Enhanced Virtual Synchronous Generator with Angular Frequency Deviation Feedforward and Energy Recovery Control for Energy Storage System

Abstract

Frequency control is one of the most important tasks in electric power systems. At the same time, in modern power systems with inertia-free converter-interfaced generation, this task has become more complex. Such an issue is especially relevant for microgrids, which are characterized by a significant increase in the rate of change of frequency and its nadir or zenith. An effective way is through the use of energy storage systems (ESSs) with a grid-forming control in microgrids. For this purpose, this paper proposes a novel structure of the control algorithm based on a current-control virtual synchronous generator (CC-VSG), in which the damping is performed using a feedforward controller. In addition, a simple proportional–integral controller is added to the CC-VSG structure to control the state of charge of the ESS. The performed frequency analysis proves the independence of the different control loop operations within the developed CC-VSG. At the same time, a methodology based on the bandwidth separation of different control loops is proposed for the CC-VSG tuning, which allows for the achievement of the desired quality of frequency regulation in the microgrid, taking into account both the energy recovery and the permissible frequency variation. Finally, the time-domain simulation using PSCAD/EMTDC is performed to confirm the obtained results.

1. Introduction

Currently, the penetration of renewable energy sources (RES) into modern power systems continues. The advantages of RES applications are well-known [1,2]. A promising direction is the use of RES together with conventional fossil-fuel-based generation, usually diesel generators (DG), as part of microgrids that can operate in isolation and in parallel with the centralized power systems [3,4]. However, it is well-known that RES penetration results in a reduction of the overall system inertia, leading to an increase in the rate of change of frequency (RoCoF) and its nadir or zenith after small or large disturbances [5,6]. Moreover, this problem is especially severe for microgrids [7,8,9].

Considering the mentioned challenges, the key direction of development for modern power systems is the transition from the conventional grid-following control to the grid-forming control strategy [10,11,12]. One of the promising solutions for the implementation of this strategy is the use of inverter control algorithms based on a virtual synchronous generator (VSG) [13]. Currently, there are a number of proposed VSG algorithms and their various modifications [14]. The comparative studies that have been performed have shown that the most effective frequency control in low-inertia power grids is achieved via cascade structures of VSG based on the conventional swing equation with a damping coefficient and a virtual governor [15,16]. The most suitable primary source for the VSG control is energy storage systems (ESSs) due to the controllability of power generation in a wide range and the ability to operate in the generation or consumption mode [17,18]. However, in this case, it becomes necessary to take into account the technological features of an ESS within the VSG algorithm [19].

Firstly, ESSs have limited power and energy capacity that can be used to provide inertial response and frequency control. In this regard, an estimation of the energy consumed at different stages of the dynamic response is proposed in [20] with both droop control and VSG-based control algorithms. Due to the limited amount of stored energy, the ESS cannot always provide the desired value of the inertial response formed by the VSG control within the set parameters. Unless this feature is controlled, there may be significant voltage sags in the DC circuit, leading to instability. The necessity of such control is shown in [21,22]. Likewise, in [23], it is proposed that an output power limiter and an additional proportional–integral (PI) controller be added to the grid inverter control system. Similar principles are applied to provide an inertial response through VSG using a DC capacitor, but the contribution to frequency control with this approach remains limited [24,25]. However, in these studies, the conventional VSG algorithm with a damping coefficient in the swing equation is used [26,27].

Secondly, in order to ensure acceptable operating conditions for ESSs and minimize the process of their degradation, it is critical to control their current state of charge (SoC) level. Different approaches for controlling the SoC of ESSs are proposed for the VSG algorithms. In the first case, the commands to charge or discharge the storage device are given under predetermined conditions, e.g., through the control of the reference current value formed by the frequency control loop [28]. A similar approach is used for the ESS in the DC circuit of wind turbine generators [29]. There is also a possibility of controlling the SoC level through a feedback loop and disabling the VSG operation in case of reaching the set limits [30]. Such control can also be carried out with the use of upper-level control systems [31]. The continuous SoC control involves the use of control loops with continuous static characteristics. For example, a combined method of the ESS control for both the parallel operation with the external grid and autonomous operation is proposed in [32]. Another approach is to use the weight coefficients for SoC and frequency deviation signals in forming the VSG active power reference [33]. In this case, a trade-off between the quality of frequency and SoC control needs to be found. An effective and easy-to-implement method is the continuous SoC control of the ESS using a PI controller. The formed signal can be used to change the droop coefficient [34], to form the power reference [29], and to control the DC/DC converter [35,36,37]. A group of studies is devoted to the application of adaptive methods for the SoC control [38,39], including fuzzy-based controllers [40,41].

The analysis of the existing studies leads to the conclusion that it is necessary to consider the features of ESSs within the VSG algorithm operations, primarily the SoC-level control. However, in this case, there is a negative impact on the frequency profile in the power grid, and a challenging trade-off between the efficiency of the frequency control and the energy recovery needs to be found [42]. The possible solutions for the above problem are not clearly presented in the research area. Likewise, the question of the most appropriate VSG structure for the ESS grid inverter that fulfills the indicated features remains outstanding. In this regard, the contribution of this paper is as follows:

• A linearized model in the form of a transfer function and a state-space model have been developed. The necessity of a significant enhancement of damping properties of the conventional VSG structure to suppress the emerging resonance is proved, which also leads to a significant increase in the ESS nominal energy capacity;
• The structure of the modified current-controlled VSG (CC-VSG) with a feedforward controller and an energy recovery control loop for the battery energy storage system (BESS) is proposed;
• The methodology of tuning the modified CC-VSG on the basis of bandwidth separation of different control loops is formed.

This paper is organized as follows. Section 2 describes the topology of the considered microgrid and its simplified scheme used for further analysis. A detailed description of the developed modified CC-VSG structure is also presented. In Section 3, the frequency analysis of the linearized model reflecting the processes of active power and frequency changes in the test scheme is performed. Based on this analysis, the provisions of the CC-VSG tuning methodology for the ESS and approaches for determining the energy consumption of the ESS at each of the frequency control stages are developed. By means of the linearized state-space model, the conclusions are confirmed. Section 4 presents the simulation results. Finally, Section 5 provides the conclusion and outlines the future work in this paper.

2. Description of the Test System and the Proposed CC-VSG Control Algorithm for BESS

2.1. Microgrid Topology

For theoretical analysis of the key features of microgrids consisting of a DG, RES, BESS, and local load with no connection to the external power grid, it is possible to use a simplified representation, highlighting only a DG and a BESS grid inverter with a VSG-based control algorithm, which operates in parallel (Figure 1). Figure 1 uses the following notations: L_gi are the inductances of the i-th (i = 1,2) section of the electric network, including a filter inductance for the grid inverter; U_i ∠ δ_i are the voltage and the phase at the point of connection of the i-th element to the electric network.

2.2. Structure of the Proposed CC-VSG for BESS

The inertial response, the primary and secondary frequency control in the microgrid using BESS is possible with the grid-forming inverters. For that purpose, in this paper, the developed control strategy based on the CC-VSG algorithm is used [43]. This algorithm is based on the VSG model first proposed in [44]. Further, it has been proven that CC-VSG can be used for the grid-forming control [45], allows for the synchronization of the converter without a phase-locked loop (PLL) controller, and contributes to the voltage and frequency regulation of the microgrid, providing the inertial response, etc. In addition, it was shown that the CC-VSG algorithm, in comparison with the conventional voltage-controlled VSG structures, ensures the stability of power converter operation in both ultra-weak and ultra-stiff grid conditions [46]. However, the damping in the CC-VSG model is carried out due to the simplified virtual damper winding. This approach leads to the necessity to consider the third-order system, which affects the complexity of adequate tuning of control loops, e.g., using Vyshnegradskii’s stability diagram [47]. To reduce the order of the CC-VSG model, this paper considers the application of the classical damping coefficient in the active power control loop instead of the virtual damper winding equations or, what is most promising, the feedforward controller, which improves the control performance due to the direct disturbance compensation [48,49]. The proposed damping controller is based on a simple proportional loop with gain K_FF, whose input signal is the VSG frequency deviation, and the output signal is added to the VSG phase angle. The structure of the modified CC-VSG with different damping approaches is shown in Figure 2.

The model of the virtual synchronous machine within the CC-VSG algorithm is described by Equations (1)–(4):

$$\left\{\begin{array}{c}{\displaystyle \frac{d{\psi }_d}{dt}}={\omega }_b\left(u_{1d}+R_v{i}_{d,VSG}+{\omega }_{VSG}{\psi }_q\right)\\ {\displaystyle \frac{d{\psi }_q}{dt}}={\omega }_b\left(u_{1q}+R_v{i}_{q,VSG}-{\omega }_{VSG}{\psi }_d\right)\end{array}\right.$$

(1)

$${\displaystyle \frac{d{\psi }_{fd}}{dt}}=K_u\left({\displaystyle \frac{Q_{ref,VSG}-Q_{VSG}}{U_o}}\right)$$

(2)

$$\left\{\begin{array}{c}{i}_{d,VSG}={\displaystyle \frac{{\psi }_{fd}-{\psi }_d}{L_v}}\\ i_{q,VSG}=-{\displaystyle \frac{{\psi }_q}{L_v}}\end{array}\right.$$

(3)

$$\left\{\begin{array}{c}{P}_{VSG}=u_{1d}{i}_{d,VSG}+u_{1q}{i}_{q,VSG}\\ Q_{VSG}=u_{1q}{i}_{d,VSG}-u_{1d}{i}_{q,VSG}\end{array}\right.$$

(4)

where ψ_d and ψ_q are the virtual stator flux linkages in the dq-axes, R_v and L_v are the virtual stator impedance, ω_VSG is the VSG angular frequency, ψ_fd is the virtual field winding flux linkage, Q_ref,VSG is the VSG reactive power reference, P_VSG and Q_VSG are the VSG output active and reactive power, K_u is the virtual field winding gain, u_1d and u_1q are the voltages in the dq-axes, U_o is the rms voltage at PCC, and i_d,VSG and i_q,VSG are the VSG virtual currents in the dq-axes.

The modified CC-VSG control structure shown in Figure 2 can be divided into three parts: (i) inner control level, (ii) outer control level, and (iii) inverter output control level. This CC-VSG specificity provides the flexibility to add additional control loops to any of the control levels, depending on their functions and control efficiency. The key feature of the considered CC-VSG structure is that the virtual synchronous generator operates at no load (P_ref,VSG = Q_ref,VSG = 0) and, accordingly, with zero internal load angle. Due to this, the inner level operates only during transients and does not affect the steady-state operation of the power converter. The latter significantly improves the stability of the considered system, which was shown in detail in [43]. To ensure the possibility of the BESS participation in the primary frequency control, the CC-VSG structure includes a virtual governor with a configurable frequency droop K_ω,VSG. The inner current control loop is implemented using PI controllers for dq-components with gains K_p,CC and K_i,CC, according to Equations (5) and (6):

$$u_{cvdref}=\left(K_{p,CC}+{\displaystyle \raisebox{1ex}{$K_{i,CC}$}\!\left/ \!\raisebox{-1ex}{$s$}\right.}\right)\left(i_{cvdref}-i_{cvd}\right)-i_{cvq}{\omega }_{VSG}{L}_f$$

(5)

$$u_{cvqref}=\left(K_{p,CC}+{\displaystyle \raisebox{1ex}{$K_{i,CC}$}\!\left/ \!\raisebox{-1ex}{$s$}\right.}\right)\left(i_{cvqref}-i_{cvq}\right)+i_{cvd}{\omega }_{VSG}{L}_f$$

(6)

where i_cvd and i_cvq are the inverter output currents in the dq-axes, and L_f is the filter inductance.

As mentioned earlier, it is necessary to add an energy recovery control to the CC-VSG active power control loop. This loop is based on a PI controller for the continuous SoC-level control with parameters K_p,SoC and K_i,SoC (highlighted in green in Figure 2). In order to perform a comparison, two previously outlined damping approaches are considered within the CC-VSG structure based on the use of (i) a damping coefficient D_VSG (highlighted in red in Figure 2) and (ii) a feedforward controller with K_FF (highlighted in blue in Figure 2). As a result, the equations of the active power and frequency control loop are written as (7)–(14), where Equations (7) and (8) correspond to the model with D_VSG, and Equations (9)–(11)—with K_FF.

$${\displaystyle \frac{d{\omega }_{VSG}}{dt}}={\displaystyle \frac{1}{2H_{VSG}}}\left(P_{ref,VSG}-P_{VSG}-P_D-P_{gov}+P_{\mathrm{SoC}}\right)$$

(7)

$${\displaystyle \frac{d{\theta }_{VSG}}{dt}}={\omega }_b{\omega }_{VSG}$$

(8)

$${\displaystyle \frac{d{\omega }_{VSG}}{dt}}={\displaystyle \frac{1}{2H_{VSG}}}\left(P_{ref,VSG}-P_{VSG}-P_{gov}+P_{\mathrm{SoC}}\right)$$

(9)

$${\displaystyle \frac{d{\theta }_s}{dt}}={\omega }_b{\omega }_{VSG}$$

(10)

$${\theta }_{VSG}={\theta }_s+K_{FF}\left({\omega }_{VSG}-{\omega }_{ref}\right)$$

(11)

$$P_D=D_{VSG}\left({\omega }_{VSG}-{\omega }_{ref}\right)$$

(12)

$$P_{gov}=K_{\omega ,VSG}\left({\omega }_{VSG}-{\omega }_{ref}\right)$$

(13)

$$P_{\mathrm{SoC}}=\left({\mathrm{SoC}}_{meas}-{\mathrm{SoC}}_{ref}\right)\left(K_{p,\mathrm{SoC}}+{\displaystyle \raisebox{1ex}{$K_{i,\mathrm{SoC}}$}\!\left/ \!\raisebox{-1ex}{$s$}\right.}\right)$$

(14)

where H_VSG is the VSG inertia constant, P_ref,VSG is the VSG active power reference, P_D is the damping power, P_gov is the virtual governor power, P_SoC is the SoC controller power, θ_VSG is the VSG phase angle, ω_b is the base angular frequency, θ_s is the swing-equation-regulated phase angle, SoC_meas is the actual battery SoC, and SoC_ref is the SoC reference value.

The study of two damping approaches will prove that the use of the damping coefficient does not allow to separate the dynamic response from different control loops, as a result of which the ability of the BESS to maintain the required SoC level or frequency profile in the microgrid deteriorates. On the contrary, by adding the phase feedforward damping control to the CC-VSG structure, it is possible to independently change the inertial response, the damping ratio, and the contribution to the microgrid frequency control, as well as to maintain the SoC of ESS in order to achieve the desired control objectives.

3. Theory Description

3.1. Modeling a Two-Unit System with a Common Load

In the studied microgrid, the processes related to the frequency deviation mainly depend on the active power balance between the power sources (P_DG and P_VSG) and the load (P_Load); therefore, for the analysis performed in this section, the voltage and reactive power control loops of the DG and VSG can be neglected. The VSG inner current control loop and electromagnetic transients in the network are also excluded since their time scale is much lower than the range considered for frequency control processes. In this connection, the equation of output electric power for the i-th source in linearized form is described by Equation (15) [50]:

$$\mathsf{\Delta }{P}_i={\displaystyle \frac{S_i{S}_j}{S_i+S_j}}\left(\mathsf{\Delta }{\delta }_i-\mathsf{\Delta }{\delta }_j\right)+{\displaystyle \frac{S_i}{S_i+S_j}}\mathsf{\Delta }{P}_{Load}$$

(15)

where i and j denote the corresponding source, ΔP_Load is the magnitude of load change per unit, and S_i is the synchronizing power of the i-th source. The latter is determined for the DG and VSG according to Equations (16) [50] and (17) [47], respectively:

$$S_{DG}={\displaystyle \frac{U_{1,0}\cdot U_{bus,0}}{L_{g1}}}\cos {\delta }_{DG,0}$$

(16)

$$S_{VSG}={\displaystyle \frac{U_{2,0}\cdot X_{C_f{L}_g}}{L_v+L_{g2}\cdot X_{C_f{L}_g}}}\cos {\delta }_{VSG,0}$$

(17)

where $X_{C_f{L}_g}={\displaystyle \frac{1}{1-L_{g2}{C}_f}}$, the subscript 0 is used to denote the steady-state value of the parameter at the linearization point.

Taking into account the obtained dependencies, Figure 3 presents a structural diagram that reflects the processes of the active power and frequency changes during a parallel operation of the DG and VSG in an isolated microgrid. The active power control loop of the CC-VSG is presented earlier in Figure 2. Such a control loop for the DG is based on the well-known swing equation for the synchronous generators [51], the standard governor, and the first-order lag function with a time-constant T_DG to reproduce delays in the DG’ mechanical parts [52].

In the obtained structural diagram (Figure 3), the dependences of the output mutual angle Δδ_i on the input active power ΔP_i for each source (18)–(19) are highlighted, which are defined as open-loop transfer functions (superscript ‘ol’):

$$\begin{array}{l}{G}_{\delta P,DG}^{ol}\left(s\right)={\displaystyle \frac{\mathsf{\Delta }{\delta }_{DG}}{\mathsf{\Delta }{P}_{DG}}}=\\ ={\displaystyle \frac{{\omega }_b\left(T_{DG}s+1\right)}{2H_{DG}{T}_{DG}{s}^3+\left(2H_{DG}+D_{DG}{T}_{DG}\right)s^2+\left(D_{DG}+K_{\omega ,DG}\right)s+K_{i,DG}}}\end{array}$$

(18)

$$G_{\delta P,VSG}^{ol}\left(s\right)={\displaystyle \frac{\mathsf{\Delta }{\delta }_{VSG}}{\mathsf{\Delta }{P}_{VSG}}}={\displaystyle \frac{{\omega }_b+K_{FF}s}{2H_{VSG}{s}^2+\left(D_{VSG}+K_{\omega ,VSG}\right)s}}$$

(19)

The battery SoC control loop is described by a PI controller whose open-loop transfer function G^ol_SoC(s) is rather simple. As a result, the closed-loop transfer function G_P,SoC(s) of the active power and frequency control loop, taking into account the energy recovery control, is described by Equation (20).

$$\begin{array}{l}{G}_{P,SoC}\left(s\right)={\displaystyle \frac{\mathsf{\Delta }{P}_{out}}{\mathsf{\Delta }{P}_{Load}}}=\\ ={\displaystyle \frac{S_{DG}{S}_{VSG}\left[G_{\delta P,VSG}^{ol}\left(s\right)\cdot S_{VSG}\cdot \left(1+G_{SoC}^{ol}\left(s\right)\right)-G_{\delta P,DG}^{ol}\left(s\right)\cdot S_{DG}\right]}{\left(S_{DG}+S_{VSG}\right)\left(S_{DG}+S_{VSG}+S_{DG}{S}_{VSG}\cdot \left[G_{\delta P,DG}^{ol}\left(s\right)+\left(1+G_{SoC}^{ol}\left(s\right)\right)\cdot G_{\delta P,VSG}^{ol}\left(s\right)\right]\right)}}\end{array}$$

(20)

3.2. Sizing of BESS for VSG

The BESS energy capacity must be sufficient to achieve the desired control objectives with the required quality. Therefore, it is necessary to take into account the amount of energy (ΔE) that is consumed at each of the control stages and, based on this, to choose the nominal energy capacity (E_nom). Moreover, it is obvious that the control algorithms used should provide the lowest possible energy consumption while ensuring the required response quality to increase the technical and economic efficiency of the BESS application as a whole. On this basis, the BESS nominal energy capacity can be found from Equation (21):

$$\mathsf{\Delta }E=E_{nom}-E_{\min }=E_{nom}{\displaystyle \frac{\mathsf{\Delta }E}{E_{nom}}}=E_{nom}\mathsf{\Delta }{E}_{\max ,\%}\Rightarrow E_{nom}={\displaystyle \frac{\mathsf{\Delta }E}{\mathsf{\Delta }{E}_{\max ,\%}/100}}$$

(21)

where ΔE_max,% is the maximum permissible value of energy deviation from the nominal value consumed for regulation processes in %, and E_min is the remaining value of BESS energy capacity after performing regulation functions.

The energy consumed by the BESS can be conditionally divided into two stages based on the duration of the transients’ stages [53,54]. At the first moment of time after the disturbance, the energy is consumed to ensure the inertial response. This value (ΔE_inertia) can be determined on the basis of the known formula for the kinetic energy (22) [55]:

$$\begin{array}{l}\mathsf{\Delta }{E}_{inertia}={\displaystyle \frac{1}{2}}{J}_{VSG}\left({\omega }_{nom}^2-{\omega }_{nadir}^2\right)\approx J_{VSG}{\omega }_{nom}\left({\omega }_{nom}-{\omega }_{nadir}\right)=\\ =J_{VSG}{\omega }_{nom}^2{\displaystyle \frac{\mathsf{\Delta }{\omega }_{\max }}{{\omega }_{nom}}}=H_{VSG}{S}_{nom}{\displaystyle \frac{\mathsf{\Delta }{\omega }_{\max }}{{\omega }_{nom}}}\end{array}$$

(22)

where J_VSG is the VSG moment of inertia, ω_nadir is the minimum value of frequency reached during the transient period, ω_nom and S_nom are the nominal grid frequency and ESS power capacity, and Δω_max is the maximum deviation of frequency.

The second stage of the transients is related to the primary regulation and frequency damping and is much slower. Therefore, the structural diagram in Figure 3 can be simplified by assuming equality of frequencies at all points of the grid Δω_DG = Δω_VSG = Δω, which allows for the exclusion of the equations of synchronizing powers. The resulting scheme is shown in Figure 4, and the closed-loop transfer function describing it is represented by Equation (23).

$$\begin{array}{l}{G}_{\omega }\left(s\right)={\displaystyle \frac{\mathsf{\Delta }\omega }{\mathsf{\Delta }{P}_{Load}}}=\\ =-{\displaystyle \frac{s\left(T_{DG}s+1\right)}{s\left[T_{DG}s+1\right]\left[2\left(H_{DG}+H_{VSG}\right)s+D_{DG}+D_{VSG}+K_{\omega ,VSG}\right]+K_{i,DG}+K_{\omega ,DG}s}}\end{array}$$

(23)

Given the frequency response to a load change in the presented diagram (Figure 4), the corresponding VSG power response (ΔP_VSG,ω) can be determined using Equation (24). By applying the final value theorem to Equation (24), which describes the coupling of frequency and time domains at t → ∞, it is possible to determine the quantity of consumed energy (ΔE_freq) for this response (25).

$$\mathsf{\Delta }{P}_{VSG,\omega }=\left(H_{VSG}s+D_{VSG}+K_{\omega ,VSG}\right)\mathsf{\Delta }\omega$$

(24)

$$\mathsf{\Delta }{E}_{freq}=\underset{s\to 0}{\lim } s{E}_{freq}\left(s\right)={\left.\mathsf{\Delta }{P}_{VSG,\omega }\right|}_{s\to 0}={\displaystyle \frac{D_{VSG}+K_{\omega ,VSG}}{K_{i,DG}}}\mathsf{\Delta }{P}_{Load}$$

(25)

Thus, using the simple Equations (22) and (25), it is possible to determine approximately the amount of BESS energy that is consumed at each stage of the transient process and the corresponding impact of the VSG parameters. To assess the validity of the simplified equations obtained, they can be compared with the value of the total BESS energy consumed, which is found as a time-domain dependence Δe(t) using the inverse Laplace transform. The resulting equation has the following form (26):

$$\mathsf{\Delta }e\left(t\right)=L^{-1}\left[\mathsf{\Delta }E\left(s\right)\right]=L^{-1}\left[{\displaystyle \frac{1}{s}}\mathsf{\Delta }{P}_{VSG,\omega }\right]$$

(26)

For the quantitative comparison, it is necessary to determine the tuning parameters of the system based on the permissible ranges of frequency change. We assume the permissible RoCoF of not more than 1 Hz/s and the DG inertia constant equal to H_DG = 2.5 s. Then, the VSG inertia constant is equal to H_VSG = 5 s, which is found by the known equation RoCoF = (ω_nomΔP)/2(H_VSG + H_DG), where ΔP = 0.3 pu [55]. Based on the assumed frequency deviation Δω_max of not more than 0.06 pu, the damping and droop coefficients can be found by the following equation Δω_max = ΔP/(D_DG + D_VSG + K_ω,DG + K_ω,VSG) [56]. We suppose that D_DG = 0, D_VSG = 0, K_ω,DG = 3, K_ω,VSG = 2, then the frequency deviation, considering the secondary control K_i,DG = 2, will be Δω = 0.039 pu, which is obtained by plotting in the time domain the dependence (23) after the inverse Laplace transform. Substituting the selected parameters into Equations (22) and (25), the following results are obtained: ΔE_inertia = 0.195 pu∙s, ΔE_freq = 0.3 pu∙s, and, accordingly, the total amount of consumed energy ΔE = 0.495 pu∙s. At the same parameters, the dependence Δe(t) is plotted in Figure 5 (brown curve in Figure 5). It follows from the obtained curve that the value of energy at time t → ∞ is equal to 0.301 pu∙s, and the value of energy consumed on the inertial response is equal to 0.184 pu∙s. As a result, the total change in energy is 0.485 pu∙s. Thus, the obtained simplified equations for finding the energy consumption at each stage of the transient process coincide with the detailed dependence Δe(t). It should be noted that with the adopted parameters, a significant contribution to the energy deviation is made by the inertial response. In this case, there are oscillations in frequency, which indicate an insufficient damping factor. In addition, it is necessary to aim at minimizing the DG usage for the primary frequency control to ensure its optimal operation without rapid changes in the output active power. In this regard, it is proposed to take K_ω,DG = 0 and K_ω,VSG = 10, which corresponds to the maximum frequency deviation Δω_max = 0.03 pu with ΔP = 0.3 pu. Thus, we find that the energy capacity of BESS is mainly consumed to provide primary regulation and damping with a reduction of the oscillations in the system (ΔE_inertia = 0.1 pu∙s, ΔE_freq = 1.5 pu∙s, blue curve in Figure 5). Taking the permissible value of energy capacity deviation ΔE_max,% = 30%, the necessary BESS nominal energy capacity can be obtained by Equation (21)—E_nom = 16.6 pu∙s. At the same time, the allowable energy capacity depends on the reference SoC_ref. In this paper, it is assumed that the value of SoC_ref is equal to 0.5 pu (or 50%). In practice, the value of the SoC_ref depends on the purposes for which the BESS is used. As a rule, most of the power capacity of BESS is reserved to provide a power supply to the load in case of a shutdown of the power source in the microgrid. Accordingly, the SoC_ref is set based on this capacity, which must be permanently stored in the BESS. The larger the load power to be supported by the BESS in the event of a power failure, the larger the SoC_ref must be set. Consequently, the smaller value of power capacity remains available for the frequency control.

3.3. Frequency-Domain Performance Assessment

For a comprehensive assessment of the dynamic response provided by the developed CC-VSG algorithm with different damping approaches (D_VSG or K_FF), the Bode plots of the full closed-loop transfer function of the active power and frequency control loop with and without the energy recovery control loop (G_P,SoC(s) and G_P(s), respectively) are considered. The transfer function G_P(s) is obtained by excluding G^ol_SoC(s) in Equation (20). The analysis of Bode plots allows us to simultaneously identify the areas of low-frequency processes, corresponding to slow processes of frequency control and oscillation damping, and high-frequency processes, which correspond to the inertial response. Due to this, it is possible to assess the impact of various factors and parameters at each stage of the transient process and to coordinate the operation of different control loops within the DG and VSG between each other to avoid negative mutual coupling.

Figure 6 shows the Bode plots of G_P(s) with the tuning parameters adopted in Section 3.2 in the case of different coefficients for the secondary frequency control K_i,DG (blue and brown curves in Figure 6). The presence of the secondary control affects only the low-frequency range and reduces the steady-state frequency deviation to zero. In the high-frequency range, a resonant peak occurs. This resonance manifests itself due to the consideration of synchronizing power feedback in the G_P(s) transfer function. The latter is confirmed by the fact that in the case of the Bode plot for the simplified loop G_ω(s), which considers only the frequency change processes, under similar conditions, the above resonance peak is absent (yellow and purple curves in Figure 6). This resonance represents a problem of oscillations occurrence under stiff grid conditions due to insufficient damping properties of control loops [46]. In spite of the fact that this resonance is not related to the frequency control loops, it is necessary to take it into account because the required increase in the damping properties of the system can affect the low-frequency range, which is already related to the frequency change processes.

As mentioned earlier, it is possible to enhance the damping properties of CC-VSG using the conventional approach by increasing the damping coefficient D_VSG in the swing equation (Figure 7a) or by using the gain K_FF in the proposed feedforward controller (Figure 7b). From the dependencies obtained, it is evident that both of the indicated approaches resulted in a significant reduction of the resonant peak. However, in the case of D_VSG, a decrease in the overshoot during the secondary frequency control is observed, which indicates the impact of D_VSG on the high-frequency and low-frequency range of the processes. On the contrary, the K_FF usage affects only the resonant peak, and the low-frequency range of the processes does not change, which is confirmed by the maintenance of the overshoot value during the secondary frequency control.

After the improvement of CC-VSG damping properties, an energy recovery control loop is added. The obtained Bode plots of G_P,SoC(s) for two damping approaches at randomly adopted PI controller tuning parameters (K_p,SoC = 0.5 and K_i,SoC = 0.014) are shown in Figure 8. In the case of D_VSG, the addition of a PI controller to recover the SoC level in the presence of secondary frequency control leads to the occurrence of a resonant peak at low frequency and, correspondingly, low-frequency oscillations with increasing magnitude, leading to instability (Figure 8a). It should also be noted that setting D_VSG = 100 pu leads to an increase in the energy ΔE_freq according to Equation (25), which is spent on primary regulation and damping. It should also be noted that in the case of D_VSG = 100 pu, there is an increase in the energy ΔE_freq according to Equation (25), which is consumed for primary regulation and damping. In this connection, an increase in the BESS nominal energy capacity is inevitable, which for the adopted parameters will be E_nom = 183 pu∙s. Otherwise, the SoC of ESS exceeds the permissible ranges (at the adopted ΔE_max,% = 30% and SoC_ref = 0.5 pu, the permissible values are SoC_min = 0.2 pu and SoC_max = 0.8 pu). Despite this significant change, the resonant peak remains—there is only a decrease in its frequency (yellow curve in Figure 8a). On the contrary, in the case of K_FF, the addition of a SoC control does not lead to a resonance despite the presence or absence of the secondary regulation. In the latter case, an additional overshoot occurs. It should be noted that the most significant in this case is the possibility of preserving the selected BESS nominal energy capacity with the use of the proposed feedforward controller.

Summarizing the above, it follows that the addition of an appropriate energy recovery control loop to the CC-VSG structure leads to the occurrence of mutual coupling with the secondary frequency control loop. Moreover, depending on the damping approach used, the nature of mutual coupling differs. Thus, it is necessary to coordinate and tune these loops with each other.

3.4. Determination of BESS Energy Recovery Control Loop Parameters by Bandwidth Separation of Control Loops

An effective approach to tuning different loops operating in the same frequency range but performing different functions is to coordinate their bandwidths ω_bw [34,57]. For the considered microgrid within the assumed assumption of frequency equality in all nodes, as well as the exclusion of delay in the DG mechanical part, the nature of the frequency change is related to the primary and secondary frequency control, as well as the recovery of the SoC level. Moreover, given the evident objectives at each of these stages, the sequence of their performance should be exactly as presented. In this regard, the corresponding bandwidths should be similarly arranged: from the fastest for the primary regulation loop to the slowest for the energy recovery control loop. If the tertiary regulation is present in the system, it will also be necessary to coordinate this bandwidth with the indicated ones.

Considering the assumed conditions, the transfer function G_prim(s) and the bandwidth ω_bw,prim of the primary frequency control loop are found using Equation (27) by excluding the integral controller of the secondary frequency control K_i,DG/s in the simplified structural diagram (Figure 4). The transfer function G_sec(s) and the bandwidth ω_bw,sec for the secondary control are found with the integral controller K_i,DG/s. For the determination of ω_bw,sec, it is assumed that the inertial component is decayed at this stage of the process, i.e., H_VSG = 0 and H_DG = 0. The resulting equations have the form (28). To obtain the transfer function of the SoC controller G_SoC(s) and its bandwidth ω_bw,SoC, it is assumed that the integral coefficient K_i,SoC is equal to zero since the process of energy recovery is rather slow. As a result, Equation (29) is obtained.

$$\left\{\begin{array}{l}{G}_{prim}\left(s\right)={\displaystyle \frac{\mathsf{\Delta }\omega }{\mathsf{\Delta }{P}_{ref}}}={\displaystyle \frac{1}{2\left(H_{VSG}+H_{DG}\right)s+D_{VSG}+D_{DG}+K_{\omega ,VSG}+K_{\omega ,DG}}}\\ {\omega }_{bw,prim}={\displaystyle \frac{D_{VSG}+D_{DG}+K_{\omega ,VSG}+K_{\omega ,DG}}{2\left(H_{VSG}+H_{DG}\right)}}\end{array}\right.$$

(27)

$$\left\{\begin{array}{l}{G}_{sec}\left(s\right)={\displaystyle \frac{\mathsf{\Delta }\omega }{\mathsf{\Delta }{\omega }_{ref}}}={\displaystyle \frac{1}{1+\frac{\left[2\left(H_{DG}+H_{VSG}\right)s+D_{VSG}+D_{DG}+K_{\omega ,VSG}+K_{\omega ,DG}\right]s}{K_{i,SG}}}}\\ {\omega }_{bw,sec}\approx {\displaystyle \frac{K_{i,SG}}{D_{VSG}+D_{DG}+K_{\omega ,VSG}+K_{\omega ,DG}}}\end{array}\right.$$

(28)

$$\left\{\begin{array}{l}{G}_{\mathrm{SoC}}\left(s\right)={\displaystyle \frac{\mathsf{\Delta }\mathrm{SoC}}{\mathsf{\Delta }{\mathrm{SoC}}_{ref}}}={\displaystyle \frac{1}{1+\frac{E_{nom}s}{K_{p,\mathrm{SoC}}+\frac{K_{i,\mathrm{SoC}}}{s}}}}\\ {\omega }_{bw,\mathrm{SoC}}\approx {\displaystyle \frac{K_{p,\mathrm{SoC}}}{E_{nom}}}\end{array}\right.$$

(29)

For the obtained transfer functions and their bandwidths (27)–(29), taking into account the indicated simplifications, the Bode plots are shown for the conventional approach to damping by D_VSG within the proposed CC-VSG (Figure 9a).

It can be seen from the obtained dependencies that the bandwidth ω_bw,prim is significantly higher than all others. Consequently, the regulators providing primary control and inertial response are fast-acting, and their dynamics of operation do not interfere with other loops, which corresponds to the assumed logic of coordination and tuning. On the contrary, the processes of secondary control and energy recovery control have comparable operating speeds, but ω_bw,sec < ω_bw,SoC. This circumstance leads to the fact that after the primary frequency control, the SoC recovery is initiated, and only then is the secondary frequency control performed. Due to this performance, a resonant peak occurs in the low-frequency range. At the same time, when the proposed feedforward controller is used for damping, the bandwidths are arranged according to the specified requirement: ω_bw,SoC < ω_bw,sec < ω_bw,prim (Figure 9b). Due to this, it is possible to exclude the occurrence of resonance phenomena in the low-frequency range.

It should be noted that in the case of D_VSG, compliance with the bandwidth ratios of the control loops is possible by increasing ω_bw,sec or decreasing ω_bw,SoC. It follows from Equation (28) that to increase ω_bw,sec, either K_i,DG should be increased or D_VSG and K_ω,VSG should be decreased. However, in the first case, the DG does not always have sufficient technological capabilities to provide secondary control with the required speed, and in the second case, the damping factor will be reduced, leading to an increase in the resonant peak. In this connection, the only solution is to change ω_bw,SoC. In the case of decreasing K_p,SoC, the low-frequency resonance is leveled, but there is a decrease in damping properties in the SoC control loop, and an overshoot occurs (brown curve in Figure 10a). Therefore, there remains only the case with increasing the BESS nominal energy capacity up to E_nom = 183 pu∙s. In such a case, by significantly reducing ω_bw,SoC, it is possible to increase K_p,SoC and obtain an acceptable quality of the dynamic response within the necessary compliance with the bandwidth separation (Figure 10b). Thus, the obtained results prove for the second time the inevitability of a significant increase in the BESS nominal energy capacity when the conventional approach to damping with D_VSG is used.

The obtained theoretical results prove that the proposed feedforward controller within the CC-VSG structure provides a wide control range for tuning parameters of different control loops within the bandwidth separation. On this basis, it is possible to tune the energy recovery control loop of the BESS. It is known that the transient process duration is approximately from 2 to 4 times relative to 1/ω_bw. Within the adopted approach, it is necessary that the secondary frequency control is completed, and only then should the energy recovery be initiated. Thus, the bandwidth ω_bw,SoC can be taken equal to 0.1 pu, which is twice less than the bandwidth ω_bw,sec = 0.2 pu. The latter, in turn, is determined on the basis of ensuring both the effective damping and permissible frequency deviation Δω, which is equal to 0.1 pu for the assumed K_ω,VSG = 10 (see Section 3.2). Based on Equation (29), the coefficient K_p,SoC = 1.77 can be determined. Due to the fact that the energy recovery control loop is a typical second-order function, the integral coefficient K_i,SoC can be found using the well-known equation for the damping factor ζ [58]. In this case, the value of ζ should be taken in the range from 0.707 to 1 pu. Taking ζ = 1, the coefficient K_i,SoC = 0.05 can be obtained from Equation (30). The final parameters after tuning for the considered microgrid are summarized in

Table 1. Network and control system parameters.

Parameter	Value	Parameter	Value
fb, Hz	50	Kω,DG, pu	0
Cf, pu	0.112	HVSG, s	5
Lv, pu	0.105	DVSG, pu	0
Lg1, pu	0.155	Kω,VSG, pu	10
Lg2, pu	0.05	KFF, pu	20
HDG, s	2.5	Enom, pu∙s	16.6
DDG, pu	0	Ki,SoC, pu∙s	0.05
TDG, s	1	Kp,SoC, pu	1.77
Ki,DG, pu∙s	2	SoCref, pu	0.5

.

$$K_{i,\mathrm{SoC}}={\displaystyle \frac{K_{p,\mathrm{SoC}}^2}{4{\zeta }^2{E}_{nom}}}$$

(30)

To verify the quality of the transient process after the performed tuning, the transient characteristics of the frequency and SoC deviation under the step load increase have been plotted using the detailed closed-loop transfer function (Figure 3) in MATLAB/Simulink (Figure 11). From the plots obtained, it can be seen that the addition of a tuned SoC control loop to the CC-VSG structure allows us to achieve a lower deviation in the SoC level while not exceeding the allowable frequency deviation Δω_max = 0.03 pu (brown curves in Figure 11) compared to the case without this loop (blue curves in Figure 11). As a result, the BESS energy saving can be achieved, which allows reducing the nominal energy capacity by 18% while maintaining the transient performance (yellow curves in Figure 11).

3.5. State-Space Model

In order to comprehensively analyze the proposed CC-VSG structure with the SoC control, a linearized state-space model for the structural diagram reflecting the processes of active power and frequency changes in an isolated microgrid (Figure 3) is formed. In general, the obtained state-space model can be represented by Equation (31):

$$\mathsf{\Delta }\stackrel{•}{x}=A\left(x_0\right)\mathsf{\Delta }x+B\left(x_0\right)\mathsf{\Delta }u$$

(31)

The state vector Δx and the input vector Δu are represented by Equation (32), given the parameters of the system and control loops:

$$\begin{array}{l}\mathsf{\Delta }x={\left[\begin{array}{ccccccccc}{\gamma }_{DG}& \mathsf{\Delta }{P}_{gov}& \mathsf{\Delta }{\omega }_{DG}& \mathsf{\Delta }{\delta }_{DG}& {\beta }_{\mathrm{SoC}}& \mathsf{\Delta }\mathrm{SoC}& \mathsf{\Delta }{\omega }_{VSG}& \mathsf{\Delta }{\delta }_s& \mathsf{\Delta }\delta \end{array}\right]}^{ \mathrm{T}}\\ \mathsf{\Delta }u={\left[\begin{array}{ccc}\mathsf{\Delta }{P}_{Load}& {\mathrm{SoC}}_{ini}& {\mathrm{SoC}}_{ref}\end{array}\right]}^{ \mathrm{T}}\end{array}$$

(32)

The formed CC-VSG state-space model with parameters from Table 1 has nine eigenvalues, among which the conjugate pairs of roots λ_3,4 and λ_7,8 are of the greatest interest. The sensitivity coefficients to the damping coefficients D_VSG and K_FF, as well as the participation factors of the state variables, are calculated for these eigenvalues. The obtained results are presented in

Table 2. Eigenvalue analysis.

	ωn, Hz	Sensitivity of λ to DVSG	Sensitivity of λ to KFF	Mainly Related State Variable
λ3,4	≈2.3	−0.016 ∓ j0.015	−0.113 ∓ j0.024	ΔωDG, ΔωVSG, Δδ
λ7,8	<0.05	0.021 ∓ j0.021	0	βSoC, ΔSoC

.

It follows from the obtained values that the K_FF parameter affects only λ_3,4, which characterizes the occurrence of resonance in the stiff grid, as identified in Section 3.2 (Figure 6). This fact is clearly confirmed by the plotted root locus in Figure 12. As can be seen, with the increase in K_FF, the roots λ_3,4 move monotonically to the left, which indicates the improvement of CC-VSG damping properties—the damping factor ζ increases from 0.02 to 0.404 pu.

On the contrary, the D_VSG parameter affects not only λ_3,4, but also the low-frequency range characterized by λ_7,8. Moreover, based on the calculated values of the sensitivity coefficients, this impact is significant (Figure 13). The latter proves the conclusions drawn that the use of the conventional damping coefficient D_VSG in the CC-VSG structure affects both the low-frequency range, associated with the secondary frequency control and the energy recovery control, and the high-frequency range, associated with the primary frequency control and the inertial response. This results in an excessive limitation of frequency deviation and a need to significantly increase the BESS nominal energy capacity to provide a required dynamic response.

By calculating the participation factors, it was found that the roots λ_3,4, in addition to the frequency deviation of the diesel generator and VSG, are also affected by the mutual angle Δδ, which is equal to the difference of the angles Δδ_DG and Δδ_VSG. Thus, due to the affecting only the VSG phase angle, it becomes possible to improve the damping properties toward the resonance in the stiff grid without affecting the other eigenvalues, which confirms the validity of the application of the phase feedforward damping control within the proposed CC-VSG structure.

4. Simulation Results and Discussion

To verify the results obtained during the analysis of the transfer functions and the state-space model, a detailed time-domain mathematical modeling of the microgrid shown in Figure 1 was performed in PSCAD/EMTDC v5.0.2 software. The parameters of the synchronous generator of the DG and its control systems are given in Appendix A in

Table A1. Diesel generator parameters.

Parameter	Value	Parameter	Value
Snom, MVA	1	R1d, X1d, pu	0.9029, 7.911
Unom, V	600	R1q, X1q, pu	0.05203, 0.3047
fnom, Hz	50	uref, ωref, pu	1
HDG, s	2.5	KU, pu	15
Rs, Xs, pu	0.01727, 0.07	Tf, s	0.3
Xad, Xaq, pu	2.83, 2.37	Emax, Emin, pu	2.1, 0
Rfd, Xfd, pu	0.006871, 0.5956	Tmax, Tmin, pu	1.1, −0.02

. Structural diagrams of the automatic voltage regulator and the governor of the DG are shown in Figure 14.

The battery was modeled in the form of the classical Shepherd model [59], which is generally described by Equation (33) considering the current SoC level of the storage device:

$$V_{batt}=E_0-R_{int}\cdot i_{batt}-K\cdot {\displaystyle \frac{1}{\mathrm{SoC}}}+A\cdot e^{-B\cdot Q_{rated}\cdot \left(1-\mathrm{SoC}\right)}$$

(33)

where V_batt is the battery voltage (V), E₀ is the battery constant voltage (V), R_int is the internal resistance (Ohm), K is the polarization voltage (V), A is the exponential zone amplitude (V), B is the exponential zone time constant inverse (1/(Ah)), i_batt is the battery current (A), and Q_rated is the battery rated capacity (Ah).

The calculation of the current SoC value for the battery was performed according to Equation (34), taking into account the conversion of simulation time from seconds to hours. According to the existing classification of SoC estimation techniques [60], this calculation method can be classified as an indirect model-based method. Considering a practical BESS, there is always an issue of SoC calculation errors, which are related to several reasons [61]. However, their analysis is out of the scope of this paper.

$${\mathrm{SoC}}_{meas}={\mathrm{SoC}}_{init}-{\displaystyle \frac{1}{3600}}\cdot {\displaystyle \frac{1}{Q_{rated}}}{\displaystyle \int i_{batt}dt}$$

(34)

where SoC_init is the initial SoC of the battery.

The connection of the BESS to the AC network was performed through a detailed model of the grid inverter, the control system of which is based on the CC-VSG structure proposed in this paper with a feedforward controller and energy recovery control. The main parameters of the battery and inverter control system are summarized in

Table A2. Battery and inverter parameters.

Parameter	Value	Parameter	Value
Unom, V	3.123	UDC, V	1500
Qnom, A∙h	2.761	fsw, kHz	4.5
Uexp, V	3.301	Kp,CC, pu	0.052
Qexp, A∙h	2.592	Ki,CC, pu∙s	13.2
Ufull, V	4.135	Rv, pu	0
Qrated, A∙h	2.998	Lv, pu	0.105
R, Ohm	0.025	Ku, pu	1

in Appendix A.

4.1. Analysis of VSG-Based BESS Operation with and without SoC-Level Control

In the first group of experimental studies, the microgrid frequency profile under the step load change in the microgrid (ΔP_Load = 0.3 pu) was assessed. At the same time, a number of control scenarios were considered (Figure 15):

(1)

Case 1—operation of the DG without BESS;

(2)

Case 2—operation of the DG with BESS and CC-VSG algorithm, but without the energy recovery control;

(3)

Case 3—operation of the DG with BESS and CC-VSG algorithm, including the energy recovery control loop, the parameters of which are coordinated with the bandwidth of the secondary frequency control (Table 1);

(4)

Case 4—the same as Case 3, but the bandwidth of the energy recovery control loop is increased and equal to the bandwidth of the secondary frequency control (K_p,SoC = 4.07, K_i,SoC = 0.26), while the damping factor for this loop remains the same (ζ ≈ 1).

Figure 15a clearly demonstrates that in the case of operation of only DG on the load, there is a maximum frequency deviation (Δf_max ≈ 1.8 Hz) and RoCoF (≈ 3.8 Hz/s), as well as continuous frequency oscillations with a damping time of about 13 s. Due to the installation of BESS with the proposed CC-VSG algorithm in the microgrid, it becomes possible to significantly reduce both the magnitude of frequency deviation by almost one-third to Δf_max = 1.22 Hz and the RoCoF by more than three times to ≈ 1.1 Hz/s (blue curves in Figure 15b). However, the disturbance results in a consistent discharge of the battery (ΔSoC_meas = 11.41%) without recovery of the SoC level to the previous value.

Through the addition of energy recovery control to the CC-VSG structure, the maintaining of the storage SoC at a reference level (SoC_ref = 50%) is achieved after the disturbances (brown curves in Figure 15b). As a result, the BESS is always able to participate in the microgrid frequency control. In this case, the value of Δf_max increases slightly to 1.48 Hz relative to the previous case. However, as can be seen, by applying the proposed recommendations for the tuning of the energy recovery control loop, it is possible to fulfill the accepted requirements for the microgrid frequency deviation range (±1.5 Hz relative to the nominal value). In addition, based on the waveform for the current SoC level, the maximum ΔSoC is reduced to 6.77% after the disturbance due to the SoC control operation, allowing a reduction of the nominal energy capacity, which was proved earlier in Section 3.4.

In the fourth case, the bandwidth of the energy recovery control loop is increased and almost equal to the bandwidth of the secondary frequency control. This results in a resonant peak similar to the one shown in the Bode plots in Figure 9a in Section 3.4. The reason for the latter is the emergence of mutual impact of the processes of the energy recovery control and the secondary frequency control, which are opposite in their functions within the BESS and microgrid in terms of the power flow direction. This is expressed in the emergence of undamped frequency and power oscillations of significant magnitude; for example, the range of the microgrid frequency variation is from 47.51 Hz to 52.22 Hz. In case of more severe disturbance, oscillations with increasing magnitude may occur, which will lead to instability in the microgrid. Thus, the simulation results prove the applicability of the developed methodology for the bandwidth separation of different control loops.

4.2. Analysis of BESS Performance with CC-VSG Algorithm in the Case of Using the Damping Coefficient and the Proposed Feedforward Controller

The next stage of experimental research was to assess the efficiency of the proposed phase feedforward controller within the CC-VSG structure in comparison with the conventional damping coefficient with respect to the quality of frequency regulation in the microgrid. The simulation results are shown in Figure 16.

As can be seen from Figure 16, due to the use of the conventional swing equation with a damping coefficient within the CC-VSG structure (blue curves in Figure 16), the best frequency profile is achieved—the maximum deviation Δf_max is only 0.17 Hz. However, such characteristics of the processes are possible only due to the high value of D_VSG = 100 pu, which is caused by the necessity of ensuring the stable operation of the BESS grid inverter. In addition, in this case, the required nominal energy capacity is significantly increased (E_nom = 183 pu∙s). As a result, the processes of both frequency and SoC level recovery are significantly delayed.

In the case of the application of the feedforward controller with the K_FF coefficient (brown curves in Figure 16), the maximum frequency deviation is significantly higher (Δf_max = 1.48 Hz), but it is within the desired range of ±1.5 Hz, taking into account the setting of a low value of K_ω,VSG and a much lower nominal energy capacity (E_nom = 16.6 pu∙s). At the same time, due to the feedforward controller, the stability of the BESS operation with necessary stability margins is provided. Moreover, the flexible BESS contribution to the microgrid frequency control is possible due to the virtual governor. Consequently, by K_ω,VSG tuning and an appropriate increase in the nominal energy capacity, it is possible to achieve results similar to the case with D_VSG (yellow and purple curves in Figure 16).

4.3. Analysis of the Microgrid Operation with Variable PV Generation under Different Approaches to BESS Control

In this case, a microgrid with DG, photovoltaic (PV) generation, BESS, and local load was simulated, as shown in Figure 17. The grid and PV array parameters are summarized in

Table A3. Grid and PV parameters.

Parameter	Value	Parameter	Value
Rg1, mOhm	0.03	Rs, Ohm	0.02
Lg1, mH	0.114	Rsh, Ohm	100
Rg2, Ohm	0.005	UOC(cell), V	0.835
Lg2, mH	0.335	ISC(cell), A	2.5
Rg3, Ohm	0.023	Ns × Np per module	36 × 1
Lg3, mH	0.074	Ns × Np per array	22 × 250
Gref, W/m2	1000	UDC, V	1000
Tref, °C	25	fsw, kHz	8
n	1.5	Kp,CC, pu	1
UG, eV	1.103	Ki,CC, pu∙s	50
IOR, A	1 × 10−9	Kp,PLL, pu	950
α, A∙K−1	0.001	Ki,PLL, pu∙s	1900

in Appendix A. The following variants of BESS control were considered: (i) a droop-based grid-forming control [62] (Figure 18a), (ii) a conventional grid-following control with PLL (Figure 18b), (iii) a CC-VSG-based grid-forming control without the energy recovery control (Figure 18c), and (iv) a CC-VSG-based grid-forming control with the energy recovery control (Figure 18d).

Comparing the results presented in Figure 18a,b, it can be seen that the application of the grid-forming control within the BESS allows for a significant reduction in the frequency oscillations in the microgrid, which in general has a positive impact on the electric power quality and the DG operation. The latter lies in the fact that the BESS mainly responds to changes in the power balance due to variable PV generation. Otherwise, when the BESS operates with the grid-following control and is only used to load peak shaving with a relatively constant active power output at certain time intervals (Figure 18b), only the DG responds to the changes in the PV generation, which leads to power and frequency oscillations in the microgrid. At the same time, due to the absence of the energy recovery control, the battery is gradually discharged naturally in both cases.

In the case of application of the proposed modified CC-VSG structure (Figure 18c) within the BESS, the processes are similar to the case of BESS with droop control (Figure 18a). It should be noted that the droop setting was performed for a specific case to approximate the quality of the transient performance to the case with CC-VSG and cannot be directly applied to other operating conditions. At further addition of the energy recovery control loop to the CC-VSG structure (Figure 18d), the waveforms of the DG and BESS active power, as well as the microgrid frequency, are unchanged. At the same time, the SoC level of the battery is controlled around the reference value SoC_ref = 50%. The obtained results prove the efficiency of the solutions proposed in this paper for the BESS control, including the real dynamics of the microgrid operation.

5. Conclusions and Prospects

In this paper, the necessity of developing a novel VSG structure for the ESS grid inverter served as motivation since, in the case of using the VSG algorithm with the conventional damping coefficient in the swing equation, a significant overestimation of its value is inevitable to suppress the resonance that occurs in the microgrid. This results in an increase in the ESS nominal energy capacity for two reasons: (i) the necessity to fulfill the limitations on the SoC level deviation beyond the permissible values and (ii) the inability to coordinate the settings of the secondary frequency control and the energy recovery control loop. In the developed CC-VSG structure, due to the use of a feedforward controller, it is possible to dampen the oscillations without affecting the low-frequency range associated with the secondary frequency control. Therefore, the addition of the PI controller to the CC-VSG structure to recover the SoC level does not lead to the occurrence of mutual coupling with the damping loop. The proposed algorithm can be a basis for the development of control systems for a practical BESS and be useful for researchers involved in the development of grid-forming control methods. In addition, the proposed tuning technique can be used in existing microgrids to coordinate the operation of different controllers.

In comparison with existing similar algorithms, two aspects should be emphasized. Firstly, a PI controller is used to maintain the SoC level. This control loop is the simplest method of continuous control. One of the main goals of this paper was to implement the simplest method so as not to complicate the whole algorithm. This was achieved by using a PI controller. Secondly, the complexity of the implementation of such control consists of the compatibility of SoC control and frequency control within the VSG-based control since these two processes have opposite effects on the output power of the BESS. Therefore, it is necessary not only to implement an effective VSG frequency control but also to achieve the compatibility of this control loop with the SoC controller. The proposed approach to VSG tuning on the basis of the bandwidth separation of different control loops provided an effective SoC recovery without a negative impact on the microgrid frequency control in general. As a result, the combination of the proposed solutions for the VSG control allowed us to reduce the BESS nominal energy capacity while maintaining the desired RoCoF and maximum frequency deviation values in the microgrid. Compared to the conventional VSG damping approach, the difference in the nominal energy capacity was 11 times.

It is worth noting that the developed algorithm has the following limitations: (i) the maximum permissible charge/discharge current of the battery is not controlled, and (ii) the reactive power of the BESS is not limited, considering the rated power of the inverter. The improvements to the algorithm to overcome these shortcomings will be carried out in a future work. In addition, future work will focus on the development of an experimental setup for a comprehensive verification of the proposed modified CC-VSG structure with a feedforward controller and an energy recovery control loop within practical energy storage systems. Such a hardware realization will allow the proposed solution to be debugged and the specifics of BESS operation in the VSG algorithm to be addressed.