Stability Control Method Utilizing Grid-Forming Converters for Active Symmetry in the Elastic Balance Region of the Distribution Grid

Abstract

The development of the elastic balance area within the distribution network places greater demands on the interaction between sources and loads, which impacts the stability of the power system. While achieving symmetry in active power is essential for stable operation, it is challenging to attain perfection due to various disruptions that can exacerbate frequency and voltage instability. Additionally, due to the inherent resonance characteristics of LCL filters and the time-varying nature of weak grid line impedance, grid-connected inverters may interact with the grid, potentially leading to oscillation issues. A grid-forming inverter control method that incorporates resonance suppression is proposed to address these challenges. First, a control model for the grid-forming inverter based on the Virtual Synchronous Generator (VSG) is established, enabling the system to exhibit inertia and damping characteristics. Considering the interaction between the VSG grid-connected system and the weak grid, sequence impedance models of the VSG system, which feature voltage and current double loops within the αβ coordinate system, are developed using harmonic linearization techniques. By combining the impedance analysis method, the stability of the system under weak grid conditions is evaluated using the Nyquist criterion. The validity of the analysis is confirmed through simulations. Finally, in order to ensure the effectiveness and correctness of the simulation, an experimental prototype of an NPC three-level LCL grid-forming inverter is built, and the experimental results have verified that the system has good elastic support capability and resonance suppression capability in the elastic region.

1. Introduction

With the advancement of new power systems, distributed energy sources such as photovoltaic and wind power generation, along with flexible loads, have been extensively integrated into the distribution system [1,2]. The randomness and volatility of outputs from distributed power sources, coupled with the extensive involvement of flexible loads in demand response, will significantly alter the operating characteristics of the power grid. Active power symmetry plays a vital role in this context. It refers to the balanced distribution and exchange of active power among different components within the power grid, including power sources, grids, loads, and energy storage. This change will shift the operating mode from a deterministic power balance—where power sources follow load demands—to a dynamic balance characterized by interactions between power sources and loads. As a result, balancing power and energy between these entities will become more challenging. Additionally, the safety and stability concerns related to large-scale interactions between power sources and loads are becoming increasingly prominent. The low inertia and weak damping characteristics of these systems exacerbate stability concerns [3,4]. Grid-forming inverters have gained significant attention because they behave like synchronous voltage sources and can provide some inertia support to the electrical grid. The most common control strategies for these inverters include droop control and Virtual Synchronous Generator control, both of which adjust power output in response to grid demand. Additionally, matching control and virtual oscillator control can enhance grid performance. Matching control optimizes the compatibility between the inverter and the grid, while virtual oscillator control introduces nonlinear oscillation mechanisms to support grid stability [5,6,7].

To improve the quality of grid-connected electrical power, an LCL-type filter needs to be connected in series with the inverter output side. However, there are significant resonance issues associated with this configuration [8]. Currently, LCL resonance suppression methods can be categorized into two primary approaches: (1) Passive damping methods involve inserting or paralleling resistors in the LCL filter to suppress resonance [9,10]. While passive damping does not introduce additional control loops to the system, it incurs considerable power losses. (2) Active damping methods mainly include the pre-filtering method and the state variable feedback method [11,12]. Reference [13] employs a notch filter in the system control loop to suppress LCL resonance using pole-zero cancelation, However, this method is sensitive to system parameters. Adaptive algorithms have been used in References [14,15] to reduce the parameter sensitivity issue, though this significantly increases the complexity of system control. Due to various drawbacks associated with the pre-filtering method, researchers have increasingly focused on the state variable feedback method. Commonly used state variables include inverter-side current, capacitor voltage, or current and grid current. When employing the grid current feedback method, the active damping controller utilizes a second-order differential component, which can effectively suppress LCL resonance [16,17]. However, these methods primarily target grid-connected inverters under conventional single-current loop control and are not readily applicable to VSG-based grid-connected systems.

Regarding system stability, the premise for VSG to actively underpin weak grids is that the VSG can function reliably when connected to the grid due to the VSG control strategy simulating the motion equation of synchronous generators; when the load changes, VSG can detect the changes and make corresponding adjustments, so that the inverter can provide good elastic support for the active frequency and reactive voltage stability of the system when facing interference in the elastic balance region [18]. References [19,20] established the state-space equation of a multi-machine parallel system using the state-space modeling method and analyzed the static stability of the grid-connected system. However, these studies primarily focus on traditional VSG control strategies, with limited research on the influence of LCL filters on the stability of VSG grid-connected systems.

Addressing the aforementioned issues, a grid-forming inverter control method with resonance suppression is discussed. In this method, VSG technology is used to make the grid-forming inverter have similar inertia and damping characteristics as the synchronous machine, ensuring that the grid provides elastic support for frequency and voltage while reducing the harmonic distortion rate of grid current. Additionally, the harmonic linearization method is used to build the sequential impedance model of the inverter system on the grid side. The stability of the system is then analyzed from an impedance perspective.

2. Topology and Control Strategies of VSG

Figure 1 illustrates the topology and control structure of the VSG-based grid-forming inverter under weak grid conditions. V_dc is the input DC voltage; C₁ and C₂ are the DC voltage divider capacitors; Point O is the midpoint of the DC side; v_ia, v_ib, and v_ic are the inverter output bridge arm voltages; C is the filtering capacitor; L₁ and L₂ are, respectively, the filter inductors on the inverter side and the grid side; i_1a, i_1b, and i_1c are the inverter-side three-phase currents; i_2a, i_2b, and i_2c are the grid-side currents; e_g is the grid voltage; and L_g is the grid inductor. By applying an appropriate control strategy, the voltage waveform output by the inverter can be controlled and, after filtering through the LCL filter, the output voltage is integrated into the grid.

2.1. P-f Control

The P-f loop of the VSG primarily exhibits inertia and damping characteristics by emulating the rotor dynamics of conventional synchronous generators. Combining with the synchronous generator’s second-order model [21,22], the P-f equation for the VSG is represented as follows (1):

$$\left\{\begin{array}{l}J{\displaystyle \frac{\mathrm{d}\omega }{\mathrm{d}t}}={\displaystyle \frac{P_{ref}}{{\omega }_n}}-{\displaystyle \frac{P_e}{{\omega }_n}}-D_p\left(\omega -{\omega }_n\right)\\ {\displaystyle \frac{\mathrm{d}\theta }{\mathrm{d}t}}=\omega \end{array}\right.$$

(1)

where, J-the VSG’s virtual inertia, P_ref is the given active power, ω and ω_n refer to the output and rated angular frequencies of the VSG, θ represents the phase angle, and P_e is the instantaneous active power.

2.2. Q-U Control

In the VSG, the Q-U loop primarily simulates the excitation controller of synchronous generators [21,22]. The Q-U equation can be expressed as (2):

$$\sqrt{2}{E}_r={\displaystyle \frac{1}{K_{q}s}}[\sqrt{2}(U_n-U)D_q+Q_{ref}-Q_e]$$

(2)

where K_q is the excitation constant, U is the effective value of the output voltage, D_q is the Q-U Loop droop constant, U_n is the rated effective voltage, Q_ref is the reference value for reactive power, Q_e is the instantaneous reactive power, and E_r represents the virtual internal voltage’s effective value.

The internal voltage of the VSG is jointly defined by the outer loop control, which is expressed as Equation (3):

$$\left\{\begin{array}{l}{e}_a=\sqrt{2}{E}_r\sin \theta \\ e_b=\sqrt{2}{E}_r\sin (\theta -{\displaystyle \frac{2}{3}}\pi )\\ e_c=\sqrt{2}{E}_r\sin (\theta +{\displaystyle \frac{2}{3}}\pi )\end{array}\right.$$

(3)

2.3. Virtual Impedance Control

According to the electrical characteristics of the stator, the internal voltage E_q, stator current I, and terminal voltage U of a synchronous generator exhibit the following relationship:

$$U=E_q-I(r_a+\mathrm{j}\omega L_d)$$

(4)

where r_a is the armature resistance of the synchronous generator and L_d is the synchronous inductance.

To achieve the stator electrical characteristics shown in Equation (4) for the grid-tied system, virtual impedance is employed in the VSG control strategy and transformed into the αβ coordinate system as shown in Equation (5):

$$\left\{\begin{array}{l}{u}_{\alpha ref}=e_{\alpha }-R_v{i}_{\alpha }+\omega L_v{i}_{\beta }\\ u_{\beta \mathrm{ref}}=e_{\beta }-R_v{i}_{\beta }-\omega L_v{i}_{\alpha }\end{array}\right.$$

(5)

where R_v denotes the virtual resistor, L_v denotes the virtual inductor; e_α and e_β are the αβ axis components of the voltages e_a, e_b, and e_c within VSG; i_α and i_β are the αβ axis components of the system grid-side currents; and the reference components of the voltage output from the virtual impedance loop in the αβ axis are u_αref and u_βref.

The voltage reference value obtained through the power outer-loop control and virtual impedance control in Section 2.1, Section 2.2 and Section 2.3 can be directly used as the modulation signal for SPWM, allowing the grid-connected system to exhibit inertia and damping characteristics as those of a synchronous generator. Nevertheless, since the relatively decelerated dynamic response of the outer loop control, the grid-tied power quality cannot be fully ensured. Therefore, it is necessary to add an inner-loop controller to accelerate the system’s dynamic response and improve power quality.

2.4. VSG Voltage Loop Control

The quasi-PR controller not only retains the advantages of the PR (Proportional-Resonant) controller but also reduces the impact of frequency deviation on the AC side. Therefore, this paper selects a voltage loop design based on the quasi-PR controller. The expression for the quasi-PR controller is shown in Equation (6).

$$G_{PR}(s)=K_p+{\displaystyle \frac{2K_r{\omega }_{i}s}{s^2+2{\omega }_{i}s+{\omega }_o^2}}$$

(6)

where K_p denotes the proportional factor, K_r denotes the resonant factor, and ω_i is the fundamental resonant bandwidth. In this paper, ω_i = πrad/s and ω_o is the angular frequency reference value.

The voltage loop expression of the VSG based on the quasi-PR controller G₁(s) is as follows:

$$\left\{\begin{array}{l}{i}_{\alpha ref}=G_1(s)(u_{\alpha ref}-u_{\alpha })\\ i_{\beta ref}=G_1(s)(u_{\beta ref}-u_{\beta })\end{array}\right.$$

(7)

where Kp1 is the proportional coefficient of G₁(s), K_r1 is the resonant coefficient of G₁(s), u_αref and u_βref are the voltage loop αβ axis reference components output by the virtual impedance, u_α and u_β represent the output voltages of the αβ axis components at the PCC of the system, and i_αref and i_βref are the current loop αβ axis reference components output by the voltage loop.

2.5. VSG Current Loop Control

After obtaining the reference value of the current loop controlled by the VSG voltage loop, it is necessary to adjust the grid side current of the grid-connected system through the current loop. Based on the proportional controller G₂(s), the expression of the VSG current loop can be expressed as (8)

$$\left\{\begin{array}{l}{u}_{i\alpha }=G_2(s)(i_{\alpha ref}-i_{\alpha })\\ u_{i\beta }=G_2(s)(i_{\beta ref}-i_{\beta })\end{array}\right.$$

(8)

where K_p2 is the proportional coefficient of G₂(s). The outputs of the current loop, u_iα and u_iβ, serve as the αβ axis components of the inverter’s modulation wave; i_αref and i_βref serve as the reference components of the current loop’s αβ axis, outputted by the voltage loop; and i_α and i_β represent the current of the αβ axis components at the grid-side in the system.

To address the resonance issues introduced by the LCL filter, an active damping component based on grid-side feedback is incorporated into the current control loop.

$$G_{ad}(s)=k_{ad}{s}^2$$

(9)

where k_ad represents the active damping coefficient.

3. Impedance Modeling of Grid-Forming Inverters Based on VSG Under Weak Grid Conditions

Considering that the three-phase grid-connected system in this paper does not control the dq coordinate system, and the advantages of harmonic linearization modeling method are clear physical meaning, convenient measurement and verification of sequential impedance model, and relatively simple stability criteria, in order to create the impedance model for the three-phase grid-connected system, this paper uses the harmonic linearization modeling technique.

3.1. Sequence Impedance Modeling

In the time domain, after injecting a small signal perturbation on the grid side, the expression for the output voltage u_a(t) of the system and the grid-side current i_2a(t) are as follows:

$$\begin{array}{ll}{u}_a(t)=& V_1\cos (2\pi f_{1}t)+V_{\mathrm{p}}\cos (2\pi f_{\mathrm{p}}t+{\phi }_{v\mathrm{p}})\\ & +V_{\mathrm{n}}\cos (2\pi f_{\mathrm{n}}t+{\phi }_{v\mathrm{n}})\end{array}$$

(10)

$$\begin{array}{ll}{i}_{2a}(t)=& I_1\cos (2\pi f_{1}t+{\phi }_{i1})+I_{\mathrm{p}}\cos (2\pi f_{\mathrm{p}}t+{\phi }_{i\mathrm{p}})\\ & +I_{\mathrm{n}}\cos (2\pi f_{\mathrm{n}}t+{\phi }_{i\mathrm{n}})\end{array}$$

(11)

where V₁, V_p, and V_n represent the amplitudes of the disturbance voltages of each sequence, respectively; I₁, I_p, and I_n represent the corresponding response current amplitudes; f₁, f_p, and f_n represent the corresponding perturbation frequencies; φ_vp and φ_vn are the initial phase angles of the positive-sequence and negative-sequence disturbance voltages, respectively; φ_i1, φ_ip and φ_in are the initial phase angles of the corresponding response currents.

Equations (10) and (11) can be transformed into the frequency domain as follows:

$$\left\{\begin{array}{l}{u}_a[f]=\left\{\begin{array}{l}{\mathit{V}}_1,f=\pm f_1\\ {\mathit{V}}_{\mathrm{p}},f=\pm f_{\mathrm{p}}\\ {\mathit{V}}_{\mathrm{n}},f=\pm f_{\mathrm{n}}\end{array}\right.\\ i_{2a}[f]=\left\{\begin{array}{l}{\mathit{I}}_1,f=\pm f_1\\ {\mathit{I}}_{\mathrm{p}},f=\pm f_{\mathrm{p}}\\ {\mathit{I}}_{\mathrm{n}},f=\pm f_{\mathrm{n}}\end{array}\right.\end{array}\right.$$

(12)

where V₁ = V₁/2; V_p = (V_p/2)e ± jφ_vp; V_n = (V_n/2)e ± jφ_vn; I₁ = (I₁/2)e ± jφ_i1; I_p = (I_p/2)e ± jφ_ip; I_n = (I_n/2)e ± jφ_in.

Since the grid-connected system studied in this paper is three-phase symmetric, Equation (12) can be transformed into the αβ coordinate system.

$$\left\{\begin{array}{l}{u}_{\alpha }[f]=\left\{\begin{array}{l}{\mathit{V}}_1,f=\pm f_1\\ {\mathit{V}}_{\mathrm{p}},f=\pm f_{\mathrm{p}}\\ {\mathit{V}}_{\mathrm{n}},f=\pm f_{\mathrm{n}}\end{array}\right.\\ u_{\beta }[f]=\left\{\begin{array}{l}\mp \mathrm{j}{\mathit{V}}_1,f=\pm f_1\\ \mp \mathrm{j}{\mathit{V}}_{\mathrm{p}},f=\pm f_{\mathrm{p}}\\ \pm \mathrm{j}{\mathit{V}}_{\mathrm{n}},f=\pm f_{\mathrm{n}}\end{array}\right.\end{array}\right.$$

(13)

$$\left\{\begin{array}{l}{i}_{2\alpha }[f]=\left\{\begin{array}{l}{\mathit{I}}_1,f=\pm f_1\\ {\mathit{I}}_{\mathrm{p}},f=\pm f_{\mathrm{p}}\\ {\mathit{I}}_{\mathrm{n}},f=\pm f_{\mathrm{n}}\end{array}\right.\\ i_{2\beta }[f]=\left\{\begin{array}{l}\mp \mathrm{j}{\mathit{I}}_1,f=\pm f_1\\ \mp \mathrm{j}{\mathit{I}}_{\mathrm{p}},f=\pm f_{\mathrm{p}}\\ \pm \mathrm{j}{\mathit{I}}_{\mathrm{n}},f=\pm f_{\mathrm{n}}\end{array}\right.\end{array}\right.$$

(14)

According to the fundamental principles of the modeling method of harmonic linearization, the expression of the function between the disturbance voltage and the response current can be obtained.

$$\left\{\begin{array}{l}{Z}_{v\mathrm{p}}=-{\displaystyle \frac{{\mathit{V}}_{\mathrm{p}}}{{\mathit{I}}_{\mathrm{p}}}}\\ Z_{v\mathrm{p}}=-{\displaystyle \frac{{\mathit{V}}_{\mathrm{n}}}{{\mathit{I}}_{\mathrm{n}}}}\end{array}\right.$$

(15)

where Z_vp(s) and Z_vn(s) represent the positive and negative sequence impedance of the VSG system. By combining the above three layers of functional relationships, the functional relationship between V_p(V_n) and I_p(I_n) can be solved, and the sequence impedance model of the VSG system can be obtained. Based on the operating principle of the system shown in Figure 1, the small signal flow chart of the system as shown in Figure 2.

Figure 2 illustrates the small signal flow diagram of the VSG system, where the main circuit is composed of filters, and the control loop consists of the VSG power loop, virtual impedance segment, voltage–current loop, and modulation segment. In the figure, the subscript x = p, n represents the positive and negative sequence disturbance component at frequency f. Based on Figure 2, the impedance modeling of the VSG system can be divided into five parts. Below, the impedance models for these five components are established.

3.2. Modeling of the Main Circuit for the VSG Grid-Connected Systems

Injecting small signal voltage disturbances in positive and negative sequences with frequencies f_p and f_n on the grid side will generate current response signals with frequencies f_p and f_n, respectively. Taking phase a as an example, Figure 3 represents the harmonic small signal equivalent model of the main circuit. In the figure, the subscript x = p indicates the positive sequence disturbance component at frequency f_p, while x = n indicates the negative sequence disturbance component at frequency f_n.

Based on Figure 3, the corresponding main circuit small signal expression is (16)

$$v_{ia}[f]=[L_1{L}_{2}C{s}^3+(L_1+L_2)s]i_{2a}[f]+(L_{1}C{s}^2+1)u_a[f]$$

(16)

3.3. VSG Power Outer Loop Modeling

It is known that the expression for the output power of the system in the αβ coordinate system can be expressed as (17)

$$\left\{\begin{array}{l}{P}_e=1.5G_f(s)(u_{\alpha }{i}_{2\alpha }+u_{\beta }{i}_{2\beta })\\ Q_e=1.5G_f(s)(u_{\beta }{i}_{2\alpha }-u_{\alpha }{i}_{2\beta })\end{array}\right.$$

(17)

where G_f(s) = ωl/(s + ωl) represents the transfer function of a low-pass filter and ωl represents the cutoff frequency of the low-pass filter.

By substituting Equations (14) and (15) into Equation (18), and combining the frequency domain convolution theorem while neglecting the effects of second-order small-signal disturbance terms, the expressions for P_e and Q_e of the system in the frequency domain can be expressed as (18)

$$P_e=\left\{\begin{array}{l}3G_f(s)({\mathit{V}}_1{\mathit{I}}_1^{*}+{\mathit{I}}_1{\mathit{V}}_1^{*}), f=\mathrm{dc}\\ 3G_f(s)({\mathit{I}}_{\mathrm{p}}{\mathit{V}}_1^{*}+{\mathit{V}}_{\mathrm{p}}{\mathit{I}}_1^{*}),f=\pm (f_{\mathrm{p}}-f_1)\\ 3G_f(s)({\mathit{V}}_1{\mathit{I}}_{\mathrm{n}}+{\mathit{V}}_{\mathrm{n}}{\mathit{I}}_1), f=\pm (f_{\mathrm{n}}+f_1)\end{array}\right.$$

(18)

$$Q_e=\left\{\begin{array}{l}3\mathrm{j}{G}_f(s)(-{\mathit{V}}_1{\mathit{I}}_1^{*}+{\mathit{I}}_1{\mathit{V}}_1^{*}),f=\mathrm{dc}\\ \pm 3\mathrm{j}{G}_f(s)({\mathit{I}}_{\mathrm{p}}{\mathit{V}}_1^{*}-{\mathit{V}}_{\mathrm{p}}{\mathit{I}}_1^{*}),f=\pm (f_{\mathrm{p}}-f_1)\\ \pm 3\mathrm{j}{G}_f(s)(-{\mathit{V}}_1{\mathit{I}}_{\mathrm{n}}+{\mathit{V}}_{\mathrm{n}}{\mathit{I}}_1),f=\pm (f_{\mathrm{n}}+f_1)\end{array}\right.$$

(19)

The expression for θ can be obtained from the VSG active power controller as (20)

$$\theta (s)=M(s)(P_{ref}-P_e+D_p{\omega }_n^2)$$

(20)

In the above formula, M(s) = 1/(J_ωns2 + D_ωns).

By substituting Equation (18) into Equation (20), the expressions about θ can be expressed as (21)

$$\theta [f]=\left\{\begin{array}{l}-3G_f(s)M(s)({\mathit{I}}_{\mathrm{p}}{\mathit{V}}_1^{*}+{\mathit{V}}_{\mathrm{p}}{\mathit{I}}_1^{*}),\hspace{1em}\hspace{1em}\hspace{1em}f=\pm (f_{\mathrm{p}}-f_1)\\ -3G_f(s)M(s)({\mathit{V}}_1{\mathit{I}}_{\mathrm{n}}+{\mathit{V}}_{\mathrm{n}}{\mathit{I}}_1),\hspace{1em}\hspace{1em}\hspace{1em}f=\pm (f_{\mathrm{n}}+f_1)\end{array}\right.$$

(21)

Regarding the output θ of the VSG active power-frequency loop, the corresponding phase angle perturbation ∆θ can be introduced by voltage perturbations, which can be expressed as (22)

$$\Delta \theta [f]=\left\{\begin{array}{l}-3G_f(s)M(s)({\mathit{I}}_{\mathrm{p}}{\mathit{V}}_1^{*}+{\mathit{V}}_{\mathrm{p}}{\mathit{I}}_1^{*}),f=\pm (f_{\mathrm{p}}-f_1)\\ -3G_f(s)M(s)({\mathit{V}}_1{\mathit{I}}_{\mathrm{n}}+{\mathit{V}}_{\mathrm{n}}{\mathit{I}}_1),f=\pm (f_{\mathrm{n}}+f_1)\end{array}\right.$$

(22)

The phase angle disturbance ∆θ yields θ = θ₁ + ∆θ, where θ₁ is the fundamental wave angle of the grid-connected system, obtaining Equation (23).

$$\begin{array}{ll}\cos \theta [f]& =\cos \left({\theta }_1[f]+\Delta \theta [f]\right)\\ & \approx \cos {\theta }_1[f]-\Delta \theta [f]*\sin {\theta }_1[f]\end{array}$$

(23)

where θ = θ₁ + ∆θ, with φ₁ denoting the power angle of the VSG, φ₁ ≈ arsin[PNω_nLv/(V_NV₁)].

In the frequency domain, the value of θ₁ can be calculated as (24)

$$\left\{\begin{array}{l}\cos {\theta }_1[f]=e^{\pm \mathrm{j}{\phi }_1}/2,\hspace{1em}\hspace{1em}\hspace{1em}f=\pm f_1\\ \sin {\theta }_1[f]=-e^{\pm \mathrm{j}({\phi }_1+\pi /2)}/2, f=\pm f_1\end{array}\right.$$

(24)

By substituting Equations (22) and (24) into Equation (23), we can obtain Equation (25).

$$\cos \theta [f]=\left\{\begin{array}{l}{\displaystyle \frac{1}{2}}{e}^{\pm \mathrm{j}{\phi }_1},\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}f=\pm f_1\\ -{\displaystyle \frac{3}{2}}{e}^{\pm \mathrm{j}{\phi }_{\mathrm{vir}}}{G}_f(s\mp \mathrm{j}2\pi f_1)M(s\mp \mathrm{j}2\pi f_1)•\\ ({\mathit{I}}_{\mathrm{p}}{\mathit{V}}_1^{*}+{\mathit{V}}_{\mathrm{p}}{\mathit{I}}_1^{*}),\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}f=\pm f_{\mathrm{p}}\\ -{\displaystyle \frac{3}{2}}{e}^{\mp \mathrm{j}{\phi }_{\mathrm{vir}}}{G}_f(s\pm \mathrm{j}2\pi f_1)M(s\pm \mathrm{j}2\pi f_1)•\\ ({\mathit{V}}_1{\mathit{I}}_{\mathrm{n}}+{\mathit{V}}_{\mathrm{n}}{\mathit{I}}_1),\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}f=\pm f_{\mathrm{n}}\end{array}\right.$$

(25)

where φ_vir = φ₁ + π/2.

From the reactive power–voltage controller Equation (2) of the VSG, the internal voltage magnitude $\sqrt{2}$E_m in the frequency domain is given by

$$\sqrt{2}{E}_r[f]=\left\{\begin{array}{l}{\mathit{V}}_1,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} f=\mathrm{dc}\\ {\displaystyle \frac{-2D_q{\mathit{V}}_{\mathrm{p}}\mp 3\mathrm{j}{G}_f(s)({\mathit{I}}_{\mathrm{p}}{\mathit{V}}_1^{*}-{\mathit{V}}_{\mathrm{p}}{\mathit{I}}_1^{*})}{K_{q}s}},f=\pm (f_{\mathrm{p}}-f_1)\\ {\displaystyle \frac{-2D_q{\mathit{V}}_{\mathrm{n}}\pm 3\mathrm{j}{G}_f(s)({\mathit{V}}_1{\mathit{I}}_{\mathrm{n}}-{\mathit{V}}_{\mathrm{n}}{\mathit{I}}_1)}{K_{q}s}},f=\pm (f_{\mathrm{n}}+f_1)\end{array}\right.$$

(26)

By combining Equations (25) and (26) with the VSG power outer loop inner voltage Equation (3), we can neglect the effects of second-order small-signal disturbance terms and transform them into the αβ coordinate system. Considering that the system is three-phase balanced and symmetrical, e_α[f] = e_a[f], the expression for e_α is obtained as (27)

$$e_{\alpha }[f]=\left\{\begin{array}{l}{H}_1(s){\mathit{V}}_{\mathrm{p}}+H_2(s)({\mathit{I}}_{\mathrm{p}}{\mathit{V}}_1^{*}-{\mathit{V}}_{\mathrm{p}}{\mathit{I}}_1^{*})+\\ H_3(s)({\mathit{I}}_{\mathrm{p}}{\mathit{V}}_1^{*}+{\mathit{V}}_{\mathrm{p}}{\mathit{I}}_1^{*}),f=\pm f_{\mathrm{p}}\\ H_1^{*}(s){\mathit{V}}_{\mathrm{n}}-H_2^{*}(s)({\mathit{V}}_1{\mathit{I}}_{\mathrm{n}}-{\mathit{V}}_{\mathrm{n}}{\mathit{I}}_1)+\\ H_3^{*}(s)({\mathit{V}}_1{\mathit{I}}_{\mathrm{n}}+{\mathit{V}}_{\mathrm{n}}{\mathit{I}}_1),f=\pm f_{\mathrm{n}}\end{array}\right.$$

(27)

where

$$\left\{\begin{array}{l}{H}_1(s)=-D_q{e}^{\pm \mathrm{j}{\phi }_1}/K_q(s\mp \mathrm{j}2\pi f_1)\\ H_2(s)=\mp 1.5\mathrm{j}{G}_f(s\mp \mathrm{j}2\pi f_1)e^{\pm \mathrm{j}{\phi }_1}/K_q(s\mp \mathrm{j}2\pi f_1)\\ H_3(s)=-1.5V_1{e}^{\pm \mathrm{j}{\phi }_{vir}}{G}_f(s\mp \mathrm{j}2\pi f_1)M(s\mp \mathrm{j}2\pi f_1)\end{array}\right.$$

(28)

3.4. Virtual Impedance Modeling of the VSG

As shown in Figure 2, the output of the VSG power outer loop, after passing through the virtual impedance stage, becomes the reference quantity for the output voltage loop. Combining Equation (5), the α-axis reference component u_αref of the voltage loop is given by (29)

$$u_{\alpha ref}[f]=\left\{\begin{array}{l}{H}_1(s){\mathit{V}}_{\mathrm{p}}+H_2(s)({\mathit{I}}_{\mathrm{p}}{\mathit{V}}_1^{*}-{\mathit{V}}_{\mathrm{p}}{\mathit{I}}_1^{*})+H_3(s)({\mathit{I}}_{\mathrm{p}}{\mathit{V}}_1^{*}+{\mathit{V}}_{\mathrm{p}}{\mathit{I}}_1^{*})-\\ R_v{\mathit{I}}_{\mathrm{p}}\mp \mathrm{j}{\omega }_n{L}_v{\mathit{I}}_{\mathrm{p}},\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}f=\pm f_{\mathrm{p}}\\ H_1^{*}(s){\mathit{V}}_{\mathrm{n}}-H_2^{*}(s)({\mathit{V}}_1{\mathit{I}}_{\mathrm{n}}-{\mathit{V}}_{\mathrm{n}}{\mathit{I}}_1)]+H_3^{*}(s)({\mathit{V}}_1{\mathit{I}}_{\mathrm{n}}+{\mathit{V}}_{\mathrm{n}}{\mathit{I}}_1)-\\ R_v{\mathit{I}}_{\mathrm{n}}\pm \mathrm{j}{\omega }_n{L}_v{\mathit{I}}_{\mathrm{n}},\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}f=\pm f_{\mathrm{n}}\end{array}\right.$$

(29)

3.5. VSG Inner Loop Modeling

Using the voltage loop reference obtained from Equation (29) and the voltage outer loop control principle in Figure 2, the α-axis current loop reference component i_αref is given by

$$i_{\alpha ref}[f]=\left\{\begin{array}{l}{G}_1\left(s\right)(u_{\alpha ref}[f]-{\mathit{V}}_{\mathrm{p}}), f=\pm f_{\mathrm{p}}\\ G_1\left(s\right)(u_{\alpha ref}[f]-{\mathit{V}}_{\mathrm{n}}),f=\pm f_{\mathrm{n}}\end{array}\right.$$

(30)

From the small-signal flowchart of the VSG system depicted in Figure 2, it can be known that the α-axis modulation wave of the current loop output is denoted as u_iα[f]:

$$u_{i\alpha }[f]=\left\{\begin{array}{l}{G}_2(s\left)\right(i_{\alpha ref}[f]-{\mathit{I}}_{\mathrm{p}})-K_{ad}{s}^2{\mathit{I}}_{\mathrm{p}}, f=\pm f_{\mathrm{p}}\\ G_2(s\left)\right(i_{\beta ref}[f]-{\mathit{I}}_{\mathrm{n}})-K_{ad}{s}^2{\mathit{I}}_{\mathrm{n}},f=\pm f_{\mathrm{n}}\end{array}\right.$$

(31)

3.6. Modulation Link Modeling

Given that the grid-connected system is balanced in three phases, u_ia[f] = u_iα[f]. The PWM modulation signal in the αβ coordinate system is derived from the output of the inner loop. By transforming it into the abc coordinate system, the response signal of the voltage of the bridge arm can be obtained as (32)

$$v_{ia}[f]=K_{\mathrm{PWM}}{u}_{ia}[f]G_d(s)$$

(32)

By combining Equations (16), (31), and (32), the sequence impedances Z_vp and Z_vn of the grid-forming inverter can be determined as follows:

$$Z_{v\mathrm{p}}(s)={\displaystyle \frac{-0.5V_1{K}_1(s)[H_2(s)+H_3(s)]+(R_v\pm \mathrm{j}{\omega }_n{L}_v)K_1(s)+K_2(s)+L_1{L}_{2}C{s}^3+(L_1+L_2)s}{0.5I_1{K}_1(s)[H_2(s)-H_3(s)]+K_1(s)[1-H_1(s)]+L_{1}C{s}^2+1}}$$

(33)

$$Z_{v\mathrm{n}}(s)={\displaystyle \frac{0.5V_1{K}_1(s)[H_2^{*}(s)-H_3^{*}(s)]+[R_v\mp \mathrm{j}{\omega }_n{L}_v]K_1(s)+K_2(s)+L_1{L}_{2}C{s}^3+(L_1+L_2)s}{-0.5I_1{K}_1(s)[H_2^{*}(s)+H_3^{*}(s)]+K_1(s)[1-H_1^{*}(s)]+L_{1}C{s}^2+1}}$$

(34)

where

$$\begin{array}{l}{K}_1(s)=K_{\mathrm{PWM}}{G}_1\left(s\right)G_2\left(s\right)G_d(s)\\ K_2(s)=K_{\mathrm{PWM}}{K}_{ad}{s}^2{G}_d(s)+K_{\mathrm{PWM}}{G}_2\left(s\right)G_d(s)\end{array}$$

(35)

$$\begin{array}{l}{K}_1(s)=K_{\mathrm{PWM}}{G}_1\left(s\right)G_2\left(s\right)G_d(s)\\ K_2(s)=K_{\mathrm{PWM}}{K}_{ad}{s}^2{G}_d(s)+K_{\mathrm{PWM}}{G}_2\left(s\right)G_d(s)\end{array}$$

(36)

4. Stability Analysis

To further analyze the stability of the VSG system, this paper analyzes the impedance proportion of grid-forming inverters and weak grids based on the Nyquist criterion. Given that the VSG system exhibits symmetrical characteristics in three-phase balance during grid-connected operation, the positive and negative sequence impedances are decoupled in the system. Therefore, the impedance proportion applied to determine the stability of the grid-connected system is calculated using Equation (36).

$$\left\{\begin{array}{l}{H}_{v\mathrm{p}}(s)={\displaystyle \frac{Z_{v\mathrm{p}}(s)}{Z_{g\mathrm{p}}(s)}}\\ H_{v\mathrm{n}}(s)={\displaystyle \frac{Z_{v\mathrm{n}}(s)}{Z_{g\mathrm{n}}(s)}}\end{array}\right.$$

(37)

Here, H_vp(s) and H_vn(s) represent the positive and negative sequence impedance proportions, respectively, between the grid-forming inverter and the weak grid. Z_gp(s) and Z_gn(s) represent the positive and negative sequence impedances of the grid, respectively, and Z_g(s) = Z_gp(s) = Z_gn(s) = sL_g.

Figure 4 shows the Nyquist diagrams of the grid-connected system’s H_vp(s) and H_vn(s) under different weak grid conditions, and

Table 1. Analysis of the Nyqist diagram in Figure 4.

Lg/mH	N + (Hvp(s))	N − (Hvn(s))	2[(N+) − (N−)]	Stability
0.5	1	1	2	stable
1.5	1	1	2	stable
3	1	1	2	stable

shows the analysis of the Nyquist diagram in Figure 4. Among them, the red solid line represents the system’s positive sequence impedance ratio H_vp(s), and the blue dashed line represents the system’s negative sequence impedance ratio H_vn(s). When the difference between the number of positive crossings N+ and the number of negative crossings N- of the open-loop frequency characteristic curve of the system in the section of the negative real axis (−1, −∞) multiplied by 2 is equal to the number of right poles of the open-loop system, the closed-loop system is stable. Both H_vp(s) and H_vn(s) have two poles in the right half-plane. Therefore, when the grid-connected system is stable, the Nyquist curves of both H_vp(s) and H_vn(s) should cross the negative real axis to the left of the point (−1, j0) positively once. It can be seen from Figure 4 that as the grid becomes weaker, the Nyquist curves of H_vp(s) and H_vn(s) each cross the negative real axis to the left of the point (−1, j0) positively once. From this, it can be concluded that the grid-connected system with the active damping link added can operate stably under weak grid conditions.

5. Simulation Analysis

To assess the efficacy of the proposed VSG control strategy for the grid-forming inverter, a simulation model based on VSG is constructed using MATLAB/Simulink 2023a under weak grid conditions. The results of this simulation are then analyzed, with the parameters outlined in

Table 2. Simulation model parameters for the system.

Parameter	Value	Parameter	Value
L1	2 mH	L2	1 mH
C	50 μF	Lg	0~3 mH
Vdc	300 V	Ug	76.7 V
Pref	2000 W	Qref	0 Var
J	0.5	Dp	5
Kq	5.8	Dq	368
Rv	0.1Ω	Lv	3 mH
Kp1	0.01	Kr1	2.1
Kp2	0.06	Kad	6*10−9
fs	10 kHz	fsw	10 kHz

.

5.1. Verification of the Dynamic Performance of the VSG

5.1.1. Participate in the Grid Frequency Modulation Capability Verification

(1) Simulation of participation in the grid frequency modulation capability verification 1: At the initial moment of 0 s, the grid-connected system operates at rated conditions with P_ref = 0 W and Q_ref = 0 Var. At 0.4 s, a sudden drop in the grid angular frequency of 0.2 π rad/s is introduced, and, at 0.75 s, the grid angular frequency is restored to its rated state. The output power, output voltage, and grid-side current of the grid-connected system are illustrated in Figure 5.

In Figure 5, at 0 s, the system operates at rated conditions, with the output P and Q tracking the given power values of P_ref and Q_ref. Following a sudden drop in the grid angular frequency of 0.2 π rad/s at 0.4 s, the output P of the system increases to 2000 W, accompanied by changes in the grid-side current. At 0.75 s, when the grid angular frequency is restored to its rated state, the output P returns to the set value, and the grid-side current also reverts to its initial condition. During the fluctuations in grid frequency, the VSG can adjust its active output through primary frequency control, ensuring the stability of the system. The variations in output voltage and grid-side current are consistent with changes in the system’s active power, indicating that the VSG control strategy proposed in this paper is effective in participating in grid frequency regulation.

(2) Simulation of participation in the grid frequency modulation capability verification 2: At the initial moment of 0 s, the grid-connected system operates at rated conditions with P_ref = 0 W and Q_ref = 0 Var. At 0.54 s, a sudden increase in the grid angular frequency of 0.2 π rad/s is introduced, and, at 0.75 s, the grid angular frequency is restored to its rated state. The output power, output voltage, and grid-side current of the grid-connected system are illustrated in Figure 6.

As shown in Figure 6, at 0 s, the grid-connected system operates at rated conditions, with the output P and Q tracking the specified power values of P_ref and Q_ref. Following a sudden increase in the grid angular frequency of 0.2 π rad/s at 0.4 s, the output P of the grid-connected system decreases to −2000 W, accompanied by corresponding changes in the grid-side current. At 0.75 s, when the grid angular frequency is restored to its rated state, the output P returns to the set value, and the grid-side current also reverts to its initial condition. During fluctuations in grid frequency, the VSG can adjust its active output through primary frequency control, ensuring the stability of the system. The variations in output voltage and grid-side current are consistent with changes in the system’s active power, further demonstrating that the VSG control strategy proposed in this paper effectively participates in grid frequency regulation.

(3) Validation of the ability to participate in frequency regulation under unbalanced grid conditions: In Figure 7, in order to verify the system’s frequency support capability under an unbalanced power grid, at 0 s, the grid-connected system operates at rated conditions, with the output P and Q tracking the specified power values of P_ref and Q_ref. Following a sudden increase in the grid angular frequency of 0.15 π rad/s at 0.4 s, the output P of the grid-connected system decreases to 15,000 W, accompanied by corresponding changes in the grid-side current. At 0.75 s, when the grid angular frequency is restored to its rated state, the output P returns to the set value, and the grid-side current also reverts to its initial condition. It can be seen that under the conditions of an unbalanced power grid, during fluctuations in grid frequency, the VSG can adjust its active output through primary frequency control, ensuring the stability of the system. The variations in output voltage and grid-side current are consistent with changes in the system’s active power, indicating that the VSG control strategy proposed in this paper can also effectively participate in grid frequency regulation under the conditions of a three-phase unbalance in the power grid.

5.1.2. Participate in the Grid Voltage Modulation Capability Verification

(1) Simulation of participation in the grid voltage modulation capability verification 1: At the initial moment of 0 s, the grid-connected system operates at rated conditions with P_ref = 0 W and Q_ref = 0 Var. At 0.4 s, a sudden drop of 2.5% in the grid voltage amplitude is introduced, and, at 0.75 s, the grid voltage amplitude returns to the set value. The output power, output voltage, and grid-side current of the grid-connected system are illustrated in Figure 7 Simulation 1 of Participation in the Voltage Regulation of the Power Grid.

From Figure 7, at 0 s, the grid-connected system operates at rated conditions, with the output P and Q tracking the specified values of P_ref and Q_ref. Following a sudden drop of 2.5% in the grid voltage amplitude at 0.4 s, the output Q of the system increases to 1000 Var, accompanied by corresponding changes in the grid-side current. At 0.75 s, when the grid voltage is restored to its rated state, the output Q returns to the set value, and the grid-side current also reverts to its initial condition. During fluctuations in grid voltage amplitude, the VSG can adjust its reactive output through voltage regulation to ensure the stability of the system. The variations in output voltage and grid-side current are consistent with changes in the system’s Q, indicating that the VSG control strategy proposed in this paper effectively participates in grid voltage regulation.

(2) Simulation of participation in the grid voltage modulation capability verification 2: At the initial moment of 0 s, the grid-connected system operates at rated conditions with P_ref = 0 W and Q_ref = 0 Var. At 0.4 s, a sudden increase of 2.5% in the grid voltage amplitude is introduced, and, at 0.75 s, the grid voltage amplitude returns to the set value. The output power, output voltage, and grid-side current of the grid-connected system are illustrated in Figure 8 Simulation 2 of Participation in the Voltage Regulation of the Power Grid.

As shown in Figure 9, at 0 s, the grid-connected system operates at rated conditions, with the output P and Q tracking the specified values of P_ref and Q_ref. Following a sudden increase of 2.5% in the grid voltage amplitude at 0.4 s, the output Q of the system decreases to −1000 Var, accompanied by corresponding changes in the grid-side current. At 0.75 s, when the grid voltage amplitude is restored to its rated state, the output Q returns to the set value, and the grid-side current also reverts to its initial condition. During fluctuations in grid voltage amplitude, the VSG can adjust its reactive output through voltage control to ensure the stability of the system. The variations in output voltage and grid-side current are consistent with changes in the system’s Q, further demonstrating that the VSG control strategy proposed in this paper effectively participates in grid voltage regulation.

(3) Validation of participation in voltage regulation capability under unbalanced grid conditions: In Figure 10, to verify the system’s voltage support capability under unbalanced grid conditions, at 0 s, the grid-connected system operates at rated conditions, with the output P and Q tracking the specified values of P_ref and Q_ref. Following a sudden drop of 1.875% in the grid voltage amplitude at 0.4 s, the output Q of the system increases to 750 Var, accompanied by corresponding changes in the grid-side current. At 0.75 s, when the grid voltage is restored to its rated state, the output Q returns to the set value, and the grid-side current also reverts to its initial condition. It can be seen that under the conditions of an unbalanced grid, during fluctuations in grid voltage amplitude, the VSG can adjust its reactive output through voltage regulation to ensure the stability of the system. The variations in output voltage and grid-side current are consistent with changes in the system’s Q, indicating that the VSG control strategy proposed in this paper can effectively participate in grid voltage regulation under conditions of three-phase unbalance in the grid.

From the above analysis, it can be seen that adopting the VSG control strategy can enable the system to have good frequency and voltage active support capabilities in the elastic region. In the case of a weak grid, the system will have good elastic support capabilities.

5.2. Verification of Steady-State Performance of the VSG

Figure 11, Figure 12 and Figure 13 present the simulation waveforms of the grid-forming inverter equipped with an additional active damping control strategy when the grid-connected system operates at rated conditions, with P_ref = 2000 W and Q_ref = 0 Var. The simulations consider grid line impedances of 0 mH, 1.5 MH, and 3 mH, respectively. Here, u_i (i = a, b, c) are the three-phase grid voltage waveforms, while i_n(n = a, b, c) denotes the three-phase grid current waveforms.

From Figure 11, Figure 12 and Figure 13 and

Table 3. Analysis of steady-state performance simulation results of VSG.

Lg/mH	THD
0	0.25%
1.5	0.36%
3	0.37%

, the grid-forming inverter with VSG control operates stably under all three grid conditions. The three-phase grid currents are sine-balanced and exhibit relatively low Total Harmonic Distortion (THD). Specifically, when L_g = 0 mH, the THD is 0.25%; when L_g = 1.5 mH, the THD is 0.36%; and when L_g = 3 mH, the THD is 0.37%. These values meet the grid connection requirements.

5.3. IEEE Equivalent Power System Model Validation

In order to verify the effectiveness and reliability of the designed system and method operating in a large power grid, this section establishes an IEEE power system equivalent model of the inverter based on the scheme designed in this paper. The specific topology diagram is shown in Figure 14, and the specific parameters are shown in

Table 4. Simulation parameter settings.

Parameters	Numeric Value
Zone 1 generator capacity (MW)	198
Zone 2 generator capacity (MW)	198
Rated power output of the PV array (KW)	500
The number of photovoltaic power station units	150
Voltage side capacitor DC voltage (V)	1500

.

5.3.1. Verification of System Frequency Regulation Capability

When the system reaches a steady state, at t = 5 s, there is a sudden increase in active load in the power grid, and the corresponding frequency fluctuation curve and inverter output active power change are shown in Figure 15. After the sudden change occurs, the inverter starts to increase power to suppress the frequency drop. Finally, at t = 12 s, the inverter increases active power by 10 MW, and the frequency stabilizes at 49.925 Hz, achieving active support for the power grid frequency.

5.3.2. Verification of System Voltage Regulation Capability

When the system reaches a steady state, at t = 5 s, there is a sudden increase in reactive load in the power grid, and the corresponding voltage fluctuation curve and inverter output reactive power change are shown in Figure 16. After the sudden change occurs, the inverter starts to increase reactive power to suppress the voltage drop. Finally, at t = 18 s, the inverter increases reactive power by 25 MVar, and the voltage stabilizes at 0.97 p.u, achieving reactive support for the power grid voltage.

In summary, after equating the system proposed in this article to the IEEE power system equivalent model, in the large-scale power grid system, when facing active and reactive load disturbances, the system will actively adjust the output of active and reactive power according to the sudden changes in power grid load to cope with changes in system frequency and voltage, thereby achieving active support for regional elastic power grids.

5.4. Experimental Validation

In order to ensure the effectiveness and correctness of the simulation, an experimental prototype of an NPC three-level LCL grid-forming inverter is built, and the corresponding experimental parameters are consistent with those of the simulation in the experimental platform.

Figure 17 shows the comparative experimental waveforms of the grid-connected system when the grid impedances are L_g = 0 mH, 1.5 mH, and 3 mH with and without the active damping loop. From Figure 17a,c,e, it can be observed that when the system does not include the active damping loop, the THD of the grid current changes little with increasing grid impedance and does not meet the grid connection requirements, exhibiting significant resonance phenomena. In contrast, Figure 17b,d,f show that after incorporating the active damping loop, the resonance phenomena of the grid-connected inverter are effectively suppressed compared to the case without it. At this point, for the three different grid impedance scenarios, the THD of the grid current is 1.518%, 1.7664%, and 1.790%, respectively, which shows significant improvement compared to the THD values under the same conditions without the active damping loop. All these values meet the grid connection requirements, indicating that the system has good current quality.

According to the comprehensive analysis, the resonance phenomenon of the grid-forming inverter is effectively suppressed under weak grid conditions by adopting the VSG control with the active damping loop proposed in this paper, and the system remains stable, has good current quality, and meets the grid-connected standard.

6. Conclusions

This paper examines the grid-side components of power systems that incorporate new energy generation sources and considers the frequency and voltage support capabilities required by elastic regional power grids. A control system based on Virtual Synchronous Generator (VSG) technology is designed. A small-signal sequential impedance model of the VSG system in the αβ coordinate frame is established using harmonic linearization methods. The stability of the system is analyzed from an impedance perspective, leading to the following conclusions, which were verified by comparative experiments.

(1) The control system based on VSG technology effectively implements grid-side management for power systems utilizing wind and photovoltaic energy. It provides essential elastic support for voltage and frequency stabilization, effectively suppresses resonance, and avoids causing oscillation, thereby enhancing power quality.

(2) The grid-connected system employing VSG technology demonstrated stable operation under varying grid impedance conditions, indicating its robustness and adaptability across different operational environments, thereby providing a resilient area for the stable operation of the power grid.