An Adaptive Control Strategy for a Better Performance of the Paralleled PV-BES-VSG Power System

Abstract

The growing integration of renewable energy sources has led to the development of virtual synchronous generator (VSG) control as a way to enhance system stability and offer primary frequency regulation. These functions of VSGs usually rely on the photovoltaic (PV) system or battery energy storage (BES), which is equipped at the DC side of the system. However, due to differences in the initial state of charges (SoCs) and uneven power distribution, the SoCs of battery energy storage systems (BESs) may become unbalanced, posing risks to the healthy operation of BESs and the overall system reliability. To realize SoC balancing, an adaptive control scheme for a paralleled PV-BES-VSG power system is presented. The adaptive SoC balancing term is applied to the active power references based on a simple segmented quadratic function. The proposed control strategy can realize optimal operation of paralleled VSGs and reduce SoC imbalance at the same time. The effectiveness of the proposed control scheme is evaluated via a case study system consisting of two paralleled PV-BES-VSG units using Matlab/Simulink R2021a.

1. Introduction

With the rapid development of industrialization, a steady increase in the energy demand has been observed [1]. For many decades, conventional reliance on fossil fuels such as oil and coal has been an integral part of providing electrical power for industrial operations. However, as a result of the over-exploitation of non-renewable energy sources, continued expansion of the global economy has created significant environmental challenges. Within the framework of global sustainability and in support of the goal of the Paris Agreement, there has been an increased emphasis on promoting the development of renewable energy sources (RES) as a key technological avenue to facilitate global energy transition [2]. As a result, there is a widespread international interest in the exploration and incorporation of RES, such as solar, wind, hydro and geothermal, into the energy infrastructure.

Photovoltaic (PV) and wind power generation, as typical types of RES, require connection to the power grid via grid-interactive power converters. As the proportion of grid-interactive power converters rises, synchronous generator-based power production diminishes [3]. Synchronous generators, characterized by significant rotational inertia, regulate the grid frequency through the rotor speed. They autonomously absorb or release energy to mitigate fluctuations caused by random disturbances, such as changes in loads. However, grid-interactive power converters lack the power-angle dynamics inherent to synchronous generators and cannot directly contribute to inertia and damping support to the grid, potentially leading to stability challenges. Additionally, the inherent stochastic, fluctuating, cyclical, and intermittent characteristics of large-scale RESs may further affect system stability [4,5].

To cope with the transition from a traditional power system relying on synchronous generators to a modern power system dominated by grid-interactive power converters, grid-forming (GFM) converters have gained more attention recently. Virtual synchronous generator (VSG) control is recognized as a representative GFM control strategy, allowing grid-interactive power converters to emulate the behavior of synchronous generators [6]. However, VSG control also inherits the oscillation characteristics inherent in synchronous generators. As a consequence, when multiple VSGs are in parallel connection, the severity of frequency and active power oscillations escalates. A lot of studies have been performed to mitigate frequency and active power oscillations, such as additional correction control loops, virtual impedance, and self-adaptive parameters. In [7], an extra damping term is introduced, composed of an acceleration control term derived from frequency feedback and a disturbance compensation acquired from an active power feedback. In [8], a reference feedforward control is proposed to mitigate oscillations without affecting the inherent inertial response. However, these methods can give rise to instability issues due to the incorporation of derivative terms, as demonstrated in [9]. Thus, a virtual impedance method is introduced in [10]. However, an accurate real impedance is not obtainable in real application. Based on that, various methods based on self-adaptive parameters are introduced in [11,12,13,14,15,16,17,18]. Some studies proposed adaptive virtual inertia methods considering the frequency deviation and the rate of change of frequency (RoCoF) [11,12]. Some studies proposed the adaptive control methods considering both self-adaptive inertia and damping coefficient [13,14,15,16,17]. To further dampen the frequency and active power oscillations, an adaptive control strategy with a mutual damping term is presented in [18].

However, in these studies, the DC-link is simplified as an ideal DC voltage source with infinite energy to provide frequency and voltage regulation. However, in practical applications, the DC side is usually equipped with capacity-limited energy storages (ES) or some other RES, such as wind turbines and PV panels. Thus, the physical constraints of the DC side need to be taken into consideration. Short-term ES systems like ultracapacitors and flywheels can offer rapid response but struggle with primary frequency regulation due to high per-kWh costs [19,20]. In contrast, battery ES (BES) can offer primary frequency regulation because of lower per-kWh costs [21,22]. Replacing the maximum power point tracking (MPPT) control with the power reserve control (PRC) can also be an effective way to provide a long-term energy supply, but it can result in a large generation loss [23,24]. Hence, a hybrid energy storage system (HESS) is preferred to distribute energy among various ES types based on their respective advantages. In [25,26,27], a battery/ultracapacitor HESS was proposed, where the battery was adopted to compensate for the slow power fluctuations to ensure the droop control and restoring the frequency to its normal value, while the ultracapacitor is used to compensate for the fast power fluctuations to mimic the inertia characteristics and mitigate the rate of frequency change. To further accelerate the process of frequency recovery and reduce the total cost of the ES, a battery-free HESS utilizing coordinated control of PV and ultracapacitor is introduced in [28]. Taking the state of charge (SoC) of BES into account, an adaptive control strategy of the VSG adopting fuzzy logic is presented in [29,30]. The adaptive adjustment of the virtual inertia and damping coefficient based on fuzzy logic is achieved in [29], which improves the system stability. In [30], a fuzzy controller is designed to make the energy storage output more flexible and reduce the frequency fluctuations caused by load disturbance. Nevertheless, these studies only focus on the system with a single VSG and do not consider the cooperated control between the multiple paralleled VSGs. In [31,32,33], a multiple-paralleled-VSG system was discussed, where each DC-side of the VSG is equipped with capacity-limited BES or PV. To fully utilize PV generation and prolong the battery life, a coordination control strategy was proposed in [31]. It divides the operation mode into five states to redistribute the power considering the load, maximum PV output power, along with the SoC of BES. Considering the converter and storage capacity limits, an adaptive VSG control was introduced in [33]. It utilizes a bang-bang control strategy to regulate inertia and damping coefficient to ensure an optimal response from VSGs. In addition, a constraint of droop coefficient is also added to keep the BES at a healthy SoC level between 20% and 80% [34,35].

Although these studies were based on multiple paralleled VSGs, and took physical constraints into account, a critical aspect of SoC balancing is ignored. The SoC of each BES unit will naturally become unbalanced without appropriate control strategies. This is due to variations in the initial SoC values and uneven power distribution in both transient and steady-state power sharing, which can result in imbalances [7]. Considering the reliability of the system, it is desirable to maintain equal SoCs among multiple BES. Otherwise, some BES with lower SoCs may not have enough energy to deal with the sudden disturbance that occurs in the system, causing the out-of-operation of BES. Conversely, some BES with higher SoCs may be fully charged, which is unhealthy for BES. Thus, SoC balancing serves to mitigate performance discrepancies, minimize premature degradation, reduce safety hazards associated with overcharging or over-discharging, and extend battery life [36]. The SoC balancing topologies can be categorized as centralized, decentralized, and distributed multiagent [37]. When communication infrastructure is available, SOC adjustment is generally performed at the secondary control stage, utilizing a combination of local and global information. A centralized secondary control strategy that modifies the droop coefficients of BES based on their SoC levels is proposed in [38]. In [39], an adaptive droop gain, along with a SoC balancing mechanism, is added to optimize power sharing among multiple BES based on a communication network. To obtain SoC balancing without communication, in [40], an adaptive cooperative terminal sliding mode strategy was proposed for distributed energy storage system, featuring a novel power allocation algorithm that ensures SoC balancing and accounts for battery capacity degradation using real-time estimation. In [41], a cooperative adaptive command filtered backstepping control strategy based on a VSG has been proposed to enable electric vehicles (EVs) to support UPS microgrids with improved dynamic performance and grid stability. A consistency algorithm-based SoC balancing scheme has been developed in [42] to manage retired lithium batteries with unequal capacities in VSG-controlled microgrids, enhancing battery utilization and system reliability. In addition, a decentralized adaptive SoC balancing control was proposed in [43]. It adopts simple linear relationships linking the SoC with both the damping coefficient and the frequency reference to ensure SoC balancing. However, these methods try to regulate the damping coefficient and inertia to realize SoC balancing, which may affect the optimal response of VSGs. Therefore, for a paralleled PV-BES-VSG power system, a method considering both optimal responses of VSGs and SoC balancing for a system with paralleled VSGs and BESs needs to be proposed.

To address the aforementioned issues, this paper presented an adaptive control strategy for a paralleled PV-BES-VSG power system. The paralleled PV-BES-VSG power system denotes an islanded system with paralleled VSGs, coupled with PV units and BES units at each DC side. The key contributions of this paper are outlined below: (1) The typical topology of a paralleled PV-BES-VSG power system and its controllers are discussed in detail. (2) An adaptive control scheme for a paralleled PV-BES-VSG power system is presented and a guideline for the parameters design is given. (3) The presented control scheme can effectively realize SoC balancing of the BESs and ensure stable operation and optimal response of paralleled VSGs simultaneously.

The rest of this paper is arranged as follows: The typical topology of a paralleled PV-BES-VSG power system and the controllers of each part are discussed in Section 2. In Section 3, the adaptive control scheme for a paralleled PV-BES-VSG power system is presented. In Section 4, the presented strategy is evaluated in Matlab/Simulink. Finally, Section 5 concludes this paper.

2. Typical Structure of a Paralleled PV-BES-VSG Power System

The structure of a paralleled PV-BES-VSG power system is depicted in Figure 1. The whole system includes multiple PV-BES-VSG units, each featuring both AC and DC sides. The DC-AC converter serves as the interface between these sides. A boost converter connects the PV module to the DC-link, while a half-bridge bidirectional DC-DC converter connects the BES to the DC-link. The VSG control is applied to the DC-AC converters, painted in orange. The PV-side DC-DC converter is equipped with the MPPT control, painted in purple. The BES-side DC-DC converter features a direct voltage control loop along with a current control loop, painted in green. The detailed control structures of these three power converters will be discussed in the following sections.

2.1. PV-Side DC-DC Converter

The structure of a boost converter connected to the output of PV modules is illustrated in Figure 2, where L_pv and C_pv represent the input filter inductor and capacitor; C_dc is the capacitor of the DC-link; P_pv, u_pv, and i_pv represent the output power, voltage, and current of the PV module, respectively.

It is commonly known that the output power of the PV modules is influenced by both irradiance and temperature. To obtain maximum power production from PV modules, an MPPT control strategy is always adopted in PV systems. The Perturb and Observe (P&O) control scheme is among the most employed MPPT control strategies, owing to its simplicity. It is based on the power–voltage (P-V) characteristics of PV modules, which is illustrated in Figure 3 [44]. The P-V curve is like a hill and the peak of the curve is defined as the maximum power point (MPP). The P&O MPPT control scheme involves iteratively adjusting the operating point of the PV module by perturbing the operating voltage and then observing the corresponding variations in output power. By continuously monitoring the output power and then comparing it to previous values, the controller detects the direction of movement towards or away from the MPP. Using this information, the controller orchestrates adjustments to the operating point to facilitate convergence towards the MPP. This iterative process continues, ensuring that the PV system operates at or near the MPP, regardless of environmental dynamics such as fluctuations in irradiance or temperature. Therefore, this paper utilized the P&O MPPT control scheme to optimize the output power from PV modules. The output of the P&O MPPT control scheme determines the reference voltage for voltage control at the boost converter input. The boost converter regulates this voltage by adjusting the duty cycle, which is modulated using pulse width modulation (PWM) to generate the gating signal S_pv for the switching device. Meanwhile, the inverter controller regulates the DC bus voltage.

2.2. BES-Side DC-DC Converter

The structure of a half-bridge bidirectional DC-DC converter fed by the BES is shown in Figure 4, where L_bes and C_bes represent the input filter inductor and capacitor; i_besref and i_bes represent the reference and output currents of the BES; u_dcref and u_dc stand for the reference voltage and voltage of the DC link; S_besu and S_besd denote the gating signals for the upper and down switching devices, respectively; P_bes is the output power of the BES.

It is noteworthy that the equivalent electrical circuit was adopted to represent the dynamic behavior of the BES [45]. The equivalent electrical circuit model is depicted in Figure 5, where u_ocv stands for the open circuit voltage of the BES; the polarization resistance and capacitance of the BES are represented by R₁ and C₁; the internal resistance of the BES is indicated by R₀.

The SoC is a crucial parameter for batteries and it can be estimated by various methods, including coulomb counting strategy, open circuit voltage measurement, impedance spectroscopy-based strategy, Kalman filter strategy, and neural network strategy. The coulomb counting strategy involves integrating the charge/discharge current over time [46]. The open circuit voltage measurement correlates the battery voltage with SoC based on established discharge curves [47]. The impedance spectroscopy-based strategy estimates the SoC by analyzing the battery’s impedance spectrum over a range of frequencies, providing insights into its electrochemical processes and internal characteristics [48]. The Kalman filter strategy offers high accuracy and robustness through a state-space model and recursive estimation [49]. The neural network strategy models the battery’s nonlinear characteristics for high accuracy, depending on training data and network design [50]. These methods are usually combined to achieve a more comprehensive and accurate estimation of SoC [51]. To simplify the computation, the coulomb counting strategy is employed, which can be given as follows:

$$SoC=So{C}_{int}-{\displaystyle \frac{1}{C_{bes}}}{\displaystyle \int i_{bes}dt}=So{C}_{int}-{\displaystyle \frac{1}{C_{bes}{u}_{bes}}}{\displaystyle \int P_{bes}dt}$$

(1)

where SoC_int represents the initial SoC of the BES; the capacity and output voltage of the BES are represented by C_bes and u_bes.

To enable bidirectional power flow between the DC link and the BES, a half-bridge bidirectional DC-DC converter was adopted. Structurally, it comprises two switching devices and corresponding diodes, configured in a manner to regulate the current flow direction. The BES-side DC-DC converter is controlled by a cascaded control system: an outer direct voltage loop along with an inner current loop. The direct voltage loop was adopted to keep the voltage of the DC-link at a rated value and gives the current references to the current loop, and then the current loop adjusts the BES current and generates the duty cycle. The PWM was adopted to modulate the duty cycle to generate the gating signals S_besu and S_besd to drive the upper and lower switching devices, respectively.

2.3. DC-AC Converter

Both PV and BES systems need to be connected to the AC side via a DC-AC converter. As RESs penetration increases, the power system may lack enough inertia to cope with the disturbances and the system stability will be challenged. To address this issue, the VSG control was introduced to enhance system stability.

The traditional topology of the VSG control is illustrated in Figure 6 [52], where u_abc and u_oabc stand for the terminal and output voltages of the converter, respectively; i_abc, i_oabc and i_Cabc denote the inductor current, converter output current, and capacitor current; the inductance and capacitance of the LC filter are represented by L_f and C_f; the line impedance between the converter and load is indicated by Z_g.

The control scheme primarily consists of three components: a power synchronization part, an excitation loop, and a cascaded voltage and current loop. The power synchronization loop and excitation loop enable the DC-AC converter to mimic the behaviors of synchronous generators. The power synchronization loop yields

$$\left\{\begin{array}{l}J{\displaystyle \frac{d\omega }{dt}}={\displaystyle \frac{P_{ref}-P_e}{{\omega }_g}}-D\left(\omega -{\omega }_g\right)\\ {\displaystyle \frac{d\theta }{dt}}=\omega \end{array}\right.$$

(2)

where the moment of inertia and damping coefficient are represented by J and D; P_ref and P_e are the reference and output active power of the converter; ω_g and ω represent the rated and actual angular frequency.

The excitation loop describes the relationship between the reactive power and voltage, which can be expressed as follows:

$$u_{odref}=E_0+k_q{\displaystyle \int \left(k_u\left(U_N-u_{od}\right)+Q_{ref}-Q_e\right)}$$

(3)

where k_u denotes the voltage droop coefficient while k_q denotes the integral coefficient; Q_ref and Q_e represent the reference and output reactive power of the converter; U_N is the rated voltage; E₀ stands for the no-load electromotive force of the converter.

The cascaded voltage and current loops are used to modulate the converter’s output voltages and currents according to the voltage reference generated by the excitation loop and power synchronization loop.

3. Adaptive Control Strategy Considering SoC Balancing

Within the PV-BES-VSG system, BES needs to provide not only short-term inertia emulation but also long-term frequency regulation. For a power system with multiple BES, the coordination between them needs to be considered. Due to the different line impedances, initial SoCs, and the capacity of each BES, the SoCs of multiple BES are not always equal. Consequently, an unfavorable situation may occur where some BES with lower SoC might fail to supply adequate energy for primary frequency regulation and will be out of operation eventually, while some other BES with higher SoC are still available. In addition, it is well known that the degradation of BESs has a strong correlation with SoC. Therefore, it is pivotal to realize SoC balancing in a system with multiple BESs.

Assuming there is no power loss in the system, in accordance with the law of conservation of energy, the power within a paralleled PV-BES-VSG power system depicted in Figure 1 satisfies the following equation:

$${\displaystyle \sum _{i=1}^n{P}_{pvi}}+{\displaystyle \sum _{i=1}^n{P}_{besi}=P_{load}=}{\displaystyle \sum _{i=1}^n{P}_{ei}}$$

(4)

where P_pvi, P_besi, and P_ei represent the output power of the PV-side DC-DC converter, the BES-side DC-DC converter, as well as the DC-AC converter of the ith PV-BES-VSG unit, respectively; P_load is the load power. Because the PV-side DC-DC converter is equipped with the MPPT control, the value of P_pvi is determined by the irradiance and temperature of each PV module.

According to the active power–frequency relationship presented in (2), it is obvious that the value of P_ei is governed by the damping coefficient, moment of inertia, and the active power reference of each VSG. Since the VSG control inherits the oscillation characteristics of synchronous generators and the oscillations become more serious in a power system with multiple paralleled VSGs, an adaptive control scheme featuring a mutual damping term is presented in [18]. In this paper, to maintain the ability to mitigate the frequency and active power oscillation as well as simultaneously realize optimal, the method presented in [18] is also kept here, which can be expressed as follows:

$${\displaystyle \sum _{i=1}^n{J}_i\frac{d{\omega }_i}{dt}=\frac{P_{re{f}_i}}{{\omega }_i}-\frac{P_{ei}}{{\omega }_i}-D_i\Delta {\omega }_i-{\displaystyle \sum _{j\in \Omega }\frac{D_m\left({\omega }_i-{\omega }_j\right)}{n}}}$$

(5)

where D_m represents the mutual damping coefficient, the set of all adjacent VSGs is represented by Ω, and J_i and D_i yield:

$$\left\{\begin{array}{l}{J}_i=J_{i0}+k_i\left({\omega }_i-{\omega }_g\right){\displaystyle \frac{d{\omega }_i}{dt}}\\ D_i=2\zeta /\sqrt{{\displaystyle \frac{X_g{\omega }_g}{3E_{i}U{J}_i}}}\end{array}\right.$$

(6)

where the ζ is set as 0.707; k_i is the adaptive coefficient.

To further consider the capacity of the DC-side and realize SoC balancing, an additional adaptive term is applied to active power reference. When all the parameters are calculated in p.u. values, the ratio of output active power between two VSGs will be nearly proportional to the ratio of their active power references. Thus, to realize SoC balancing between multiple BES, the active power reference should be set larger if the SoC of the BES is higher, while the active power reference should be set smaller if the SoC of the BES is lower. If the SoC of a BES is close to 50%, its active power reference does not need to change a lot. Based on the requirements above, the adaptive SoC balancing term can be designed as follows:

$$P_{refi}=\left\{\begin{array}{ll}-a{(So{C}_i-0.5)}^2+b& So{C}_i\le 0.5\\ a{(So{C}_i-0.5)}^2+b& So{C}_i>0.5\end{array}\right.$$

(7)

where a > 0, b > 0; a and b are the parameters to be designed.

To ensure the robustness of this method, when two BES with SoC of 20% and 80%, the ratio of active power references is set as 2%, which is based on prior testing and practical considerations. Based on this, the relationship between a and b can be obtained as

$$a=10.67b$$

(8)

The specific value of a can be arbitrary, as long as the proportional relationship between a and b is maintained. For ease of calculation, set the value of a as 10, and then the value of b can be calculated as 0.937. In summary, the adaptive term can be given as follows:

$$P_{refi}=\left\{\begin{array}{ll}-10{(So{C}_i-0.5)}^2+0.937& So{C}_i\le 0.5\\ 10{(So{C}_i-0.5)}^2+0.937& So{C}_i>0.5\end{array}\right.$$

(9)

To better explain the relationship between the active power references and SoCs under the proposed adaptive SoC balancing term, Figure 7 shows the ratio of output active power references between two BES with varying SoCs. It is obvious that, when the SoC difference between the two BES increases, the ratio between the active power references increases accordingly. Moreover, the further the SoC is away from 50%, the faster the ratio between the active power references changes, which matches the design expectations.

4. Simulation Results

To evaluate the performance of the presented method for a paralleled PV-BES-VSG power system, a case study system with two paralleled units is conducted in Matlab/Simulink. Subscript 1 represents unit 1 and subscript 2 represents unit 2. Some main parameters of the case study system are listed in

Table 1. Main parameters of the case study system.

Description	Symbol	Value
Maximum power of PV modules	PPV1, PPV2	7.6 kW, 7.6 kW
Inductor and capacitor of PV-side DC-DC converter	Lpv, Cpv	2 mH, 10 μF
Rated capacity of batteries	Cbes1, Cbes2	1 Ah, 1 Ah
Inductor and capacitor of BES-side DC-DC converter	Lbes, Cbes	2 mH, 10 μF
Initial SoCs of batteries	SoC1, SoC2	50%, 60%
DC-link capacitor	Cdc	10 mF
Rated power of VSGs	SVSG1, SVSG2	15.8 kVA, 15.8 kVA
LC filter of VSGs	Lf, Cf	3 mH, 10 μF
Load power	Pload	15 kW

. Some detailed parameters can be found in our previous research [18].

Two conditions were chosen to evaluate the presented adaptive control scheme for a paralleled PV-BES-VSG power system. One is the irradiance variation, and the other one is the load step. In addition, the performance of the presented method was compared to that of the tradition VSG control method as well as the method without the SoC balancing mechanism presented in [18]. For clarity, the definition of the three methods is shown in

Table 2. Description of three method.

Case	Description
Method 1	Traditional VSG control method with fixed moment of inertia and damping coefficient [6], as shown in (2)
Method 2	Adaptive moment of inertia and damping coefficient, with a mutual damping term [18], as shown in (5)
Method 3	The adaptive control strategy with SoC balancing mechanism proposed in this study, as shown in (9)

. The traditional VSG control method with a fixed moment of inertia and damping coefficient is defined as Method 1. The method with an adaptive moment of inertia, damping coefficient, and mutual damping term—but without an SoC balancing mechanism—as presented in (5), is defined as Method 2. The adaptive control strategy with a SoC balancing mechanism presented in this study is defined as Method 3.

4.1. Irradiance Variation

Because the MPPT control is implemented in the PV-side DC-DC converter, the change in irradiance will affect the output power of the PV modules. When there is a decrease in irradiance within unit 1 during 1 s to 3 s, the irradiance and output power of the PV modules are depicted in Figure 8. The output power of VSG₁ as well as the output power and SoC of BES₁ under the three control methods are depicted in Figure 9. The output power of VSG₂ as well as the output power and SoC of BES₂ under the three control methods are depicted in Figure 10. The reference active power of the VSGs and the difference between the SoCs under the three control methods are depicted in Figure 11.

As the irradiance of PV module 1 decreases, the output power of the PV part decreases, and the BES side generates more power to compensate for the power deficit and supply enough power to the load. For Method 1 and 2, due to the constant ratio between active power references of two VSGs, the power deficit resulting from the irradiance decrease in unit 1 is totally compensated by the BES in unit 1. However, the initial SoC of BES₁ is less than that of BES₂. If only BES₁ discharges, the difference between the SoCs of the two BESsΔSoC increase further, leading to an unfavorable situation where BES₁ is out of operation while BES₂ is still at a higher SoC.

In contrast to Method 1 and 2, Method 3’s control incorporates a SoC balancing term. This results in a decrease in the ratio between the active power references of the two VSGs, P_ref1/P_ref2. Consequently, unit 1 supplies less power while unit 2 supplies more power. As a result, the power deficit resulting from the irradiance decrease in unit 1 can be shared by both BES₁ and BES₂, which can decrease SoC unbalancing compared to Method 1. After 3 s, although the irradiance increases back to 1 kW/m², the SoCs of the two BES are still not the same. Thus, the ratio between active power references of two VSGs is not equal to 1 and the SoCs of the two BES change at different rates. As shown in Figure 11, in comparison to Method 1 and 2, the SoC difference between the two BES under Method 3 remains consistently smaller. At the end of the simulation, with the adoption of the SoC balancing term, the SoC difference decreases from 10.7% to 10.2%. Because the simulation time is only 5 s, the SoCs of the two VSGs have not achieved balancing. However, if the simulation time is extended, both the output power and SoCs of the two BES will reach the same value. The ratio between active power references of two VSGs will be equal to 1, and both VSGs’ output active power will be the same accordingly. Therefore, the presented control scheme can effectively decrease the difference of SoCs and realize the SoC balancing, thereby enhancing system reliability.

4.2. Load Step

The condition of a load step is also conducted under the two control methods. When a load step of 5 kW is added to the system at 1 s, the output power of VSG as well as the output power and SoC of BES, within two units under the three control methods, are depicted in Figure 12 and Figure 13, respectively. The reference active power of the VSGs, SoC differences, as well as the RoCoFs under the three control methods are depicted in Figure 14.

As a load step of 5 kW is added to the system, the output power from the BES side increases to provide power to the load. For Method 1 and 2, because the ratio between the active power references of the two VSGs is 1, the two VSGs share the power equally. The output active power of the BES in both units is around 2.5 kW, and the SoCs of the BES₁ and BES₂ change at the same rate. Nevertheless, the initial SoC of BES₁ is less than that of BES₂. As a result, the BES₁ will be out of operation before BES₂, which may deteriorate the reliability of the whole system.

However, when Method 3 is applied, the ratio between the active power references of the two VSGs P_ref1/P_ref2 is less than 1, causing unit 1 to provide less power compared to unit 2. Consequently, BES₁ generates less power than that of BES₂, the SoC of BES₁ decreases at a lower rate, while the SoC of BES₂ decreases at a higher rate. During the simulation time, the SoC of BES₁ is always lower than that of BES_2, but because the difference between SoC₁ and SoC₂ decreases, the ratio between active power references of two VSGs P_ref1/P_ref2 is always less than 1 but the value of P_ref1/P_ref2 increases. As shown in Figure 14, the SoC difference between two BES under the condition of Method 3 is always lower than that under the conditions of Methods 1 and 2. By employing the SoC balancing term, the SoC difference at the end of the simulation is reduced from 10% to 9.4%. Eventually, when the simulation time is extended, the two BESs will generate the same power and have the same SoC. The ratio between active power references of two VSGs P_ref1/P_ref2 will be 1 and the output active power P_e1 and P_e2 will be equal. In addition, as shown in Figure 14, under the control of Method 3, the maximum RoCoF is the smallest among the three methods. Therefore, the presented control method can slow down the change of frequency, ensuring system stability. Through the analysis above, the presented adaptive control strategy for a paralleled PV-BES-VSG power system can effectively obtain the SoC balancing and optimal operation of paralleled VSGs under the condition of a load step.

5. Conclusions

In this paper, an adaptive control strategy for a paralleled PV-BES-VSG power system was introduced to realize SoC balancing of battery energy storages and ensure optimal response of paralleled virtual synchronous generators at the same time. The typical topology and controllers of a power system with paralleled PV-BES-VSG units were discussed comprehensively. An adaptive term was applied to active power references based on a simple segmented quadratic function. Furthermore, the design guidelines for some key parameters were also provided, ensuring the sensitivity and robustness of the proposed method. Simulation results verified the effectiveness of the presented method.

Although the proposed method successfully addresses both SoC balancing and the optimal operation of paralleled virtual synchronous generators, several areas remain open for improvement and could be explored in future research. Specifically, this paper focused solely on the SoC balancing of battery energy storage systems. In practical applications, more complex hybrid energy storage systems—combining batteries with supercapacitors or other storage technologies—should be taken into account. Moreover, the integration of advanced artificial intelligence algorithms could further enhance the accuracy of SoC estimation and improve the overall control performance.