Model-Free Adaptive Cooperative Control Strategy of Multiple Electric Springs: A Hierarchical Approach for EV-Integrated AC Micro-Grid

Highlights

What are the main findings?

• Proposal of a novel hierarchical cooperative control strategy for multi-EV-ES systems, integrating directed-graph-based consensus for upper-level power distribution and model-free adaptive constrained control (MFACC) for lower-level voltage regulation.
• The MFACC framework introduces an anti-windup compensator and pseudo-partial derivative (PPD)-based output observer to suppress voltage oscillations caused by inverter saturation, achieving precise tracking with minimal harmonic distortion.
• The strategy enables distributed power decoupling without requiring precise system parameters, leveraging local agent communication to stabilize bus voltage/frequency and allocate active/reactive power proportionally.

What is the implication of the main finding?

• The approach significantly enhances micro-grid resilience against voltage fluctuations from EV/renewable integration, maintaining PCC voltage at reference value, while transferring fluctuations to non-critical loads.
• Reduced reliance on model parameters lowers implementation barriers for V2G systems, with faster stabilization time and lower overshoot compared to conventional methods.
• Theoretical support for such data-driven control architectures can increase the penetration of electric vehicles without compromising grid stability, thereby accelerating the decarbonization of the power system.

Abstract

With the aim of addressing the power quality problem associated with voltage fluctuation of multiple electric vehicles and renewable energy generation equipment integration into the AC micro-grid, a multi-agent system-based model-free adaptive constrained control method is proposed in this paper. First, a novel hierarchical control structure is developed. Therein, the upper-level cooperative controller is designed based on the directed graph and droop control strategy, enabling efficient power distribution among multiple electric vehicles. For the lower-level voltage controller, a model-free adaptive constrained control strategy is designed, incorporating a pseudo-partial derivative-based output observer, and an anti-windup compensator is designed to solve the voltage fluctuation problem, which achieves precise tracking of each electric spring output voltage. Finally, the effectiveness and superiority of the proposed control strategy is verified by the MATLAB/Simulink platform under scenarios of grid-side voltage fluctuations and load variations.

1. Introduction

Given dwindling fossil fuel reserves and pressing environmental concerns [1,2], the electric vehicle (EV) is rapidly gaining popularity in society [3], which is classified as a low-carbon and sustainable mode of transport [4,5,6]. Meanwhile, clean energy generation is progressively displacing traditional, highly polluting fossil fuel-based power generation. However, research indicates that EVs are idle for most of the day. Consequently, with high EV penetration enabled by vehicle-to-grid technology (V2G), the aggregated batteries of numerous EVs can serve as a viable distributed energy storage system [7,8]. V2G technology allows EVs to feed stored power back into the grid, enhancing grid flexibility and facilitating more effective integration with clean energy generation [9,10].

However, large-scale EV integration can seriously impact micro-grid stability [11]. Especially in the case of critical load fluctuation, micro-grids are susceptible to large voltage fluctuations, which also affects the power quality of the micro-grid [12]. To address this challenge, the electric spring (ES) technique has been proposed as a solution to mitigate voltage fluctuations. For example, in [13], as a demand-side management model, an electric spring is proposed to improve load profiles and address supply–demand imbalance. With the intensive research on electric springs, various types of electric spring configurations and management techniques have been developed [14]. ES-1 type represents a foundational first-generation electric spring configuration, and ES-2 type is an electric spring configuration that improves on the ES-1 type. By replacing the DC side of ES-1 with an energy storage unit, ES-2 achieves a wider operational range and enhanced capability. Thus, in the base ES-2, this paper proposes the EV-ES type, which incorporates the EV into the DC side, to realize the V2G function of EV through an electric spring, enhancing the stability of EVs accessed by an AC micro-grid [15], and its system diagram is shown in Figure 1.

In AC micro-grids with large-scale EV integration, the regulation capability of a single EV-ES is not enough to support the stability of the whole system [16]. When numerous distributed EV-ES units are embedded within a micro-grid [17], the system state is collectively determined by these units.

As a proven power distribution method, droop control was introduced for a distributed system. Reference [18] uses quadratic control for droop control and achieved quadratic correction of bus voltage and reactive power sharing, but quadratic correction of frequency response is not considered. In [19], a voltage regulation approach utilizing adaptive virtual impedance is introduced, which can effectively control the voltage drop as well as power distribution, and obtain the same voltage reference value using distributed virtual impedance;however, considering only the distributed generator in parallel does not guarantee that the voltage in the series case is approaching the rated value. A distributed generalized droop control that takes delay into account is employed in [20]; it effectively ensures stability in frequency and voltage while minimizing the communication overhead. however, the power distribution among the distributed generator is not considered [21]. Aiming at power distribution problems in multi-EV-ES systems, this paper integrates a multi-agent system into conventional droop control to design the upper-level controller [22,23,24]. This approach requires only narrowband communication for neighboring EV-ES units to exchange local information, achieving voltage stabilization and power distribution in the multi-EV-ES system [25].

In terms of the lower-level controller of the multi-EV-ES, considering the nonlinear, multi-variable, and strongly coupled characteristics of EV-ES, the conventional control strategy fails to achieve satisfactory control performance. Reference [26] proposed a phase-based repetitive control method to achieve periodic stability and non-periodic response. Reference [27] proposed a radial-chordal decomposition-based control for voltage and power angle control of a smart load (SL). Reference [28] employed a consensus algorithm as the upper-level controller, with a proportional-resonant (PR) controller for the lower-level control, but it only achieved stability of the bus voltage. Reference [29] proposed a model predictive control-based coordinated control strategy for multi-ES. However, most of the existing control strategies for ES are designed based on model parameters, with only a few focusing on power decoupling. Inspired by Reference [30], some researchers have applied model-free adaptive control (MFAC) to the control strategy of multi-EV-ES, but conventional MFAC encounters saturation issues [31], leading to voltage waveform oscillation and reduction in power quality. In terms of dealing with power quality problems, various anti-windup methods have been proposed to address the challenges. Reference [32] introduced a Proportional-Integral with Anti-Windup (PIAW) controller based on Lyapunov functions for DC-DC buck power converters. Reference [33] proposed an Adjustable Virtual Impedance method that employs a back-calculation anti-windup strategy. By adjusting the back-calculation gain to modify the reactance-to-resistance ratio of the equivalent virtual impedance, this approach enhances transient stability and grid-forming capability under current-limiting conditions. Reference [34] proposed a model-free cooperative sliding mode control scheme with a directed-graph-based observer for distributed energy storage systems in DC micro-grids. These studies provide diverse solutions to the input windup problem. However, there is continuing potential for further exploration of methods for integrating anti-windup and MFAC in multi-EV-ES AC micro-grid systems.

To address these challenges, this paper proposes a novel hierarchical and distributed cooperative control strategy. In this method, a directed graph-based consensus algorithm is utilized as the upper-level control of multi-EV-ES, enabling voltage and frequency regulation as well as precise distribution of active and reactive power [35]. The lower-level control employs model-free adaptive constrained control (MFACC), explicitly accounting for disturbances within the micro-grid [36]. In the base of conventional MFAC, MFACC utilizes an input-constrained anti-windup compensator and a pseudo-partial derivative (PPD)-based output observer to mitigate voltage oscillations and improve power quality; thus, a novel hierarchical control is being established. The key achievements of this paper are outlined below:

• The method employs a regulation strategy based on local information exchange among neighboring EV-ES units. This approach addresses the shortcomings of traditional droop control in maintaining voltage/frequency stability and achieving accurate power distribution within the multi-EV-ES system.
• Considering disturbances in the micro-grid, a novel estimation algorithm is proposed to establish a more accurate dynamic linearization data model for disturbances, using the compact form dynamical linearization method.
• To address voltage oscillation issues caused by inverter duty cycle saturation, MFACC is proposed. It employs an input-constrained anti-windup compensator and pseudo-partial derivative-based output observer to ensure power quality in the micro-grid.

The subsequent parts of the paper are organized in the following manner: In Section 2, the foundational working mechanisms and mathematical frameworks of EV-ES are presented. Section 3 proposes a hierarchical control scheme for multi-EV-ES, including a leader consensus algorithm based on directed graphs, the MFACC algorithm, adaptive parameter estimation, output-based adaptive observers, and input-constrained anti-windup compensators. Section 4 conducts simulation analysis to compare and validate the proposed control strategies. Finally, Section 5 presents the conclusions.

2. Topology and Mathematical Modelling of EV-ES

2.1. Topology of EV-ES

The object of this study is the EV-based ES-2; its topology is shown in Figure 2. It is clear that the system is composed of a single-phase voltage-source inverter and EV battery. Therein, non-critical loads (NCL) are placed with the EV-ES to form the SL, and the SL is placed with the critical load (CL) to form the point of common coupling (PCC) load node; $Z_1$ is the line impedance of transmission; $C_f$ and $L_f$ are the filter capacitor and inductor, respectively; $V_{\mathrm{CL}}$ represents the voltage of the CL while $V_{\mathrm{NCL}}$ denotes the voltage of the NCL; $V_{\mathrm{AB}}$ is the output voltage of the inverter; $V_G$ is the grid-side voltage; $V_{\mathrm{ES}}$ is the output voltage of the EV-ES; $I_{\mathrm{L}}$ is the output current of the inverter; $I_1$, $I_2$, and $I_3$ are the injected current at PCC, the current of CL, and NCL, respectively; when fluctuations occur on the grid-side, EV-ES transfers power and voltage fluctuations to NCL to ensure that the CL operates stably at the given reference value.

With the integration of numerous renewable energy generation devices into micro-grid, distributed embedding of multiple EV-ES units is required to regulate multiple CL and ensure micro-grid power quality. Figure 3 illustrates the multi-EV-ES structural diagram of the distributed micro-grid system. Taking four EV-ES as an example, the four buses shown in Figure 3, the buses are connected in series via line impedances, each bus segment is considered as a PCC node, and each PCC is composed of the topology structure shown in Figure 2, including EV-ES, CL, and NCL. The grid side of the system is composed of alternating current sources and intermittent renewable energy sources as wind turbines and photovoltaic panels.

2.2. The Mathematical Model of EV-ES

Reference [30] summarized the derivation of the state equations for the EV-ES. The mathematical model equations are as follows:

$$\left\{\begin{array}{cc}\hfill {\displaystyle \frac{d{I}_{\mathrm{L}}}{dt}}& ={\displaystyle \frac{V_{\mathrm{AB}}}{L}}-{\displaystyle \frac{V_{\mathrm{ES}}}{L}}\hfill \\ \hfill {\displaystyle \frac{d{V}_{\mathrm{ES}}}{dt}}& ={\displaystyle \frac{I_{\mathrm{L}}}{C}}+{\displaystyle \frac{I_3}{C}}\hfill \\ \hfill {\displaystyle \frac{d{I}_1}{dt}}& ={\displaystyle \frac{V_{\mathrm{G}}}{L_1}}-{\displaystyle \frac{V_{\mathrm{ES}}{Z}_2}{L_1(Z_2+Z_3)}}-\hfill \\ \hfill & {\displaystyle \frac{I_1(Z_1{Z}_2+Z_2{Z}_3+Z_3{Z}_1)}{L_1(Z_2+Z_3)}}\hfill \end{array}\right.$$

(1)

$$V_{\mathrm{CL}}={\displaystyle \frac{Z_2}{Z_2+Z_3}}{V}_{\mathrm{ES}}+{\displaystyle \frac{Z_2{Z}_3}{Z_2+Z_3}}{I}_1$$

(2)

For effective power control of the EV-ES, it is essential to develop a model in the d-axis and q-axis coordinate system. However, due to the absence of orthogonal signals required for rotational transformation in three-phase systems, the establishment of d-axis and q-axis models in single-phase systems is more intricate. For this purpose, a virtual quadrature signal is required to be developed to transform the single-phase system from stationary to a rotating coordinate system. In this paper, virtual quadrature current and voltage signals are constructed with the help of a second-order generalized integrator (SOGI) [37]; its topology is shown in Figure 4.

The PCC receives injected voltage and current, which can be formulated as follows:

$$v_{CL}=V_{m}cos(\omega t+\phi )$$

(3)

$$i_1=I_{m}cos(\omega t+\psi )$$

(4)

therein, $\omega$ denotes the angular frequency; $\phi$ and $\psi$ are the initial phases; $V_m$ and $I_m$ are the amplitude of $v_{CL}$ and $i_1$, respectively; $v_{CL}$ can be expressed as a synthetic vector of virtual orthogonal components in the coordinate system as

$${\mathbf{v}}_{\alpha \beta }=\left[\begin{array}{c}{v}_{\alpha }\\ v_{\beta }\end{array}\right]=\left[\begin{array}{c}{V}_{m}cos(\omega t+\phi )\\ V_{m}sin(\omega t+\phi )\end{array}\right]$$

(5)

In (5) the virtual orthogonal signals are obtained after SOGI for the axial and axial components, respectively. In a synchronous rotating coordinate system, the virtual signal is decomposed into d-axis and q-axis components by a coordinate transformation, which can be expressed as

$${\mathbf{v}}_{dq}=\left[\begin{array}{c}{v}_d\\ v_q\end{array}\right]=T\left(\widehat{\delta }\right)\left[\begin{array}{c}{v}_{\alpha }\\ v_{\beta }\end{array}\right]=\left[\begin{array}{c}{V}_{m}cos\phi \\ V_{m}sin\phi \end{array}\right]$$

(6)

$$T\left(\widehat{\delta }\right)=\left[\begin{array}{cc}cos\omega t& sin\omega t\\ -sin\omega t& cos\omega t\end{array}\right]$$

(7)

where $T\left(\widehat{\delta }\right)$ is the transformation matrix and $\delta$ is the phase angle of the PCC.

Therefore, the power transferred to the PCC can be expressed as

$$\begin{array}{c}\overline{S}={\overline{V}}_{CL}·{\overline{I}}_1\\ =V_m{I}_m(cos(\phi -\psi )+jsin(\phi -\psi ))\\ =P_{\mathrm{in}}+j{Q}_{\mathrm{in}}\end{array}$$

(8)

wherein the active power and reactive power can be represented as

$$\left\{\begin{array}{c}{P}_{\mathrm{in}}=V_{\mathrm{CL}d}{I}_{1d}+V_{\mathrm{CL}q}{I}_{1q}\hfill \\ Q_{\mathrm{in}}=V_{\mathrm{CL}q}{I}_{1d}-V_{\mathrm{CL}d}{I}_{1q}\hfill \end{array}\right.$$

(9)

where $V_{CLd}$ and $V_{CLq}$ are the voltage components along the d and q axis of the PCC; $I_{1d}$ and $I_{1q}$ are the current components along the d and q axis injected into the PCC.

Similarly, the mathematical model of ES in d-axis and q-axis coordinates can be obtained by arithmetic simplification from (1)–(7) as

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left[\begin{array}{c}{I}_{\mathrm{L}d}\\ I_{\mathrm{L}q}\end{array}\right]=\left[\begin{array}{cc}0& \omega \\ \omega & 0\end{array}\right]\left[\begin{array}{c}{I}_{\mathrm{L}d}\\ I_{\mathrm{L}q}\end{array}\right]-{\displaystyle \frac{1}{L_{\mathrm{f}}}}\left[\begin{array}{c}{V}_{\mathrm{ES}d}\\ V_{\mathrm{ES}q}\end{array}\right]+{\displaystyle \frac{1}{L_{\mathrm{f}}}}\left[\begin{array}{c}{V}_{\mathrm{AB}d}\\ V_{\mathrm{AB}q}\end{array}\right]$$

(10)

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left[\begin{array}{c}{V}_{\mathrm{ES}d}\\ V_{\mathrm{ES}q}\end{array}\right]=\left[\begin{array}{cc}0& \omega \\ \omega & 0\end{array}\right]\left[\begin{array}{c}{V}_{\mathrm{ES}d}\\ V_{\mathrm{ES}q}\end{array}\right]+{\displaystyle \frac{1}{C}}\left[\begin{array}{c}{I}_{\mathrm{L}d}\\ I_{\mathrm{L}q}\end{array}\right]-$$

$${\displaystyle \frac{1}{C(Z_2+Z_3)}}\left[\begin{array}{c}{V}_{\mathrm{ES}d}\\ V_{\mathrm{ES}q}\end{array}\right]+{\displaystyle \frac{Z_2}{C(Z_2+Z_3)}}\left[\begin{array}{c}{I}_{1d}\\ I_{1q}\end{array}\right]$$

(11)

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left[\begin{array}{c}{I}_{1d}\\ I_{1q}\end{array}\right]=\left[\begin{array}{cc}0& \omega \\ \omega & 0\end{array}\right]\left[\begin{array}{c}{I}_{1d}\\ I_{1q}\end{array}\right]-{\displaystyle \frac{Z_2}{L_1(Z_2+Z_3)}}\left[\begin{array}{c}{V}_{\mathrm{ES}d}\\ V_{\mathrm{ES}q}\end{array}\right]-$$

$${\displaystyle \frac{Z_1{Z}_2+Z_2{Z}_3+Z_1{Z}_3}{L_1(Z_2+Z_3)}}\left[\begin{array}{c}{I}_{1d}\\ I_{1q}\end{array}\right]+{\displaystyle \frac{1}{L_1}}\left[\begin{array}{c}{V}_{\mathrm{G}d}\\ V_{\mathrm{G}q}\end{array}\right]$$

(12)

In this context, the subscripts containing ‘d’ and ‘q’ denote the components of each variable that correspond to the d and q axis, where $I_1$ denotes the output current of EV-ES; $V_{ES}$ denotes the output voltage of EV-ES; $I_1$ is the transmission line input current; $V_G$ is the grid-side voltage.

3. Integrated Cooperative Control Design for Multi-EV-ES

3.1. Multi-EV-ES Distributed Upper-Level Controller Design

For the system proposed in this paper, in terms of addressing the issue of efficient power allocation between each EV-ES, this paper incorporates MAS to improve the conventional droop control [38]. Each EV-ES is treated as an individual agent, and a consensus algorithm is developed for power allocation [39]. The basic expression of droop control can be express as

$$\begin{array}{c}{V}_{ref}=V-K_n(Q-Q_{ref})\hfill \end{array}$$

(13)

$$\begin{array}{c}{f}_{ref}=f-K_m(P-P_{ref})\hfill \end{array}$$

(14)

where $Q_{ref}$, Q and $P_{ref}$, P denote the targeted and realized values of reactive power, alongside the targeted and realized values of active power at the PCC, respectively; $K_n$ and $K_m$ are the droop factor of power.

In the micro-grid system, the stability of both bus voltage and frequency holds paramount importance. However, voltage drops occur with increasing line length. If all EV-ES units adopt an identical voltage reference under these conditions, it can lead to unreasonable competition among them, potentially causing system-wide oscillations. Therefore, the outer loop of voltage control and frequency control are designed in the power decoupling-based droop control. Initially, the system bus voltage and frequency, along with the EV-ES power, must meet the following objectives:

$$\begin{array}{c}\underset{t\to \infty }{lim}{\displaystyle \sum _{i=1}^n}(V_i\left(t\right)/\phantom{V_i\left(t\right)n-V_{ref}}n-V_{ref})=0\hfill \end{array}$$

(15)

$$\begin{array}{c}\underset{t\to \infty }{lim}({\displaystyle \sum _{i=1}^n}{f}_i\left(t\right)/\phantom{f_i\left(t\right)n-f_{ref}}n-f_{ref})=0\hfill \end{array}$$

(16)

$$\begin{array}{c}\underset{t\to \infty }{lim}{\displaystyle \sum _{i=1}^n}(Q_{ES,i}\left(t\right)/\phantom{(Q_{ES,i}\left(t\right)Q_{ES,i}^{max}-Q_{ES,j}\left(t\right)/\phantom{Q_{ES,j}\left(t\right)Q_{ES,j}^{max}}{Q}_{ES,j}^{max}}{Q}_{ES,i}^{max}-Q_{ES,j}\left(t\right)/\phantom{Q_{ES,j}\left(t\right)Q_{ES,j}^{max}}{Q}_{ES,j}^{max})=0\hfill \end{array}$$

(17)

$$\begin{array}{c}\underset{t\to \infty }{lim}{\displaystyle \sum _{i=1}^n}(P_{ES,i}\left(t\right)/\phantom{(P_{ES,i}\left(t\right)P_{ES,i}^{max}-P_{ES,j}\left(t\right)/\phantom{P_{ES,j}\left(t\right)P_{ES,j}^{max}}{P}_{ES,j}^{max}}{P}_{ES,i}^{max}-P_{ES,j}\left(t\right)/\phantom{P_{ES,j}\left(t\right)P_{ES,j}^{max}}{P}_{ES,j}^{max})=0\hfill \end{array}$$

(18)

where $V_{ref}$ and V are the voltage reference and actual values of PCC, respectively; $f_{ref}$ and f represent the targeted and actual values of the frequency at the PCC, respectively; $V_{ref}$ is set to the bus desired voltage RMS 220 V; $f_{ref}$ is set to the bus desired operating frequency of 50 Hz.

To stabilize the PCC voltage and frequency, local information is exchanged among neighboring agents. This employs a leader-consensus algorithm and a leaderless-consensus algorithm; the task is to assign the EV-ES’s active and reactive power, and its control diagram is shown in Figure 5.

Remark 1.

It should be noted that the power allocation is not without any deviation in the power of all EV-ES, but it achieves a relative static error of zero. In that case, the consensus algorithm is developed. From (15)–(18) we can obtain

$$\begin{array}{c}{e}_{v,i}=K_v{\displaystyle \sum _{j\in N_i}}{a}_{ij}(V_i-V_j)+b_i(V_i-V_{ref})\hfill \end{array}$$

(19)

$$\begin{array}{c}{e}_{f,i}=K_f{\displaystyle \sum _{j\in N_i}}{a}_{ij}(f_i-f_j)+b_i(f_i-f_{ref})\hfill \end{array}$$

(20)

$$\begin{array}{c}{e}_{Q,i}=K_Q{\displaystyle \sum _{j\in N_i}}{a}_{ij}(Q_{ES,i}/\phantom{Q_{ES,i}{Q}_{ES,i}^{max}}{Q}_{ES,i}^{max}-{Q_{ES,}}_j/\phantom{{Q_{ES,}}_j{Q}_{ES,j}^{max}}{Q}_{ES,j}^{max})\hfill \end{array}$$

(21)

$$\begin{array}{c}{e}_{P,i}=K_P{\displaystyle \sum _{j\in N_i}}{a}_{ij}(P_{ES,i}/\phantom{P_{ES,i}{P}_{ES,i}^{max}}{P}_{ES,i}^{max}-{P_{ES,}}_j/\phantom{{P_{ES,}}_j{P}_{ES,j}^{max}}{P}_{ES,j}^{max})\hfill \end{array}$$

(22)

where $K_v$, $K_f$, $K_P$, and $K_Q$ are the voltage, frequency, active power, and reactive power observe gain, respectively; $e_{v,i}$, $e_{f,i}$, $e_{Q,i}$, and $e_{P,i}$ are the voltage, frequency, active power, and reactive power observe errors, respectively; and $Q_{ES}^{max}$ and $P_{ES}^{max}$ are the maximum rated output power for EV-ES. The adjustment structure is shown in Figure 6.

Remark 2.

The $b_i$ is restraining gain; the root node connected to the leader is assigned a value of 1, while the rest are assigned a value of 0.

To fulfill the goals set by the upper-level controller, the communication network structure for the multi-EV-ES is established, as shown in Figure 7. Therein, each agent only exchanges information with the connected agents in the network. Agent-ES0 serves as the virtual leader, and the Laplacian matrix of the communication network in the diagram can be represented as

$$\begin{gathered} L=D-A=\left[\begin{array}{cccc}2& -1& 0& -1\\ -1& 2& -1& 0\\ 0& -1& 2& -1\\ -1& 0& -1& 2\end{array}\right] \\ =\left[\begin{array}{cccc}2& 0& 0& 0\\ 0& 2& 0& 0\\ 0& 0& 2& 0\\ 0& 0& 0& 2\end{array}\right]-\left[\begin{array}{cccc}0& 1& 0& 1\\ 1& 0& 1& 0\\ 0& 1& 0& 1\\ 1& 0& 1& 0\end{array}\right] \end{gathered}$$

(23)

Definition 1.

Four agent $N_i=n=4$, $A={\left(a_{ij}\right)}_{4\times 4}$, D, and A are the in-degree matrix and the adjacency matrix. Integrate the observe error signals (19)–(22) to obtain the new error values on the d and q-axis as follows:

$$\begin{array}{c}{e}_{d,i}=K_m(P_i-P_{ES,i})+{\widehat{e}}_{f,i}-e_{p,i}-e_{f,i}\hfill \end{array}$$

(24)

$$\begin{array}{c}{e}_{q,i}=K_n(Q_i-Q_{ES,i})+{\widehat{e}}_{v,i}-e_{Q,i}-e_{v,i}\hfill \end{array}$$

(25)

where ${\widehat{e}}_{f,i}=f_{ref}-f_i$ and ${\widehat{e}}_{v,i}=V_{ref}-V_i$ are the relative error in frequency and the relative error in voltage, respectively. Finally, the d and q-axis virtual signals are regenerated based on the observation errors of voltage, frequency, and active and reactive power, to obtain new desired voltage waveforms as desired signals, transferring to the lower level controller.

It is obvious that the upper-level controller proposed in this paper not only decouples active power and reactive power, but also relies solely on the processing of local signals from agents; thus, it does not require specific model parameters of the EV-ES.

3.2. Analysis of Lipschitz Condition

This imposes a constraint on the system’s output rate of change. Its fundamental purpose is to require that the system’s response to input signals exhibits a finite rate of mutation, thereby preventing untraceable abrupt variations in the control effort. In other words, finite changes in the input should produce proportionally bound output variations within the system.

For the object system researched in this article, it can be seen from the topology of the left side of the EV-ES that it is partially structured as a single-phase full-bridge inverter. The single-phase full-bridge inverter uses PWM modulation to produce the output voltage $V_{AB}$ close to a sinusoidal waveform by controlling the on-off switching of the switching tubes, where the output voltage amplitude is related to the control input.

Maximum voltage change rate limited by DC bus voltage $V_{dc}$ (energy source) switching period $T_s$ (simulation sample time constraint), and physical constraint ${\left|{\displaystyle \frac{d{V}_{AB}}{dt}}\right|}_{max}\le {\displaystyle \frac{V_{dc}}{T_s}}|$ means that the maximum change in output voltage within one switching cycle is bounded. After sampling period discretization process $\Delta {\nu }_i\left(k\right)\triangleq {\nu }_i\left(k\right)-{\nu }_i(k-1)\approx {\displaystyle \frac{d{\nu }_i}{dt}}{|}_{t=k{T}_s}·T_s$ substituting the voltage rate of change constraint, we can obtain $\mid \Delta {\nu }_i\left(k\right)\mid \le \left({\displaystyle \frac{V_{dc}}{T_s}}\right)·T_s=V_{dc}$. For the duty cycle variation $\Delta m_i\left(k\right)=m_i\left(k\right)-m_i(k-1)$, when $\Delta m_i\left(k\right)=1$ holds, it means that duty cycle is from $0\%$ to $100\%$. Finally, we can obtain $\mid \Delta {\nu }_i\left(k\right)\mid \le V_{dc}·\mid \Delta m_i\left(k\right)\mid$.

It is also logical that considering the system in terms of energy calibration, it is reasonable that a finite change in energy input should produce a finite change in energy output within the system.

3.3. Lower-Level Model-Free Voltage Controller Design

Aiming at the nonlinear behavior, and the multi-variable and strongly coupled system of EV-ES [40,41], the MFACC is proposed. It is designed with an anti-windup compensator and a pseudo-partial derivative-based output observer to track the reference voltage with only slight chatter, its control diagram is shown in Figure 8. Firstly, the nonlinear system of EV-ES can be represented as

$$\begin{array}{c}{v}_i(k+1)=f_i(v_i\left(k\right),\cdots ,v_i(k-n_v),m_i\left(k\right),\\ \cdots ,m_i(k-n_m),d_i\left(k\right),\cdots d_i(k-n_d))\end{array}$$

(26)

Assumption 1.

The continuity of the partial derivatives of the nonlinear function $f_i(\cdots )$ with respect to its input variables is assured, which is a typical constraint in the design of control strategies for nonlinear systems.

Assumption 2.

The Lipschitz condition is satisfied by the nonlinear system, $\left|v_i(k+1)-v_i\left(k\right)\right|\le b\left|m_i\left(k\right)-m_i(k-1)\right|$, $k\in N^{+}$, $b>0$ is a constant.

Assumption 3.

The norm value of $\Delta u_i\left(k\right)$ is bounded, namely $∥\Delta u_i\left(k\right)∥\le \vartheta$, $\vartheta >0$. where $m_i\left(k\right)$ is the control input; $v_i\left(k\right)$ is the system output; $d_i\left(k\right)$ is the potential disturbance in the micro-grid; $n_v$ $n_m$ $n_d$ are unknown order of the system; $f_i(\cdots )$ is the unknown kinetic function of the system.

Using the compact form dynamical linearization method, the EV-ES is converted into a simplified, equivalent dynamic linearization data model; there exists a PPD when $\Delta u_i\left(k\right)\ne 0$, such that the following data model holds:

$$\begin{array}{c}{v}_i(k+1)-v_i\left(k\right)={\varphi }_i\left(k\right)(u_i\left(k\right)-u_i(k-1))\end{array}$$

(27)

where

$${\varphi }_i\left(k\right)={\left[\begin{array}{ccc}{\varphi }_1^i\left(k\right)& {\varphi }_2^i\left(k\right)& d_i\left(k\right)\end{array}\right]}^{\top },\Delta u_i\left(k\right)={\left[\begin{array}{ccc}\Delta m_i\left(k\right)& \Delta v_i\left(k\right)& 1\end{array}\right]}^{\top }$$

${\varphi }_1^i\left(k\right)$ and ${\varphi }_2^i\left(k\right)$ are the PPD corresponding to $\Delta m_i\left(k\right)$ and $\Delta v_i\left(k\right)$, respectively.

Data model can be further rewritten as

$$\begin{array}{c}\Delta v_i(k+1)={\varphi }_1^i\left(k\right)\Delta m_i\left(k\right)+{\varphi }_2^i\left(k\right)\Delta v_i\left(k\right)+d_i\left(k\right)\end{array}$$

(28)

Assumption 4.

For a given bounded expectation output signal $v_i(k+1)$, there is always a bounded $\Delta u_i\left(k\right)$; thus, driven by this control input signal, the result is $v_i(k+1)$.

Assumption 5.

$d_i\left(k\right)$ is a slow time-varying parameter with small variation at each step.

Remark 3.

It can be concluded from Equation (27) and Assumption 4, when the control input is increased, the corresponding output of the controlled system should not decrease, and the system can be regarded as having the proposed linearization characteristics.

Based on the equivalent linear data model, the PPD is an unknown time-varying parameter with complex dynamics that are difficult to model precisely. Therefore, the following estimation algorithm is designed:

$$\begin{array}{c}{\widehat{\varphi }}_i(k+1)={\widehat{\varphi }}_i\left(k\right)+{\Lambda }_i\left(k\right)\Delta u_i\left(k\right)(\Delta v_i\left(k\right)\\ -{\widehat{\varphi }}_i\left(k\right)\Delta u_i\left(k\right))\end{array}$$

(29)

where ${\widehat{\varphi }}_i\left(k\right)$ is the estimation value of PPD; ${\Lambda }_i\left(k\right)=\eta /(\mu +{∥\Delta u_i\left(k\right)∥}^2)$; $\mu >0$ is a penalty factor for constraining the variation of PPD; and $\eta \in (0,1]$ is a step size factor—its inclusion enhances the adaptability of the estimation algorithm.

Additionally, the trajectory tracking error is defined as $e_i\left(k\right)=v_d^i\left(k\right)-v_i\left(k\right)$; taking into account the adverse effects of inverter saturation on power quality, the trajectory tracking error is redefined as

$$\begin{array}{c}{e}_i\left(k\right)=v_d^i\left(k\right)-v_i\left(k\right)-{\xi }_i\left(k\right)\end{array}$$

(30)

where $v_d^i\left(k\right)$ represents the desired tracking trajectory for the system’s output; ${\xi }_i\left(k\right)$ is the compensation signal to address actuator saturation. The control input of the controlled system is subject to amplitude constraints:

$$\begin{array}{c}{m}_{min}\le m_i\left(k\right)\le m_{max}\end{array}$$

(31)

where $m_{min}$ and $m_{max}$ are the control input’s lower and upper limits, respectively. To mitigate the adverse effects of actuator saturation, an anti-windup compensator based on input constraints is designed [34], that is

$$\begin{array}{c}{\xi }_i(k+1)=\tau {\xi }_i\left(k\right)+{\varphi }_1^i\left(k\right)(m_0^i\left(k\right)-m_i\left(k\right))\end{array}$$

(32)

$\tau \in (0,1]$ is the weighting factor of the compensation signal; $m_0^i\left(k\right)$ is the constrained control input.

Considering the disturbances present in the micro-grid, and with the goal of improving the system’s stability and robustness, the design of the relative output observer is outlined as follows:

$$\begin{array}{c}{\widehat{v}}_i(k+1)={\widehat{v}}_i\left(k\right)+{\varphi }_i^{\mathrm{T}}\left(k\right)\Delta u_i\left(k\right)+K_0(v_i\left(k\right)-{\widehat{v}}_i\left(k\right))\end{array}$$

(33)

where $K_0\in (0,1]$ is the observer gain; ${\widehat{v}}_i\left(k\right)$ is the output estimation value; $e_0^i\left(k\right)=v_i\left(k\right)-{\widehat{v}}_i\left(k\right)$ is defined as the estimated value of the error in the relative output. Thereafter, this results as

$$\begin{array}{c}{e}_0^i(k+1)=\mathrm{E}{e}_0^i\left(k\right)+{\varphi }_i^{\mathrm{T}}\Delta u_i\left(k\right)\end{array}$$

(34)

where $\mathrm{E}=1-K_0$, ${\overline{\varphi }}_i\left(k\right)={\widehat{\varphi }}_i\left(k\right)-{\varphi }_i\left(k\right)$ is represented as the error in the estimation of relative parameters. To avoid the occurrence of algebraic loops, an approximate solution for $e_0^i(k+1)$ is obtained based on the two-step delayed estimation technique [42] as

$$\begin{array}{c}{e}_0^i(k+1)\approx 2e_0^i\left(k\right)-e_0^i(k-1)\end{array}$$

(35)

The new adaptive PPD estimation algorithm can alternatively rewritten as

$$\begin{array}{c}{\widehat{\varphi }}_i(k+1)={\widehat{\varphi }}_i\left(k\right)+\Delta u_i\left(k\right){\Lambda }_i\left(k\right)(2e_0^i\left(k\right)\\ -e_0^i(k-1)-\mathrm{E}{e}_0^i\left(k\right))\end{array}$$

(36)

Remark 4.

Additionally, to enhance the tracking capability of the adaptive parameter estimation algorithm, a PPD resetting algorithm was developed for slowly varying parameters: ${\widehat{\varphi }}_i\left(k\right)={\varphi }_i\left(1\right)$, if ${\widehat{\varphi }}_i^{\mathrm{T}}\left(k\right){\widehat{\varphi }}_i\left(k\right)\le \epsilon$ or $\Delta u_i^{\mathrm{T}}\Delta u_i\le \epsilon$ or $\mathrm{sgn}({\widehat{\varphi }}_1^i\left(k\right))\ne \mathrm{sgn}({\widehat{\varphi }}_1^i\left(1\right))$.

According to Equations (28)–(34), the new system output can be rewritten as

$$\begin{array}{c}{v}_i(k+1)={\widehat{v}}_i\left(k\right)+{\widehat{\varphi }}_i^{\mathrm{T}}\Delta u_i\left(k\right)+2e_0^i\left(k\right)\\ -e_0^i(k-1)+K_0{e}_0^i\left(k\right)\end{array}$$

(37)

From Equations (36) and (37), the controller can be designed as

$$\begin{array}{c}{m}_0^i\left(k\right)={\displaystyle \frac{\rho }{\lambda +{\varphi }_1^{i^2}}}(v_d^i(k+1)-{\widehat{v}}_i\left(k\right)-{\varphi }_2^i\Delta v_i\left(k\right)\\ -d_i\left(k\right)-(2+K_o)e_o^i\left(k\right)\\ +e_o^i(k-1)-\tau {\xi }_i\left(k\right))\end{array}$$

(38)

$$m_i\left(k\right)=\mathrm{Sat}\left(\right(m_0^i\left(k\right)-m_0^i(k-1\left)\right),m_{min},m_{max})$$

(39)

where $\mathrm{Sat}(\cdots )$ denotes the saturation function as

$$\mathrm{Sat}(m,m_{min},m_{max})=\left\{\begin{array}{cc}{m}_{min}& m\le m_{min}\hfill \\ m\hfill & m_{min}\le m\le m_{max}\hfill \\ m_{max}\hfill & m_{max}\le m\hfill \end{array}\right.$$

(40)

Theorem 1.

Consider the nonlinear system (26) under Assumptions 1–4. If $v_i(k+1)=\mathrm{const}$ holds, then under the controller defined by (29), (36), (37), the system tracking error converges monotonically, and the closed-loop system inputs $u_i\left(k\right)$ and outputs $v_i\left(k\right)$ are bounded.

Proof. 

The proof is organized in two parts.

Part (1): Based on Equation (29) and Assumption 5, it can be obtained that

$${\widehat{\varphi }}_i(k+1)=\Gamma \left(k\right){\widehat{\varphi }}_i\left(k\right)$$

(41)

Taking an absolute value, herein the gain $\Gamma \left(k\right)$ has the property of being monotonically decreasing as

$$∥\Gamma \left(k\right)∥=\left|1-{\displaystyle \frac{\eta {∥\Delta u_i\left(k\right)∥}^2}{\mu +{∥\Delta u_i\left(k\right)∥}^2}}\right|\le \sigma <1$$

(42)

where $\sigma$ is a small positive constant, from Equation (42) it can be further derived that

$$\begin{array}{c}∥{\widehat{\varphi }}_i(k+1)∥\le \sigma ∥{\widehat{\varphi }}_i\left(k\right)∥\le {\sigma }^2∥{\widehat{\varphi }}_i(k-1)∥\\ \le \dots \le {\sigma }^k∥{\widehat{\varphi }}_i\left(1\right)∥\end{array}$$

(43)

It is obvious from Equation (43) that ${\widehat{\varphi }}_i\left(k\right)$ is asymptotically stable.

Part (2): Firstly, to facilitate the calculation, some variables need to be combined into column vectors:

$$\begin{array}{cc}\hfill v\left(k\right)& ={[v_1\left(k\right),v_2\left(k\right),\cdots ,v_N\left(k\right)]}^{\mathrm{T}},\hfill \\ \hfill \widehat{v}\left(k\right)& ={[{\widehat{v}}_1\left(k\right),{\widehat{v}}_2\left(k\right),\cdots ,{\widehat{v}}_N\left(k\right)]}^{\mathrm{T}},\hfill \\ \hfill e_0\left(k\right)& ={[e_0^1\left(k\right),e_0^2\left(k\right),\cdots ,e_0^N\left(k\right)]}^{\mathrm{T}},\hfill \\ \hfill \widehat{\varphi }\left(k\right)& ={[{\widehat{\varphi }}_1^{\mathrm{T}}\left(k\right),{\widehat{\varphi }}_2^{\mathrm{T}}\left(k\right),\cdots ,{\widehat{\varphi }}_N^{\mathrm{T}}\left(k\right)]}^{\mathrm{T}},\hfill \\ \hfill \mathrm{H}\left(k\right)& =\mathrm{diag}(\Delta u_1^{\mathrm{T}}\left(k\right),\Delta u_2^{\mathrm{T}}\left(k\right),\cdots ,\Delta u_N^{\mathrm{T}}\left(k\right)).\hfill \end{array}$$

Then, the system output estimation error value can be further rewritten as

$$e_0(k+1)=\mathrm{E}{e}_0\left(k\right)+\mathrm{H}\left(k\right)\widehat{\varphi }\left(k\right)$$

(44)

Construct the Lyapunov function as $V\left(k\right)=e_0^{\mathrm{T}}\left(k\right)e_0\left(k\right)$; thus, we can obtain

$$\begin{array}{c}\Delta V(k+1)\le {\left|\mathrm{E}\right|}^2+2∥\mathrm{H}\left(k\right)\widehat{\varphi }\left(k\right)∥\left|\mathrm{E}\right|∥e_0\left(k\right)∥\\ ({∥\mathrm{H}\left(k\right)\widehat{\varphi }\left(k\right)∥}^2-1){∥e_0\left(k\right)∥}^2\end{array}$$

(45)

It is obvious that $\mathrm{E}<1$, namely $\Delta V(k+1)<0$ holds with the condition $∥e_0\left(k\right)∥>{\displaystyle \frac{\left|\mathrm{E}\right|}{1-∥\mathrm{H}\left(k\right)\widehat{\varphi }\left(k\right)∥}}$, which also makes $\underset{k\to \infty }{lim}∥e_0\left(k\right)∥\le {\displaystyle \frac{\left|\mathrm{E}\right|}{1-∥\mathrm{H}\left(k\right)\widehat{\varphi }\left(k\right)∥}}$. It can be concluded from part (1) that $\underset{k\to \infty }{lim}∥e_0\left(k\right)∥=0$; thus, $e_0^i\left(k\right)$ is asymptotically stable, which is decouple observer stability.

The redefined tracking error $e_i\left(k\right)={\nu }_d^i\left(k\right)-{\nu }_i\left(k\right)-{\xi }_i\left(k\right)$ embeds the anti-windup compensator signal ${\xi }_i\left(k\right)$, and the compensator dynamics (32) ensure ${\xi }_i\left(k\right)$ is bounded when $m_0^i\left(k\right)-m_i\left(k\right)$ is bounded during actuator saturation. Consequently, $e_i\left(k\right)$ converges if $v_i\left(k\right)$ tracks ${\nu }_d^i\left(k\right)-{\xi }_i\left(k\right)$, not the original ${\nu }_d^i\left(k\right)$. This shifts the control objective to a feasible trajectory during saturation. □

4. Simulation Results

In this section, the control performance of the proposed control strategy is analyzed and evaluated through simulation. A micro-grid model, as shown in Figure 3, is constructed for the multi-EV-ES grid-connected system. The multi-agent consensus algorithm (MCA) MFACC control strategy is applied in the MATLAB/Simulink (R2023a) environment, and the effectiveness and superiority of the proposed controller is compared with the MCA-MFAC [43], MCA-PR [30], and MCA-SMC [44] control strategies. MCA-MFAC builds a dynamic linearization data model of the system using the compact form dynamic linearization method through the input/output data of the system and then designs an adaptive control law based on this model; MCA-PR is a control strategy commonly used in AC systems, which is capable of static-free tracking of signals at specific frequencies. The PR controller consists of a proportional part and a resonant part. The SMC adjusts the parameters so that the system state can quickly converge to the slide mode surface when it is far away from it. The block diagram of the simulation system is shown in Figure 9, and the system operating parameters and controller parameters are shown in

Table 1. Parameters of multiple electric spring system (results of ten runs).

Parameter	Value	Parameter	Value	Parameter	Value
$R_1/\Omega$	0.1	$L_1/\mathrm{mH}$	10	${\mathrm{Z}}_{\mathrm{line}1\text{–}3}/\mathrm{mH}$	1
$V_{ev}/\mathrm{V}$	1200	$\mathrm{CL}/\Omega$	50	$\mathrm{NCL}/\Omega$	10
$\mathrm{L}/\mathrm{mH}$	3	$\mathrm{C}/{}^{-}\mathrm{F}$	60	T/s	$5\times {10}^{-5}$

and

Table 2. Parameters of the proposed MCA-MFACC strategy (results of ten runs).

Parameter	Value	Parameter	Value	Parameter	Value
$\tau$	${10}^{-6}$	$k_{\mathrm{p}}$	0.1	$k_{\mathrm{i}}$	5
$m_{min}$	−1	$m_{max}$	1	$\lambda$	${10}^{-6}$
$\rho$	1.5	$\mu$	${10}^6$	$\eta$	0.9
${\widehat{\varphi }}_i^1\left(1\right)$	0.002	${\widehat{\varphi }}_i^2\left(1\right)$	0.01	${\widehat{\varphi }}_i^3\left(1\right)$	0

; while PSO [45] and GWO [46] offer efficiency, manual scanning was prioritized for interpretability in this initial study.

4.1. Grid-Side Voltage Fluctuation Test

The operating condition is set as follows: when t = 0–1 s, the grid-side voltage of the micro-grid is maintained at a normal value of 220 V. When t = 1 s, the grid-side voltage increases from the initial value of 220 V to 230 V. When t = 2 s, it drops from 230 V to 210 V, to simulate a significant fluctuation in the grid-side voltage. Simultaneously, when t = 3 s, the consensus regulation algorithm is enabled. After installing EV-ES at each bus of the micro-grid, with the fluctuation of the grid-side voltage, the control strategy proposed in this paper effectively maintains the bus voltage near the standard value of 220 V which realizes the stabilization of the bus voltage. Therefore, each EV-ES transfers the voltage fluctuation from the PCC to NCL, and the fluctuation of power in NCL is very similar to the grid-side voltage fluctuation.

As shown in the Figure 10, the control strategy proposed in this paper effectively stabilizes the bus voltage and transfers the fluctuation to NCL. When t = 3 s in Figure 11, the changes in power of each EV-ES validate the effectiveness of the consensus algorithm in avoiding unreasonable power distribution among the EV-ESs. Based on the reasonable power distribution, the algorithm ensures the stability of the global average voltage and the voltage of each bus.

To demonstrate the advantages of the proposed control strategy, simulations are conducted comparing MCA-MFACC with MCA-MFAC, MCA-PR, and MCA-SMC. Figure 11d,f,h show the PCC voltage and the fluctuation of NCL power under the MCA-MFAC, MCA-PR, and MCA-SMC control strategies, and the summary of control performance values is shown in

Table 3. Performance in four methods (results of ten runs).

Performance	MCA-SMC	MCA-PR	MCA-MFAC	MCA-MFACC
Stabilization time/s	0.15	0.24	0.12	0.11
Overshoot/%	1.45	1.91	1.25	1.13
THD %	3.56	7.54	2.13	1.01

. In order to provide statistically significant comparative data analysis, a t-test analysis (the number of simulation runs used for t-tests is 10) and confidence intervals based on these metrics were used to quantitatively compare the performance metrics of the control methods, the content of which is presented

Table 4. p value (t-test) and confidence intervals (95%) for MFACC to three methods (results of ten runs).

Performance	MCA-SMC	MCA-PR	MCA-MFAC
Stabilization time	8.609 $\times {10}^{-5}$ [0.1469 s, 0.1511 s]	1.428 $\times {10}^{-6}$ [0.2351 s, 0.2429 s]	0.0083 [0.1140 s, 0.1240 s]
Overshoot	3.1939 $\times {10}^{-12}$ [1.45%, 1.45%]	1.3157 $\times {10}^{-14}$ [1.90%, 1.91%]	3.148 $\times {10}^{-8}$ [1.24%, 1.25%]
THD	3.8269 $\times {10}^{-12}$ [3.56%, 3.56%]	3.8336 $\times {10}^{-17}$ [7.54%, 7.55%]	2.7005 $\times {10}^{-10}$ [2.12%, 2.13%]

.

Comparing Figure 12 and Figure 13, it can be observed that the lower-level control strategy using MFACC exhibits faster stabilization of bus voltage and frequency, rapid transfer of fluctuations, and quicker frequency response compared to MFAC, PR, and SMC control strategies. Specifically, MFACC achieves the shortest stabilization time and minimum overshoot for bus voltage, as well as minimal harmonic content.

In this study, a nonlinear baseline is implemented in MATLAB/Simulink using the ode45 solver and the full circuit model (battery, EV-ES, line impedance, and duty cycle saturation). The linearized model corresponds to the CFDL-based voltage estimates in the proposed MFACC controller.

Table 5. Max absolute error and max relative error analysis (results of ten runs).

	PCC1	PCC2	PCC3	PCC4
Max Absolute Error %	1.2011	1.2535	1.3291	1.6465
Max Relative Error %	1.4968	1.5524	1.8671	2.1081

and Figure 14a demonstrate the error analysis. As well, the variation of PPD parameters and disturbance estimate value are shown in Figure 14b–d, which represent the slow time-varying characteristics of PPD. The PPD is a slow time-varying parameter, and bounded estimate rather than a parameter that converges to a fixed value. Since there is no unique steady-state value, we interpret the true convergence criterion as a manifestation of the voltage tracking performence, as can be seen in Figure 11a.

4.2. Micro-Grid Topology Changing Test

The initial value of the grid-side voltage is set to 210V below the rated voltage state. When t = 1 s, the NCL on PCC4 changes from 10 to 20, and when t = 2 s, it changes from 20 to 10, to simulate the fluctuation of NCL under the condition of constant grid-side voltage. The reason for this is to validate the insensitivity of the proposed control strategy to changes in system topology parameters.

Figure 15 shows the RMS voltage of each PCC under the NCL fluctuation under the MCA-SMC, MCA-PR, MCA-MFAC, and MCA-MFACC control strategies. It is evident that when fluctuations occur, all three control strategies can eventually maintain the bus voltages near their nominal values. Among them, MCA-MFACC exhibits the fastest response speed, minimal overshoot, and the best power quality compared to the MCA-MFAC, MCA-PR, and MCA-SMC control strategies.

Figure 16 shows the system frequency variation during the NCL fluctuation. It can be observed that compared to MCA-SMC, MCA-PR, and MCA-MFAC, the MCA-MFACC exhibits smaller frequency fluctuations and can quickly stabilize near 50 Hz. Additionally, MCA-SMC, MCA-PR, and MCA-MFAC show small-range fluctuations under normal operation.

4.3. Power Fluctuation Test

In this section, the control performance of the control strategy proposed in this paper is tested in the presence of renewable energy power fluctuations. As can be seen in Figure 17b, set the predefined injected power fluctuation, and its dynamics is characterized as 1100 W from t = 0–1 s, falling to 600 W in t = 1–2 s, and increasing to 1100 W in t = 2–4 s; the grid-side voltage is set to 220 V.

In Figure 17, the (a) RMS voltage values of PCC, (b) predefined injection power graphs and PCC measured power graphs, (c) power graphs of NCL, and (d) power graphs of EV-ES are shown, respectively. In (a), it can be seen that the voltage stabilization ability for PCC is the same as the tests in the previous sections, maintaining an excellent tracking ability; in (b), the predefined inject power lines are grey dashed lines, and the active power of each PCC achieves a strong tracking of the power fluctuations; in (c), it can be seen that when the predefined injected power fluctuates, the active power of NCL also fluctuates with it; and in (d), the reactive power of EV-ES also fluctuates with power fluctuations.

As can be seen from the above changes, as the predefined injected power fluctuates, the power of the NCL and EV-ES changes in the same way, but at the same time the voltage to the PCC is always effectively stable. From this, it can be concluded that EV-ES not only effectively stabilizes the CL voltage, but also acts as a power manager to transfer the input power fluctuations to the NCL.

4.4. Controller Parameter Test

The parameters defined in the controller proposed in this paper have a great impact on the control performance, especially the control law weighting factor $\rho$. Several values of $\rho$ are provided, which leads to different tracking performance as shown in Figure 18. The results show that the control performance of (a) $\rho =1.5$ is better than the controllers of (b) $\rho =0.1$ and (c) $\rho =5$, due to having faster response and smaller overshoot.

In addition, several simulations of the controller parameters are simulated by means of parameter scanning, and the following

Table 6. Controller performance (stabilization time/overshoot/THD) for different values of $\eta$ and $\mu$ (results of ten runs).

$\mathit{\eta }\setminus \mathit{\mu }$ Values	${10}^4$	${10}^5$	${10}^6$	${10}^7$	${10}^8$
1.1	0.15 s 1.20% 1.54%	0.15 s 1.20% 1.53%	0.14 s 1.20% 1.53%	0.16 s 1.20% 1.54%	0.16 s 1.20% 1.54%
1.0	0.15 s 1.20% 1.21%	0.14 s 1.19% 1.19%	0.14 s 1.19% 1.19%	0.14 s 1.19% 1.20%	0.16 s 1.19% 1.22%
0.9	0.14 s 1.16% 1.08%	0.12 s 1.15% 1.07%	0.11 s 1.13% 1.01%	0.13 s 1.15% 1.06%	0.16 s 1.15% 1.10%
0.8	0.16 s 1.19% 1.18%	0.15 s 1.19% 1.13%	0.15 s 1.19% 1.06%	0.14 s 1.19% 1.11%	0.16 s 1.19% 1.17%
0.7	0.16 s 1.19% 1.28%	0.15 s 1.19% 1.27%	0.15 s 1.19% 1.24%	0.15 s 1.19% 1.24%	0.16 s 1.19% 1.25%

describes the control performance for different values of $\eta$ and $\mu$, in order to discuss and illustrate the effect of different values on the performance of the controller.

In terms of stabilization time, as the value of $\mu$ increases from ${10}^4$ to ${10}^6$, the stabilization time generally decreases, indicating that increasing $\mu$ helps to accelerate the system’s response speed. For instance, when $\eta =1.1$, the stabilization time reduces from 0.15 s to 0.14 s as $\mu$ increases from ${10}^4$ to ${10}^6$. However, when $\mu$ further increases to ${10}^7$ and ${10}^8$, the stabilization time exhibits an upward trend, suggesting that excessively high $\mu$ values may compromise system stability. The variation in stabilization time also differs for different $\eta$ values. When $\eta$ is low ($\eta =0.7$), the stabilization time is relatively long and changes more gently with increasing $\mu$.

Regarding overshoot, the overshoot first decreases and then increases as $\mu$ increases. Taking $\eta =1.0$ as an example, the overshoot declines from 1.20% to 1.19% as $\mu$ increases from ${10}^4$ to ${10}^6$, yet rises back to 1.19% when $\mu$ reaches ${10}^8$. This shows that while increasing $\mu$ within a certain range can reduce overshoot, excessively high $\mu$ values may lead to an increase in overshoot. The impact of different $\eta$ values on overshoot is also significant. When $\eta$ is low ($\eta =0.7$), the overshoot is relatively large and exhibits smaller fluctuations with increasing $\mu$.

As for THD, it first decreases and then increases with rising $\mu$ values. For example, when $\eta =0.9$, the THD drops from 1.08% to 1.01% as $\mu$ increases from ${10}^4$ to ${10}^6$, but climbs to 1.10% when $\mu$ reaches ${10}^8$. This implies that increasing $\mu$ within a certain range can reduce THD, but excessive $\mu$ values may result in higher THD. The influence of different $\eta$ values on THD is also considerable. When $\eta$ is low ($\eta =0.7$), the THD is relatively large and fluctuates more significantly with increasing $\mu$.

It is obvious that the controller stabilization time increases more, the amount of overshoot increases less, and the THD increases less when $\eta$ is taken at a lower value; the controller stabilization time increases less, the amount of overshoot increases more, and the THD increases less when $\mu$ is taken at a lower value. Where $\mu$ is a penalty factor for constraining the variation of PPD, and $\eta$ is a step size factor, its inclusion enhances the adaptability of the estimation algorithm. The sensitivity test results obtained through parameter scanning show that $\eta$ has a greater impact on convergence speed, while $\mu$ mainly affects steady-state oscillation suppression.

To further quantify the role of the anti-windup weighting factor $\tau$, set $\tau$ to ${10}^{-4}$ and ${10}^{-8}$ while the remaining parameters hold consistent with Table 2. The results are summarized in Figure 19. It is obvious from Figure 19b that excessive compensation levels increase the overshoot to 1.34% and extend the stabilization time to 0.15 s; Figure 19c shows that the compensator responds too sluggishly to the accumulated saturation error, leading to a significant overshoot of 1.67% and a prolonged stabilization time of 0.2 s. The specific performance is show in

Table 7. Control performance via several values (results of ten runs).

Parameter Value	Overshoot	Stabilization Time
${10}^{-6}$	1.13%	0.11 s
${10}^{-4}$	1.34%	0.15 s
${10}^{-8}$	1.67%	0.2 s

.

4.5. Charging Mode Switching Test

In this section, the performance is tested in multiple EV charging mode and V2G mode switching scenarios. As can be seen from Figure 20, at t = 1 s, the multi-electric vehicle switches from discharge (V2G) mode to charging mode; at t = 2 s, the multi-electric vehicle switches from charging mode to discharging mode; the grid-side voltage is set to the standard value of 220 V during this process, and the rest of the controller parameter settings are all the same as those of the tests in Section 4.1.

In Figure 20, the (a) RMS voltage values of PCC, (b) battery State of Charge (SoC), (c) active power of EVs, and (d) active power of NCL are shown, respectively. In (a), at the instant when the multiple EVs is switched from grid-connected charging to V2G discharging, the PCC voltage is kept near the standard voltage reference value of 220 V (±1 V), and is attributed to the fast response of the control strategy proposed in this paper; the voltage is kept stable within 0.25 s.; combining (b) and (c), the active power of multiple EVs changes from +400 (charging) to −400 (V2G discharging) at the switching instant, with no overshooting, and the power distribution ratio of each vehicle is inversely proportional to the SoC, which is in line with the established strategy; from (d), it can be seen that due to the switching of the charging and discharging modes, the corresponding NCL generates different degrees of active power changes to transfer the power fluctuations accompanying the generation of Cl to NCl, among which the active power of NCL4 has the largest change, which is due to the fact that it is located the closest to the grid, and it needs to bear the largest fluctuations. With the boost of the power allocation mechanism of the control strategy proposed in this paper, the unreasonable output of the whole system is effectively suppressed in 4th part alone.

4.6. Saturation Test

In this section, the controller performance under inverter saturation triggering scenarios is simulated, and a closed-loop comparison is made between the MFACC and the standard integral anti-windup (back-calculation). The simulation results are summarized in Figure 21, which clearly demonstrate the effectiveness of the proposed control strategy.

In Figure 21, (a) is the duty cycle under saturation condition, (b) is the PCC voltage under the control strategy proposed in this paper, (c) is the PCC voltage under the traditional standard integral anti-windup and the (d) THD value. From the figure, it can be observed that compared with the conventional standard integral anti-windup, the stabilization time of the proposed control strategy is reduced from 0.16 s to 0.11 s; the overshoot is reduced from 1.37% to 1.13%, and the THD value is reduced from 1.62% to 1.01%. The MFACC compensator outperforms the conventional integral anti-windup under saturated condition, which is reflected in the lower overshoot, faster recovery, and smaller harmonics, which further validates the practicality of the proposed strategy. The practicality and robustness of the proposed strategy are verified.

5. Conclusions

This paper proposed an innovative hierarchical and distributed cooperative control method for multi-EV-ES. The upper-level control employs a consensus algorithm based on directed graphs, whereas the lower-level controller utilizes MFACC. Based on simulation experiments, the following conclusions have been derived:

• In the upper-level controller, the consensus algorithm based on directed graphs effectively achieves cooperative adjustments of voltage, frequency, and power by exchanging local information with adjacent EV-ES.
• In the lower-level controller, compared to PR control, both MFAC and MFACC can achieve satisfactory control performance. Furthermore, as an improvement over MFAC, MFACC exhibits superior response speed, smaller overshoot, and lower harmonic content in simulation analysis. This enhancement contributes to the improvement of micro-grid power quality.
• The topology of multi-EV-ES is flexible, and the proposed hierarchical control system can adapt flexibly to accommodate more EV-ES embedded in the micro-grid for cooperative control.

In the future research, we will focus on the following aspects: Firstly, we will carry out tests on the hardware aspects, such as experiments using the dSPACE platform, to verify the feasibility and effectiveness of the proposed method in real micro-grid scenarios [26,29,30,44]; second, for the increasing number of large-scale micro-grid deployments, we will conduct in-depth research on quantitative assessment methods for computational complexity and communication bandwidth requirements, clarify the constraints of their practical applications, and explore optimization strategies to ensure that the control performance of micro-grids remains reliable under imperfect communication conditions; third, we plan to incorporate intelligent optimization methods like PSO [45] and GWO [46] to automatically adjust the parameters of key control coefficients. Below is a roadmap (Figure 22) of the future experimental work.

These research directions will help to promote the further development and improvement of the control strategy proposed in this paper, making it more adaptable to the complexities and challenges of practical applications.