Seasonal Load Statistics of EV Charging and Battery Swapping Stations Based on Gaussian Mixture Model for Charging Strategy Optimization in Electric Power Distribution Systems

Abstract

The rapidly growing demand of electric vehicle (EV) charging is one of the main challenges to modern electrical distribution systems. Accurate modelling of the EV charging load is crucial for charging load prediction and optimization. However, previous methods based on the charging behaviors of private EVs are hard to collect user’s private data. In this study, charging load data from 962 charging and battery swapping stations (CBSSs), classified into dedicated charging stations, public charging stations, and battery swapping stations, collected during 2021–2022, are analyzed to investigate seasonal variations in the charging coincidence factor. A data-driven probabilistic model of charging load, based on the Gaussian Mixture Model, is developed to address various scenarios, including new station construction, capacity expansions, and optimized charging strategies. This model is applicable to different types of CBSSs. A real-world 10 kV feeder system is employed as a case study to validate the model, and a delayed charging strategy is proposed. The results demonstrate that the proposed model accurately estimates charging load peaks after new construction and expansion in 2023, with an error rate under 3%. Furthermore, the delayed charging strategy achieved a 24.79% reduction in maximum load and a 31.96% decrease in the peak–valley difference. Its implementation in the real-world feeder significantly alleviated nighttime overloading in 2024.

1. Introduction

Electric vehicles (EVs) are the main way for carbon dioxide emission reduction in transportation and have attracted significant attention due to their environmental benefits. With the performance improvement and price dropping, EV sales are growing in recent decades. According to the statistics, EV sales reached 10 million units in China, constituting about 45% of the total vehicle sales in 2024 [1]. Global EV sales are predicted to reach 18 million in 2025. Most EVs are accessed to MV (Medium Voltage) electrical power distribution networks to charge. With the rapid growth of EV adoption, the increasing EV charging load has emerged as a remarkable increment in electrical power distribution networks which have been heavy-loaded [2]. Thus, it is necessary to accurately predict the EV charging loads for evaluating the available capacity.

In the last decade, EV charging load modelling methodologies were proposed with the increasing of EV penetration. In 2010, a composite Gaussian model was developed to estimate the EV charging load, considering the statistical distribution of the initial state of charge and the stochastic characteristics of charger activation and deactivation times [3]. In 2014, a spatial-temporal model was proposed to quantify the influence of EV charging on urban distribution networks [4]. Origin–destination data from the EU MERGE database were analyzed using Monte Carlo simulation to model EV mobility, thereby estimating each vehicle’s daily travel distance and the corresponding charging start time and location. Then, the EV charging branch load in a day can be calculated. In 2017, a stochastic approach was used to forecast charging load, accounting for various factors and their impact on residential load profiles [5]. Liu et al. [6] introduced a semi-Markov random trip chain generation method by analyzing the probability distributions of departure times, arrival times, and dwell times for small EVs. Wu et al. [7] developed an EV load forecasting model based on multivariate kernel density estimation by analyzing historical EV trajectory data along with energy consumption per kilometer. Ul-Haq et al. [8] employed a stochastic approach to develop a charging load model, accounting for factors such as vehicle type, battery capacity, state of charge, and driving habits. The model was designed to calculate the daily charging load of EVs across different months.

Recently, with the increasing of EV charging data, the data-drive model and model validation are available. Bollerslev et al. [9] combined travel survey data with EV charging records. This study developed a model for EV charging simultaneity that accounts for both driving and charging behaviors. In 2018, Cui et al. [10] analyzed the usage patterns of private vehicles and taxis by mining data related to EV charging behavior, including charging start times, the state of charge, and charging duration. The study provides a comprehensive summary of the usage pattern characteristics for ten distinct categories of EV users. In 2022, Powell et al. [11] investigated EV users’ charging preferences by analyzing actual charging data. The study used a real-world charging dataset to calibrate the model and developed a correlation model linking charging behavior clusters to variables such as EV user income, housing characteristics, driving distance, and charging options. Wang et al. [12] proposed a novel data-driven approach for modelling EV charging loads, recognizing that different types of EVs exhibit distinct charging load distributions. In this study, a mathematical framework was established to describe the flexibility of EV charging demand, and advanced simulation techniques were employed to calibrate and validate different EV load models across diverse operating scenarios. Zhang et al. [13] analyzed the travel data of electric vehicle users, user characteristics across different city regions. This study establishes typical daily charging load characteristic models for different functional areas. However, the charging behavior data of EVs involve personal privacy, which makes their collection particularly challenging. And most of these studies ignore the constraints of the capacity and distribution of charging stations. This could lead to models that fail to adequately consider infrastructure limitations, affecting the accuracy of spatial distribution of charging load prediction [14]. Zhu et al. [15] applied deep learning approach to model EV charging demand, exploring the influence of variables like charging time, historical load, and overall power consumption. Wang et al. [16] proposed a LSTM for the load prediction of charging stations. However, most existing load forecasting models are limited by the lack of extensive historical data for training at newly established charging stations. To solve this problem, Tian et al. [17] introduced a hybrid prediction framework that integrates transfer learning to estimate large-scale electric vehicle power demand. Gilanifar et al. [18] developed a clustered Gaussian process framework capable of modeling and learning data from several charging stations. Huang et al. [19] developed a meta-learning method that combines a Transformer architecture and probabilistic techniques to forecast charging loads for stations with limited historical data. Shi et al. [20] introduced a forecasting framework for multiple charging stations that leverages geographical correlations to model varying relationships among stations. Compared with EV charging load modelling based on individual EV behavior, analyzing the charging loads of charging stations provides a better representation of aggregated demand patterns and station-level dynamics [21].

A comparison of recent literature related to charging load modelling was conducted, as summarized in

Table 1. Comparison of the existing literatures. (× refers to ‘not considered’, while √ refers to ‘considered’.)

Literatures	Data Types	Subjects	Consider Seasonal Differences	Consider New Station Constructions	Consider Station Capacity Expansions
[4,6,9]	Data of EVs travel and charging behavior	Dedicated EV, Personal EV	×	×	×
[10]		Personal EV, Electric Taxies, Dedicated EV	√	×	×
[13,16]		Personal EV	×	×	×
[15]	Data of EV charging and battery swapping stations load	PCS	×	×	×
[18]		PCS	×	√	×
[19]		PCS	√	√	×
[20]		PCS	×	×	√

. Previous studies have made valuable contributions to the field of EV load modelling. However, due to the difficulty in collecting large-scale EV data in the early stages, most existing studies primarily focused on public charging stations (PCSs) serving taxis and personal vehicles, without considering the load modelling of bus charging stations and battery swapping stations, the seasonal variability in charging demand, or overlooked future infrastructure development scenarios.

To accurately estimate the seasonal charging load of multiple charging and battery swapping stations (CBSSs), this paper first classifies the stations into dedicated charging stations (DCSs), public charging stations (PCSs), and battery swapping stations (BSSs). It then introduces the charging coincidence factor (CCF) and employs the Gaussian Mixture Model (GMM) to investigate the dynamic patterns in seasonal charging loads across different station types. Based on the GMM, a data-driven probabilistic load model is developed to account for various scenarios, including new station construction, capacity expansions, and optimized charging strategies. The model is used seasonal charging load data from 962 stations collected on typical days during 2021–2022 and verified by the data of 2023. The key contributions of this paper are summarized as follows:

• Classification and seasonal characterization of CBSSs: The 962 CBSSs are categorized into three types—dedicated charging stations, public charging stations, and battery swapping stations. The seasonal characteristics and variations of the charging coincidence factor among the three station types are comprehensively analyzed.
• Development of a statistical model using real-world data: In contrast to previous studies that rely on assumed private EV user behavior, this work utilizes historical charging load data from numerous stations. To capture the multi-modal probability density distribution of the seasonal CCF across different time periods, GMM is employed. The model provides a more accurate representation of temporal distribution and dynamic patterns in the charging loads of different station types.
• Proposal and validation of a data-driven probabilistic load model: Based on the GMM, a data-driven probabilistic model is developed to predict charging loads under multiple scenarios, including new station construction, capacity expansion, and optimized charging strategies. A delayed charging strategy is proposed and validated through a real-world feeder system, effectively mitigating overload issues caused by charging demand.

The remainder of the paper is organized as follows. Section 2 presents the overall research framework of this study. Section 3 provides the data and method. Section 4 presents the results and discussion. Section 5 presents the conclusions and findings. Section 6 provides the limitations and future works.

2. Overview of the Research Framework

The coincidence factor is a critical parameter for quantifying potential grid impacts. According to the International Electrotechnical Commission, the coincidence factor (CF) is defined as “the ratio of the simultaneous maximum demand of a group of electrical appliances or consumers during a specified period to the sum of their individual maximum demands within the same period” [22]. A lower CF indicates reduced grid capacity requirements to accommodate the same set of loads, thereby improving operational efficiency. Conversely, a higher CF may pose risks to grid stability, potentially necessitating early grid reinforcement [23]. In this work, the charging coincidence factor (CCF) is introduced to quantify the correlation between charging load and the capacity of CBSSs. The CCF is defined as the ratio of the station’s instantaneous charging load to its rated capacity. The factor represents the utilization rate of station’s charging equipment at any time.

This paper first classifies 962 CBSSs into dedicated charging stations (DCSs), public charging stations (PCSs), and battery swapping stations (BSSs). It then employs the Gaussian Mixture Model (GMM) to investigate the dynamic patterns in seasonal charging loads across different station types. Based on the GMM, a data-driven probabilistic load model is developed to account for various scenarios, including new station construction, capacity expansions, and optimized charging strategies. The model is validated using a real-world distribution feeder system in Wuhan, demonstrating its effectiveness in accurately estimating seasonal charging load peaks. A delay strategy is proposed to avoid feeder cable overload, offering a robust theoretical foundation for managing EV charging loads and evaluating capacities of CBSSs. The research framework is illustrated in Figure 1.

3. Data and Methodology

This section presents a comprehensive analysis of charging load data from 962 charging and battery swapping stations in Wuhan, based on typical days of the four seasons from 2021 to 2022. We investigate the seasonal variations in the charging coincidence factor (CCF) across the three types of stations and develop a probabilistic load model for these stations. The dataset and methodologies employed in this analysis are described in detail below.

3.1. Data Collection and Preprocessing

The 962 charging and battery swapping stations (CBSSs) include 219 dedicated charging stations (DCSs), 615 public charging stations (PCSs), and 128 battery swapping stations (BSSs). DCSs are designed to exclusively buses, sightseeing vehicles, and vans, with charging access prohibited for other types of vehicles. PCSs and BSSs are located at commercial buildings, hospitals, shopping malls, residential complexes, highways, and other accessible sites. They provide charging services to a wide range of users, including private EVs, electric taxis, and others.

Consequently, the sample of 962 CBSSs is deemed to be adequately representative. Charging session data were collected through on-line metering, with each record containing a unique identifier corresponding to the respective charging pile and station. To protect user privacy associated with travel patterns during charging, all data in this study were processed and presented in an anonymized format.

We collect 704,184 seasonal charging load profiles from 962 CBSSs in 2021–2022. Figure 2 presents the charging load dataset in tabular form, providing a detailed overview of the structure of the original data. Daily charging load profile of the CBSSs is segmented into 15-min intervals, so each profile has 96 data points.

During data preprocessing, the first step is to filter out invalid charging load profiles. These include sessions prematurely terminated by charger faults or inaccurately logged due to recording device errors. Furthermore, the charging load profiles with low variations are deleted, referring to Equation (1).

$${\Delta }_k={\displaystyle \frac{\max \left(X_k\right)-\min \left(X_k\right)}{\max \left(X_k\right)}}$$

(1)

In which, Δ_k represents the threshold for a low-variation charging load profiles, which is set to 5% in the analysis, X_k is the k-th charging load profile. Finally, 171,835 data points were removed, resulting in 532,349 valid charging load profiles for the CBSSs.

In the context of charging and battery swapping stations (CBSSs), the charging coincidence factor (CCF) is a key parameter, defined as the ratio of the station’s instantaneous charging load to its rated capacity. This factor represents the utilization rate of station’s charging equipment at a specific point in time. Using Equation (2), the charging load data is transformed into a charging coincidence factor.

$${\gamma }_{ij}(t)={\displaystyle \frac{P_{ij}(t)}{S_{ij}}}$$

(2)

where γ_ij(t) represents the charging coincidence factor of the j-th charging and battery swapping station of the i-th type at time t, P_ij(t) denotes the load of the j-th station of the i-th type at time t, S_ij refers to the installed capacity of the j-th station of the i-th type.

3.2. Seasonal Charging Coincidence Factor Distribution Fitting Based on Gaussian Mixture Model

Previous research commonly presumes that EV charging loads conform to a Gaussian distribution [24]. However, this assumption neglects the intrinsic variability and fluctuations of the daily charging demand. Empirical evidence suggests that charging loads at stations often do not adhere to standard distributions, especially when analyzed across varying time periods. Instead, a Gaussian Mixture Model (GMM) provides a flexible framework for approximating arbitrary probability density function (PDF) by representing them as a weighted combination of multiple Gaussian functions [25].

The GMM facilitates the characterization of the PDF for the charging coincidence factor (CCF) at charging and battery swapping stations, providing a robust foundation for developing probabilistic models of charging loads across different time periods. We model the probability distribution of the CCF with a GMM to estimate the charging demand accurately. The function is described as Equation (3):

$$f_{it}\left(\gamma \right)={\displaystyle \sum _{k=1}^K{A}_k{N}_k\left({\mu }_{it},{\sigma }_{it}^2\right)}$$

(3)

$$N_k\left({\mu }_{it},{\sigma }_{it}^2\right)={\displaystyle \frac{1}{\sqrt{2\pi }{\sigma }_{it}}}\exp \left(-{\displaystyle \frac{{\left(\gamma -{\mu }_{it}\right)}^2}{2{\sigma }_{it}^2}}\right)$$

(4)

In which, f_it(γ) is denoted as the PDF of the CCF for the i-th type of charging and battery swapping station at time t; K represents the number of Gaussian components; N_k(μ_it,${\sigma }_{it}^2$) is the k-th Gaussian component within the PDF of the CCF for the i-th type of station at time t; A_k indicates the weight of the k-th Gaussian component; μ_it is the mean of the random variable associated with the k-th Gaussian component for the CCF of the i-th type of station at time t; ${\sigma }_{it}^2$ is the variance of the random variable associated with the k-th Gaussian component for the CCF of the i-th type of station at time t.

This section identifies the time intervals from the 96 daily periods for three types of CBSSs under typical seasonal conditions. These 962 CBSSs adopt a standardized time-of-use electricity price that excludes any additional charges, as shown in Figure 3. In 2023, the peak price (20:00–22:00) is set at 0.1875 $/kWh; the high-demand price (9:00–15:00) is 0.1414 $/kWh; the standard price (7:00–9:00, 15:00–20:00, and 22:00–23:00) is 0.0971 $/kWh; and the off-peak price (23:00–7:00) is 0.0473 $/kWh. In comparison, the residential electricity price remains at 0.0781 $/kWh throughout the day.

The probability distribution of the CCFs at any time on typical days of the four seasons for DCSs, PCSs, and BSSs is shown in Figure 4. The charging coincidence factor of CBSSs across the four seasons shows a multimodal distribution at various times. Over time, the probability distribution fluctuates in a wave-like pattern, regularly shifting between low and high CCFs.

The probability distribution of the CCF at DCS consistently exhibits a trend of higher values at night and lower values during the day across all seasons. In every season, the CCFs rapidly transition from low to high at 23:00, primarily because most buses, tourist vehicles, and vans served by DCSs cease operations at 23:00 and begin charging. However, the duration of high CCFs during the night shows significant seasonal variations: it lasts until 5:00 in spring, 6:00 in summer, 4:00 in autumn, and 3:00 in winter. The longest duration occurs in summer, reflecting the highest charging demand. This is primarily due to the higher temperatures in Wuhan during summer, which increase energy consumption for air conditioning in buses, tourist vehicles, and vans. In contrast, winter has the shortest duration, which can be attributed to shorter daylight hours, fewer bus services, and reduced travel demand caused by colder temperatures and inclement weather (such as freezing rain) [26]. As a result, the number of passengers on buses, sightseeing vehicles, and vans decreases significantly.

The CCF at PCSs typically exhibits a triple-peak pattern throughout the four seasons, with most of the high CCF values concentrated during 23:00–01:00, 06:00–07:00, and 15:00–16:00. This phenomenon can be explained by the fact that, in Wuhan, taxis and private cars—the primary users of public charging stations—tend to charge predominantly during these three periods [27]. Compared to spring and autumn, the proportion of charging coincidence factors in the high coincidence factor region is more pronounced in both summer and winter. This discrepancy is primarily due to the high temperatures in summer and the low temperatures in winter in Wuhan, which result in taxis and private cars significantly increasing their power consumption to operate air conditioning [28].

Compared to traditional charging stations, the charging coincidence factor of battery swapping stations fluctuates more steadily, primarily due to the significant differences in charging modes between battery swapping and charging stations. Overall, the charging coincidence factor of battery swapping stations is higher in summer and winter, while it remains lower during spring and autumn. The peak coincidence factors are mainly concentrated during the time periods of 15:00–17:00 and 23:00–00:00, which closely align with the peak charging periods observed at public charging stations. In Wuhan, battery-swapped vehicles are predominantly taxis. Energy consumption significantly increases during the summer and winter due to the use of air conditioning. Additionally, these two periods coincide with lower electricity price, making them peak times for taxi battery swapping and charging.

To illustrate the time-varying dynamic process of the GMM for the PDF of CCF at different time points, GMM models of the charging coincidence factor distributions are presented for three peak charging periods (06:00–07:00, 15:00–16:00, and 23:00–00:00) at public charging stations, the nighttime peak charging period at dedicated charging stations, and the afternoon and nighttime peak charging periods at battery swapping stations, as shown in Figure 5, Figure 6 and Figure 7.

The overall trend at public charging stations during the three peak charging periods demonstrates a gradual shift of the Gaussian component from the left to the right, exhibiting a wave-like progression. Specifically, as time progresses, the charging coincidence factor gradually transitions from the region below 0.2 to above 0.4. Furthermore, during the transitional moments when the charging coincidence factor increases, the number of Gaussian components also increases by one or two, as observed at time points such as 06:00, 06:45, and 15:00.

During the nighttime peak charging period from 23:00 to 02:00 at dedicated charging stations, the charging coincidence factor begins to transition from below 0.2 to above 0.4 around 23:15, with a gradual decline back to lower coincidence factor regions starting around 00:30. Similar to public charging stations, during the transitional moments when the charging coincidence factor increases, the number of Gaussian components increases by 1 to 2, observed at time points such as 23:15 and 01:15. Conversely, during the transitional moments when the charging coincidence factor decreases, the number of Gaussian components decreases by 1 to 2, as seen at 00:30 and 01:00.

During the charging peak periods at the battery swapping stations (15:00–17:00 and 23:00–00:00), the Gaussian components of the charging coincidence factor demonstrate a transition from lower to higher coincidence rate regions, with peak values observed at 16:45 and 23:45, respectively. Consistent with traditional charging stations, during transitional moments when the charging coincidence factor increases, the number of Gaussian components also increases by one or two, such as at 16:30. Conversely, during transitional moments when the charging coincidence factor decreases, the number of Gaussian components decreases by one or two, such as at 16:00.

3.3. Probabilistic Load Model of Charging and Battery Swapping Station

The GMM is utilized to approximate the probability density function by constructing a linear combination of multiple Gaussian distributions. The number of components is determined by referencing clustering algorithms from unsupervised learning, with the number of Gaussian components predefined. Parameter estimation within the GMM framework was conducted using the Expectation-Maximization (EM) algorithm to optimize the model parameters [29]. The probability density function of the charging coincidence factors dataset {γ₁, γ₂, …, γ_n} at time t is approximated using a GMM. The parameters of the GMM are estimated through the Expectation-Maximization algorithm, with the detailed procedure outlined as follows:

• Initialization: Set the initial estimates $A_k^{\left(0\right)}$, ${\mu }_k^{\left(0\right)}$, ${\sigma }_k^{\left(0\right)}$, k = 1, …, K. Calculate the initial log-likelihood function using Equation (5).

$$L^{(0)}={\displaystyle \frac{1}{n}}{\displaystyle \sum _{j=1}^n\log \left({\displaystyle \sum _{k=1}^K{A_k}^{(0)}}N\left({\gamma }_j|{{\mu }_k}^{(0)},{{\sigma }_k}^{(0)}\right)\right)}$$

(5)

2.

E-Step: Calculate the posterior probability ${\zeta }_{jk}^{\left(h\right)}$ for each data point γ_i belonging to each Gaussian distribution k, representing the probability that the i-th data point is generated by the k-th Gaussian component. Additionally, compute the total posterior probability $n_k^{\left(h\right)}$. The formulas are as follows:

$${{\zeta }_{jk}}^{(h)}={\displaystyle \frac{{A_k}^{(h)}N\left({\gamma }_j|{{\mu }_k}^{(h)},{{\sigma }_k}^{(h)}\right)}{{\displaystyle \sum _{l=1}^K{A_l}^{(h)}N\left({\gamma }_j|{{\mu }_l}^{(h)},{{\sigma }_l}^{(h)}\right)}}},j=1,\dots ,n,\hspace{1em}k=1,\dots ,K$$

(6)

$${n_k}^{(h)}={\displaystyle \sum _{j=1}^n{{\zeta }_{jk}}^{(h)}},k=1,\dots ,K$$

(7)

3.

M-Step: Using the posterior probabilities calculated in the E-step, update the parameters by maximizing the log-likelihood estimate. The updated parameters are obtained as follows:

$${A_k}^{(h+1)}={\displaystyle \frac{{n_k}^{(h)}}{n}},k=1,\dots ,K$$

(8)

$${{\mu }_k}^{(h+1)}={\displaystyle \frac{1}{{n_k}^{(h)}}}{\displaystyle \sum _{j=1}^n{\gamma }_j{{\zeta }_{jk}}^{(h)}},k=1,\dots ,K$$

(9)

$${{\sigma }_k}^{(h+1)}=\sqrt{{\displaystyle \frac{1}{{n_k}^{(h)}}}{\displaystyle \sum _{j=1}^n{\left({\gamma }_j-{{\mu }_k}^{(h+1)}\right)}^2{{\zeta }_{jk}}^{(h)}}},k=1,\dots ,K$$

(10)

4.

Convergence Check: Compute the new log-likelihood function L^(h+1), If |L^(h+1) − L^(h)| > δ, return to Step 2 (E-Step) and continue iterative parameter estimation. If |L^(h+1) − L^(h)| ≤ δ, output the final parameters $A_k^{(h+1)}$, ${\mu }_k^{(h+1)}$ and ${\sigma }_k^{(h+1)}$, and terminate the algorithm. The new log-likelihood function is calculated as follows:

$$L^{(h+1)}={\displaystyle \frac{1}{n}}{\displaystyle \sum _{j=1}^n\log \left({\displaystyle \sum _{k=1}^K{A_k}^{(h+1)}}N\left({\gamma }_j|{{\mu }_k}^{(h+1)},{{\sigma }_k}^{(h+1)}\right)\right)}$$

(11)

Using the EM algorithm to estimate the parameters of the GMM model, the probability density function (PDF) of the charging coincidence factor for charging and swapping stations can be determined across different times and seasons. Based on this PDF, a probabilistic model describing the charging load of these stations is established, as shown in the following Equation (12).

$$P_{ij}(t)={\displaystyle {\int }_{{\gamma }_{i\min }}^{{\gamma }_{i\max }}{S}_{ij}}\cdot {\gamma }_i\cdot f_{it}\left({\gamma }_i\right)d{\gamma }_i$$

(12)

where P_ij(t) denotes the charging load of the j-th charging and swapping station of the i-th category at time t, S_ij represents the installed capacity of the j-th charging and swapping station the i-th category, γ_i(t) refers to the charging coincidence factor of the i-th category of charging and swapping stations at time t, and f_it(γ_i) represents the probability density function (PDF) of the charging coincidence factor γ_i(t) for the i-th category of charging and swapping stations at time t.

When the j-th charging and swapping station of the i-th category undergoes a capacity expansion by ΔS_ij, the charging load of the expanded station can be transformed from Equation (12) into Equation (13) as follows:

$$P_{ij}(t)={\displaystyle {\int }_{{\gamma }_{i\min }}^{{\gamma }_{i\max }}\left(S_{ij}+\Delta S_{ij}\right)}\cdot {\gamma }_i\cdot f_{it}\left({\gamma }_i\right)d{\gamma }_i$$

(13)

When multiple types of charging and swapping stations are connected to the distribution system, the total charging load of all charging and swapping stations on the distribution system can be expressed by Equation (14):

$$P_T(t)={\displaystyle \sum _{i=1}^3{\displaystyle \sum _{j=1}^{N_i}{P}_{ij}(t)}}$$

(14)

In which, P_T(t) represents the total charging load of all charging and swapping stations connected to the distribution system. The index i refers to the type of charging and swapping stations, with three types in total. The index j denotes the j-th station within the i-th type. N_i indicates the total number of i-type charging and swapping stations.

At time t, the total load of the feeder after the integration of charging and swapping stations can be calculated using Equation (15). In which, P_L(t) represents the total load of the distribution system, and P₀(t) denotes the base load of the distribution system.

$$P_L(t)=P_T(t)+P_0(t)$$

(15)

To mitigate the risk of overloading due to the new construction and expansion of charging stations on the feeder, it is essential to optimize the charging strategy for charging and battery swapping stations. The main focus is on personal economic benefit, with secondary focus on reduced the risk of overloading. Therefore, the optimization charging strategy discussed in this paper primarily focuses on the delayed charging strategy and the intelligent charging strategy.

The objective of the optimization is to minimize the total cost for both the EV charging user and the power supply company. The objective function for optimization is shown in Equation (16), where C_charging represents the charging cost for EV users, C_time denotes EV users’ the time cost due to the optimized charging strategy, and C_power represents the maintenance cost of the grid. In Equation (17), TOU(t) refers to the time-of-use electricity price. In Equation (18), $t_{ij}^{opt}$ denotes the charging end time after optimization, $t_{ij}^{org}$ refers to the original charging end time, and c_t is the time cost coefficient per unit time. In Equation (19), P_LP represents the maximum total load of the distribution system, P_LPV refers to the peak-valley load difference of the distribution system, and c_p is the maintenance cost coefficient per unit of power. In Equation (21), P_LV represents the minimum total load of the distribution system.

$$\min C={\omega }_1{C}_{\mathrm{charging}}+{\omega }_2{C}_{\mathrm{time}}+{\omega }_3{C}_{\mathrm{power}}$$

(16)

$$C_{\mathrm{charging}}={\displaystyle {\int }_0^{24}{P}_T(t)\times TOU(t)dt}$$

(17)

$$C_{\mathrm{time}}=c_{\mathrm{t}}{\displaystyle \sum _{i=1}^3{\displaystyle \sum _{j=1}^{N_i}\left(t_{ij}^{opt}-t_{ij}^{org}\right)}}$$

(18)

$$C_{\mathrm{power}}=c_{\mathrm{p}}\left(P_{LP}+P_{LPV}\right)$$

(19)

$$P_{LP}=\max P_L\left(t\right)$$

(20)

$$P_{LV}=\min P_L\left(t\right)$$

(21)

$$P_{LPV}=P_{LP}-P_{LV}$$

(22)

The delayed time should not exceed the allowable delayed time, Δt_ijmax, to ensure the normal operation of each charging and battery swapping station. When the j-th charging and swapping station of the i-th category implements a delayed charging strategy, the charging load of the station at time t, after a delay of Δt_ij, can be transformed from Equation (12) into Equation (23) as follows:

$$P_{ij}(t)={\displaystyle {\int }_{{\gamma }_{i\min }}^{{\gamma }_{i\max }}{S}_{ij}\cdot {\gamma }_i\cdot f_{i{t}_d}\left({\gamma }_i\right)d{\gamma }_i}$$

(23)

$$t_d=t-\Delta t_{ij}$$

(24)

$$\Delta t_{ij}\le \Delta t_{ij\max }$$

(25)

The intelligent charging strategy dynamically allocates charging power to each charging and battery swapping station, ensuring that the distribution network avoids overload, while also minimizing the charging costs for EV users. In Equation (26), P_Lmax represents the maximum power that a 10 kV three-core cable can safely operate, ${P^{\prime }}_T$(t) denotes the charging load after using the intelligent charging strategy, and ${P^{\prime }}_{T\max }$(t) represents the maximum charging load of time t. In Equation (29), t₀ represents the start time of the intelligent charging strategy, and t_max represents the latest end time of the intelligent charging strategy.

$$P_{L\max }(t)\ge P_T^{\prime }(t)+P_0(t)$$

(26)

$$P_T^{\prime }(t)={\displaystyle \sum _{i=1}^3{\displaystyle \sum _{j=1}^{N_i}{P}_{ij}^{\prime }(t)}}$$

(27)

$$P_T^{\prime }(t)\le P_{T\max }^{\prime }(t)$$

(28)

$${\displaystyle {\int }_{t_0}^{t_{\max }}{P}_{ij}^{\prime }(t)dt}\ge {\displaystyle {\int }_{t_0}^t{P}_{ij}(t)dt}$$

(29)

The specific execution steps of the proposed method are outlined in Figure 8. First, historical charging load data and capacity information from CBSSs are input. The charging load data is then converted into the CCF, and GMM is applied to fit the PDF of the seasonal CCF. The EM algorithm is employed to estimate the parameters of the GMM and derive the PDF of the CCF.

Next, based on the derived PDF of the CCF, the proposed probabilistic load model is utilized to calculate the charging load and total load following new construction and expansion. The model also assesses whether the total load complies with the maximum load limit of the transmission lines. If the total load exceeds the limit, the charging strategy is optimized iteratively until the load meets the maximum limit. Once the limit is satisfied, the process outputs the original load, the optimized charging strategy, and the resulting load after implementing the strategy, thus completing the procedure.

4. Results and Discussion

A generic urban power distribution system in Wuhan is presented in Figure 9a. The system consists of a 220/110/10 kV, 240 MVA transformer and 12 outgoing 10 kV feeders (F₁–F₁₂). Feeders F₁, F₂, and F₃, comprising the three-phase 10 kV cable system, utilize YJV22 3 × 400 mm² cross-linked polyethylene cables installed through a ducted laying method for efficient power distribution. The 10 kV feeder supplies five charging and swapping stations: Public Charging Station 1 (630 kVA, put into operation in late 2022), Public Charging Station 2 (upgraded from 2000 kVA to 2500 kVA by the end of 2022), Public Charging Station 3 (3600 kVA), the Bus Charging Station (expanded from 2000 kVA to 3600 kVA by the end of 2022), and the Battery Swapping Station (1250 kVA). Figure 9b illustrates the seasonal typical daily base load of the 10 kV feeder.

Parameters of probabilistic load model are estimated utilizing charging load data from 962 stations in Wuhan during 2021–2022 to estimate seasonal peak charging loads in 2023, following new construction and capacity expansions. The seasonal charging load, estimated using a probabilistic model, is presented in Figure 10. In 2023, the daily maximum charging loads exhibit a distinct triple-peak pattern, occurring at 23:00–01:00, 06:00–07:00, and 15:00–17:00, with a consistent surge around 23:00. The annual peak load occurs at 23:30 in summer, reaching 5564.4 kW, which is 28.63% higher than the lowest nighttime peak observed in winter. Daytime peak loads occur at 16:45 in summer and 15:30 in winter, surpassing those in spring and autumn afternoons by 17–20%. These findings highlight significant seasonal variations in EV charging load, emphasizing their critical importance for effective grid planning and load management.

This study employs a probabilistic model based on charging coincidence factors to estimate seasonal charging load peaks following new construction and expansion in 2023. The relative errors of the estimated seasonal charging load are presented in

Table 2. Relative error of estimated seasonal charging load on maximum load days in 2023.

Season	Time	Estimated Charging Power Peak	Actual Charging Power Peak	Relative Error
Spring	23:15	4648.60 kW	4594.27 kW	1.18%
	06:30	2055.40 kW	2024.63 kW	1.52%
	16:00	2254.52 kW	2298.94 kW	1.93%
Summer	23:30	5564.40 kW	5598.90 kW	0.62%
	06:15	2609.83 kW	2547.60 kW	2.44%
	16:45	2747.16 kW	2688.95 kW	2.16%
Autumn	23:30	4435.60 kW	4489.61 kW	1.20%
	06:30	1960.70 kW	2006.93 kW	2.31%
	15:30	2291.27 kW	2327.86 kW	1.57%
Winter	23:30	3921.28 kW	3988.56 kW	1.69%
	06:30	1450.40 kW	1491.72 kW	2.77%
	15:30	2493.17 kW	2536.43 kW	1.71%

. With estimation errors as low as 0.62% and consistently below 3%, the findings validate the proposed model’s high precision in predicting the loads of the charging and battery swapping stations.

Using charging load data from 962 charging and battery swapping stations in Wuhan, the study compares the performance of three models: a Gaussian model based on charging load, a Gaussian model (GM) based on charging coincidence factor, and the GMM based on charging coincidence factor proposed in this work as shown in Figure 11.

Overall, the charging load demonstrates a seasonal variation, being highest in summer, followed by spring, autumn, and winter. The GMM exhibits significantly higher accuracy than the GM in estimating seasonal charging loads. However, as charging loads decrease, the accuracy of the GM improves, further supporting the observation that early charging load distributions were well-aligned with Gaussian characteristics. During the initial stages of EV adoption, when market penetration was still low, charging loads at specific time points approximately followed a Gaussian distribution, making the GM a reasonable choice for early research. However, the reliance on public charging load data during that period limited the applicability and accuracy of GM in estimating loads of diverse types of charging and battery swapping stations.

As shown in

Table 3. Comparison of three charging load models.

Season	MAPE of GM Based on Power	MAPE of GM Based on CCF	MAPE of GMM Based on CCF
Spring	133.79%	49.48%	6.29%
Summer	138.13%	53.46%	6.80%
Autumn	36.59%	31.58%	4.36%
Winter	35.46%	51.63%	5.19%

, the mean absolute percentage error (MAPE) of GM is significantly larger than GMM, indicating that the GM is no longer suitable for estimating large-scale charging loads of diverse types of electric vehicles. The limitation arises from its inability to accurately capture the time-varying probability distributions of charging loads at bus charging stations and battery swapping stations. By contrast, the GMM, with its enhanced flexibility and adaptability, effectively addresses these challenges and demonstrates superior performance in modelling the complexities of modern EV charging load scenarios.

The 10% and 90% quantiles of the charging coincidence factor are used as the bounds of the probability domain. The analysis focuses on the summer season, which experiences the highest charging load, as illustrated in Figure 12. The fluctuation patterns of charging loads across three types of charging and swapping stations with varying capacities closely correspond to the trends observed in their respective charging coincidence factor probability domains.

As presented in Figure 12, the loads of different types of CBSSs exhibit considerable variability across capacity levels. However, for each type, the charging loads consistently lie within the probability domain defined by the charging coincidence factor, with their fluctuation trends closely aligning with those of the charging coincidence factor. In contrast to direct analysis of charging loads, which is often complicated by the influence of capacity expansions, the analysis method based on charging coincidence factors demonstrates strong generalizability.

The probabilistic model based on charging coincidence factors is applicable to charging and swapping stations of varying types and capacities and holds considerable potential for broader application in diverse scenarios, including capacity expansion, new station construction, and infrastructure planning.

By combining the estimated seasonal charging loads with the baseline seasonal load profiles, the maximum daily total load for the feeder in 2023, illustrated in Figure 13, peaked at 6234.7 kW at 23:30 during the summer, exceeding the capacity of the YJV22 3 × 400 mm² cross-linked polyethylene cable [30]. Following the construction and expansion of CBSSs, two overload-induced outages were actually recorded around 23:00 on summer nights in 2023. These results highlight the efficacy of the proposed model as a robust tool for assessing the capacity adequacy of upgraded feeder infrastructure.

To mitigate cable overload resulting from the construction and expansion of CBSSs, upgrading the feeder infrastructure offers a long-term solution but is limited by significant costs and extended implementation timelines. As more feasible short-term measures, the delayed charging strategy for controllable charging loads and intelligent charging strategy have been proposed [31].

Owing to the centralized management of bus charging stations by the bus company, the implementation of a delayed charging strategy is both practical and effective. The bus charging stations operate daily from 5:30 to 23:00, with centralized charging starting thereafter and extending until 3:00. To ensure uninterrupted bus operations, the maximum allowable charging delay is limited to 2.5 h. The proposed load probability model and delay strategy are evaluated using 15-min intervals as the fundamental time unit. As illustrated in Figure 14, the optimization results recommend the following delayed charging strategies: a delay of 1.25 h in spring, 2.5 h in summer, 2 h in autumn, and 2.5 h in winter.

The implementation of the optimized delayed charging strategy results in significant decrease in seasonal feeder charging loads, as illustrated in Figure 15a. The maximum charging load decreases from 5564.4 kW to 4021.8 kW, representing a 27.72% decrease. On maximum load days, peak loads decrease by 16% in spring, 17.1% in autumn, and 29.79% in winter. In Wuhan, where extreme temperatures drive increased residential electricity demand, delayed charging for bus charging stations effectively reduces load during critical periods, mitigating strain on the distribution network and preventing feeder cable overloads. Figure 15b presents the seasonal total load of the feeder. The maximum load peak is reduced from 6234.7 kW to 4689.4 kW, representing a 24.79% decrease, effectively alleviating cable overloading during summer nights. Additionally, the peak-to-valley load difference decreases by 31.96%, demonstrating a significant peak-shaving and valley-filling effect in summer. For spring, autumn, and winter, the maximum load peaks decrease by 0.99%, 14.68%, and 0.86%, respectively, while the peak-to-valley load differences are reduced by 15.29%, 20.05%, and 13.87%. These results highlight the effectiveness of the delayed charging strategy in reducing load variations and enhancing distribution network stability.

Compared to the delayed charging strategy, the intelligent charging strategy requires the installation of a dynamic power distribution module for the charging equipment. The charging equipment adjusts the charging power in real-time based on the load of the system and the electricity price. The charging power is reduced during periods of peak electricity prices and high grid load, while increased during off-peak prices and low load periods. This approach not only reduces charging costs for EV users but also effectively mitigates the overload issues in the distribution network. The implementation of the intelligent charging strategy results in significant reductions in seasonal feeder charging loads, as illustrated in Figure 16a. The maximum charging load decreases from 5564.4 kW to 3321.99 kW, representing a 40.30% reduction. Figure 16b presents the seasonal total load of the feeder. The maximum load peak is reduced from 6234.7 kW to 4227.93 kW, representing a 32.19% decrease, effectively alleviating cable overloading during summer nights. Additionally, the peak-to-valley load difference decreases by 47.62%, demonstrating a significant peak-shaving and valley-filling effect in summer. These results also highlight the effectiveness of the intelligent charging strategy in reducing load variations and enhancing distribution network stability.

Compared with existing strategies, the proposed charging strategy is developed based on the classification and seasonal analysis of CBSSs. By considering the distinct charging characteristics of different types of stations across various seasons, our approach formulates targeted charging strategies that better align with the operational features of each station type. In contrast, existing strategies are largely constrained by data collection challenges and typically focus only on public charging stations, with limited attention to battery swapping stations and dedicated charging stations (such as bus charging stations).

A comparative analysis of the results for the three charging strategies is shown in the

Table 4. Comparison of the seasonal metrics for three charging strategies. (× indicates no Equipment Upgrade required; √ indicates Equipment Upgrade required.)

Season	Charging Strategy	Increased Charging Time/Hours		Typical Daily Charging Cost/$	Total Load Peak/kW	Total Load Peak-Valley Difference/kW	Equipment Upgrade
Spring	Original charging strategy	/	/	2540.38	4963.20	3204.80	×
	Delayed charging strategy	PCS	/	2413.46	4473.10	2714.70	×
		BCS	1.25
		BSS	/
	Intelligent charging strategy	PCS	0.50	2259.01	3469.15	1710.75	√
		BCS	1.00
		BSS	0.25
Summer	Original charging strategy	/	/	3214.28	6276.70	4302.30	×
	Delayed charging strategy	PCS	/	3148.80	4689.40	2927.40	×
		BCS	2.50
		BSS	/
	Intelligent charging strategy	PCS	1.25	3032.82	4227.93	2253.53	√
		BCS	1.75
		BSS	0.50
Autumn	Original charging strategy	/	/	2599.15	4835.20	3132.80	×
	Delayed charging strategy	PCS	/	2544.65	4125.50	2504.70	×
		BCS	2.00
		BSS	/
	Intelligent charging strategy	PCS	0.75	2322.24	3652.80	1950.40	√
		BCS	1.25
		BSS	0.25
Winter	Original charging strategy	/	/	2604.91	4605.92	2870.56	×
	Delayed charging strategy	PCS	/	2497.79	4209.97	2472.51	×
		BCS	2.50
		BSS	/
	Intelligent charging strategy	PCS	1.25	2338.57	3685.44	1950.08	√
		BCS	1.00
		BSS	0.75

. The analysis indicates that intelligent charging strategy demonstrates the best performance in optimizing load management and reducing EV users’ charging costs. However, the intelligent charging strategy is implemented at public charging stations (PCSs), bus charging stations (BCSs), and battery swapping stations (BSSs), covering a broad range. While most EV users accept intelligent charging strategy due to its effective reduction in charging costs, some are reluctant to extend their charging time, which may pose challenges for the widespread adoption of the intelligent charging strategy. Furthermore, the intelligent charging strategy requires operators of charging and battery swapping stations to upgrade their equipment, which will increase the cost of equipment upgrades. Therefore, the intelligent charging strategy presents greater challenges in terms of equipment upgrades and practical deployment, making it better suited as a mid-term or long-term solution.

In contrast to the intelligent charging strategy, the delayed charging strategy is exclusively implemented at BCSs during their nighttime rest periods. This method not only avoids disruption to the normal operations of bus companies but also contributes to a reduction in charging costs and alleviates grid overload in the short time. In Wuhan, compared to PCSs and BSSs managed by different companies, BCSs are centrally operated by the bus company, facilitating the implementation of optimized charging strategies. And BCSs are staffed with dedicated charging management personnel, ensuring that delayed charging strategies do not disrupt the normal rest periods of bus drivers while also reducing the company’s charging costs. A comprehensive comparison of the delayed charging strategy and the intelligent charging strategy shows that the delayed charging strategy is more convenient and practical for reducing charging costs and alleviating grid overload in the short term in Wuhan.

Figure 17 presents the actual total feeder load on a typical summer day from 2022 to 2024. During the summer of 2022, the total feeder load attained its maximum value of 4726.4 kW at 23:30. In the summer of 2023, the total feeder load reached a maximum of 6234.7 kW at 23:30, following the new construction and expansion of CBSSs, resulting in an overload of the feeder due to charging loads. In the summer of 2024, the proposed delayed charging strategy for bus charging stations was applied to this feeder. On a typical summer day, the feeder’s actual peak load occurred at 01:45, reaching a maximum value of 4780.8 kW, effectively alleviating the cable overloading caused by the new construction and expansion of CBSSs in 2023. This value differed by only 1.91% from the peak load of 4689.4 kW estimated by the proposed model. These findings highlight the practicality and effectiveness of the proposed delayed charging strategy while further validating the accuracy of the methodology developed in this study.

5. Conclusions and Findings

The charging load data from 962 charging and battery swapping stations in Wuhan, China during 2021–2022 are analyzed. The charging coincidence factor is introduced to investigate the seasonal variations of the charging coincidence factor across three types of charging and battery swapping stations. A data-driven probabilistic model of charging load, based on the GMM, is developed to address various scenarios, including new station construction, capacity expansions, and optimized charging strategies. To mitigate cable overload caused by high charging demand, a practical and effective delayed charging strategy is proposed. The conclusions are drawn as follows:

5.1. Key Conclusions

• The charging coincidence factor of charging and battery swapping stations exhibits distinct seasonal variations. And there are notable differences in the charging coincidence factor across different types of stations.
• By using the GMM, the multi-modal probability density distribution of the seasonal charging coincidence factor across different time periods can be effectively captured. The model provides a more accurate representation of the temporal distribution and dynamic patterns in the charging loads of different types of stations.
• A probabilistic load model based on GMM is proposed and validated though a real feeder system. The model estimates seasonal feeder charging loads after new station constructions and expansions in 2023, with the summer peak reaching 5564.4 kW at 23:30, only 0.62% below the actual value of 5598.9 kW.
• A delayed charging strategy is proposed to effectively alleviate nighttime overloading without additional operational costs. Implementation on a real feeder in 2024 resulted in a 23.32% reduction in summer peak load compared with 2023, confirming the practical effectiveness of the proposed strategy.

5.2. Main Findings

• Classification and seasonal characterization of CBSSs: Dedicated charging stations experience the longest duration of high coincidence factors in the summer and the shortest in winter, with the peak charging period mainly concentrated between 23:00 and 01:00. Public charging stations exhibit higher coincidence factors in both summer and winter, displaying three peak periods at 6:00–7:00, 15:00–17:00, and 23:00–00:00. Battery swapping stations maintain relatively stable charging coincidence factors, with the highest values occurring in both summer and winter. The peak charging periods occur between 15:00–17:00 and 23:00–00:00, with the summer peak concentrated between 23:00–00:00 and the winter peak occurring between 15:00–17:00.
• Temporal dynamics captured by GMM: During off-peak periods, the charging coincidence factors are low, with most stations clustering below 0.4. In contrast, during peak charging periods, these stations gradually shift to higher coincidence factors, with the majority surpassing 0.6 at peak times. Additionally, during the transition from off-peak to peak periods, the number of Gaussian components in the model increases significantly by 1 to 3, while during the transition from peak to off-peak periods, the number of Gaussian components decreases by 1 to 2.
• Validation and prediction of the probabilistic load model: The probabilistic load model, constructed using 2021–2022 charging load data, was validated on a real 10 kV feeder in Wuhan. The model accurately predicted the summer 2023 peak load (5564.4 kW at 23:30) with only a 0.62% deviation from the actual value (5598.9 kW). After implementing the delayed charging strategy, the deviation of the predicted peak load remained within 1.91%, demonstrating the model’s high predictive accuracy in load forecasting.
• Effectiveness of the delayed charging strategy: An optimized delay strategy was formulated based on the seasonal CCF distribution of different station types. The nighttime peak loads were reduced by 24.79%, and the peak-to-valley difference decreased by 31.96%, without incurring any additional operational costs. Implementation on a real feeder in 2024 resulted in a 23.32% reduction in summer peak load compared with 2023, confirming the practical effectiveness of the proposed strategy.

6. Limitations and Future Works

6.1. Limitations

The proposed methodology, while effective in estimating EV charging loads in distribution systems and optimizing charging strategies, has certain limitations. The study incorporates seasonal and temporal variations in charging coincidence factors and utilizes regional historical data to estimate loads and design optimization strategies. However, additional factors—such as electric vehicle penetration, ambient temperature, service fees, and network fees—may also influence the charging coincidence factors and warrant further investigation.

Moreover, this study is limited to data collected from Wuhan, which is located in the subtropical monsoon zone. Nonetheless, the proposed data-driven probabilistic load model is expected to be broadly applicable to regions with similar climatic conditions. For other climate zones, the framework can be adapted through recalibration of model parameters using local charging and battery swapping station data. Further validation across diverse geographical and climatic regions is necessary to ensure the model’s generalizability.

In addition, although the GMM enhances the accuracy of charging load estimation, its computational complexity may pose challenges for real-time applications. The current study focuses on offline analysis, where such computational demands are acceptable. Nonetheless, real-time implementation and scalability remain potential constraints for practical integration into utility management systems.

6.2. Future Works

Future research will focus on further investigating the influence of multiple factors on the charging coincidence factors, including EV penetration, ambient temperature, and electricity pricing—particularly final electricity prices including additional charges (e.g., taxes, network fees, etc.). These analyses will support the exploration of optimal planning and pricing strategies within a multi-agent game framework, considering the interactions among EV users, power supply companies, and charging station operators.

Furthermore, to enhance the practicality of the proposed model, future work will explore lightweight or incremental learning variants of the GMM to improve computational efficiency and assess the model’s adaptability for real-time integration into distribution and energy management systems.

In addition, the increasing frequency of extreme events, such as ice storms and heatwaves, poses significant challenges to the secure and reliable operation of distribution networks. During extreme heat, the combined load from large-scale EV charging and residential consumption may exceed the network’s safe operating capacity. To mitigate this risk, future work will explore system reinforcement strategies, including upgrading transformers and distribution cables to higher-capacity configurations. Conversely, in extremely cold conditions, elevated energy demand and reduced renewable generation could result in short-term power shortages. Under such circumstances, the potential for large-scale EVs to provide vehicle-to-grid (V2G) support will be investigated. The application of probabilistic models for EV charging coincidence factors will continue to play a critical role in ensuring the long-term reliability and resilience of distribution networks.