In recent years, a low-carbon society has become a common goal for many countries, which promotes energy transition (Wang et al., 2021b); (Qin et al., 2024b); (Sun et al., 2023). Moreover, large-scale renewable energy resources (RES) have been witnessed in distribution networks, such as wind power, solar power, hydropower, etc., (Qin et al., 2024c); (Jia et al., 2020); (Liu et al., 2022); (Qin et al., 2024a). These RES are much more variable and unpredictable than conventional generations and loads, posing a great challenge to the stable operation of the modern network (Guo et al., 2022); (Shuai et al., 2021). Improper regulation may lead to voltage fluctuation and instability, which will degrade the performance of the network, even resulting in potential danger (Li et al., 2017); (Wang Z. et al., 2020); (Zhang et al., 2024). To address the issue, there is a growing need for a flexible and responsive voltage regulation strategy. One promising approach is the utilization of electric vehicles (EVs) for voltage regulation (Yang et al., 2024); (Wang et al., 2024); (Yin et al., 2023); (Yang et al., 2023). The network operator compensates EVs for their participation in voltage regulation. In this paper, we investigate how to utilize EVs to realize voltage regulation in distribution networks at a relatively low cost.

Voltage regulation by EVs in distribution networks aims to minimize the voltage mismatch by regulating the charge/discharge power of EV clusters. Model predictive control (MPC) based methods are widely considered in this area (Li et al., 2019); (Wang et al., 2022); (Hu et al., 2022); (Feng and Liu, 2023). In reference (Li et al., 2019), a model predictive control (MPC) method is proposed to allow EVs to participate in the grid voltage regulation as a reactive power compensation device to maintain the grid voltage within a stable range. Authors in Wang et al. (2022) developed an MPC-based decentralized algorithm to solve the voltage-related problem using the respective PVs and EVs while ensuring that the EV charging demand is satisfied while contributing to the voltage regulation. Reference (Hu et al., 2022) established a distributed MPC strategy for EV chargers to exploit reactive power vehicle-to-grid (V2G) abilities and participate in real-time voltage regulation of both balanced and unbalanced distribution networks without intervening in the active power exchange. The adaptive primary control and the distributed MPC-based secondary control were combined in Feng and Liu (2023) to maintain accurate dynamic power sharing and stable frequency and voltage.

While MPC-based methods are powerful for voltage regulation, optimization-based methods may handle complex objectives and constraints and scale better with problem size. Many existing works have investigated optimization-based methods (Beaude et al., 2013); (Wu et al., 2016); (Wang S. et al., 2020); (Zhao et al., 2020); (Hu et al., 2021); (ur Rehman, 2022); Yumiki et al. (2022); (Wang et al., 2023). Authors in Beaude et al. (2013) investigated a decentralized optimization methodology to coordinate EV charging in order to contribute to the voltage control on a residential electrical distribution feeder. Reference Wu et al. (2016) used the EVs to regulate the voltage of the smart grid and propose an algorithm to maximize the revenue of the parking lot in a line distribution grid considering the EV users’ charging demand and the voltage fluctuations. A game-theoretic machine learning framework was proposed in Wang S. et al. (2020) to utilize EV charging stations and achieve decentralized Volt-Var control. Reference Zhao et al. (2020) studied an incentive mechanism based on Nash bargaining theory to solve the voltage regulation problem in a distributed network integrated with EVs. Reference Hu et al. (2021) proposed a distributed voltage regulation scheme for dominantly resistive distribution networks through a coordinated EV charging/discharging process. A novel and robust central aggregation hierarchical V2G optimization algorithm is proposed in ur Rehman (2022) to provide voltage and frequency regulation services to the grid. Reference Yumiki et al. (2022) developed a system-level design for the control of electric power grids by EVs to realize multi-objective ancillary service including primary frequency control and voltage amplitude regulation. Authors in Zhang et al. (2018) proposed a novel EV charging scheduling mechanism by controlling the active and reactive charging power of EVs to provide joint voltage and frequency regulation. They formulate a non-convex optimization problem and solve it by transforming the problem into a second-order cone program problem. Reference Wang et al. (2023) proposed an incentive-based control strategy to encourage EVs in desired locations to participate in grid voltage regulation.

In most aforementioned optimization-based works, the main focus is on problem formulation and modeling. The optimization problem is usually formulated with power flow constraints of the network model. Few works intend to investigate how to solve the problem effectively. Commonly, gradient-based methods to find the optimal solution are widely utilized. However, accurate model information is hard to obtain, e.g. resistance values of electrical lines, due to environmental influences. How to realize voltage regulation with EVs in distribution networks without accurate model information remains unsolved.

In this paper, we formulate a voltage regulation strategy in distribution networks provided by EV clusters with an online parameter estimation method. Our main contributions are as follows:

• The voltage regulation problem with EV clusters is formulated. Both power flow constraints and economic cost are considered in the optimization problem. The operator aims to eliminate voltage mismatch as well as minimize the cost of compensation for EV clusters.

The rest of this paper is organized as follows. In Section 2, preliminaries are presented and the distribution network is built. Section 3 formulates the problem of voltage regulation with EV clusters in distribution networks. An optimal seeking algorithm with online parameter estimation is proposed in Section 4. The convergence proof of the algorithm is also given in this section. Numerical results are demonstrated in Section 5. Finally, Section 6 concludes the paper.

In this section, we formulate an optimization problem to realize voltage regulation in the distribution network by EV clusters. For EV cluster i , the discharge power is limited, denoted by g ̲ i ≤ g i ≤ g ̄ i , ∀ i ∈ N (6) where g ̲ i and g ̄ i represent the lower and upper bounds of discharge power, respectively.

When EV clusters discharge to help realize voltage regulation, we compensate for EV clusters according to their discharge power. The compensation function for EV cluster i is denoted by f i c g i = a i g i 2 + b i g i (7) where a i and b i are positive constants (Wang et al., 2021a).

To realize voltage regulation, we aim to maintain voltage at each node as a predetermined value. We use the node voltage as a feedback value and define the voltage mismatch as the following function: V i v i = v i − v 0 2 (8)

The goal for the network operator is to lower voltage mismatch while reducing compensation costs, which is a combination of (Equation 7) and (Equation 8). Therefore, the regulation problem can be summarized as the following optimization problem: min g Φ g = α ∑ i = 1 N f i c g i + 1 − α ∑ i = 1 N V i v i (9a) s . t . g ̲ i ≤ g i ≤ g ̄ i , ∀ i ∈ N (9b) v i = π i g i , g − i (9c) where 0 < α < 1 is a trade-off constant. By tuning the value of α , we can achieve different balances between compensation cost and regulation performance.

Lemma 1

The cost function Φ ( g ) (Equation 9a) is μ -strongly convex.

Proof

The cost function Φ ( g ) (Equation 9a) can be expressed as: Φ g = α g T A g + α b T g + ‖ R g + X q − R d + 1 v s − v 0 1 ‖ 2

For a positive μ satisfying μ < 2 σ min ( A ) , we have Φ g − μ 2 ‖ g ‖ 2 = g T α A − μ 2 I g + α b T g + ‖ R g + X q − R d + 1 v s − v 0 1 ‖ 2 .

Obviously, Φ ( g ) − μ 2 ‖ g ‖ 2 is still a convex function. Therefore, the cost function Φ ( g ) is μ -strongly convex with 0 < μ < 2 α σ min ( A ) .

Remark 1. For the optimization problem (Equation 9a–c), the cost function (Equation 9a) is a strictly convex function subject to g . The constraints (Equation 9b) and (Equation 9c) form a non-empty convex compact set on g . Therefore, there exists a unique optimal solution for the problem (Equation 9a–c).

The optimization problem (Equation 9a–c) is a constrained convex optimization problem. One commonly used algorithm is the projected gradient descent (PGD) method, which is demonstrated in (Equation 10). g k + 1 = P G g k − τ ∇ Φ g k (10) where τ is the step size and the set G = ∏ i = 1 N G i . The gradient is computed as follows: ∇ Φ g k = 2 α A g k + α b + 1 − α R T v k − v 0 1 (11) where the matrix A and vector b are defined as follows: A = a 1 0 ⋯ 0 0 a 2 ⋯ 0 ⋮ ⋮ ⋱ ⋮ 0 0 ⋯ a N , b = b 1 b 2 ⋮ b N The key step in the iteration (Equation 10) is the gradient computation (Equation 11). Obviously, it relies on the network resistance parameter R , which is difficult to obtain due to two reasons. On the one hand, the precise resistance value is usually not available because of wire aging and environmental influences. On the other hand, not all lines are exactly measured. Thus the resistance matrix R in the linear equation given in (Equation 5) is not known. To solve the problem, we utilize recursive least squares (RLS) to help estimate R for the gradient computation. The RLS algorithm acquires new knowledge during the iteration and uses it to update the estimation of R online.

For the RLS design, we first transform (Equation 5) into a simpler form. Let v ̄ = X q − R d − 1 v s , then we express (Equation 5) as: v k = R g k + v ̄ (12) Since we only need to estimate R in gradient computation (Equation 11), we introduce a difference form of (Equation 12) to eliminate the constant term v ̄ . In PGD method (Equation 10), the variable g k changes for each iteration until the algorithm converges. We use Δ g k = g k + 1 − g k to denote the iteration difference of g , leading to a voltage difference defined as Δ v k = v k + 1 − v k . Then we have the difference form of (Equation 12): Δ v k = R Δ g k (13)

To utilize RLS, we need to transform the matrix R into a vector. By introducing the regressor matrix ψ k = Δ g k ⊗ I N and parameter vector θ = v e c ( R ̃ ) , we transform (Equation 13) into: Δ v k = ψ k T θ (14)

Now we utilize the RLS method to obtain the recursive estimation of θ , denoted by θ k ̂ . The RLS design is as follows: θ ̂ k + 1 = θ ̂ k + P k − 1 ψ k D k − 1 Δ v k − ψ k ⊤ θ ̂ k (15a) P k = λ − 1 I − P k − 1 ψ k D k − 1 ψ k ⊤ P k − 1 (15b) where D k = λ T + ψ k ⊤ P k − 1 ψ k with T ∈ R N × N being a positive definite matrix and constant real forgetting factor 0 < λ < 1 . P k ∈ R N 2 × N 2 is the covariance matrix with the initial matrix P − 1 being positive definite. This RLS algorithm (Equations 15a, b) is for the multiple-output system like (Equation 14), which is an extension to the traditional single-output RLS algorithm Johnstone et al. (1982).

To learn θ by the RLS method (Equations 15a, b), we need to get enough information, i. e, utilize different ψ k to explore different v k and then infer the elements of θ . This can be described as the persistency of excitation condition Brüggemann and Bitmead (2021): The matrix sequence { ψ k } is said to be persistently exciting if for some S ≥ 0 and all j ≥ 0 there exist positive reals, β and γ , such that 0 < β I ≤ ∑ i = j j + S ψ i ψ i ⊤ ≤ γ I < ∞ .

Using the estimation parameter θ k vector to recover R k , we may compute the gradient (Equation 11) as: ∇ Φ ̂ k g k = 2 α A g k + α b + 1 − α R k T v k − v 0 1 (16) Then the operator may use (Equation 16) to update g k in (Equation 10). The PGD-RLS algorithm is outlined in Algorithm 1. Our proposed algorithm consists of two steps. Step 1 involves the PGD (Equation 10), where the gradient is computed using the estimated parameter R k from the previous iteration instead of the exact value. Step 2 employs an RLS algorithm (Equations 15a, b) for online parameter estimation. It makes use of the information about voltage and power during iterations to update the estimation R k . The online estimation does not rely on prior knowledge of the network resistance. Step 1 provides sufficient input-output information for Step 2 to refine the estimation of R online. Meanwhile, Step 2 provides the updated estimation result for Step 1, allowing for the consistent update of g k until convergence is achieved.

Algorithm 1

PGD-RLS.

Initialization for EV clusters: g i , 0 ∈ G i

Next, we will analyze the performance of the proposed algorithm, which includes the estimation accuracy of RLS method (Equations 15a, b) and the convergence of decision variable g k . We first show the exponential convergence of RLS method (Equations 15a, b).

Theorem 1

For any initial estimation θ ̂ 0 , the estimation error ‖ θ ̂ k − θ ‖ 2 converges exponentially to zero, i.e. there exists a ν > 0 such that ‖ θ ̂ k − θ ‖ 2 ≤ ν λ k ‖ θ ̂ 0 − θ ‖ 2 , for all k ≥ S , where ν = λ − ( S + 1 ) − 1 σ min ( T − 1 ) β ( λ − 1 − 1 ) σ max P − 1 − 1 with P − 1 being a positive definite matrix.

Following Lemma 2 and Theorem 3 in Brüggemann and Bitmead (2021), Theorem 1 is easy to prove, which is omitted here. With Theorem 1 guaranteeing the convergence of estimation error, we now intend to prove the convergence of the PGD-RLS Algorithm 1. First, we have a property about the gradient of the cost function (Equation 9a).

Lemma 2

The gradient of the cost function ∇ Φ ( g ) (9a) is L -Lipschitz continuous.

Proof. For any g 1 , g 2 ∈ G , we have ‖ ∇ Φ g 1 − ∇ Φ g 2 ‖ = 4 α 2 g 1 − g 2 T A T A g 1 − g 2 + 1 − α 2 g 1 − g 2 T R T R 2 g 1 − g 2 .

If we choose L ≥ σ max ( 4 α 2 A T A + ( 1 − α ) 2 ( R T R ) 2 ) , then ‖ ∇ Φ ( g 1 ) − ∇ Φ ( g 2 ) ‖ ≤ L ‖ g 1 − g 2 ‖ . Therefore, the gradient of the cost function is L -Lipschitz continuous.

With Theorem 1 and Lemma 1,2, we have the following theorem about the convergence of the PGD-RLS Algorithm 1.

Theorem 2

With step sizes chosen to satisfy 0 < τ < 2 μ L 2 , the decision variable g k in the proposed PGD-RLS Algorithm 1 converges to the optimal solution to the optimization problem (Equations 9a–c), denoted by g * .

Proof. ‖ g k + 1 − g * ‖ 2 = ‖ P G g k − τ ∇ Φ ̂ k g k − P G g * − τ ∇ Φ g * ‖ 2 ≤ ‖ g k − τ ∇ Φ ̂ k g k − g * − τ ∇ Φ g * ‖ 2 ≤ τ 2 ‖ ∇ Φ g k − ∇ Φ ̂ k g k ‖ 2 + ‖ g k − τ ∇ Φ g k − g * − τ ∇ Φ g * ‖ 2 The first inequality holds because the projection operator is nonexpansive while the second inequality holds because of the triangle inequality. For the term τ 2 ‖ ∇ Φ ( g k ) − ∇ Φ ̂ k ( g k ) ‖ 2 , we have τ 2 ‖ ∇ Φ g k − ∇ Φ ̂ k g k ‖ 2 = 1 − α 2 τ 2 ‖ R k − R v k − v 0 1 ‖ 2 ≤ 1 − α 2 τ 2 ‖ R k − R ‖ 2 ‖ v k − v 0 1 ‖ 2 = 1 − α 2 τ 2 σ max R k − R 2 ‖ v k − v 0 1 ‖ 2 ≤ 1 − α 2 τ 2 ∑ i = 1 N σ i R k − R 2 ‖ v k − v 0 1 ‖ 2 = 1 − α 2 τ 2 ‖ R k − R ‖ F 2 ‖ v k − v 0 1 ‖ 2 = 1 − α 2 τ 2 ∑ i , j | R k i , j − R i , j | 2 ‖ v k − v 0 1 ‖ 2 = 1 − α 2 τ 2 ‖ θ ̂ k − θ ‖ 2 ‖ v k − v 0 1 ‖ 2 ≤ 1 − α 2 τ 2 ν λ k ‖ θ ̂ 0 − θ ‖ 2 M 2 where ν is the constant in Theorem 1 and M is the maximum of the function ‖ v k − v 0 1 ‖ 2 . ‖ v k − v 0 1 ‖ 2 is a quadratic function with respect to v k , which is an affine function on g k . Since all the elements of g k are bounded by (Equation 6), the output of function ‖ v k − v 0 1 ‖ 2 is bounded. Therefore, the maximum M exists. Since λ < 1 , the convergence of τ 2 ‖ ∇ Φ ( g k ) − ∇ Φ ̂ k ( g k ) ‖ 2 is guaranteed.

For the second term ‖ g k − τ ∇ Φ ( g k ) − ( g * − τ ∇ Φ ( g * ) ) ‖ 2 , we have ‖ g k − τ ∇ Φ g k − g * − τ ∇ Φ g * ‖ 2 = ‖ g k − g * ‖ 2 − 2 τ ⟨ g k − g * , ∇ Φ g k − ∇ Φ g * ⟩ + τ 2 ‖ ∇ Φ g k − ∇ Φ g * ‖ 2 ≤ 1 − 2 τ μ + τ 2 L 2 ‖ g k − g * ‖ 2

The inequality holds because Φ ( g k ) is μ -strongly convex with the gradient being L -Lipschitz continuous. Since 0 < τ < 2 μ L 2 , then ρ = ( 1 − 2 τ μ + τ 2 L 2 ) < 1 . So the convergence of ‖ g k − τ ∇ Φ ( g k ) − ( g * − τ ∇ Φ ( g * ) ) ‖ 2 is guaranteed.

Therefore, we have proved that the decision variable g k in the PGD-RLS Algorithm 1 converges to the optimal solution g * to the optimization problem (Equations 9a–c).

In this section, we validate the PGD-RLS algorithm through simulation results on the 33-bus distribution network shown in Figure 2.

In this work, we investigate voltage regulation in distribution networks using EV clusters. By utilizing a feedback-based optimization approach, we formulate a voltage regulation problem to minimize voltage mismatch while keeping the cost of services provided by EV clusters relatively low. To find the optimal solution without precise knowledge of network resistance, we propose a PGD algorithm with an RLS method to enable online estimation of resistance. The accuracy of the RLS online estimation is proved. The convergence of the proposed PGD-RLS algorithm is rigorously guaranteed. Numerical results in the IEEE 33-bus verify the effectiveness of the proposed method for voltage regulation. Future research directions include the decentralization of the proposed framework to allow each EV cluster to individually implement the update of discharge power.