An Overview of Electric Vehicle Load Modeling Strategies for Grid Integration Studies

Abstract

The adoption of electric vehicles (EVs) has emerged as a solution to reduce greenhouse gas emissions in the transportation sector, which has motivated the implementation of public policies to promote their use in several countries. However, the high adoption of EVs poses challenges for the electricity sector, as it would imply an increase in energy demand and possible impacts on the power quality (PQ) of the power grid. Therefore, it is important to conduct EV integration studies in the power grid to determine the amount that can be incorporated without causing problems and identify the areas of the power sector that will require reinforcements. Accurate EV load patterns are required for this type of study that, through mathematical modeling, reflect both the dynamic behavior and the factors that influence the decision to recharge EVs. This article aims to present an overview of EVs, examine the different factors considered in the literature for modeling EV load patterns, and review modeling methods. EV load modeling methods are classified into deterministic, statistical, and machine learning. The article shows that each modeling method has its advantages, disadvantages, and data requirements, ranging from simple load modeling to more accurate models requiring large datasets.

1. Introduction

As the Earth experiences a steady rise in temperature due to the accumulation of greenhouse gases in the atmosphere, it has become imperative that we seek solutions to reduce emissions of large amounts of carbon dioxide (C${\mathrm{O}}_2$). The transportation sector is currently one of the largest contributors to global C${\mathrm{O}}_2$ emissions [1]. In this context, electric vehicles (EVs) have emerged as a promising alternative that could lead to a more sustainable future. Because, unlike traditional cars that run on internal combustion engines (ICEs) that use fossil fuels, EVs employ electric batteries to store energy and electric motors [2]. For this reason, several countries are promoting the use of EVs by introducing economic subsidies for purchase, the development of charging stations, and other economic instruments [3]. For example, in European countries such as Germany, Austria, the Netherlands, and the United Kingdom, EVs receive purchase subsidies and are exempt from property taxes [4]. In the United States, California has the Zero-Emissions Vehicle Action Plan program, which includes rebates for EV purchase or lease, as well as access to dedicated lanes, and charging corridors [3].

Despite the various benefits and efforts for EV adoption, the increased adoption of EVs poses new challenges to the power sector due to increased power demand, which could lead to power quality (PQ) problems in the grids [5]. PQ problems can include the introduction of harmonics, undervoltage, phase unbalance, and increased power loss in the transformer and distribution lines [6]. Moreover, these problems are not only dependent on the level of EV adoption but are also related to factors such as EV charging technology, grid voltage level, charging time and schedule, location of charging stations, battery state of charge, and driving habits [7]. For example, Ref. [8] analyzes the effect of EV state of charge on harmonic distortion during the charging process. It was found that the total harmonic distortion (THD) increases as the EV battery approaches 100% charge. On the other hand, in [6] they identified that the increased penetration of EVs at the distribution level in Bangladesh is related to the occurrence of harmonics, voltage fluctuations, and power losses.

Several studies have combined EV-related factors to establish EV load patterns. This makes it possible to analyze how the incorporation of EVs into the electric grid can influence PQ. As an example, in [9], the PQ issues posed by EV charging are studied. During the analysis, the authors recreate EV load patterns over a day, taking into account factors such as load current level, harmonic distortion caused by the load, and temperatures in winter and summer. When comparing the load patterns, a large difference is observed due to the factors considered. Similarly, in [10] they present load patterns for a commercial, residential, and public area, showing how the habits of the inhabitants determine these patterns. To identify and search for solutions to PQ problems, the literature proposes the analysis of the hosting capacity (HC) of electric vehicles (EVs). HC is generally defined as the amount of new load or generation that can be connected on a feeder without jeopardizing the reliability or PQ of the power system [11]. The research conducted in [12,13] presents a comprehensive review of the various methods for analyzing EV hosting capacity. A methodology for evaluating EV hosting capacity in a 123-node IEEE system, taking into account EV load patterns in commercial, residential, and public locations, is shown in [10]. The hosting capacity study helps elucidate the problems that may arise due to the increase in EV penetration and proposes solutions based on this. The literature considers solutions ranging from grid reconfiguration, use of renewable resources such as photovoltaic and wind systems, and improved EV chargers. For example, in [14], it is proposed to optimize feeder reconfiguration by using genetic algorithms (GAs) to maximize EV hosting capacity.

In order to carry out all the studies mentioned above, it is essential to understand and have EV load patterns. As it is known, EVs have a dynamic behavior, influenced by various factors that determine when an EV is charged. In addition, the driving patterns, the uncertainty as to the time and place of charging, and the total energy demand of EVs in a given area are constantly developing in a random manner [15]. Figure 1 shows this random behavior of EVs interacting with the power grid. To address these problems, current EV research focuses on model-based impact analyses of load patterns and control strategies. Therefore, the development of mathematical modeling of EV load patterns becomes a key aspect that will allow scientists to conduct more realistic studies to forecast energy demand and propose solutions to problems that may arise [5]. Various methods for modeling EV charging demand are proposed in the literature, and each of them has individual characteristics.

For instance, in [16] a modeling approach based on Markov chains and probability distributions is used with historical vehicle charging data. On the other hand, in [17] they investigate the performance of static and dynamic charging models used in electrical system studies for EV load modeling. The results show that static modeling adequately represents the steady-state load of EVs, while dynamic modeling fails to adequately capture their behavior in the presence of disturbances. In [18], they investigate how EV charging demand affects the power grid and examine the factors influencing this demand, considering EV charging models through the Monte Carlo method, using the 2009 U.S. National Travel Survey as a database. However, the authors in [5] combine artificial neural networks, recurrent neural networks, and short- and long-term memories to model EV loads, using a dataset from a total of 20,562 random transactions at a charging station. Accordingly, each method requires varied initial data, where different factors can be considered and classified according to criteria, such as the temporal/spatial dimension or the nature of the input data, as discussed in [15,19].

In this overview, data influencing the modeling of EV load patterns are addressed. Then, existing EV load modeling methods are reviewed and classified into three main groups: deterministic, statistical, and machine learning techniques. The rest of the paper is organized as follows: Section 2 contextualizes EVs, while Section 3 presents the different input factors used for EV load modeling, and Section 4 classifies, describes, and compares the different EV load modeling methods. Finally, Section 6 presents the conclusions.

2. Background on EVs

The history of EVs dates to the 19th century, when the first electric automobiles prototypes emerged thanks to Robert Anderson [3]. However, their use and development declined with the growing popularity of vehicles powered by ICEs. For much of the 20th century, EVs were relegated to limited roles, such as golf carts and delivery vehicles [20]. However, in 2010 Nissan Leaf, the first mass-produced electric car, was launched. Tesla, founded in the same period, marked a major milestone in electric mobility, launching its Roadster and Model S models, backed by a network of superchargers [3]. Today, EVs have become an increasingly popular alternative to conventional vehicles, driven by the need to reduce pollution and lower acquisition costs.

There are four types of EVs on the market, as shown in Figure 2, although only two of them require a connection to the electric grid for operation. Hybrid electric vehicles (HEVs) are configured with a combination of an ICE, a battery, and an electric motor [2]. The battery of this type of vehicle is charged by regenerative braking so it does not require charging. For this reason, these types of vehicles do not generate any negative impact on the electrical grid [21].

Plug-in hybrid electric vehicles (PHEVs) have an ICE, a battery, and an electric motor similar to HEVs. The batteries can be charged in the same way as HEVs or from an external source [22]. This type of EV usually has a smaller battery, so its impact and ability to provide grid services is limited [23]. However, higher penetration levels of PHEV charging may result in PQ impacts.

Battery electric vehicles (BEVs) are fully electric vehicles and have no ICE; they are propelled by one or more electric motors powered by a series of batteries [21]. The range of BEVs depends on the capacity of their batteries. Additionally, they can be charged during EV deceleration and braking [24]. These types of EVs have the largest battery capacity and their charging can have severe negative impacts on the electrical system. However, their ability to provide services to the electrical grid is greater than other EVs [25].

Fuel-cell electric vehicles (FCEVs), on the other hand, use hydrogen as fuel. In it, the hydrogen releases electrons that circulate through fuel cells, thus providing energy to the engines [2]. This type of EV will not impact the electrical system since it does not require recharge from the distribution system [21].

PHEVs and BEVs require electric chargers to charge their batteries. Chargers consist of an AC/DC converter, power factor correction elements, and a DC/DC converter [26]. The most common standard for EV charger classification is the SAE J1772 [27] system, which defines three charging levels for both AC and DC, levels I, II, and III, as shown in

Table 1. Load-level characteristics according to SAE J1772 [28,29].

Type of Charge	Supplied Voltage Range	Output Power Level	Estimated Charge Time
AC Level I	120 V	<1.92 kW	7–17 h
AC Level II	208–240 V	<19.2 kW	0.4–7 h
AC Level III	208–240 V	<96 kW	<0.5 h
DC Level I	200–450 V	<36 kW	0.4–1.2 h
DC Level II	200–450 V	<90 kW	0.2–0.4 h
DC Level III	200–600 V	<240 kW	0.1–0.2 h

. Each of them has different power levels and charging times, making them suitable for different situations and user needs [28,29].

The current ratings of all DC charge levels and AC level III are very high (greater than 80 A). Therefore, the current cannot be supplied by the distribution grid. This charging level is mostly used for public charging stations. AC level I and II charging stations are suitable for residential applications [24].

3. Input Data Used for EV Load Modeling

This section details the input factors in EV load modeling. The inherent complexity in EV load modeling stems from the existence of random spatial and temporal input variables [15]. Figure 3 summarizes these input variables found in the literature for EV load modeling.

The relevant factors in EV load modeling can be classified into two groups: direct and indirect factors. The direct factors are intricately linked to electric vehicles, charging infrastructure, and users [15]. Individual EVs have technological variations, such as EV model, battery capacity, charging power, and range, among others [20]. These individual EV particularities directly influence the scale of charging demand. In addition, the charging and mobility patterns of users have a significant influence on EV load modeling. The charging infrastructure also plays a crucial role, with differences in charging time depending on whether charging takes place at residences (where it tends to be slower) or at charging centers (where charging is faster, but with specific times and locations).

On the other hand, indirect factors are related to external events that are beyond the control of EV owners. Examples of these factors include weather conditions, where extreme temperatures may increase the use of heating or air conditioning, thus impacting the consumption of energy stored in EVs. Other indirect factors include government incentives and environmental regulations, which may influence EV usage decisions [30].

In summary, EV charging is influenced by factors that result in dynamic characteristics in load demand modeling. Compared to conventional loads, modeling EV loads presents unique particularities and specific challenges. These challenges focus on accurately representing the relationships between factors, such as distance traveled and EV battery state of charge (SoC), or the mathematical expressions that relate these factors. It aims to explore the dynamics of these factors that influence load modeling. Traditional methods often have significant limitations in representing these factors with their dynamic characteristics [15]. Some of the most common factors used in the literature are detailed below.

3.1. Battery

The EV battery is closely related to the energy demand during its charging process. The literature on EV load pattern modeling contains information on battery capacity and other technical specifications of EV batteries. For example, in [31], they use and present a table with various EV models intended for different uses, along with their respective battery capacities and charging powers. Lithium-ion (Li-Ion) is the most commonly used battery type in EVs because it offers greater safety, long life, and stable charge/discharge cycles [32]. EV batteries are still being developed and improved, and other materials are being explored. For example, lithium metal batteries can store more energy in the same volume or weight as Li-Ion, but they are less safe than Li-Ion batteries [33]. In addition, some EVs combine technologies, such as the Toyota Bz4x, which has nickel–metal hydride batteries with Li-Ion.

Table 2. Battery characteristics of different EV models on the market [24,32].

Brand	Model	Battery Type	Battery Capacity (kWh)	Range (km)	Charging Time (0–80%)	Type of Charge
Renault	Twizzy	Li-Ion	6.1	80	3.5 h	220 V AC
Renault	Zoe	Li-Ion	52	395	1 h	DC fast
Hyundai	IONIQ 5	Li-Ion	72.6	451	61 min	DC fast
Hyundai	Kona Electric	Li-Ion	64	484	54 min	DC fast
Nissan	Leaf	Li-Ion	30	270	40–60 min	DC fast
VW	E-Golf	Li-Ion	24.2	150	45 min	DC fast
Tesla	S	Li-Ion	100	610	30 min	Supercharger
Chevrolet	Bolt EV	Li-Ion	66	416	1 h	DC fast
Toyota	Bz4x	Nickel–metal hydride/Li-Ion	71.4	450	30 min	DC fast

presents different models of electric vehicles along with a description of the characteristics of the batteries they use.

On the other hand, one of the most commonly employed factors for modeling EV load patterns is the state of charge (SoC). The SoC plays a crucial role in energy demand, as the recharge time and the amount of energy required upon arrival at a charging station are directly dependent on this state. SoC is a factor that evolves over time or as miles are driven. These factors are interrelated and can be derived from mathematical formulas. For example, in [34] the authors calculate the battery SoC of an EV after one day of driving as follows:

$${SoC}_t={(SoC}_{t-1}-\frac{d}{D}) \times 100$$

(1)

where ${SoC}_t$ and ${SoC}_{t-1}$ are the SoC of the battery at the end of the day and start of the day, respectively, d is the distance traveled during the day, and D is the maximum distance the EV can travel.

The SoC can also be calculated by [35]

$$\begin{array}{c}\hfill {SoC}_{desired}={SoC}_{init}+\frac{E_{demand}}{C_B}\end{array}$$

(2)

where ${SoC}_{desired}$ refers to the desired battery state to travel a certain number of kilometers, ${SoC}_{init}$ is the SoC at the start of charging, $C_B$ is the EV battery capacity, and $E_{demand}$ refers to the energy demanded during charging by the EV. The energy demand can also be determined by [36]

$$\begin{array}{c}\hfill E_{demand}=P_C\eta \Delta t\end{array}$$

(3)

where $P_C$ is the charging power capacity, $\eta$ represents the charging efficiency, and $\Delta t$ is the charging time. Another important parameter is the battery charging time, discussed in [31,36]. It is calculated by

$$\begin{array}{c}\hfill \Delta t=\frac{({SoC}_{out}-{SoC}_{in})C_B}{P_C}\end{array}$$

(4)

where ${SoC}_{in}$ and ${SoC}_{out}$ are the SoC of the battery at the beginning and end of the charge.

The charging power ($P_C$) can also be represented by the battery charging current, $I_b$, as exemplified in [37]. This is because the battery capacity can be expressed in kW/h or ampere-hours (Ah).

Additionally, the SoC can be derived from a set of assumptions, as described in [16], where the maximum depth of discharge of the battery is stated to be 30%, a constraint to limit the EV state of charge at the start of charging.

3.2. Distance Traveled

EV user behavior plays a crucial role in modeling EV charging, and this approach is widely employed in research. Information on travel patterns can be obtained through travel surveys or estimated data (which often require additional statistical analyses) and worldwide traffic patterns (which are obtained from pilot experiments or GPS navigation data), as mentioned in [5]. For the most part, the studies use variables such as daily distances traveled, energy consumption per mile, and trip length, either from estimated data or actual records. When accurate data are not available, it is common to resort to travel surveys as a surrogate source of information. For example, Ref. [38] reports that in Austria, an EV typically travels an average of 32 km per day.

On the other hand, it is possible to determine the energy demand per mile driven through historical data or surveys that suggest a maximum energy consumption of 17 kWh/100 km in the winter and a minimum of 15 kWh/100 km in the summer. A similar approach is found in [16], which considers a constant energy consumption in the range of 0.15–0.30 kWh/km, depending on the distance traveled.

3.3. Weather

The charging demand of EVs can be correlated with temperature due to their thermal sensitivity, which implies that fluctuations in temperature impact the energy demand in the grid [5]. Therefore, this indirect factor is considered when modeling EV load. For example, in [9], different demand profiles of an EV in winter and summer are presented. In addition, maximum energy consumption data of 17 kWh/100 km in winter and a minimum of 15 kWh/100 km in summer are used to estimate the electrical demand of EVs [38].

3.4. Day

In many studies, time variables are an important factor. The literature often focuses on modeling EV charging demand for a weekday, although some compare differences between weekdays and weekends, as in [39]. On the other hand, in [5], the modeling and prediction of the weekly demand profile is addressed, highlighting the daily variations in EV charging. In summary, it is possible to obtain different EV load demand models by considering different time intervals. A comprehensive approach could involve modeling EV charging demand over a year.

4. EV Load Modeling Methods

This section addresses EV load modeling. The results of these models can be based on measurements at EV charging stations, or linked to factors such as EV arrival/departure times, charging duration, and distance traveled, among others. With this information, it is possible to reconstruct the demand required by EVs. Mathematical modeling of any event, such as EV demand, follows a formulation process or model building, depicted in Figure 4.

In addition, details on the strengths, weaknesses, and requirements of each modeling approach are provided. According to the articles included in this review, EV load models can be grouped into three categories: deterministic methods, statistical methods, and artificial intelligence-based methods.

4.1. Deterministic Methods

Deterministic methods for EV load modeling assume that specific EV parameters are known. Simple mathematical models that relate power and voltage at the load bus are employed. These mathematical models are divided into static and dynamic models [41]. The methodology used for deterministic modeling of EVs loads is shown in Figure 5.

Static modeling expresses the load characteristics by algebraic functions at any instant of time. These load models are generally used to calculate the steady-state behavior of loads [17]. On the other hand, dynamic load modeling is an extension of static load modeling used to capture and represent disturbances similar to faster phenomena, e.g., analysis of power system behavior after small or large disturbances. Dynamic models express active and reactive powers as a union of voltage and time [17,41].

The most commonly used types of static and dynamic load modeling in EV modeling include exponential static load models, polynomial static load models, and exponential recovery dynamic loading models.

• The exponential static load model (EXP) can be represented by

  $$\begin{array}{c}\hfill \frac{P}{P_0}={\left(\frac{V}{V_0}\right)}^{\alpha }\end{array}$$

  (5)

  $$\begin{array}{c}\hfill \frac{Q}{Q_0}={\left(\frac{V}{V_0}\right)}^{\beta }\end{array}$$

  (6)

  where the variables $P_0$, $Q_0$, and $V_0$ are rated active power, rated reactive power, and rated voltage, respectively, while $\alpha$ and $\beta$ are the unknown model parameters.
  There is a modification of this equation, known as constant power plus exponential model (PEXP), where a constant variable is added to the exponential load model, as follows:

  $$\begin{array}{c}\hfill \frac{P}{P_0}=p_1+p_2{\left(\frac{V}{V_0}\right)}^{\alpha }\end{array}$$

  (7)
  Another modification of the exponent model is the linear exponential model (LEXP), as shown:

  $$\begin{array}{c}\hfill \frac{P}{P_0}=p_1\left(\frac{V}{V_0}\right)+p_2{\left(\frac{V}{V_0}\right)}^{\alpha }\end{array}$$

  (8)
• The polynomial static load model (ZIP) can be expressed as

  $$\begin{array}{c}\hfill \frac{P}{P_0}=p_1{\left(\frac{V}{V_0}\right)}^2+p_2\left(\frac{V}{V_0}\right)+p_3\end{array}$$

  (9)

  $$\begin{array}{c}\hfill \frac{Q}{Q_0}=q_1{\left(\frac{V}{V_0}\right)}^2+q_2\left(\frac{V}{V_0}\right)+q_3\end{array}$$

  (10)

  where $p_1$, $p_2$, and $p_3$ are the unknown parameters of the model as well as $q_1$, $q_2$, and $q_3$.
• The exponential recovery dynamic load model (ERL) represents the active and reactive power responses to staggered bus voltage disturbances. This model is commonly used to represent loads that recover slowly over time, ranging from several seconds to tens of minutes [41].

  $$\begin{array}{c}\hfill T_p\frac{d{P}_r}{dt}+P_r=P_0{\left(\frac{V}{V_0}\right)}^{a_s}-P_0{\left(\frac{V}{V_0}\right)}^{a_t}\end{array}$$

  (11)

  $$\begin{array}{c}\hfill P_1=P_r+P_0{\left(\frac{V}{V_0}\right)}^{a_t}\end{array}$$

  (12)

  where $P_r$ is a state variable related to active and reactive power dynamics, $T_p$ is the exponential recovery response time constant, while $a_s$ and $a_t$ are exponents related to the transient load response.

In [42], a static modeling approach is discussed based on current and active power measurements at an electric vehicle (EV) charging station. The ZIP, EXPO, PEXPO, and LEXP models are evaluated to accurately represent EV charging, showing that the LEXP and EXP models are the most suitable. In [43], the impact of EV charging on the power grid is explored by evaluating loads with voltage variations under different ZIP, PEXP, and EXP charging models. The results indicate that each EV charging model have a different impact on the IEEE 34-bar study system. The ZIP, PEXP, and EXP methods have a higher, medium, and lower impact, respectively. This highlights the importance of selecting more realistic EV load models, as this significantly influences the network analysis results. On the other hand, in [17], static and dynamic models describe the behavior of EV chargers. Initial simulations used static charging models, such as ZIP and EXP, demonstrating their effectiveness. However, simulations with the ERL dynamic charging model highlighted the significant influence of control parameters on the dynamic response of EV chargers, questioning the ability of standard models to adequately capture the dynamic behavior of EVs. In [44], the PEXP model is applied in power flow analysis, revealing a direct correlation between EV charging and voltage stability as the rated charging power of EVs increases.

From the literature, it can be concluded that deterministic methods can be valuable for infrastructure planning and power flow analysis in situations with robust data and accurate steady-state assumptions. However, such methods do not address the variability and uncertainty inherent in human behavior and electric vehicle adoption.

4.2. Statistical Methods

Statistical methods for EV load modeling are used to understand and predict how EV load demand varies as a function of the factors described in Section 3. These methods leverage statistical techniques, such as probabilistic distributions and stochastic processes, to analyze historical data and generate demand models that can aid in EV charging infrastructure planning and grid management [19,24].

In the statistical methods approach, probability density functions (PDFs) can be applied to represent and quantify uncertainty in EV-related data or events, such as charging duration, distance traveled, and availability of charging points [35]. In [18,31], researchers represent daily distance traveled and charging start time with a normal distribution. Similarly, in [38], a normal distribution is used to represent the start time of the first and second trip, assuming an average of two trips per day. On the other hand, in [31], the maximum distance traveled is represented by a log-normal distribution. However, Ref. [45] indicates that standard single-mode probability distributions (e.g., normal distribution) are unrepresentative of real situations. Therefore, it is more appropriate to formulate a multi-mode representation, such as a beta probability distribution, to represent the charging start time. Figure 6 briefly describes the most commonly used probability distributions in EV load modeling.

Probability distributions rely heavily on historical data to determine which best represent observed behavior. For instance, historical data are used in a study that relies on motor vehicle traffic surveys in Australia [38], while the Elaad database provides aggregated data from public, private, and workplace charging stations in the Netherlands [45]. Likewise, the KOSTAT database provides EV information for South Korea [46].

Stochastic methods are based on stochastic process theory, which means that they consider the randomness and variability of events over time [19]. These methods may include event simulation and modeling of evolving systems with random components. Often based on event simulations, they do not necessarily require detailed historical data as they make use of PDFs [35]. They can be useful when limited information is available or when different future scenarios are to be explored. The most commonly used stochastic modeling methods include Monte Carlo, Markov chain theory, and autoregressive integrated moving average (ARIMA).

4.2.1. Monte Carlo

The Monte Carlo method (MCS) is a stochastic simulation technique that employs random numbers to address practical problems with the primary purpose of obtaining numerical solutions. This approach can generate random results that follow a probability distribution by simulating a process, and then, estimating the numerical properties of the model using statistical methods [47]. Within the MCS method, the EV load estimation process is carried out by a large number of samples generated using PDFs based on the input data [24]. The MCS is especially effective in scenarios characterized by high variability in loading patterns. Its accuracy can reach very high levels with an adequate number of simulations, although this may imply an increase in computational costs. In addition, to improve the accuracy of the MCS, it is important to ensure that the input data are accurate and up to date. This encompasses detailed information on user charging behavior, grid conditions, and other relevant factors influencing EV load modeling [15].

In [46], researchers use the MCS method to model EV loading patterns in New Zealand, considering EV plug-in time for charging and miles driven as input, characterized by normal probability distributions. From the miles driven and charging time, the EV SoC and charging duration can be calculated. The methodology used for modeling EV loading using the MCS method is shown in Figure 7.

In [48], a case study based on the Java–Bali Indonesian electric system is considered to study different realistic scenarios of EV adoption, using probabilistic models and an MCS modeling approach. The results of the study show that since EVs overload the electricity system and incur high electricity production costs, adopting a proposed new rate scheduling strategy can alleviate these problems. In [34], the Bass forecasting model is employed using historical EV growth data between 2013 and 2018 to estimate the number of EVs through 2022. In addition, the authors resort to MCS simulations to estimate EV loading demand. The loading characteristics for different EVs (private, cab, and buses) are analyzed, considering factors such as charging time, starting time, daily mileage, charging method, and charging power. Similarly, Ref. [18] analyzes the factors influencing the EV load distribution and formulates the corresponding PDF, based on variables such as charging mode, SoC, load demand, and initial charging time specific to each EV type. The daily charging profile of the different EV types is calculated using an MCS simulation. Ultimately, the total EV load distribution curve is obtained by superimposing the individual contributions of the different EV types.

4.2.2. Markov Chain Theory

Markov chain theory uses historical data to investigate the future state of the system. What is particularly noteworthy is its efficiency in both the statistical analysis and the temporal aspect of the datasets [24]. The charging behavior of EVs depends mainly on their mobility characteristics and the available charging facilities. Even though it is difficult to model the load behavior, some random methods, such as state, dwell time, and state transition probability can describe the expected behavior [15]. The procedure of modeling the EV load behavior using the Markov chain theory method adopts the following methodology. First, all the values of the studied phenomena are dispersed over several states (Figure 8).

In Figure 8, $P_{i,j}$ denotes the transfer probability from interval i to interval j, which can be expressed in terms of conditional probabilities as Equation (13) [36]:

$$\begin{array}{c}\hfill P_{S_i\to S_j}=P\left(S_j|S_i\right)=P_{i,j}\end{array}$$

(13)

Then, considering that the series of states are aligned by a homogeneous Markov chain, a transition probability matrix of these states is determined, represented by Equation (14).

$$\begin{array}{c}\hfill {{P=(P}_{ij})}_n=\left[\begin{array}{cc}\begin{array}{cc}{P}_{11}& P_{12}\\ P_{21}& P_{22}\end{array}& \begin{array}{cc}\cdots & P_{1n}\\ \cdots & P_{2n}\end{array}\\ \begin{array}{cc}⋮& ⋮\\ P_{n1}& P_{n2}\end{array}& \begin{array}{cc}⋮& ⋮\\ \dots & P_{nn}\end{array}\end{array}\right]\end{array}$$

(14)

This matrix is then applied to create a new chain of states. Finally, each state in this new chain is transformed into an EV parameter value with a firm random generator. The predicted values from the Markov method are based on the probabilities obtained from the historical EV data [24].

Markov chain theory can analyze the relationship between the starting point, destination, and arrival time of EVs. For example, in [16], researchers use traffic data to model urban areas in Nanjing, identifying five zones, and adjusting travel times using a Weibull PDF and Markov chain theory to characterize traffic flow patterns and electric vehicle charging demand, considering variables such as distance traveled and battery SoC. The analysis shows that peak demand tends to occur at 6 a.m., midday, and evening for various types of EVs, except for electric cabs because of variability in the distances traveled. On the other hand, the theory of Markov chains is also used to represent the evolution of the battery SoC over time. The work in [36] focuses on an aggregation model of EVs and highlights the use of higher-order Markov chains to define the charging and discharging states of EVs. In addition, it uses the Poisson distribution to predict the EV charging start time. This approach reduces the complexity of the state space. The model is validated in MATLAB with data from a charging station in China and its ability to accurately predict EV charging is demonstrated. Hence, methods based on Markov chain theory have a large memory and carefully examine the problem space, so are appropriate for modeling the behavior of EVs.

4.2.3. Autoregressive Integrated Moving Average (ARIMA)

Autoregressive integrated moving average is a statistical technique used in time-series analysis to model and predict patterns in sequential data, such as EV charging station loading [49]. The ARIMA method is represented by a mathematical equation that describes the relationship between the values of a time series and its past values, as well as past errors. The general equation of an ARIMA model can be expressed as

$$\begin{array}{c}\hfill y_t=\mu +\sum _{i=1}^p{\gamma }_i{y}_{t-1}+{\epsilon }_t+\sum _{i=1}^q{\theta }_i{\epsilon }_{t-1}\end{array}$$

(15)

where $y_t$ is the present value, $\mu$ is a constant, $\gamma$ and ${\theta }_i$ are the undetermined coefficients and ${\epsilon }_t$ is the error. The AR(p) model expresses the relationship between the present value and the historical data. The MA(q) model focuses on the error accumulation of the AR(p) model, which can effectively eliminate fluctuations. The construction of the ARIMA model usually consists of three steps. The first step includes pattern recognition and order determination, while the second step involves parameter estimation, and the third step conducts model validation [50]. The authors in [49] employ an ARIMA method for modeling and forecasting conventional electric load and EV parking demand. ARIMA parameters are adjusted to minimize the mean square error and improve accuracy by decoupling daily and seasonal load profiles. This approach shows significant error reduction and is used in a daily scheduling problem with random constraints. Simulation results demonstrate daily cost savings of 2.9% and 23% on 6-bus and 24-bus systems, respectively. On the other hand, Ref. [50] focuses on modeling and short-term prediction of vehicle flexibility and participation in real-time energy markets. Predictions are studied for both one hour and 15 min ahead. Due to the complexity of the load data and their lack of continuity, the ARIMA method combined with a Gaussian filter is used. It is found that using the ARIMA method for EV load forecasting yields values close to the actual values. Both studies highlight the usefulness of ARIMA in EV load forecasting and demand response flexibility, with a focus on optimizing accuracy and reducing costs in energy operations.

4.2.4. Fuzzy Logic

The fuzzy logic modeling method uses fuzzy triangular numbers to model uncertainty in EV demand, allowing for network component planning without the need for precise data. This is achieved by assigning fuzzy numbers to EV power profiles based on approximate scenarios. In Figure 9, the minimum (at1), average (at2), and maximum (at3) values of EV power at different times are represented as triangular fuzzy numbers, with at2 being the most likely value. Fuzzy logic is employed to classify key factors in EV charging, such as SoC, distance traveled, and parking duration, allowing for flexible calculations of demand and charging times [24].

In [51], the impact of EV battery charging on the electric grid is examined. The researchers use data from driving cycles and parking patterns to model driver behavior. A fuzzy inference system is used to model drivers’ recharging decisions, representing the SoC, parking duration time, and charging probability as triangular fuzzy numbers. Hourly charges are estimated for various vehicle types and different battery capacities, considering charging regimes and home and work charging scenarios. The results show the impact on the electric grid with the adoption of EVs.

Fuzzy logic can also be used in conjunction with the MCS. This hybrid method combines probabilistic and fuzzy techniques to model EV loading and address uncertainty in the input data. PDFs or datasets are required to extract EV loading. This hybrid approach considers both spatial and temporal uncertainty of EVs, unlike most methods that focus only on temporal uncertainty and assume uniform load locations throughout the network [24]. In [52], the demand for EVs on the electric grid is addressed by using a fuzzy logic model to simulate EV users’ decision making about charging. Key factors, such as EV autonomy (AEV), battery state of charge (SoCEV), and daily travel distance (Dd), are considered. These factors are described in terms of low, medium, and high, and are used to determine charging decisions. A Monte Carlo simulation is employed to analyze EV charging over multiple days, considering multiple factors. EV type and charging power are selected based on market contribution and geographic location. The battery SoC on arrival is estimated and the charging period is calculated based on daily travel distance and charging power. The EV charging decision is determined through the function C(AEV, Dd, SoCEV). With a similar approach, Ref. [53] proposes a novel method based on a fuzzy inference algorithm to predict the EVs’ load distribution in time and space. A travel chain model is established to describe the dynamic process of EVs by considering traffic factors and modeling PDFs of the spatiotemporal variables in the travel chain. A fuzzy inference system with three inputs and one output is used to calculate the load probability, instead of assuming specific load conditions. Load distribution curves are obtained by Monte Carlo simulation, confirming the validity and accuracy of the method.

4.3. Machine Learning Methods

Machine learning is a sub-discipline of artificial intelligence that focuses on the creation of algorithms and models capable of enabling computers to learn and make data-driven decisions without requiring explicit programming. Its purpose is to automatically discover principles and patterns from collected data or interactions, using a trial-and-error approach [54]. For applications in EV demand modeling, machine learning sub-methods called random forest and neural networks are mostly used.

4.3.1. Random Forest (RF)

Random forest is a set of decision trees used in machine learning. Each tree is trained with a random selection of samples and can be used for both regression predictions and classification [55]. The output of the RF is the average of the values or the top-ranked classification of the decision trees. Each RF tree is trained independently and has its own training dataset. The results are then combined to obtain a more robust and accurate prediction (Figure 10 and Figure 11) [56]. RF is known for its ability to overcome the limitations of individual decision trees and its good scalability [57].

The researchers in [55] focus on predicting the spatiotemporal distribution of EV load to optimize the power grid using an improved random forest (IRF)-based prediction model. The results demonstrate that IRF provides higher accuracy compared to methods such as support vector machine (SVM), back propagation neural network (BPNN), and general RF. Python is used to implement these algorithms, based on EV data collected over five weeks. Similarly, the authors in [58] present an EV charging forecasting model that combines the sparrow search algorithm (SSA) and improved random forest regression (RFR). The SSA-RFR model significantly improves the accuracy of load prediction compared to other models due to parameter optimization and better generalization capability. It focuses on the load prediction of 12,450 EVs in a specific region. EV load data were collected at 15 min intervals by MCS simulation and divided into training and test sets. On the other hand, in [57], a load forecasting method for EV charging stations is proposed that combines generative adversarial networks (GANs) and the RF algorithm. The GAN-RF model shows high accuracy and generalization when considering load variability and human behavior. The study collected data from charging stations in a specific region of Xian Yang for 15 days, with sampling every half hour. The last day was used for validation. In [56], the EV charging prediction of the seasonal autoregressive integrated moving average (SARIMAX) and RF models are compared using real data from 1700 charging stations in the Netherlands. The SARIMAX models outperformed RF in predicting charging for different time horizons. Although the machine learning models outperformed a persistence approach, they could not match the accuracy of SARIMAX. The authors conclude that this is possibly due to the limited size of the training dataset.

These studies highlight the diversity of approaches in EV load prediction and underscore the need to consider factors such as dataset size and problem complexity to achieve accurate and effective predictions in power grid optimization.

4.3.2. Artificial Neural Networks (ANNs)

Artificial neural networks are a group of algorithms that fall within the field of machine learning and form the basis of deep learning algorithms. The design and operation of ANNs resemble the functioning of the human brain, where interconnected processing units work together to transmit and process information, like neurons in the brain. Prediction techniques based on neural networks such as recurrent neural networks (RNNs), convolutional neural networks (CNNs), long short-term memory (LSTM), and recurrent unit gating (GRU) are widely used in the field of load prediction [59]. EV load modeling requires handling large volumes of data, which makes ANNs and machine learning valuable. ANNs adapt to the complexity of the phenomenon under study, considering factors such as arrival time, departure time, and variability in driver behavior. The use of deep ANNs is essential to address this complexity [24]. Figure 12 shows the general structure of ANNs for EV modeling.

An example of ANN application is discussed in [61], where a new approach to EV charging station occupancy prediction is employed, highlighting its potential in efficient EV fleet management. A mixed LSTM neural network that combines historical data and temporal features to obtain accurate predictions is used. The authors mention that this method outperforms conventional approaches and offers potential for improving smart charging strategies. The accuracy of the model is high for short-term predictions, decreasing as the prediction horizon is extended, but is still adequate for windows of less than 60 min. Similarly, a novel charging forecasting method for EV charging stations that uses Bayesian deep learning and LSTM to address uncertainty in predictions is introduced in [60]. Experiments are conducted on real data from a charging station at the Caltech California campus. The method used shows superior performance compared to other methods, suggesting great potential for practical applications. A similar study is performed in [62], focusing on accurate prediction of electric vehicle charging. It uses the k-means algorithm to cluster electric vehicle charging and the multiresolution discrete wavelet transform to extract features. Then, it employs LSTM deep learning prediction model to make accurate predictions of EV charging. The effectiveness of the method is supported through a case analysis with more than 1300 charging stations in Zhejiang. On the other hand, the authors in [63] address the growing concern of predicting EV charging at charging stations, given their non-stationary nature and erratic charging procedures. The study uses a deep learning-based CNN to predict traffic flow and evaluates uncertainties to establish prediction intervals. Then, it calculates EV arrival rates and studies the charging process using a probabilistic queuing model that considers service constraints and driver behaviors. It is tested with real UK traffic data, demonstrating its potential for practical applications, and improving the accuracy and reliability of EV charging demand prediction, which is beneficial for charging station operators and traffic management.

Artificial neural networks offer a powerful tool for EV charging prediction, as they can capture complex relationships and adapt to variability in charging behavior. Although they may require large datasets and computational complexity, they offer great potential for improving efficiency and accuracy in EV charging management.

4.4. Comparison of Methods

Each of the EV load modeling methods analyzed has distinctive characteristics that make them suitable for specific applications or studies. Deterministic modeling uses mathematical formulas that relate energy demand to current or voltage to model EV charging. It assumes that the parameters related to EVs are known, such as EV arrival and departure time and distance traveled, in other words, the EV is seen as a stationary storage system [24]. There are studies where they rely on real measurements or charging station simulations to find the parameters of ZIP or polynomial charging models [19]. Its advantage is that it is easy to understand and apply because it is based on mathematical equations and is useful for stationary studies [41]. Additionally, it is easy to apply and implement computationally [5]. Its main limitation is that it does not adequately represent the random loading behavior of EVs and is limited when attempting to capture complex or irregular patterns. The amount of data required is minimal since as mentioned it is mostly based on assumptions, predefined parameters, or measurements. The accuracy of this model is low, depending on the accuracy of the assumptions used [41].

On the other hand, statistical methods use data analysis techniques and probabilities to model or predict EV load profiles. This approach is based on historical load data collection, pattern analysis, and trend analysis [19]. Its main advantage is that it can capture variations in EV user behavior over time [5]. Generally, these models are more constrained than deterministic models, since they are based on historical data. The main limitation they presents is the need for historical data as they require a significant amount of data. Another limitation is the complex analysis that requires specialized knowledge in statistics [64]. Additionally, they have a moderate computational cost [20]. The amount of data required is moderate to high, depending on the desired accuracy and variability of the data. Its accuracy is high, especially if a large amount of high-quality data is available [65].

However, machine learning-based modeling and prediction of EV load patterns utilizes advanced algorithms in learning patterns and characteristics of EV load data [66]. This approach includes various techniques such as neural artificial network models and decision trees [5]. Its main advantage is that it can provide very accurate predictions by learning complex nonlinear relationships from the input data [67]. It fits well with evolving data and can learn from new trends and patterns. Its main limitation is its large data requirements as it requires large amounts of data to train models effectively [19]. Additionally it requires advanced computational skills in data science and machine learning [20]. The accuracy is very high provided sufficient high-quality data is available and appropriate algorithms are used [65].

Table 3. List of advantages and disadvantages of all methods [15,20,24,41].

Approach	Method		Advantages	Disadvantages	Application	References
Deterministic	Static Models (EXP, ZIP)		Very simple, historical data not needed, very low computational cost	Output not accurate, uncertainty and driving patterns not considered	Appropriate for research aiming to examine the approximate impact of EVs	[17,42,43,44]
Statistical	Monte Carlo		High accuracy, models data uncertainty well	Does not account for data correlation, accuracy contingent on quantity of historical data and sample size	Prediction of EV load demand with high variability in user behavior	[18,34,46,48]
	Markov Chain Theory		Considers all events of the transition matrix with high accuracy, moderate computational cost	Performance tied to dimensionality of input data, high computational cost for large state transition matrices, suboptimal with low input data dimension	Time-series modeling of EV scenarios	[16,36]
	ARIMA		High accuracy in time series, models uncertainty adequately	Requires computational effort, experience, many input data samples, performance degrades beyond short forecast horizons	Optimizing and reducing costs in energy operations	[49,50]
	Fuzzy Logic		Can be modeled without historical data, models load uncertainty well, combines with other methods (e.g., MCS) for accuracy	Accuracy depends on rule configuration, based on investigator’s experience	Sufficient when historical data are not available	[51,52,53]
Machine Learning	Random Forest		Versatile, no prior assumptions on data shape, considers data correlation	Requires large data volumes for generalization, accuracy depends on data quality, weak interpretability, no ability to extrapolate	Focus on various patterns in EV dynamics	[55,56,57,58]
	ANN	RNN	Good for sequential and time-series data, models complex time dependencies	Issues with long-term dependencies, high computational cost, medium accuracy compared to other ANN methods	Prediction of load patterns based on historical data	[5,60,61,62,63,68,69]
		CNN	Useful for extracting spatial features and patterns in 2D data, high accuracy on spatial data	Ineffective for pure sequential data, high computational cost	Analysis of spatial data, e.g., location of parking
		LSTM	Captures long-term dependencies in sequential data, high precision	Very high complexity, higher computational cost than other ANN methods	Long-term demand forecasting, complex time series
		GRU	Computationally efficient compared to LSTM, captures long-term dependencies in sequential data	High accuracy but slightly lower than LSTM	Time-series load prediction with lower computational cost

shows the advantages and disadvantages of each method.

In [68], the predictive performance of the RF and ANN methods at different spatial levels for predicting EV load is studied. In this study, they concluded that the RF model was found to be more robust and accurate at different spatial levels and in case studies of different sizes compared to the ANN model. On the other hand, in [69], they present an investigation of nine diverse methodologies for forecasting EV load curves, encompassing statistical, machine learning (ML), and deep learning (DL) techniques. The methodologies are evaluated using four public and real EV datasets, with models incorporating online and offline historical data for different scenarios and exploring seasonal variations through annual simulations. The findings in this research suggest that ML models are the most suitable due to their higher accuracy in forecasting EV load across different datasets compared to DL and statistical models. However, the models studied demonstrated the ability to predict EV load hourly, maintaining their accuracy even in the presence of outlier data.

5. Future Trends in EV Load Modeling Methods

EV load modeling is constantly evolving due to the growing adoption of EVs and the need to efficiently and sustainably integrate them into the electric grid. One of the main trends is the use of data-driven modeling methodologies and machine learning. Collecting and analyzing large volumes of data generated by EVs and charging stations enables the development of more accurate models. These are used to predict load patterns, optimize load distribution, and manage demand in real time [20]. However, one of the challenges is the availability and accessibility of specific data, such as battery-related data, traffic or trip data, and station charging session records [5]. Many studies do not clearly show their data nor the provenance or how to obtain it, which is an obstacle to continuous improvement in the models. It is essential to find a balance that allows for the protection of EV users’ information and, at the same time, transparency in the data needed. The creation of real-time management algorithms is another key trend, as these can balance energy demand and supply, optimize EV charging costs, and minimize the impact on the grid [20]. Consideration of government regulations or policies also plays a crucial role, as emissions regulations, incentives for EV adoption, and infrastructure guidelines significantly influence EV charging modeling strategies [70]. Likewise, understanding EV user behavior is crucial for charging center location planning and demand management. Another future trend is for models of EV load to incorporate factors such as usage habits, travel patterns, and charging preferences of EV users. On the other hand, EV-related technology is constantly evolving. For example, fast charging infrastructure is changing EV charging dynamics, so models must consider the impact of these new technologies on the grid and charging patterns. Another emerging trend is the vehicle-to-grid (V2G) concept [15]. This approach allows EVs to not only draw power from the grid but also return energy stored in their batteries back to the grid. This would help to balance energy demand, during consumption peaks and provide ancillary services to the grid. However, for EV integration to be truly beneficial for environmental preservation, it is crucial that EV charging does not rely on fossil fuel-based power generation [70]. The increasing adoption of EVs is promoting the integration of renewable energy. For this reason, modeling is being carried out where EV charging is combined with renewable sources, such as photovoltaic systems, to maximize sustainability and reduce carbon emissions [5]. Important to developing more accurate EV charging modeling methods is the integration into the model of new technologies that emerge with respect to EVs, real-time data collection and analysis, and collaboration between different sectors. These methods will make it possible to forecast the impact of EV integration into power grids and seek effective solutions.

6. Conclusions

The increasing adoption of EVs in the power grid is an inescapable trend that will have a significant impact on PQ, stability, and economic efficiency of the grid. This shift poses both opportunities and challenges in the optimal management of EV charging as a flexible energy source. Therefore, it is essential to develop accurate and efficient models for understanding and managing EV charging, which will affect both its future development and the overall charging and electrical infrastructure. Accuracy in quantifying the scale and evolutionary characteristics of EVs is critical to addressing the challenges in planning and operating future energy systems.

In this overview, the examination initially focuses on various factors, both direct and indirect, influencing EV charging modeling. In most of the papers reviewed, it was observed that the main factors considered are SoC, average daily distance traveled, and EV charging time. We then delved into EV charging modeling methods, dividing them into three categories: deterministic, statistical, and machine learning modeling methods. Deterministic modeling methods are characterized by their simplicity, do not require historical data, and have low computational time. However, their main disadvantage is that they provide accurate modeling only for specific points in time and do not consider the uncertainty of EVs. These methods may be appropriate for studies that seek to assess the impact of EVs in a steady state. On the other hand, statistical methods offer high modeling accuracy and consider the uncertainty of EVs, and can model EV load demand both spatially and temporally. They require historical data, which can be collected through surveys or traffic reports, and mostly the input data for these models are presented in PDFs. However, their main disadvantage is that they involve higher computational costs. Finally, machine learning methods can more accurately predict EV demand by considering the correlation between input and output data. These methods can model EV demand both spatially and temporally, for periods ranging from 15 min to several days. However, they are highly dependent on the amount of historical data available, and their computational cost is high. This review highlights the importance for researchers or network planners to identify the amount of available data and their accuracy requirements before selecting the most appropriate modeling method for EV loading.