Hosting Capacity Evaluation Method for Power Distribution Networks Integrated with Electric Vehicles

Wei Dai; Cheng Wang; Hui Hwang Goh; Jingyi Zhao; Jiangyi Jian

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Hosting Capacity Evaluation Method for Power Distribution Networks Integrated with Electric Vehicles PDF

- ORCID：
Wei Dai (Member, IEEE)
✉
- ORCID：
Cheng Wang
✉
- ORCID：
Hui Hwang Goh (Senior Member, IEEE)
✉
- ORCID：
Jingyi Zhao
✉
- ORCID：
Jiangyi Jian
✉

the School of Electrical Engineering, Guangxi University, Nanning 530000, China

Updated：2023-09-20

DOI：10.35833/MPCE.2022.000515

OUTLINE

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.

Keywords

Capacity evaluation; demand response (DR); electric vehicle (EV); electric vehicle chargeable area (EVCA).

I. Introduction

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 [

1]. The elimination of energy shortage issues, the reduction of carbon emissions, and much more, are highly anticipated with the deployment of electric vehicles (EVs) [2]. However, the integration of EVs at a massive scale would inevitably pose a severe challenge to the secure and economic operation of power distribution networks (PDNs) [3], [4]. Particularly, EV charging loads may cause serious network conditions such as voltage deviations, line overloading, and substation overburdening [5]. Specifically, uncontrolled EV charging may significantly introduce high load peaks, thereby increasing energy losses. This leads to a surge in the operation costs for optimal scheduling and future reinforcement in distribution networks [6].

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 [

7], [8]. In these methods, the hosting capacity of a PDN for EV integration is considered an essential precondition for determining the boundary conditions and checking feasibility [9]. Moreover, these methods can be divided into comprehensive indicator analysis and quantitative capacity evaluation. In the comprehensive indicator analysis method, the hosting capacity of PDNs is evaluated using certain system indices, which are commonly applied to the selection of planning strategies. References [10]-[12] show that the hosting capacity indices include the profile of power loss, undercurrent/overcurrent, undervoltage/overvoltage, and voltage imbalance. Specifically, these indices suggest the risk and economic situation of a PDN with an uncertain EV charging demand. In the quantitative capacity evaluation method, the maximum EV charging load is calculated considering the security constraints of the distribution network [13], [14]. In particular, this method is used to quickly check the feasibility of scheduling results and hence, formulate regulation strategies to deal with the unexpected EV charging demand. In view of this, the quantitative capacity evaluation of EV integration from a visualization perspective is extended in this paper to calculate the acceptable range of the EV charging load. More precisely, a secure region for the hosting capacity is obtained to accommodate EV charging in the PDN.

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 [

15] and [16], iterative methods are applied to calculate the hosting capacity, where renewable energy sources of different sizes and different combinations are increased stepwise until the operational constraints of the PDN are violated. However, these methods are computationally intensive owing to numerous iterations and abundant combinations of sizes of renewable energy sources. Heuristic methods based on artificial intelligence, such as particle swarm optimization [17], firefly algorithm [18], and genetic algorithm [19], can be used to quickly estimate the hosting capacity. However, these methods cannot guarantee the accuracy of the evaluation results because they use random variables without mathematical certainty. In addition, in [20] and [21], multiple optimizations are used to calculate the maximum and minimum capacities of renewable energy consumption without violating the operational constraints of power systems. However, the use of extreme values fails to reflect the effect of coupling between renewable generators or EVs. For instance, the maximum capacity of charging stations A and B is calculated as 1 MW in the multiple optimization method. Whereas, there is uncertainty about whether station A or B can independently accommodate 0.8 MW charging load. In other word, only a maximum capacity is determined but the respective capacity of charging stations A and B is not clear. This also bring difficulty to operators in checking the security of operation point of charging stations.

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 [

22], [23]. Therefore, all the feasible points of the renewable energy generators form the security region. However, this enumeration method that solves for the OPF is computationally intensive. Nevertheless, the analytical methods in [24]-[26] are applied to determine an approximate security region, thereby reducing the computational demand. In particular, the optimization problems are simplified for quick and easy resolution. Moreover, the security region is characterized by a multiparameter programming method in [27], [28], which requires a linear power system model. In addition, the vertex search method in [29] characterizes the feasible region of EV charging by characterizing vertices. However, this method requires the linear model, which omits system information and reduces the accuracy of the feasible region without ensuring the feasibility of evaluation results.

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 [

30], [31]. As an incentive payment to use less electric power, a DR program can smoothen the profile of the system load and thus improve the hosting capacity [32], [33]. However, to the best of our knowledge, there has been no research on the characterization of the feasible region for EV integration coupled with a DR program.

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.

II. Mathematical Model

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.

A. Modeling of AC OPF

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 [

34] can be modeled as follows, in which

t \in 𝒯

is introduced to represent the time intervals.

m i n (\sum_{t \in 𝒯} \sum_{i \in P_{N}} (a_{1} (p_{i, t}^{g})^{2} + a_{2} p_{i, t}^{g}) + \sum_{t \in 𝒯} ρ_{t} P_{0 s, t}^{l})

(1)

p_{i, t}^{g} + p_{i, t}^{r f} - p_{i, t}^{d c} = v_{i, t}^{2} G_{i i} + \sum_{l \in P_{L}} P_{i j, t}^{l} \forall i, l, t

(2)

q_{i, t}^{g} - q_{i, t}^{d} = - v_{i, t}^{2} B_{i i} + \sum_{l \in P_{L}} Q_{i j, t}^{l} \forall i, l, t

(3)

P_{i j, t}^{l} = v_{i, t} v_{j, t} (G_{i j}^{l} c o s (θ_{i, t} - θ_{j, t}) + B_{i j}^{l} s i n (θ_{i, t} - θ_{j, t}))

(4)

Q_{i j, t}^{l} = v_{i, t} v_{j, t} (G_{i j}^{l} s i n (θ_{i, t} - θ_{j, t}) - B_{i j}^{l} c o s (θ_{i, t} - θ_{j, t}))

(5)

\{\begin{array}{l} {\underset{̲}{p}}_{i}^{g} \leq p_{i, t}^{g} \leq {\bar{p}}_{i}^{g} \\ {\underset{̲}{q}}_{i}^{g} \leq q_{i, t}^{g} \leq {\bar{q}}_{i}^{g} \\ {\underset{̲}{v}}_{i} \leq v_{i, t} \leq {\bar{v}}_{i} \end{array} \forall i, t

(6)

\{\begin{array}{l} I_{i j, t}^{l} = \sqrt[]{\frac{(P_{i j, t}^{l})^{2} + (Q_{i j, t}^{l})^{2}}{v_{i, t}^{2}}} \\ I_{i j}^{l} \leq I_{l}^{r} \end{array} \forall i, l, t

(7)

\sqrt[]{(P_{i j, t}^{l})^{2} + (Q_{i j, t}^{l})^{2}} \leq {\bar{S}}_{i j} \forall l, t

(8)

R D \leq p_{i, t}^{g} - p_{i, t - 1}^{g} \leq R U \forall i, t

(9)

where $P_{L}$ is the set of lines in the PDN; $P_{N}$ is the set of buses in the PDN; $𝒯$ is the set of time slots; $a_{1}$ and $a_{2}$ are the energy production cost coefficients at bus i; $p_{i, t}^{g}$ and $q_{i, t}^{g}$ are the active and reactive power generated at bus i, respectively; $ρ_{t}$ is the contract electric price with the main grid; $p_{i, t}^{r f}$ is the forecasted active power of the renewable DGs; $P_{0 s, t}^{l}$ is the active power transferred from slack bus to PDN; $p_{i, t}^{d c}$ and $q_{i, t}^{d}$ are the fixed active and reactive power loads, respectively; $G_{i j}^{l}$ and $B_{i j}^{l}$ are the entries in the i^th row and j^th column of the conductance and susceptance matrices of the nodes in the PDN G and B, respectively; $P_{i j, t}^{l}$ and $Q_{i j, t}^{l}$ are the active and reactive power flows in line l, respectively; $v_{i, t}$ and $v_{j, t}$ are the voltage magnitudes at buses i and j, respectively; $θ_{i, t}$ and $θ_{j, t}$ are the voltage phase angles at buses i and j, respectively; ${\bar{p}}_{i}^{g}$ and ${\bar{q}}_{i}^{g}$ are the active and reactive generation capacities of the dispatchable DGs, respectively; ${\underset{̲}{p}}_{i}^{g}$ and ${\underset{̲}{q}}_{i}^{g}$ are the lower limits of active and reactive generation, respectively; ${\bar{v}}_{i}$ and ${\underset{̲}{v}}_{i}$ are the upper and lower boundaries of the voltage magnitude, respectively; $I_{i j, t}^{l}$ is the magnitude of the current in line l; $I_{l}^{r}$ is the thermal limit of the current of line l; and $R U$ and $R D$ are the maximum ramp-up and ramp-down rates of the energy production, respectively.

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 (2) and (3) are the balance equations for the active power and reactive power at each bus $i \in P_{N}$ , where i and j are the beginning and ending buses of line $l \in P_{L}$ . The power flow in the distribution line is modeled using (4) and (5), expressed in terms of the voltage magnitude and phase angle. The typical nonlinearity of (4) and (5) in the PDN model can be solved using nonlinear programming methods. The constraint in (6) provides bounds on the voltage magnitudes and generator output. Equation (7) expresses the relationship between the distribution line current and the power flow, and the thermal limit $I_{l}^{r}$ is imposed on the current magnitude $I_{i j}^{l}$ . Constraint (8) is a conventional complex power flow limit. Constraint (9) presents the ramp-up and ramp-down limits of the generator, which embody the effect of coupling between two adjacent time periods in the PDN.

B. Model of EV Charging Demand

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 [

13]. A charging demand scenario can be obtained using the summarized value at each bus in the PDN. Various scenarios can be generated by repeating the process; then, the aggregated EV charging demand is obtained as:

p_{i, t}^{c, a v} = \frac{1}{|N_{i, t}|} \sum_{s \in N_{i, t}} p_{i, t, s}^{c} i \in 𝒞_{N}, s \in 𝒮, \forall t

(10)

where $𝒞_{N}$ is the set of buses connected to the EVCSs; $𝒮$ is the set of samples; $p_{i, t, s}^{c}$ and $p_{i, t}^{c, a v}$ are the aggregated charging demand in scenario s and its average value, respectively; $| \cdot |$ is the number of elements in the set; and $N_{i, t}$ is the set of charging demand scenarios. The uncertainty in the charging demand can be properly modeled for a sufficiently large sample size $𝒮$ .

Considering the EV charging demand, the power load at the corresponding bus is given as:

p_{i, t}^{d} = p_{i, t}^{d c} + p_{i, t}^{c, a v} + p_{i, t}^{c, a v a i l} i \in 𝒞_{N}

(11)

p_{i, t}^{c, a v a i l} - p_{i, t - 1}^{c, a v a i l} < \sum_{i} (p_{i, t}^{g} - p_{i, t - 1}^{g})

(12)

p_{i, t}^{c, a v} + p_{i, t}^{c, a v a i l} \leq p^{c, m a x} i \in 𝒞_{N}

(13)

where $p_{i, t}^{d}$ is the total active power load at bus i; $p_{i, t}^{c, a v a i l}$ is the available charging power for the upcoming EVs at bus i; and $p^{c, m a x}$ is the upper bound of the EV charging power.

Equation (11) indicates that the power load at bus $i \in 𝒞_{N}$ of the PDN consists of three parts, i.e., $p_{i, t}^{d c}$ , $p_{i, t}^{c, a v}$ , and $p_{i, t}^{c, a v a i l}$ . Formula (12) represents the temporal coupling of EV charging. The variation of charging power cannot exceed the ramp capacity of the generators. In (13), the upper bound $p^{c, m a x}$ is imposed on the EVCSs owing to the finite number of charging facilities.

C. Modeling of Renewable DGs

Considering that renewable generation deviates from the forecasted value [

35], renewable DGs are modeled as in (14). For the renewable DGs in (2) and (3), their active power generation can be reformulated as:

p_{i, t}^{r} = p_{i, t}^{r f} + ξ

(14)

where $p_{i, t}^{r f}$ and $ξ$ are the forecasted output and forecast errors, respectively, and $ξ$ follows a Gaussian distribution with a known covariance matrix $Σ$ .

D. DR Constraints

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 $i \in D (B)$ , its price elasticity and responsive loads are described by the DR constraints in (15)-(19) [

36]. In particular, a specific set of demands participating in the DR program are considered by the introduction of a binary variable [37].

μ_{t τ} = \frac{ρ_{τ}}{p_{i, t}^{d}} \frac{Δ p_{i, t}^{D R}}{Δ ρ_{τ}^{D R}} t, τ \in 𝒯

(15)

[\begin{matrix} \frac{Δ p_{i, 1}^{D R}}{p_{i, 1}^{d}} \\ \frac{Δ p_{i, 2}^{D R}}{p_{i, 2}^{d}} \\ ⋮ \\ \frac{Δ p_{i, T}^{D R}}{p_{i, 𝒯}^{d}} \end{matrix}] = [\begin{matrix} μ_{11} & μ_{12} & . . . & μ_{1 𝒯} \\ μ_{21} & μ_{22} & . . . & μ_{2 𝒯} \\ ⋮ & ⋮ & ⋮ \\ μ_{𝒯 1} & μ_{𝒯 2} & . . . & μ_{𝒯 𝒯} \end{matrix}] [\begin{matrix} \frac{Δ ρ_{1}^{D R}}{ρ_{1}} \\ \frac{Δ ρ_{2}^{D R}}{ρ_{2}} \\ ⋮ \\ \frac{Δ ρ_{𝒯}^{D R}}{ρ_{𝒯}} \end{matrix}]

(16)

(1 - ε Λ_{i}) p_{i, t}^{d c} \leq p_{i, t}^{d c} \leq (1 + ε Λ_{i}) p_{i, t}^{d c} i \in 𝒟_{N}

(17)

\sum_{i \in D (B)} Λ_{i} \leq \bar{Λ} i \in 𝒟_{N}

(18)

q_{i, t}^{d c} + Δ q_{i, t}^{D R} = \frac{q_{i, t}^{d c}}{p_{i, t}^{d}} (p_{i, t}^{d c} + Δ p_{i, t}^{D R}) i \in 𝒟_{N}

(19)

where $μ_{t τ}$ is the price elasticity of the $t^{t h}$ period versus the $τ^{t h}$ 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; $Δ p_{i, t}^{D R}$ and $Δ ρ_{t}^{D R}$ are the variations of the power demand and electricity price with perturbations in the DR, respectively; $Δ q_{i, t}^{D R}$ is the variation of reactive power load in DR; $ε$ is the limitation on the variation of the power load; $𝒟_{N}$ is the set of buses participating in DR; $Λ_{i}$ is the binary variable indicating the participation of a DR program at bus i, and $Λ_{i} = 1$ indicates that bus i participates in a DR program and vice versa; and $\bar{Λ}$ is the number of buses in a DR program. Equation (15) represents the responsive power demands with respect to the variation of the $𝒯^{t h}$ electricity price. For a scheduling period of $𝒯$ , the elasticity coefficients can be represented by a $𝒯 \times 𝒯$ matrix, as shown in (16). The diagonal and off-diagonal elements of this matrix represent the self- and cross-elasticities, respectively, for which a detailed explanation is defined in [

31]. Constraint (17) reflects the degree of flexibility in the power demands. Constraint (18) represents the total number of buses participating in the DR program. The demand pattern of the reactive power with and without DR perturbation is presented in (19). In addition, when considering the charging load participated in a DR program,

p_{i, t}^{d c}

is replaced as

p_{i, t}^{c, a v}

III. Hosting Capacity Evaluation Method for PDNs

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 as an illustrative example, where an EVCA with two EVCSs (EVCS1 and EVCS2) is characterized. P_EVCS1 and P_EVCS2 are the available charging power of EVCS1 and EVCS2, respectively. Specifically, the operation points denote the charging load of the EVCSs and can be divided into feasible, infeasible, and boundary points. At the boundary, the charging demand is maximally satisfied without the violation of operational constraints. In addition, the DSO can conveniently manage EV charging by determining whether an operation point is beyond the boundary. Consequently, this provides decision support for PDN operation by adequately monitoring the safety margin of the operation points and boundary.

Fig. 1 Diagram of an EVCA with two EVCSs.

A. Problem Formulation for EVCA Characterization

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:

\{\begin{array}{l} m i n \sum_{t} a^{T} x_{t} \\ s . t . h_{t} (x_{t}, y_{t}, P_{t}^{c}) = 0 \\ g_{t} (x_{t}, y_{t}, P_{t}^{c}) \leq 0 \end{array}

(20)

where a is a vector denoting the cost coefficient; h_t and g_t denote the power balance in (2)-(5) and the operational constraints in (6)-(9) at time period t, respectively; x_t is a vector denoting the power injections from generators and substations; y_t is a vector denoting the state variables in the PDN; and $P_{t}^{c}$ is a vector denoting the EV charging power maximally accepted by the PDN, which consists of the charging load $p_{i, t}^{c, a v}$ and the available charging power.

The EVCA during any time period can be viewed as a projection from a constraint space (x, y, $P^{c}$ ) to $P^{c}$ , where the operation point of the EVCS can satisfy the operational constraints in (20). The projected area $Ω$ can be expressed as:

Ω = {D \subset R^{n} | h (x, y) - P^{c} = 0, g (x, y, P^{c}) \leq 0}

(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 [

38]. As a consequence, the EVCA is generated as the hull of the obtained boundary points. Furthermore, an optimization model for searching for boundary points during a certain period is formulated in (22), where the subscript t is neglected for simplicity.

\{\begin{array}{l} m a x w_{k}^{T} P_{k}^{c} \\ s . t . h (x, y) - P_{k}^{c} = 0 \\ g (x, y, P_{k}^{c}) \leq 0 \end{array}

(22)

where the subscript k represents the k^th boundary point of the EVCA; w_k is the stress direction vector of the PDN; and $P_{k}^{c} = (p_{i, t}^{c})$ is the maximum charging power along the direction w_k. In this case, the aggregated charging demand $p_{i, t}^{c, a v}$ in the constraints (11) and (12) is contained in the variable $p_{i, t}^{c}$ . Consequently, the feasible region can be denoted as a polytope $Ω ≜ \{A P_{k}^{c} \leq H\}$ , where A is the coefficient matrix for different stress directions; and H is a coefficient vector that describes the boundary of the EVCA.

In addition, the power flow and voltage constraints in $g (x, y, z) \leq 0$ are reformulated as chance constraints owing to the uncertainty in the renewable DGs:

P r \{c o s (m \frac{π}{l}) P_{i j, t}^{l} + s i n (m \frac{π}{l}) Q_{i j, t}^{l} \leq {\bar{S}}_{i j}\} \geq 1 - α_{S}

(23)

P r \{c o s (m \frac{π}{l}) P_{i j, t}^{l} + s i n (m \frac{π}{l}) Q_{i j, t}^{l} \geq - {\bar{S}}_{i j}\} \geq 1 - α_{S}

(24)

P r {v_{i, t} \leq {\bar{v}}_{i}} \geq 1 - α_{V}

(25)

P r {v_{i, t} \geq {\underset{̲}{v}}_{i}} \geq 1 - α_{V}

(26)

where l is a preset number; $m = 0,1, . . ., l$ ; $P r {\cdot}$ denotes the probability operator; and $α_{S}$ and $α_{V}$ are the probabilities of violating the constraints, which are set as $α_{S} = α_{V} = 0.05$ . 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 [

39].

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 [

40]:

- {\bar{S}}_{i j} \leq p P_{i j, t}^{l} + q Q_{i j, t}^{l} \pm Ω_{P Q_{l}} \leq {\bar{S}}_{i j}

(27)

v_{i, t} + Ω_{v_{i}} \leq {\bar{v}}_{i}

(28)

v_{i, t} - Ω_{v_{i}} \geq {\underset{̲}{v}}_{i}

(29)

where $Ω_{P Q_{l}}$ and $Ω_{v_{i}}$ are the uncertainty margins that tighten the constraint bounds to guarantee the secure operation of the PDN; $p = c o s (m π / l)$ ; and $q = s i n (m π / l)$ .

Considering the constraints (27)-(29) and replacing the corresponding inequality constraints in g, the problem in (22) can be reformulated as:

\{\begin{array}{l} m a x w_{k}^{T} P_{k}^{c} \\ s . t . h (x, y) - P_{k}^{c} = 0 \\ g^{P r} (x, y, z) \leq 0 \\ (15) - (19), (27) - (29) \end{array}

(30)

where g^Pr denotes the reformulated operational constraints. The DR constraints are considered when exploring the impact of the DR on the hosting capacity.

A physical interpretation of the solution to the problem in (30) is as follows. For the obtained boundary point z_k with the corresponding search direction w_k, there exists an operation point $(x, y, z)$ in which the operational constraints are satisfied. For any other point beyond z_k along the same search direction w_k, there is no such operation point that satisfies all constraints.

B. Search Process of Multipoint Approximation Method

It should be noted that $P_{k}^{c}$ consists of the charging demand $P_{0}^{c}$ and the available charging power $Δ P_{k}^{c}$ . Considering the charging demand obtained in (10), the boundary points will be determined if the available charging power $Δ P_{k}^{c}$ 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, and the details are given as follows.

Fig. 2 Flowchart of multipoint approximation method.

1)　Initialization

It is known that $P_{0}^{c} = (p_{i}^{c, a v})$ 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 $P_{0}^{c}$ 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 $P_{0}^{c}$ and the available charging power $Δ P_{k}^{c}$ , where the value of $Δ P_{k}^{c}$ is negative if the charging demand is beyond the EVCA.

\{\begin{array}{l} m a x w_{k}^{T} Δ P_{k}^{c} \\ s . t . h (x, y) - (Δ P_{k}^{c} + P_{0}^{c}) = 0 \\ g^{P r} (x, y, z) \leq 0 \\ (27) - (29) \end{array}

(31)

In initialization, we set each element of the direction vector w_k to be 1 to construct an n unit direction vector, where n denotes the dimension of w_k. 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 Fig. 2, where the base and boundary points are displayed in the $p_{i, t}^{c} - p_{j, t}^{c}$ plane. We denote $w_{k}$ $(k = 1,2)$ as the initial directions and $z_{k}$ ( $k = 1,2$ ) as the corresponding boundary point coordinates.

2)　Determination of New Boundary Points

In each round of searching, for any pair of obtained boundary points z_i and z_j, we can calculate the average value of their corresponding directions as a new search direction, as shown in Fig. 2, e.g., $w_{3} = (w_{1} + w_{2}) / 2$ . Following this, a new boundary point $z_{m}$ can be characterized by solving the problem in (31). We define the search distance d_m as the shorter distance between the new boundary point $z_{m}$ and its adjacent points z_i and z_j.

d_{m} = m a x {| | z_{m} - z_{i} | |_{2}, | | z_{m} - z_{j} | |_{2}}

(32)

In a certain round of searching, if $d \leq \partial$ , 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 $\partial$ 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 Algorithm 1.

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 $k = 0$ , convergence threshold, the time period t, corresponding EV charging demand $P_{0}^{c}$ , and initial direction set $W_{0} = {w_{1}, w_{2}, w_{3}}$

2: Solve the problem in (31) with $W_{0}$ and $P_{0}^{c}$ to obtain the solution set $Z_{0} = {z_{1}, z_{2}, z_{3}}$ , i.e., the EVCA boundary point set

3: Repeat

4: $k = k + 1$

5: Loop

6: Calculate a new search direction $w_{m} = w_{i} + w_{j}$ with a pair of adjacent directions and solve the problem in (31) with the new obtained set $W_{k} = {w_{1}, w_{2}, . . ., w_{n}}$ to obtain the new solution set $Z_{k} = {z_{1}, z_{2}, . . ., z_{n}}$ .

7: Calculate d_n for each new point using (32)

8: $d_{k} = m a x {d_{i} | i = 1,2, \dots, n}$ , $Z_{k} = Z_{k} ⋃ Z_{k - 1}$

9: End loop

10: Until $d_{k} \leq \partial$

11: Return Z_k

C. Illustrative Example

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, consisting of two EVCSs and one reactive power compensator. A detailed description of the system is available in [

41].

Fig. 3 Modified IEEE 4-bus distribution network.

Here, we set the initial directions $w_{11} = (1,0)$ and $w_{12} = (0,1)$ and the initial charging demand $P_{0}^{c} = (100,100)$ . Subsequently, we solve the problem in (31) for $Δ P_{k}^{c}$ and obtain two boundary points, $(400,0)$ and $(0,400)$ , by calculating $z = Δ P_{k}^{c} + P_{0}^{c}$ . According to Algorithm 1, we obtain a new direction $w_{21} = (1,1)$ . Next, we obtain the directions for the third search, $w_{31} = (2,1)$ and $w_{32} = (1,2)$ . The process of searching for boundary points is repeated until the search stops. The characterization of the EVCA is shown in Fig. 4, where P_c1 and P_c2 are the available charging power of two EVCSs.

Fig. 4 Characterization of EVCA of two EVCSs.

IV. Case Studies

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 [

42], which is an open-source software package that provides an interior point method for nonlinear optimization.

A. Simulation Setup and Comparison Methods

The modified 33-bus distribution network in Fig. 5 is adopted to validate the effectiveness of the proposed method. Here, three nodes are connected to the EVCS, two nodes are connected to the renewable DGs, and two nodes are connected to the dispatchable DGs containing voltage regulator facilities. Following this, 24-hour scheduling periods are set, and the aggregated capacity of each EVCS is 400 kW. Further details are available in [

41].

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 [

22].

M1: characterize the EVCA based on the multiple optimization method [

21].

M2: characterize the EVCA based on the vertex search method in the linear model [

28].

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 $N = N_{f}^{M 0} + N_{i n f}^{M 0}$ is obtained, where $N_{f}^{M 0}$ and $N_{i n f}^{M 0}$ are the numbers of feasible and infeasible points in M0, respectively. Correspondingly, $N = N_{f}^{M i} + N_{i n f}^{M i}$ ( $i = 1,2, 3$ ) 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:

\{\begin{array}{l} e_{f 1}^{M i} = N_{f}^{M 0} - N_{f}^{M i} \\ e_{i n f 1}^{M i} = N_{i n f}^{M 0} - N_{i n f}^{M i} \\ e_{f 2}^{M i} = \frac{N_{f}^{M 0} - N_{f}^{M i}}{N_{f}^{M 0}} \\ e_{i n f 2}^{M i} = \frac{N_{i n f}^{M 0} - N_{i n f}^{M i}}{N_{i n f}^{M 0}} \end{array} i = 1,2, 3

(33)

where $e_{f 1}^{M i}$ and $e_{f 2}^{M i}$ are the absolute and relative errors denoting the difference in feasible points between methods M0 and Mi, respectively; and $e_{i n f 1}^{M i}$ and $e_{i n f 2}^{M i}$ are the absolute and relative errors denoting the difference in infeasible points between methods M0 and Mi, respectively.

B. Results from Evaluation Methods

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 11^st hour and the 17^th hour) as examples for adequate analysis, as shown in Figs. 6 and 7, where Fig. 6(a) and Fig. 7(a) are regarded as the actual chargeable areas for reference. The red points in Fig. 6(a) and Fig. 7(a) are the sampling points of Monte Carlo method. P_c3 is the available charging power of the EVCS3. In practice, the EV charging power cannot be negative. In general, the coordinates denote the nodal active power injected for EV charging. In comparison with the EVCAs at the 17^th hour, the EVCAs at the 11^st hour have a larger range because the load level during the morning peak hour is lower, thereby increasing the hosting capacity of the PDN for the charging load. Conversely, the EVCAs are reduced during periods with higher load levels owing to the decrease in the hosting capacity. According to the definition of the EVCA, the excessive integration of EVs beyond the hosting capacity leads to a violation of the operational constraints. Thus, an accurate evaluation of the capacity is critical to guide EV charging, especially during peak hours when EVCA insufficiency is evident.

Fig. 6 EVCAs at the 11^st hour for methods M0-M3. (a) M0. (b) M1. (c) M2. (d) M3.

Fig. 7 EVCAs at the 17^th hour for methods M0-M3. (a) M0. (b) M1. (c) M2. (d) M3.

By comparing Figs. 6 and 7, it is evident that the EVCAs for M0 and M3 are almost identical. A comparison of the visualizations shows that M3 is more accurate than M1 and M2. To further illustrate this conclusion, the EVCAs at the 11^st hour are quantified and are compared with the volume and previously mentioned indices, as summarized in Table I. More precisely, with the base value set to be 10⁶ kW³, the unit value of volumes of EVCA for M0-M3 are 4.5671, 4.0785, 5.9694, and 4.5203, respectively. In addition, the performance indices demonstrate that the proposed method can provide an accurate approximation. Furthermore, the indices reveal that the EVCA for M1 contains fewer feasible points and more infeasible points than those for M0. In other words, many operation points of the EVCSs are misinterpreted as infeasible for M1. Although there are no operational constraint violations for M1, the hosting capacity of the PDN is not fully utilized. In contrast, more feasible points are contained in the EVCA for M2 compared to that for M0 because the infeasible points are misinterpreted as feasible. Although the hosting capacity can be maximally exploited, the security risks are still significantly increased. Therefore, the boundary points of the EVCAs for M1 and M2 are overly conservative and optimistic. For the EVCA for M3, the omission of feasible points is acceptable, with $e_{f 2}^{M i}$ being only 0.82%. Secure operation of the PDN is also ensured because there is no misinterpretation of the feasible points.

TABLE I Comparison of Accuracy and Calculation Time

Method	Volume (p.u.)	$N_{f}^{M i}$	$e_{f 1}^{M i}$	$e_{f 2}^{M i}$ (%)	$N_{i n f}^{M i}$	$e_{i n f 1}^{M i}$	$e_{i n f 1}^{M i}$ (%)	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 17^th hour as an example and sample the operation points of the EVCSs in and outside the EVCA as inputs to solve the AC OPF problem. In this case, we observe the variations in the power flow and voltage magnitude in the PDN. The profiles of the power flows S_f and S_inf at the feasible and infeasible points and the difference $∆ S$ between S_f and S_inf are shown in Fig. 8. The profiles of the voltage magnitudes V_f and V_inf at the feasible and infeasible points and the difference $∆ V$ between V_f and V_inf are shown in Fig. 9. It is evident that excessive charging loads require more power transmission from the main grid when the EVCSs operate at an infeasible point. As a consequence, the lines and substations may overload in this situation. The voltage magnitude of a substation node cannot maintain its given value with an excessive charging load. Compared with the operation status at a feasible point, the intensity of the voltage deviation sharply increases, resulting in a reduction in the power quality. However, the power flow and voltage magnitude are both within the acceptable range when the EVCSs operate at a feasible point. The results show that the obtained EVCA can provide better decision support for maintaining PDN operation by restraining the charging load in an EVCS.

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

Fig. 9 Profiles of voltage magnitudes and voltage magnitude variation ∆V.

C. Efficiency of Evaluation Method

The computational time of M0-M3 for the modified IEEE 33-bus system is listed in Table I. Specifically, M0 can construct an actual EVCA by large-scale sampling of operation points. However, as previously stated, Monte Carlo sampling requires more computation as the number of sampling points increases. For M1-M3, the computational time is 27.4 s, 1.8 s, and 38.3 s, respectively. Hence, M1-M3, contrary to M0, all obtain efficiency gains in characterizing the EVCA. The computational efficiency of M3 is not outstanding and may limit its application in real-time operation, especially when compared with that of M2. Considering the duration of the charging load, the proposed method, M3, is still preferable when evaluating the security of the operation of PDNs during mild timescales. Consequently, a tradeoff between the accuracy and the efficiency is achieved using M3.

D. Impact of DR Program on EVCA

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. Owing to the increasing price, the power demand is restrained, and consumers tend to utilize cheaper power. By comparing Cases 1 and 2 with Case 0, it is observed that partial loads during peak hours are transferred to periods with lower prices. Nevertheless, the total variation in the power demand in Case 1 is lower because fewer nodes participate in the DR.

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. 11 and 12. Specifically, in Fig. 11, it is observed that the EVCAs in Cases 1 and 2 are larger than those in Case 0. This is because more power is available for EV charging in the PDN owing to the reduction in the power demand when using the DR program. In Case 2, all nodes participate in the DR, resulting in a considerable hosting capacity. In Fig. 12, the EVCA in Case 2 decreases, and the EVCA in Case 1 remains unchanged compared with that in Case 0. It is intuitive that the peak load in Case 2 is transferred to a period with a lower price, resulting in a shrunken EVCA. Owing to the less responsive load in Case 1, the impact of the demand variation on the hosting capacity is minimal. The results show that the DR program during peak hours can improve the hosting capacity for EV integration, whereas it may have a negative impact during off-peak hours, depending on the extent of the nodes participating in the DR. Hence, the DR program can be flexibly used by DSOs to coordinate the secure operation of PDNs considering the EV charging demand. For example, a period with decreasing EVCAs is matched with a period with a low charging demand.

Fig. 11 EVCAs at the 11^th hour in Cases 0-2. (a) Case 0. (b) Case 1. (c) Case 2.

Fig. 12 EVCAs at the 22^nd hour in Cases 0-2. (a) Case 0. (b) Case 1. (c) Case 2.

E. Selection of Predefined Threshold

In the proposed method, the convergence criterion is given by $d \leq \partial$ . The impact of the threshold $\partial$ on the search process of the EVCA is evaluated via numerical tests, as shown in Fig. 13. Specifically, the number of boundary point searches increases with smaller thresholds. It is observed that the EVCA does not significantly change when the threshold is smaller than 0.001 p.u.; a tighter search criterion leads to increased computational demand and few precision improvements. Thus, we set $\partial$ to be 0.001 p.u. to achieve a good approximation of the EVCA with a tradeoff between the accuracy and the efficiency.

Fig. 13 Impact of threshold $\partial$ on search process of EVCA.

V. Conclusion

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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