EVs classification and mobility characterization
EVs can be broadly classified into two main categories: passenger cars and commercial vehicles. Passenger cars are primarily designed and engineered for transporting passengers and their luggage, with a seating capacity of up to nine, including the driver. Commercial vehicles, on the other hand, encompass all goods vehicles and buses with more than nine seats. Given the current landscape of electric vehicle development and the distinct functions of these vehicles types, passenger cars are typically subdivided into three categories: private cars, taxis and official vehicles. Commercial vehicles can be further classified into buses, express logistics vehicles, sanitation vehicles, and engineering repair vehicles (which specifically include emergency repair vehicles for utilities, such as electricity and drainage, but excluding large-scale engineering and freight vehicles). Unlike passenger cars, commercial vehicles tend to follow more fixed routes and have predictable charging locations. Therefore, this study focuses on the analyzing the travel behavior and predicting the charging loads for electric vehicles within the passenger car category.
Characterization of commuter trips by private EVs
The demand for user charging is closely linked to travel patterns. Predicting the expected electricity demand for EVs based on the travel characteristics of EV users has been demonstrated to be an effective approach24. A robust characterization of user travel demand within urban transportation networks provides a critical foundation for accurately forecasting the charging loads.
For EVs, a trip typically comprises three key events – departure, destination parking, and a potential charging event at the destination or parking lot25. The routes taken by private EVs are generally more predictable, primarily revolving around commutes between residential and work locations26. EV usage tends to peak during morning and evening commuting hours, with significantly lower activity throughout the rest the day. Private EVs are often parked at workplaces or offices during the day on weekdays, and at home during the evenings. This results in extended parking periods, particularly overnight when charging opportunities are more readily available. Additionally, weekends or holidays may see an increase in travel frequency and longer trips, including the possibility of occasional long-distance travel.
The trip chain analysis method can effectively simulate the daytime travel behavior of private EV users, closely mirroring actual user routines. A Trip chain refers to the sequence of travel activities an individual undertakes, delineated by changes in time and location. It involves the process by which an individual departs from an origin point, at a specific time, uses a particular mode of transportation, visits multiple destinations in a defined order, and eventually reaches the final destination, completing the journey27. The National Household Travel Survey (NHTS) data, which provide comprehensive transportation statistics, including details on the date, origin, destination, and purpose of each trip. According to NHTS2022 data, residential travel is dominated by simple chains, where both origins and destinations are primarily home locations, and the longest chain length is three. Given the primary purpose of many trips, it is reasonable to assume that brief stops are less likely to result in charging events.
The primary travel behaviors of private EVs and the proportion of trips associated with each activity are outlined in Table 128,29. Based on the functional characteristics of urban areas, five types of travel destinations can be classified: home (H), work (W), shopping malls (SR), leisure places (SE) and other locations (O).
$$f({t_s})=\frac{1}{{\sigma \sqrt {2\pi } }}{e^{ – \frac{{{{({t_s} – \mu )}^2}}}{{2{\sigma ^2}}}}}$$
(1)
Where,\(\mu\)and\(\sigma\)represent the mean and variance of the initial moments corresponding to distinct trip chains, respectively. The specific parameters for these distribution are provided in Table 228,29:
Electric taxis travel characterization
Electric taxis operate in shifts, providing near-continuous service throughout the day. Due to their intensive operational nature, they accumulate significantly higher daily mileage compared to private cars, making more trips and covering longer distances. Stops are typically brief, occurring primarily when passengers are picked up, dropped off, or while waiting for new passengers. The high degree of randomness in taxi destinations, driven by unpredictable passenger demand, makes route forecasting challenging. Charging is generally required during service breaks or at shift changeovers, creating a limited window for recharging during operational hours. Consequently, fast charging solutions are often necessary, while slow charging usually takes place during shift transitions in the early morning hours.
Electric taxis primarily transport passengers during the day, with peak operating hours occurring between 6:00 p.m. and 8:00 p.m. Based on the fitted data presented in the literature30, the initial travel times follow the probability distribution in Eq. (2):
$$f({t_s})={\lambda _1}{e^{ – {{(\frac{{{t_s} – {\alpha _1}}}{{{\beta _1}}})}^2}}}+{\lambda _2}{e^{ – {{(\frac{{{t_s} – {\alpha _2}}}{{{\beta _2}}})}^2}}}$$
(2)
Where, \({\lambda _1}\)= 0.389, \({\alpha _1}\)= 7.046, \({\beta _1}\)= 1.086, \({\lambda _2}\)= 0.066, \({\alpha _2}\)= 15.610, \({\beta _2}\)= 9.667.
Official EVs travel characterization
Official EVs serve a wide range of purposes, encompassing government, commercial and specialized. The routes, origins, and destinations of these EVs are more varied, as they often accommodate both work-related official travel and personal use. Official EVs typically make frequent trips with irregular driving patterns, often entering and leaving different locations multiple times throughout the day. Due to the diverse nature of their tasks, the timing and location of their recharging are more dispersed. This can result in charging requirements conflicting with the need for immediate availability during emergency tasks, thus placing higher demands on the strategic placement and scheduling of charging infrastructure.
The majority of initial travel times for official EVs are concentrated during two distinct periods: 7:00–9:00 in the morning and 18:00–20:00 in the evening, exhibiting a typical double-peak pattern. In this study, the statistical data of official car travel times have been analyzed and fitted, and the probability distribution for the initial travel times is expressed in Eq. (3):
$$\begin{aligned} f(t_{s} ) & = a_{0} + a_{1} \cdot \cos (w \cdot t_{s} ) + b_{1} \cdot \sin (w \cdot t_{s} ) \\ & \quad + a_{2} \cdot \cos (2 \cdot w \cdot t_{s} ) + b_{2} \cdot \sin (2 \cdot w \cdot t_{s} ) \\ \end{aligned}$$
(3)
Where, \({a_0}\)= 24.58, \({a_1}\)=-16.06, \({b_1}\)=-14.28, \({a_2}\)= 2.119, \({b_2}\)=-10.88, w = 0.2887.
Trip path analysis based on modified OD probability matrix
Due to the high uncertainty in the routes of electric taxis and official cars, the Origin-Destination (\(OD\)) analysis method, which can effectively capture the road peer flow information, is utilized for path planning. The\(OD\)matrix represents the traffic flow between all origins and destinations within the transportation network. Bu using traffic flow observation data provided by the traffic department, it is possible to determine the volume of electric vehicle traffic on the roads during different time periods. With the aid of TransCAD, inverse traffic modelling is employed to derive \(OD\) matrices for varying time periods31. By dividing a day into m time periods, the \(OD\) matrix consists of m submatrices \(B_{{n \times n}}^{{t,t+1}}\), where n is the number of nodes in the region. Each submatrix contains details about the probability of travelling between an origin and destination for each road during the time period t to \(t+1\). According to Eq. (4), the probability of a vehicle travelling from node i to node j at time t is calculated32:
$$C_{{ij}}^{{t,t+1}}=\frac{{b_{{ij}}^{{t,t+1}}}}{{\sum\nolimits_{{j=1}}^{n} {b_{{ij}}^{{t,t+1}}} }},\quad \left( {1 \leq i \leq n,1 \leq j \leq n,i \ne j} \right)$$
(4)
Where, \(b_{{ij}}^{{t,t+1}}\) denotes the vehicle traffic between the origin node i and the destination node j during the time period t to \(t+1\); \(C_{{ij}}^{{t,t+1}}\) represents the trip probability of an EV starting at node i and selecting node j as its destination within the same time period. The \(OD\) probability matrix K for the period t to \(t+1\) is then defined by Eq. (5):
$$K=\left[ {\begin{array}{*{20}{c}} {C_{{11}}^{{t,t+1}}}&{C_{{12}}^{{t,t+1}}}& \cdots &{C_{{1n}}^{{t,t+1}}} \\ {C_{{21}}^{{t,t+1}}}&{C_{{22}}^{{t,t+1}}}& \cdots &{C_{{2n}}^{{t,t+1}}} \\ \cdots & \cdots & \cdots & \cdots \\ {C_{{n1}}^{{t,t+1}}}&{C_{{n2}}^{{t,t+1}}}& \cdots &{C_{{nn}}^{{t,t+1}}} \end{array}} \right]$$
(5)
EVs generate destinations through random sampling based on \(OD\) probability matrix according to their type and the specific time period.
In this study, the original \(OD\) matrix is dynamically adjusted. \(g_{{ij}}^{{t,t+1}}\) denote the difference in traffic flow from node i to node j during the time period t to \(t+1\), calculated as:
$$g_{{ij}}^{{t,t+1}}=\left| {b_{{ij}}^{{t+1}} – b_{{ij}}^{t}} \right|$$
(6)
The diagonal elements of the corrected \(OD\) matrix represent the adjusted traffic flow from the current node i to itself. Let \(B_{{ij}}^{t}\) denote the corrected vehicle traffic between the origin node i and the destination node j during the time period from t to \(t+1\), calculated as follows:
$$B_{{ij}}^{t}=\sum\limits_{{j=1}}^{n} {g_{{ij}}^{{{t_1},{t_2}}}}$$
(7)
After the dynamic correction of the \(OD\) matrix, the trip probability of selecting a destination can be calculated using Eq. (8):
$$C_{{ij}}^{{t,t+1}}=\frac{{B_{{ij}}^{{t,t+1}}}}{{\sum\nolimits_{{j=1}}^{n} {B_{{ij}}^{{t,t+1}}} }},{\text{ }}\left( {1 \leq i \leq n,1 \leq j \leq n,i \ne j} \right)$$
(8)
The traffic situation undergoes dynamic changes throughout the day, particularly during the morning and evening rush hours as well as the midday lull, where traffic flows vary significantly. By dynamically correcting the \(OD\) matrix to account for the possibility of vehicles remaining stationary at a given location, the model’s accuracy can be significantly enhanced, making it more reflective of the real traffic conditions.
EVs power consumption calculation
Velocity-flow model
In the urban traffic system, the travel speed of EVs is primarily influenced by road capacity and traffic flow. To accurately simulate the EV travel process, this study adopts the practical speed-flow model described in the literature33. \({v_{ij}}(t)\), the speed of a vehicle travelling on road segment \(R(i,j)\) at time t, can be described by Eq. (9):
$$\left\{ \begin{gathered} {v_{ij}}(t)=\frac{{{v_{ij}}(m)}}{{1+{{[\frac{{{q_{ij}}(t)}}{{{c_{ij}}}}]}^\beta }}} \hfill \\ \beta =a+b \cdot {[\frac{{{q_{ij}}(t)}}{{{c_{ij}}}}]^\theta } \hfill \\ \end{gathered} \right.$$
(9)
Where, \({v_{ij}}(t)\) represents the free flow speed for road\((i,j)\), and\({c_{ij}}\) is the maximum capacity of road\((i,j)\),which is related to the road class; \({q_{ij}}(t)\) is the volume of traffic on road\((i,j)\) at time t; The ratio of \({q_{ij}}(t)\) and \({c_{ij}}\) is the road saturation at moment t. The term \(\beta\) is an empirical coefficient; while a, b, \(\theta\) are the adaptive coefficients for different road classes, respectively.
In this paper, roads are classified into two categories. For Class I (Arterial Roads), the adaptive coefficients a, b, \(\theta\) are 1.726, 3.15, and 3, respectively. For Class II (Secondary Arterials), the corresponding values are 2.076, 2.870, and 3.
Energy consumption calculation
The energy consumption per unit mile of an Ev’s battery is influenced by several factors, including ambient temperature, traffic conditions and the use of the air conditioning systems34,35,36,37. These factors, in turn, affect the vehicle’s charging load. While many studies focus on the chemical and physical properties of the batteries when predicting charging loads, they often overlook the real-world driving conditions of EVs. Since the average driving speed varies significantly across the road classes, it impacts the energy consumption of EVs. The energy consumption per unit mile for EVs across various road classes can be expressed as38:
$$\left\{ \begin{gathered} {E_f}=0.247 – \frac{v}{{250}}+1.520/v+2.992 \times {10^{ – 5}}v \hfill \\ {E_a}= – 0.179+0.004v+5.492/v \hfill \\ {E_s}=0.210 – 0.001v+1.531/v \hfill \\ {E_b}=0.208 – 0.002v+1.553/v \hfill \\ \end{gathered} \right.$$
(10)
Where \({E_f}\), \({E_a}\), \({E_s}\) and \({E_b}\) denote the energy consumption per unit mile for Expressways, Arterial Roads, Secondary Arterials and Collector Roads, respectively.
In this paper, we focus on the energy consumption per unit mile for EVs under varying temperature conditions, considering the impact of air conditioning usage. This is expressed as the ratio of the energy consumption per unit mile with the air conditioning turned on to that with the air conditioning off. The higher this ratio, the greater the energy consumption per unit mile of the EV when air conditioning is in use. After fitting the data, the temperature-energy ratio coefficient of the battery \({K_{temp}}\) is calculated using Eq. (11)39:
$${K_{temp}}=0.0001 \times {(T – 11)^2}+1.1894$$
(11)
Where T is the current ambient temperature.
According to the curve fitting results in Eq. (11), the actual unit energy consumption E of an EV travelling on the current road r at the current temperature T with the air conditioning on is given by:
$$E={K_{temp}} \cdot {E_r}$$
(12)
Where, \({E_r}\) is the standard energy consumption for the specific road class r, taking the values of \({E_f}\), \({E_a}\), \({E_s}\) and \({E_b}\), respectively.
EVs charging behavior model
The charging choice model formulates the decision-making process of EV user when selecting a specific charging location, taking into account the flexibility in charging time, power using time, spatial factors, and energy consumption parameters to predict the demand for EV charging10. In this study, a charging demand model is constructed based on a detailed analysis of the charging characteristics of different types of EVs. This model aims to accurately describe the charging behavior of EV users at specific nodes.
Analysis of EVs charging behaviors
The travelling patterns of electric private cars are characterized by long parking duration, sufficient charging windows, and relatively fixed driving destinations. Key charging locations for these vehicles include office areas, residential zones, large-scale superstores, and integrated commercial-residential areas. Given that high-power fast charging may lead to fast battery degradation3, slow charging is typically preferred by private car users. Numerous studies on EV charging behavior and travel data suggest that the battery range of EVs is generally sufficient for most users’ urban travel needs, making emergency charging a rare occurrence. Therefore, private EV users tend to charge at home or at work rather than at public charging stations40,41.
Private car owners usually weigh the advantages and disadvantages of charging at different locations. For example, public charging stations typically charge fees, which may be higher than those incurred at home or workplaces, but they often come with perks such as reserved parking spaces at leisure ord entertainment venues42. In general, most private EV charging events occur at home43, followed by workplaces or commuting locations40. Nighttime charging at night is preferred by users due to its greater convenience and flexibility than daytime charging. Therefore, in this paper, the charging behavior of private cars is modeled by considering the need to charge upon arrival at a destination, while slow charging is assumed to take place at home at the end of the day’s travel if the SOC falls below a set psychological threshold.
Electric taxis, primarily engaged in passenger transport during the day, face more urgent charging needs due to the demand for profitability. Consequently, it is assumed that electric taxis will chose the nearby fast charging station when the remaining SOC drops below a set threshold during operational hours, while slow charging is concentrated during the shift change window from midnight to 6:00 a.m.
Official EVs include private official EVs, business EVs and functional EVs. Their travel behavior is intermittently distributed throughout the day, and their destinations exhibit a certain degree of randomness44. Depending on the remaining power, the distance to the next destination, and the travel time interval between trips, official EVs may either opt for fast-charging at a nearby node when the remaining SOC drops below the threshold, ensuring they meet the mileage requirements for the next trip, or choose fast-charging or slow-charging based on the available dwell time at the destination. Considering the typical daily patterns of urban residents, Official EV travel is assumed to conclude by 10:00 p.m.
Charging decision model
Previous studies have often simplified the charging behavior of an individual EV users by assuming one of two decision strategies: either ‘charge immediately after the trip’ or ‘charge after the trip only if the energy required for the next trip cannot be met’. However, this approach does not accurately reflect real-world charging behavior, as it overlooks the inherent uncertainty in the charging demand of private EVs.
In practical, a user’s charging demand depends on whether the current SOC can meet the requirement of the next trip. If the SOC is insufficient, charging is inevitable. However, if the SOC is adequate, the need to charge becomes uncertain and tends to diminish as the available SOC increasingly exceeds the energy demands of the upcoming trip. To capture this complexity, a charging demand model is established, triggered by the user’s arrival at a destination, and accounts for both the necessity and the uncertainty of charging based on the current SOC and future travel needs.
In categorizing private EV users based on the current SOC, they can be divided into elastic and inelastic users. This classification hinges on the elasticity coefficient, denoted as \({u_1}\). Let\({A_s}\) represent the SOC at the current moment. if \({A_s} , the current SOC cannot meet the demand of the next trip, resulting in mileage anxiety and necessitating immediate charging. These users are defined as inelastic. Conversely, if \({A_s} \geq {u_1}\), the current SOC is sufficient for the upcoming journey, categorizing these users as elastic.
Fuzzy theory is utilized in the analysis of charging demand for elastic users. In this framework, a fuzzy set M represents the presence of charging demand, with the membership degree represented by an membership function \(M({A_s})\). This function assigns values within the interval [0,1]45. The membership degree is given by Eq. (13):
$$M(A_{s} ) = \left\{ {\begin{array}{*{20}l} {1,} \hfill & {A_{s}
(13)
Where, \(M({A_s})\) is the membership degree of \({A_s}\) to M, \({u_1}\) is the elasticity coefficient and \({u_2}\) is the fuzzy coefficient. If \({A_s} \geq {u_2}\), the current SOC can meet the next trip, indicating no need for charging; if \(u_{1} \le A_{s} , the closer \({A_s}\) is to \({u_2}\), the closer \(M({A_s})\) is to 0, indicating a weaker need for charging; the closer \({A_s}\) is to \({u_1}\), the closer \(M({A_s})\) is to 1, indicating a stronger need for charging. If \(A_{s} , \(M({A_s})\) is 1 and the EV user must chose to charge. In this analysis, \({u_1}\) is set to 0.3 and \({u_2}\) is set to 0.8.
For electric taxis and official EVs, charging demand is triggered when \({A_s}\) is below the threshold while driving. In such cases, the charging load is assigned to the nearest charging node The decision condition for triggering this charging demand is represented by Eq. (14):
$${A_s} \leq \varepsilon {C_s}$$
(14)
Where \({C_s}\) is battery capacity of EV;\(\varepsilon\) is charging threshold. To account for the user’s range anxiety, the value of \(\varepsilon\) is set to 0.3.
To mitigate the detrimental effects of overcharging on battery life, this study sets the post-charging SOC to 90% of the battery capacity. The method for calculating the charging time of EVs is presented by Eq. (15):
$${T_e}=\frac{{0.9 – {A_s}}}{{{P_t}}}$$
(15)
Where \({T_e}\) is the charging time of EV and \({P_t}\) denotes the charging power, which varied based on the charging type selected by the user.
For a private EV with a specific parking during, if charging to a full SOC is not feasible within the available parking time at the destination, the vehicle is charged up until the start of next trip. The SOC of the EV at the initial time of the next trip is calculated using Eq. (16):
$$SO{C_s}=\frac{{{P_t} \times {t_d}}}{{\eta {C_s}}} – {A_s}$$
(16)
Where \(SO{C_s}\) is the SOC of the EV at the start of the next trip, \({t_d}\) is the EV’s parking time at the current destination, and \(\eta\) represents the charging efficiency of the EV, which ranges from 0.9 to 1.
For the official EV, if the next destination, as determined by the modified \(OD\) probability matrix, is the node where the vehicle is currently located, it is assumed that the driver chooses to stop at the current node. Depending on the functional area to which the node belongs, the parking time at this location with different probability distributions can be modeled as Eq. (17)-Eq. (19) .
$$f\left( {{t_m}} \right)=\frac{{1.153}}{{195.787}} \cdot {\left( {\frac{{{t_m}}}{{195.787}}} \right)^{1.153 – 1}}{e^{ – {{\left( {{{{t_m}} \mathord{\left/ {\vphantom {{{t_m}} {195.787}}} \right. \kern-0pt} {195.787}}} \right)}^{1.153}}}}$$
(17)
$$\left\{ \begin{gathered} \frac{{{t_m} – 438.445}}{{164.506}} \hfill \\ f\left( z \right)\frac{1}{{164.506}}{e^{ – {{\left( {1 – 0.234z} \right)}^{{{ – 1} \mathord{\left/ {\vphantom {{ – 1} {\left( { – 0.234} \right)}}} \right. \kern-0pt} {\left( { – 0.234} \right)}}}}}} \cdot {\left( {1 – 0.234z} \right)^{{{ – 1 – 1} \mathord{\left/ {\vphantom {{ – 1 – 1} {\left( { – 0.234} \right)}}} \right. \kern-0pt} {\left( { – 0.234} \right)}}}} \hfill \\ \end{gathered} \right.$$
(18)
$$\left\{ \begin{gathered} z=\frac{{{t_m} – 68.520}}{{41.761}} \hfill \\ f(z)=\frac{1}{{41.761}}{e^{ – {{\left( {1+0.657z} \right)}^{{{ – 1} \mathord{\left/ {\vphantom {{ – 1} {0.657}}} \right. \kern-0pt} {0.657}}}}}} \cdot {\left( {1+0.657z} \right)^{{{ – 1 – 1} \mathord{\left/ {\vphantom {{ – 1 – 1} {0.657}}} \right. \kern-0pt} {0.657}}}} \hfill \\ \end{gathered} \right.$$
(19)
Where, the parking time in the H region follows the Weibull distribution described by Eq. (17), the parking time in the W region adheres to the generalized extreme value distribution shown in Eq. (18), and the parking time in the SR/SE/O region follows the generalized extreme value distribution specified in Eq. (19).
After generating the appropriate parking time based on these distribution functions, if the parking time is insufficient to achieve a full charge, the actual charging time is calculated using Eq. (20):
$${t_p}=\hbox{min} \left( {{t_d},\frac{{0.9 – {A_s}}}{{\eta {P_t}}}} \right)$$
(20)
Where \({t_p}\) is the charging time of the official EV at the current node.
Transportation-power coupled network (TPCN)
Transportation network
The graph-theoretical analysis method is employed to abstract the transportation network into an empowered directed graph, which describes the topological features of the network. In this graph, road sections with identical road attributes are represented as directed edges, while the connection points between these road sections are modeled as nodes The dynamic transportation network model, based on time segmentation, is described as follows:
$$\left\{ \begin{gathered} G=(N,R,T,W) \hfill \\ N=\left\{ {{n_i}\left| {i=1,2,3 \cdots ,n} \right.} \right\} \hfill \\ R=\left\{ {{n_{ij}}\left| {{n_i} \in N,} \right.{n_j} \in N,i \ne j} \right\} \hfill \\ {T_m}=\left\{ {{t_m}\left| {t=1,2,3, \cdots ,m} \right.} \right\} \hfill \\ W=\left\{ {{w_{ij}}\left| {{n_{ij}} \in M} \right.} \right\} \hfill \\ \end{gathered} \right.$$
(21)
Where G denotes the transportation network of the region; N is the set of all node information in the graph G; while R represents the set of all roads in the graph Gand \({T_m}\) denotes the set of time segments, dividing the entire day into m time segments, wit \({t_m}\) representing the \(mth\) time segment. The road weights are denoted by W, of which the element \({w_{ij}}\) refers to the weight between node i and node j. The weight represents the expenditure for the EV while travelling on the road and can include factors such as road length passage speed, travel cost, etc., which are essential for EV path planning.
As shown in Eq. (22), the element \({d_{ij}}\) represents the connection relationship between node i and node j in the transportation network. The connection relationships of all nodes in the transportation network set G are described by the adjacency matrix D, which consists of the element \({d_{ij}}\) as defined in Eq. (23):
$${d_{ij}}=\left\{ \begin{gathered} {w_{ij}}{\text{ }}{n_{ij}} \in R \hfill \\ 0{\text{ }}{n_i}={n_j} \hfill \\ Inf{\text{ }}{n_{ij}} \notin M \hfill \\ \end{gathered} \right.$$
(22)
$$D=\left[ {\begin{array}{*{20}{c}} 0&{{k_{12}}}&{{k_{13}}}&{{k_{14}}} \\ {{k_{21}}}&0&{{k_{23}}}&{Inf} \\ {{k_{31}}}&{{k_{32}}}&0&{{k_{34}}} \\ {{k_{41}}}&{Inf}&{{k_{43}}}&0 \end{array}} \right]$$
(23)
Where, \({w_{ij}}\) denotes the weight of the road between node i and node j. This weight is assigned a value of 0 when \(i=j\), indicating that the weight of a road connecting a node to itself is zero. The term \(Inf\) indicates that there is no direct connectivity between node i and node j.
Power distribution network
In the TPCN, the power distribution network nodes are spatially coupled with the transportation network nodes, necessitating the establishment of a distribution network model that aligns with the transportation network. For the modeling of the power distribution network, it is primarily divided into two components: the network topology model and power flow calculations. Let the set of network nodes be \({N_p}\), the set of branches be \({M_p}\), and the matrix of branch electrical parameters be \({\varphi _p}\). Together, these components form the network \({G_p}=\left( {{N_p},{E_p},{\varphi _p}} \right)\). The network topology can be expressed as follows:
$$\left\{ \begin{gathered} {G_p}=\left( {{N_p},{E_p},{\varphi _p}} \right) \hfill \\ {N_p}=\left\{ {{n_{pi}}\left| {i=1,2, \cdots ,{n_{pg}}} \right.} \right\} \hfill \\ {M_p}=\left\{ {\left( {{n_{pi}},{n_{pj}}} \right)\left| {{n_{pi}},{n_{pj}} \in {N_p}} \right.} \right\} \hfill \\ {\varphi _p}=\left\{ {\left( {{r_i},{x_i},{c_i},P_{i}^{m}} \right)\left| {\left( {{n_{pi}},{n_{pj}}} \right) \in {M_p}} \right.} \right\} \hfill \\ \end{gathered} \right.$$
(24)
Where, \({n_{pg}}\) is the number of nodes, and the parameters in the branch parameter matrix \({\varphi _p}\) represent the resistance \({r_i}\), reactance \({x_i}\), conductance \({c_i}\), and line power flow \(P_{i}^{m}\), respectively.
Power flow calculation is a fundamental method for analyzing steady-state power system operation. It involves calculating the nodal voltages and power distributions based on the given nodal line parameters to estimate the system’s operating state. For each node in the network, the nodal voltage value is determined using the nodal power equation in the power system’s power flow calculation:
$$\left\{ \begin{gathered} {P_i}={V_i}\sum\limits_{k} {\left( {{G_{ik}}{V_k}\cos \left( {{\delta _i} – {\delta _k}} \right)+{B_{ik}}{V_k}\sin \left( {{\delta _i} – {\delta _k}} \right)} \right)} \hfill \\ {Q_i}={V_i}\sum\limits_{k} {\left( {{G_{ik}}{V_k}\sin \left( {{\delta _i} – {\delta _k}} \right) – {B_{ik}}{V_k}\sin \left( {{\delta _i} – {\delta _k}} \right)} \right)} \hfill \\ \end{gathered} \right.$$
(25)
Where, \({P_i}\) and \({Q_i}\) are the injected active power and injected reactive power at node i, respectively; \({V_i}\) and \({V_k}\) are the voltage magnitudes at node i and node k, respectively; \({\delta _i}\) and \({\delta _k}\) are the phase angles of node i and node k, respectively; \({G_{ik}}\) and \({B_{ik}}\) are the conductance and the susceptance between node i and node k.
The solution to Eq. (25) can be obtained through iterative computation using the Newton-Raphson iteration method. The results from the power flow calculation can be used to assess the voltage stability of the electrical system.
Finally, the charging load demand generated by EVs on the transportation network, as partially obtained in Sect. 2.1 to 2.3, is mapped to the corresponding nodes within the distribution network based on the established correspondence. These loads are then aggregated to determine the total charging load at each node of the distribution network, as illustrated in Eq. (26):
$${P_d}\left( t \right)=\sum\limits_{{i=1}}^{m} {\left[ {{P_d}\left( t \right)+{P_{di}}(t)} \right]}$$
(26)
Where, \({P_g}\left( t \right)\) represents the total charging load demand at node d at time t for EVs under uncoordinated charging, \({P_d}\left( t \right)\) denotes the base load demand at node d at time t, and \({P_{di}}(t)\) is the charging load demand generated by the ith EV at this node at time.
