{b}\mathrm{u}\mathrm{s},j}\left(t\right))·\tan {\phi }_j\left(t\right)\end{array}$$

(23)

Next, the big-M method and the second-order cone relaxation (SOCR) method are adopted to convert the power flow constraints and the voltage–current security constraints into the following forms:

$$\begin{array}{c}{v}_i\left(t\right)-v_j\left(t\right)-2\left(R_{ij}{P}_{ij}\left(t\right)+X_{ij}{Q}_{ij}\left(t\right)\right)+Z_{ij}^2{I}_{ij}^2\left(t\right)+M\left(1-{\alpha }_{ij}\left(t\right)\right)\ge 0\end{array}$$

(24)

$$\begin{array}{c}{v}_i\left(t\right)-v_j\left(t\right)-2\left(R_{ij}{P}_{ij}\left(t\right)+X_{ij}{Q}_{ij}\left(t\right)\right)+Z_{ij}^2{I}_{ij}^2\left(t\right)+M\left(1-{\alpha }_{ij}\left(t\right)\right)\le 0\end{array}$$

(25)

$$\begin{array}{c}{‖{\left[2P_{ij}\left(t\right),2Q_{ij}\left(t\right){,l}_{ij}\left(t\right)-v_i\left(t\right)\right]}^T‖}_2\le l_{ij}\left(t\right)+v_i\left(t\right)\end{array}$$

(26)

$$\begin{array}{c}{V}_{i,\min }^2\le v_i\left(t\right)\le V_{i,\max }^2\end{array}$$

(27)

$$\begin{array}{c}0\le l_{ij}\left(t\right)\le {{\alpha }_{ij}\left(t\right)I}_{ij,\max }^2\end{array}$$

(28)

where $M$ is a sufficiently large number (needing only to be two orders of magnitude larger than $v_i\left(t\right)$), and ${\alpha }_{ij}\left(t\right)$ denotes the state variable for branch switching—taking 0 when the branch is disconnected and 1 when closed.

The SOCR applied to the DistFlow model is known to be exact (i.e., the relaxation is tight) for radial distribution networks under typical operating conditions [29,30]. The key conditions guaranteeing tightness are as follows:

(1)

The network topology is radial.

(2)

The objective function is monotonically increasing with respect to branch power losses.

(3)

The system does not operate at the boundary of the feasible region (e.g., no line power reaches its limit, and node voltages are far from the upper and lower bounds).

In the context of the proposed optimization model for evaluating the regulation potential of flexible load, the aforementioned conditions for tightness are satisfied for the following reasons:

(1)

The IEEE 33-bus and PG&E 69-bus test systems used in the case studies are radial in structure and operate within normal voltage and current limits, thus fulfilling conditions (1) and (3).

(2)

The primary objective functions defined in Equations (12) and (14), which aim to minimize load fluctuation and peak-valley difference, respectively, inherently discourage excessive power flow and associated losses. Since branch power losses $\left(R_{ij}{l}_{ij}\left(t\right)\right)$ are monotonically increasing functions of the squared current magnitude $l_{ij}\left(t\right)$, minimizing system-level power variations implicitly creates pressure to reduce $l_{ij}\left(t\right)$. Therefore, these objective functions are consistent with the requirement of being non-decreasing with respect to branch losses, ensuring the relaxation’s exactness. For the objective of maximizing renewable consumption (Equation (13)), the relaxation remains exact in radial networks as the solution typically lies within a non-boundary operating point.

Therefore, the solutions obtained from the relaxed SOCR model (Equations (22)–(28)) are globally optimal for the original non-convex problem, ensuring the accuracy of evaluation results.

3.2.4. Contribution Degree Indexes

To quantitatively analyze the effect of flexibility load on smoothing fluctuation of system load, enhancing consumption of renewable energy and minimizing the load peak-valley difference, three contribution degree indexes are set as follows:

(1)

Contribution degree to the average fluctuation of system load:

$$\begin{array}{c}{D}_{\mathrm{a}}={\displaystyle \frac{F_{\mathrm{a}}^{\left(0\right)}-F_{\mathrm{a}}^{\ast }}{F_{\mathrm{a}}^{\left(0\right)}}}\times 100\%\end{array}$$

(29)

(2)

Contribution degree to renewable energy consumption:

$$\begin{array}{c}{D}_{\mathrm{c}}={\displaystyle \frac{F_{\mathrm{c}}^{\ast }{-F}_{\mathrm{c}}^{\left(0\right)}}{F_{\mathrm{c}}^{\left(0\right)}}}\times 100\%\end{array}$$

(30)

(3)

Contribution degree to the peak-valley difference in system load:

$$\begin{array}{c}{D}_{\mathrm{p}}={\displaystyle \frac{F_{\mathrm{p}}^{\left(0\right)}-F_{\mathrm{P}}^{\ast }}{F_{\mathrm{p}}^{\left(0\right)}}}\times 100\%\end{array}$$

(31)

where $D_{\mathrm{a}}$, $D_{\mathrm{c}}$, and $D_{\mathrm{p}}$ represent the contribution degrees of flexible load to minimizing the average fluctuation of system load, maximizing renewable energy consumption, and minimizing the peak-valley difference in system load, respectively; $F_{\mathrm{a}}^{\left(0\right)}$, $F_{\mathrm{c}}^{\left(0\right)}$, and $F_{\mathrm{p}}^{\left(0\right)}$ denote the average fluctuation of system load, renewable energy consumption, and peak-valley difference in system load without considering the regulation potential of flexible load, respectively; and $F_{\mathrm{a}}^{\ast }$, $F_{\mathrm{c}}^{\ast }$, and $F_{\mathrm{P}}^{\ast }$ stand for the average fluctuation of system load, renewable energy consumption, and peak-valley difference in system load when the regulation potential of flexible load is considered, respectively.

4. Case Study Analysis

4.1. Case Introduction

A modified IEEE 33-bus distribution network, the structure of which is shown in Figure 3, is used for a case study to verify the effectiveness of the evaluation framework proposed in this paper. This system consists of 33 buses and 34 branches, with a base voltage of 12.66 kV and a total maximum system load of (7.617 + j3.473) MVA. Distributed wind power sources with maximum capacities of 1.7 MVA and 2.0 MVA are connected to bus 5 and bus 13 in the distribution network, respectively.

The historical load data used herein is from the annual electricity load dataset of 2009 users in a certain region, provided by the National Basic Discipline Public Science Data Center in China, with a time resolution of 15 min. In this case study, flexible load users are connected to buses 9, 11, 16, 20, 24, and 29, while the load at the remaining buses are rigid load. The number of flexible load users connected to each bus is presented in

Table 1. Number of flexible load users connected to each bus.

Bus	9	11	16	20	24	29
Number of flexible load users	11	9	12	10	11	9

.

This case study aims to verify the following two aspects:

(1)

The stability and robustness of the proposed ensemble clustering algorithm in the quantitative evaluation of user-level regulation potential;

(2)

The effectiveness of the proposed system-level optimization model in quantitatively evaluating the system-level contribution of flexible loads to the distribution network.

4.2. Performance Verification of Ensemble Clustering

(1)

Stability analysis of ensemble clustering

Accurate and stable clustering of user load curves is the cornerstone for precise quantification of user-level regulation potential. In view of this, this section conducts a systematic comparison between the ensemble clustering algorithm proposed in this paper and three single clustering algorithms, i.e., K-Means, hierarchical clustering, and density-based spatial clustering of applications with noise (DBSCAN), to verify the stability of the proposed algorithm in load curve clustering.

The DBI is used as the criterion for evaluating clustering quality. The verification is conducted from two perspectives: the stability under repeated clustering for a single user and the stability across different users. First, the K-Means clustering, hierarchical clustering, DBSCAN, and ensemble clustering algorithms are used to conduct multiple clustering experiments on the load data of the same user in the dataset. The boxplots of DBI are shown in Figure 4a. The box height corresponding to the ensemble clustering algorithm is the smallest, indicating that its clustering results have the lowest volatility and the best stability for the specific user. To further validate its generalizability, we extend the experiment to 100 users from the dataset. The DBI distributions of the clustering results across these users are illustrated in Figure 4b. Similarly, the box corresponding to the ensemble clustering algorithm exhibits the smallest height and the lowest position among all methods. This demonstrates that the proposed algorithm not only maintains minimal volatility for a single user but also achieves the most consistent and superior clustering performance across a diverse user population, thereby confirming its exceptional stability.

(2)

Robustness analysis of ensemble clustering

To verify the robustness of the ensemble clustering algorithm in resisting noise interference, simulated load data with known category labels are used to conduct clustering performance tests. The simulated dataset contains 4 categories of daily load curves of users with distinct patterns, with 200 load curves in each category. By injecting noise curves of different noise content into the data, the impact of noise content on the performance of four algorithms—K-Means, hierarchical clustering, DBSCAN and the ensemble clustering algorithm—is evaluated from two aspects: the clustering accuracy rate and DBI. The experimental results are shown in Figure 5.

As can be seen from Figure 5, with the increase in noise content, the clustering accuracy rates decrease and the values of DBI increase, indicating that noise could greatly degrade the clustering effect. However, the accuracy rate and the value of the DBI by ensemble clustering are still the highest and smallest, respectively, which verifies that the ensemble clustering algorithm is the least affected by noise among the four algorithms and has good robustness when facing noise disturbances.

4.3. Evaluation of User-Level Regulation Potential

The above experiments confirm the stability and robustness of the ensemble clustering algorithm in the clustering of load curves, which provides the foundation for the accurate quantitative evaluation of user-level regulation potential. Figure 6 shows the results of clustering the load curves of a certain user at bus 9 by the ensemble clustering algorithm. It can be observed that the load curves are divided into three clusters with significant differences, and each cluster represents a distinct electricity consumption behavior pattern, which intuitively reflects the diversity of user load characteristics.

Based on the aforementioned clustering results, Figure 7 presents the quantitative results of the regulation potential for this user. The figure clearly shows the upper and lower limits of the user’s load power, the upper and lower limits of regulation potential, and the temporal distribution characteristics of the user’s load power and regulation potential, providing data support for the subsequent bus-level and system-level evaluations.

4.4. Evaluation of System-Level Regulation Potential

Figure 8, Figure 9 and Figure 10 show the regulatory effects of flexible load by three optimization objectives: minimizing the average fluctuation of system load, maximizing the consumption of renewable energy, and minimizing the peak-valley difference in system load. The results indicate that flexible load can play an effective regulatory role in reducing the fluctuation of system load, promoting wind power consumption, and decreasing the peak-valley difference in system load.

Figure 11 presents the contribution degree of flexible loads to each operational objective. As can be seen from the figure, in this case study, the contribution of flexible loads to minimizing the average fluctuation of system load is the most prominent, while they also make significant contributions to maximizing wind power consumption and minimizing the peak-valley difference in system load.

The above results verify that by solving and analyzing the system-level flexible load regulation potential evaluation model proposed in this paper, the contribution degree of flexible loads to the distribution network system in aspects such as minimizing the average fluctuation of system load, maximizing the accommodation of renewable energy, and minimizing the peak-valley difference in system load can be effectively quantified.

4.5. Cross-System Validation Based on the PG&E 69-Bus System

This section aims to further validate the effectiveness of the proposed multi-level evaluation framework by applying it to a larger and more complex US PG&E 69-bus distribution system. Compared to the IEEE 33-bus system, the PG&E 69-bus system, which is a simplification of a real-world grid, features a larger scale and more representative topological structure, enabling more effective simulation of real-world distribution networks.

4.5.1. Case Introduction for the Modified PG&E 69-Bus Distribution System

Figure 12 illustrates the structure of the modified PG&E 69-bus distribution system. The modified PG&E 69-bus distribution system consists of 69 buses and 73 branches, with a base voltage of 12.66 kV and a total maximum system load of (18.063 + j7.874) MVA. Distributed wind power sources with maximum capacities of 1.7 MVA and 2.0 MVA are connected to bus 20 and bus 30 in the distribution network, respectively, and a distributed photovoltaic (PV) system with a maximum capacity of 1.7 MVA is connected to bus 5.

In the PG&E 69-bus system, a larger number of flexible load users are integrated. The specific buses to which they are connected are marked in blue in Figure 12, with an average of 10 flexible load users per bus, while the loads at the remaining buses are rigid. The historical load data for these users still originates from the aforementioned dataset.

4.5.2. Evaluation of System-Level Regulation Potential for the Modified PG&E 69-Bus Distribution System

Figure 13, Figure 14 and Figure 15 show the regulatory effects of flexible loads under the three optimization objectives in the PG&E 69-bus system. It can be observed that flexible loads also play an effective regulatory role in this system by reducing load fluctuation, promoting renewable energy consumption, and minimizing the peak-valley difference in system load. Furthermore, due to the integration of a larger number of flexible loads, the regulatory effects are more significant.

Table 2. Comparison of system contribution degree indexes.

System Contribution Degree Indexes	IEEE 33-Bus System	PG&E 69-Bus System
System load fluctuation	18.12%	55.96%
Renewable energy consumption	8.31%	24.45%
Peak-valley difference in system load	4.74%	24.84%

compares the system contribution indexes between the two test systems. The results clearly show that the PG&E 69-bus system achieves more pronounced regulatory effects across all three aspects. This significant enhancement is attributable to the substantially larger aggregate capacity of flexible loads integrated into the PG&E 69-bus system, which is approximately 2.4 times that of the IEEE 33-bus system. This successful application in a significantly different and larger system demonstrates the method’s independence from a specific test case and its effectiveness in quantifying the system-level regulation potential of flexible loads in networks of varying scale and structure.

5. Conclusions

To conduct quantitative evaluation on the regulation potential of flexible loads in distribution networks, this paper proposes a multi-level evaluation method for flexible load regulation potential based on ensemble clustering. This method constructs a multi-level evaluation framework ranging from the user-level to the bus-level and then to the system-level. By leveraging the ensemble clustering algorithm, the stability and robustness of load clustering are enhanced, thereby ensuring the quality of quantitative evaluation results. Through case analysis, the main conclusions are drawn as follows:

(1)

Compared to some traditional single clustering algorithms, the ensemble clustering algorithm adopted in this paper can effectively improve the stability and robustness of load clustering, thus providing a guarantee for the stability of the quantitative evaluation results of flexible load regulation potential.

(2)

The “user–bus–system” three-level evaluation framework constructed in this paper enables quantitative evaluation of flexible load regulation potential at multiple levels. Firstly, the power regulation range of each user can be accurately characterized based on the ensemble clustering algorithm. Then, the regulation potential of each bus in the distribution network can be obtained through aggregation calculation. Finally, by solving multiple system-level optimization models, the regulation effects of flexible loads in aspects such as minimizing the average fluctuation of system load, maximizing renewable energy consumption, and minimizing the peak-valley difference in system load can be achieved. Moreover, the system-level regulation effects in various aspects can be accurately quantified through multiple system contribution indexes.

The method proposed in this paper still has certain limitations. For instance, the system-level optimization model does not explicitly incorporate the uncertainties associated with renewable energy generation and load forecasts, which are inherent in real-world power system operations. Addressing these uncertainties is crucial for further enhancing the robustness and practical applicability of the evaluation results. Future research will focus on integrating uncertainty modeling techniques, such as stochastic optimization or robust optimization, into the proposed framework to account for forecast errors in renewable energy output and load demand. Additionally, the method may be extended and combined with refined modeling approaches to support the quantitative evaluation of regulation potential for emerging types of flexible loads, such as green hydrogen facilities, 5G base stations, and data centers.