Abstract
The large-scale deployment of electric vehicles (EVs) poses critical challenges to the secure and economic operation of power distribution networks (PDNs). Therefore, a method for evaluating the hosting capacity that enables a PDN to determine the EV chargeable area (EVCA) to satisfy the charging demand and ensure the secure operation is proposed in this paper. Specifically, the distribution system operator (DSO) serves as a public entity to manage the integration of EVs by determining the presence of the charging load in the EVCA. Hence, an EVCA optimization model is formulated on the basis of the coupling effect of the charging nodes to determine the range of the available charging power. In this model, nonlinear power flow equations and operational constraints are considered to maintain the solvability of the power flow of the PDN. Subsequently, a novel multipoint approximation technique is proposed to quickly search for the boundary points of the EVCA. In addition, the impact of the demand response (DR) mechanism on the hosting capacity is explored. The results show that the presence of the DR significantly enlarged the EVCA during peak hours, thus revealing the suitability of the DR mechanism as an important supplement to accommodate the EV charging load. The examined case studies demonstrate the effectiveness of the proposed model and show that the unmanaged allocation of the charging load impedes secure operation. Finally, the proposed method provides a reference for the allocation of the EV charging load and a reduction in the risk of line overloading.
IN recent years, the electrification of vehicles has attracted considerable interest worldwide owing to the environmental issues and dwindling supplies caused by the excessive consumption of fossil fuels [
The widely-adopted methods that aim to mitigate the adverse impacts of EV integration in PDNs mainly stem from the EV perspective. These methods involve the optimal planning of charging facilities and EV charging guidance [
Furthermore, we observe that few studies have adopted a quantitative capacity evaluation method to calculate the hosting capacity for EV integration compared with that for renewable energy integration. For example, in [
Thus, to address this defect, methods for evaluating the security region have been proposed to characterize the feasible region of renewable energy generators instead of the maximum value. The simplest method involves sampling numerous outputs of renewable energy generators and subsequently solving for the optimal power flow (OPF) to examine the feasible sample outputs [
In summary, iterative methods are time consuming, whereas heuristic methods lack precision. The multiple optimization method is unable to characterize the boundary of a security region. Enumeration methods precisely map security regions using all feasible points; however, they are computationally intensive. To overcome these limitations, the multiparameter programming and the vertex search method are applied to efficiently characterize the security region. Instantaneous linearized technical constraints are required; thus, the multiperiod ramp constraint is ignored. In addition, there still exist certain difficulties in implementing a balance between speed and precision in the above methods. Interestingly, the demand response (DR) has drawn a significant amount of attention in improving the hosting capacity for EV and renewable energy integration [
Hence, a novel evaluation method is proposed in this paper to characterize the feasible region of the hosting capacity. Specifically, a multiperiod capacity evaluation model is realized considering nonlinear technical constraints and the effect of coupling between EV charging nodes to improve the model precision. A multipoint approximation method is adopted to promptly approximate the feasible region to target the coordination of the solution speed and precision. The proposed method can efficiently obtain the feasible region of the charging load to provide decision support and a feasibility analysis related to the charging schedule and PDN operation for distribution system operators (DSOs). Further, the capacity evaluation integrated with the DR program is studied to examine the mechanism of the variation in the feasible region and the potential to promote EV accommodation.
The remainder of this paper is organized as follows. Section II describes the mathematical model, including the alternating current (AC) OPF, EV charging demand, renewable distributed generators (DGs), and DR. The hosting capacity evaluation method for PDNs is introduced in Section III. In Section IV, the results of various simulations to characterize the feasible region are discussed to verify the effectiveness and benefits of the proposed method. Finally, the conclusions are drawn in Section V.
To describe the proposed multipoint approximation method, the multiperiod operation model for a PDN is first mathematically formulated. Thereafter, the uncertainty in the EV charging demand, the uncertainty in the renewable energy, and the DR mechanism for power demand regulation are modeled.
EV charging facilities with a refueling service are connected to a PDN via a node. The PDN, generally, a radial network, can be characterized by a graph with a tree topology. Without loss of generality, it is assumed that each bus connects to one generator and serves one power load with a deterministic demand. The power supply or demand is set to be zero if no generator or load exists at the corresponding bus. The multiperiod AC OPF [
(1) |
(2) |
(3) |
(4) |
(5) |
(6) |
(7) |
(8) |
(9) |
where is the set of lines in the PDN; is the set of buses in the PDN; is the set of time slots; and are the energy production cost coefficients at bus i; and are the active and reactive power generated at bus i, respectively; is the contract electric price with the main grid; is the forecasted active power of the renewable DGs; is the active power transferred from slack bus to PDN; and are the fixed active and reactive power loads, respectively; and are the entries in the
The objective in (1) minimizes the production cost, where the first term is the production cost of the dispatchable DGs in the PDN, and the second term is the cost of purchasing electricity from the main grid. Equations (
The EV charging demand, which varies over time and location, can be modeled using the travel behavior of the EV. It is assumed that the charging facilities are aggregated as an EV charging station (EVCS) served by a bus in the PDN. To model the aggregated charging demand at a PDN bus, sampling with a replication method is used to randomly select samples from the EV charging demand database [
(10) |
where is the set of buses connected to the EVCSs; is the set of samples; and are the aggregated charging demand in scenario s and its average value, respectively; is the number of elements in the set; and is the set of charging demand scenarios. The uncertainty in the charging demand can be properly modeled for a sufficiently large sample size .
Considering the EV charging demand, the power load at the corresponding bus is given as:
(11) |
(12) |
(13) |
where is the total active power load at bus i; is the available charging power for the upcoming EVs at bus i; and is the upper bound of the EV charging power.
Considering that renewable generation deviates from the forecasted value [
(14) |
where and are the forecasted output and forecast errors, respectively, and follows a Gaussian distribution with a known covariance matrix .
In implementing a time-based DR program, the power demanded by consumers is stimulated by the electricity price that changes during different time periods. For the power demand at bus , its price elasticity and responsive loads are described by the DR constraints in (15)-(19) [
(15) |
(16) |
(17) |
(18) |
(19) |
where is the price elasticity of the period versus the period, reflecting the power demand sensitivity towards the electricity price, and the elasticity is regarded as the amount of change in the power demand when the electricity price changes by one unit; and are the variations of the power demand and electricity price with perturbations in the DR, respectively; is the variation of reactive power load in DR; is the limitation on the variation of the power load; is the set of buses participating in DR; is the binary variable indicating the participation of a DR program at bus i, and indicates that bus i participates in a DR program and vice versa; and is the number of buses in a DR program.
The hosting capacity is evaluated to characterize an EV chargeable area (EVCA). The EVCA is defined as a feasible region containing all operation points of charging stations that guarantee the operational constraints of the PDN. Consider

Fig. 1 Diagram of an EVCA with two EVCSs.
As previously mentioned, it is challenging to characterize the EVCA owing to the nonlinear information of the PDN. Therefore, a multipoint approximation method is proposed to search for the boundary points of the EVCA and subsequently the bounded area is constructed via the boundary points, approximating the actual EVCA.
Before a comprehensive description of the multipoint approximation method, we detail the EVCA using the AC OPF model. Considering the EV charging demand and the uncertainty in renewable energy, a compact description of the AC OPF consisting of (1)-(14) is modeled as:
(20) |
where a is a vector denoting the cost coefficient; ht and gt denote the power balance in (2)-(5) and the operational constraints in (6)-(9) at time period t, respectively; xt is a vector denoting the power injections from generators and substations; yt is a vector denoting the state variables in the PDN; and is a vector denoting the EV charging power maximally accepted by the PDN, which consists of the charging load and the available charging power.
The EVCA during any time period can be viewed as a projection from a constraint space (x, y, ) to , where the operation point of the EVCS can satisfy the operational constraints in (20). The projected area can be expressed as:
(21) |
where D is a vector denoting the descriptor variables, which reflect the most influential combinations of parameters that influence the PDN security boundary, such as the line current, the power flow on a certain line, and the bus voltage; and n is the dimension of D. Thus, the boundary of the EVCA is the maximum operation point of EVCSs without the violation of security constraints and power equilibrium.
As stated previously, the direct calculation of the EVCA boundary is difficult because of the high computational demand. Therefore, the proposed multipoint approximation method searches for boundary points by utilizing EVCS coupling, which characterizes an approximate feasible region. Specifically, the coupling represents the interaction that exists among the available charging power at different charging nodes, i.e., the maximum operation status of the EVCSs in different growth directions of the charging load. Thus, the boundary points can be explored by stressing the charging load along a given stress direction until a violation of the security constraints occurs [
(22) |
where the subscript k represents the
In addition, the power flow and voltage constraints in are reformulated as chance constraints owing to the uncertainty in the renewable DGs:
(23) |
(24) |
(25) |
(26) |
where l is a preset number; ; denotes the probability operator; and and are the probabilities of violating the constraints, which are set as . These chance constraints ensure that the probabilities of violating the constraints hold for a prescribed value. It is noteworthy that the quadratic constraint in (8) leads to intractability of the chance constraint problem. Thus, (8) is replaced by l linear constraints using the linearization method described in [
However, the problem in (22) remains difficult to solve because of the presence of the chance constraints (23)-(26). Thus, the chance constraints are reformulated into tractable forms [
(27) |
(28) |
(29) |
where and are the uncertainty margins that tighten the constraint bounds to guarantee the secure operation of the PDN; ; and .
Considering the constraints (27)-(29) and replacing the corresponding inequality constraints in g, the problem in (22) can be reformulated as:
(30) |
where
A physical interpretation of the solution to the problem in (30) is as follows. For the obtained boundary point zk with the corresponding search direction wk, there exists an operation point in which the operational constraints are satisfied. For any other point beyond zk along the same search direction wk, there is no such operation point that satisfies all constraints.
It should be noted that consists of the charging demand and the available charging power . Considering the charging demand obtained in (10), the boundary points will be determined if the available charging power is calculated. Thus, the charging demand is set as the new starting point to search for the boundary points. The flowchart of the multipoint approximation method is shown in

Fig. 2 Flowchart of multipoint approximation method.
It is known that at a certain time period, which is obtained by (10), is closer to the EVCA boundary than the base point O. Thus, we regard the charging demand as the new base operation point. Thereafter, a boundary point can be obtained as long as the distances between the base point and the boundary points are calculated. Therefore, the problem in (30) is reformulated as (31) based on the new base point and the available charging power , where the value of is negative if the charging demand is beyond the EVCA.
(31) |
In initialization, we set each element of the direction vector wk to be 1 to construct an n unit direction vector, where n denotes the dimension of wk. Subsequently, the initial boundary points of the EVCA can be determined by solving the problem in (31) with the initial directions. Furthermore, a simple example is also shown in
In each round of searching, for any pair of obtained boundary points zi and zj, we can calculate the average value of their corresponding directions as a new search direction, as shown in
(32) |
In a certain round of searching, if , the search is stopped, and no new boundary points will be generated. In this case, the line segment passing through any two adjacent points is sufficiently accurate to approximate the EVCA boundary. Otherwise, the search will continue until the boundary approximation is sufficiently accurate. It should be noted that is a predefined convergence threshold fulfill the tradeoff between the accuracy and the efficiency of EVCA characterization. The pseudocode of the multipoint approximation method is presented in
The EVCA during a single time period can be characterized using the proposed multipoint approximation method. However, there exists a temporal coupling of power generation between adjacent periods in the PDN, which is reflected by the ramp constraint in (9). For intraday PDN operation, the balance between the power supply and the demand is maintained by the DGs and main grid. Owing to the ramp constraint, the EVCA reflecting the range of charging loads is coupled during different periods. In this case, the charging demand at the EVCA boundary may result in difficult adjustments to the PDN power generation due to the ramp constraints. Thus, we apply multiperiod AC OPF and charging demand to obtain the optimal operation state of the PDN during each scheduling period. The obtained state is regarded as the reference for the ramp constraints in the multipoint approximation for the EVCA.
Algorithm 1 : multipoint approximation method |
---|
1: Initialization: input the number of iterations , convergence threshold, the time period t, corresponding EV charging demand , and initial direction set 2: Solve the problem in (31) with and to obtain the solution set , i.e., the EVCA boundary point set 3: Repeat 4: 5: Loop 6: Calculate a new search direction with a pair of adjacent directions and solve the problem in (31) with the new obtained set to obtain the new solution set . 7: Calculate dn for each new point using (32) 8: , 9: End loop 10: Until 11: Return Zk |
A simple example is presented in this subsection to illustrate the proposed method for obtaining the EVCA. A modified IEEE 4-bus distribution network is shown in

Fig. 3 Modified IEEE 4-bus distribution network.
Here, we set the initial directions and and the initial charging demand . Subsequently, we solve the problem in (31) for and obtain two boundary points, and , by calculating . According to

Fig. 4 Characterization of EVCA of two EVCSs.
Numerical results are obtained for a PDN test system. All simulations are implemented in MATLAB on a laptop equipped with an Intel Core i7-10700U CPU running at 2.90 GHz and 16 GB of RAM. The proposed model with nonlinearity is solved using Interior Point Optimizer [
The modified 33-bus distribution network in

Fig. 5 Modified IEEE 33-bus distribution network.
To illustrate the effectiveness of the proposed method for approximating the EVCA, four evaluation methods for characterizing EVCAs are set up as follows, where M0 serves as a reference for comparison.
M0: sample and choose feasible points to construct the EVCA using Monte Carlo sampling [
M1: characterize the EVCA based on the multiple optimization method [
M2: characterize the EVCA based on the vertex search method in the linear model [
M3: the proposed multipoint approximation method for the EVCA.
For the analysis of results, several performance indices and parameters should be introduced. The total number of sampled points N in M0 is set to be 20000. After Monte Carlo sampling in M0, the sampled points can be divided into two types: feasible and infeasible points. If a sampled point is a boundary point or in the obtained EVCA, it is feasible and vice versa. Afterward, the numerical relation is obtained, where and are the numbers of feasible and infeasible points in M0, respectively. Correspondingly, () can be obtained by determining whether the sampled points are in the EVCA, as obtained by M1-M3. Consequently, four indices are introduced to evaluate the accuracy of different evaluation methods:
(33) |
where and are the absolute and relative errors denoting the difference in feasible points between methods M0 and Mi, respectively; and and are the absolute and relative errors denoting the difference in infeasible points between methods M0 and Mi, respectively.
Given the daily power load in 24 1-hour intervals, the EVCAs are characterized by methods M0-M3. However, the secure operation of the PDN is possibly impeded during peak hours owing to the simultaneous occurrence of high system and charging loads. Therefore, we choose EVCAs at two typical peak hours (the 1

Fig. 6 EVCAs at the 1

Fig. 7 EVCAs at the 1
By comparing Figs.
Method | Volume (p.u.) | (%) | (%) | Time (s) | ||||
---|---|---|---|---|---|---|---|---|
M0 | 4.5671 | 14253 | - | - | 5747 | - | - | 64843.0 |
M1 | 4.0785 | 12782 | 1471 | 10.320 | 7218 | -1471 | -25.60 | 27.4 |
M2 | 5.9694 | 18475 | -4222 | -29.602 | 1525 | 4222 | 73.46 | 1.8 |
M3 | 4.5203 | 14136 | 117 | 0.820 | 5864 | -117 | -2.03 | 38.3 |
Note: “-” indicates that M0 does not participate in the error calculation.
In addition, the effectiveness of the obtained EVCA is analyzed using the AC power flow. Specifically, we take the EVCA at the 1

Fig. 8 Profiles of power flows and power flow variation ∆S.

Fig. 9 Profiles of voltage magnitudes and voltage magnitude variation ∆V.
The computational time of M0-M3 for the modified IEEE 33-bus system is listed in
The load profile characteristics are improved using the DR program. For the safe and economic operation of the PDN, it is necessary to characterize how the DR program affects the EVCAs. Considering different nodes participating in the DR, we compare the following three cases.
1) Case 0: none of the nodes participate in the DR.
2) Case 1: nodes connected to the EVCAs participate in the DR.
3) Case 2: all nodes participate in the DR.
It should be noted that the DR electricity price is regulated, and the power demanded by consumers changes with the price variation. The profiles of the electricity price and day-ahead power demand in Cases 0-2 are shown in

Fig. 10 Profiles of electricity price and day-ahead power load in Cases 0-2.
Taking two typical periods as examples, the corresponding EVCAs are shown in Figs.

Fig. 11 EVCAs at the 1

Fig. 12 EVCAs at the 2
In the proposed method, the convergence criterion is given by . The impact of the threshold on the search process of the EVCA is evaluated via numerical tests, as shown in

Fig. 13 Impact of threshold on search process of EVCA.
In this paper, the hosting capacity of a PDN is evaluated to accommodate the EV charging load. More precisely, the hosting capacity is defined as the EVCA with the maximum available charging power, ensuring the solvability of the power flow and non-violation of the operational constraints.
Specifically, the voltage and power flow constraints are tightened to lower the risk of constraint violation in the presence of uncertainties in the renewable energy. Furthermore, the EV charging demand is modeled by sampling from historical travel data and used as an initial point in the fast approximation process of the EVCA. The EVCA is characterized by the proposed multipoint approximation method, which characterizes boundary points using the coupling effect of EVCSs. Subsequently, case studies are examined to illustrate the effectiveness of the proposed method to accurately evaluate the hosting capacity and decision support of PDN operation. Additionally, the impact of the DR program on the hosting capacity is investigated. Specifically, the EVCAs at peak hours can be increased via the DR program to accommodate more charging demand. Finally, the operation of the EVCSs is restricted within a secure range with the aid of the proposed method for EVCA characterization. Hence, the violation of operational constraints is avoided, and the EV charging demand is satisfied to the largest extent in the PDN.
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