Large-Signal Impedance Modeling and Stability Analysis of a Grid-Connected Inverter Considering the Influence of a Limiter in Different Control Links

Abstract

The limiter in grid-connected inverter control may cause sustained oscillation in the system. The large-signal impedance model is provided since the traditional small-signal impedance model cannot accurately describe the characteristics of the limiter. In this paper, there are three established large-signal impedance models of grid-connected inverters that take into account the limiter in the phase-locked loop (PLL), the limiter in the current control loop and the limiter in the pulse width modulation (PWM). Based on the established models, the influences of these different limiters on the output impedance characteristics of the grid-connected inverter are discussed. Furthermore, the stability of the system considering the influence of different limiters is analyzed and the oscillation frequency and amplitude of the system are predicted. The simulation verifies the accuracy of the large-signal impedance model and the predicted oscillation frequency and amplitude.

1. Introduction

With the scarcity of fossil fuels and the worsening of environmental degradation, renewable energy power production technology has advanced significantly [1]. However, as the types and installed capacity of renewable energy generation expand, it is more challenging to deal with a series of oscillation issues produced by the connection of a large number of grid-connected inverters to a weak grid [2,3]. Actual engineering data show that the wind power grid-connected system may have non-growing and persistent constant amplitude oscillations if it reaches saturation or control limits [4]. When operational circumstances vary, oscillations continue to exist in the system, appearing and disappearing. In some cases, this damages the equipment or eventually triggers certain protective measures that trip the system. In order to design protection control and avoid equipment damage and accurately estimate the amplitude and frequency of oscillation, the influence of nonlinear limiters in the control circuit of the grid-connected inverter must be considered when analyzing the system stability [5,6].

The impedance model-based stability analysis considers the grid and the grid-connected inverter as two independent systems, and the structure or parameter changes in one part will not affect the impedance model in the other part, reducing the difficulty of system analysis. As a result, it is the preferred method for studying the interaction between the grid-connected inverter and the grid [7]. At the moment, the majority of impedance model research is concentrated on small-signal impedance models based on harmonic linearization. The small-signal impedance model mainly includes the DQ impedance model [8,9,10], sequence impedance model [11,12,13] and phasor domain impedance model [14,15]. However, the stability analysis based on the small-signal impedance model can only evaluate the system dynamics under the small disturbance close to the static operating point. It is unable to analyze the dynamic characteristics of the oscillation throughout the entire process or to predict the amplitude of the oscillation [16]. Moreso, it ignores the nonlinear limiter in practical systems and fails to analyze the constant amplitude oscillations induced by the limiter. Transient stability analysis techniques are frequently utilized in large-signal disturbance and limiter analysis [17,18,19].

In order to compensate for the defects of the small-signal impedance model and to analyze the impedance characteristics of the grid-connected system with the participation of the limiter, the large-signal impedance is defined in reference [20] as the impedance response of the converter with different magnitudes of injected disturbance. Based on the small-signal impedance and the PWM limiter’s description function, the large-signal impedance model is created. This model contributes to the improvement of impedance amplitude and phase characteristics in the process of nonlinear oscillation. Moreover, a prediction approach of oscillation amplitude of a grid-connected system based on large-signal impedance is proposed. Ref. [21] measures the large-signal impedance of a 1 MW VSC inverter and a 4 MW medium voltage doubly fed induction generator. The experiments show that the theoretical value of the large-signal impedance model is consistent with the measured value, and it is confirmed that the injected disturbance amplitude has an impact on the impedance response. Additionally, it is discovered that the trip protection triggers as the injection disturbance rises, resulting in a narrowing of the impedance scan’s frequency range. Ref. [16] suggests a novel approach to forecast the oscillation frequency and amplitude of grid-connected converters based on a large-signal impedance model. Ref. [22] establishes the large-signal impedance model of the grid-connected inverter, and the effects of PLL bandwidth and current loop control bandwidth are investigated. The accuracy of the large-signal impedance model decreases as PLL bandwidth increases, making it impossible to disregard the nonlinearity generated by PLL. Ref. [23] analyzes the impact of sampling delay on large-signal impedance, demonstrating how the sampling delay contributes to the expansion of the phase’s negative resistance region. Ref. [24] contrasts the large-signal impedance analysis approach with other large-signal analysis methods, demonstrating how the large-signal impedance method simplifies the study and assesses the converter impedance’s sensitivity to disturbance amplitude. Ref. [25] establishes two occurrence criteria of equal-amplitude oscillation based on a large-signal impedance model, using the criteria to assess the amplitude of the oscillation as well as the effects of grid strength and load level.

In summary, the advantages of the large-signal impedance model over the small-signal impedance model are shown in

Table 1. Comparison of the small-signal impedance model with the large-signal impedance model.

Comparison	The Small-Signal Impedance	The Large-Signal Impedance
analyzing the stability of the system	✓	✓
predicting oscillation frequency	✓	✓
considering limiter	✗	✓
describing the impedance change as Vp changes	✗	✓
analyzing the dynamic properties of oscillations	✗	✓
predicting oscillation amplitude	✗	✓

. At present, the limiter in PWM is taken into account by the majority of large-signal impedance models of grid-connected inverters. However, the actual system may include a limiter in different control loops, which might cause sustained oscillation in various frequency bands. In this paper, based on the existing large-signal impedance model considering the PWM limiter, the large-signal impedance models considering the limiter in the current control loop and the limiter in the PLL of the grid-connected inverter are developed for the first time. Three large-signal impedance models’ impedance properties are compared and analyzed, along with the similarities and distinctions between large-signal and small-signal impedance, as well as the effects of various limiters on the grid-connected inverters’ output impedance properties. When the limiter in the current control loop is taken into account, Bode and Nyquist plots are utilized to forecast the oscillation frequency and amplitude.

This paper is organized as follows: Section 2 introduces the topology of the grid-connected inverter system, Section 3 establishes three large-signal impedance models, and Section 4 verifies the accuracy of these models. The oscillation frequency and amplitude are predicted in Section 5, and Section 6 concludes the paper.

2. Structure and Control of Grid-Connected Inverter System

Figure 1 shows the structure and control diagram of the grid-connected inverter system, which mainly includes the grid impedance, filter circuit, grid-connected inverter, and grid-connected inverter control circuit. V_dc is the DC side voltage of the inverter, S₁~S₆ are the control switch tubes of the inverter, L_f, C_f and R_f are the filter inductor, filter capacitor and damping resistance, respectively, L_g is the equivalent inductance of the grid, v_a, v_b and v_c are the three-phase voltage of the inverter union point and i_a, i_b and i_c are the three-phase current of the inverter union point. Z_g and Z_p are the grid impedance and the grid-connected inverter positive sequence impedance, respectively.

The control circuit of the grid-connected inverter mainly includes three parts: phase-locked loop control, current control and PWM modulation. H_PLL(s) is the PLL’s PI controller, and its value is (K_{p_PLL} + K_{i_PLL}/s), K_{p_PLL} and K_{i_PLL} are the proportional and integral coefficients of the PLL’s PI controller, respectively. K_d is the current loop decoupling coefficient; K_f is the voltage feedforward coefficient; H_i(s) is the PI controller of the current loop, and its value is (K_{p_I} + K_{i_I}/s). K_{p_I} and K_{i_I} are the proportional and integral coefficients of the current loop’s PI controller, respectively. limiter₁, limiter₂ and limiter₃ in the yellow box are the limiters in the PLL, the limiter in the current control loop and the limiter in the PWM, respectively. In this paper, when the large-signal impedance model considering the limiter in different control steps is established and the system stability is analyzed, only one of the corresponding limiters is selected. i_max and i_min are the upper and lower limits of the current loop limiter, PLL_max and PLL_min are the upper and lower limits of the PLL limiter, and PWM_max and PWM_min are the upper and lower limits of the PWM limiter.

3. Large-Signal Impedance Modeling of Grid-Connected Inverter Considering the Influence of Limiter in Different Control Links

To study sustained oscillation, Shahil Shah proposed the large-signal impedance model (LSIM) in 2017. The large-signal impedance is a representation of the converter impedance for various injections of disturbance along the nominal operating trajectory. The large-signal impedance response has changed as a result of the addition of limiters to the converter’s many control loops.

3.1. Limiter Description Function

The approximate equivalent frequency characteristic of the nonlinear link in the control system under the action of a sinusoidal signal is called the describing function of the nonlinear link. When the input of the nonlinear link is a single sinusoidal signal, the describing function can be expressed as the complex ratio of the first harmonic component of the steady-state output of the nonlinear link to the input sinusoidal signal:

$$N(A)=\frac{Y_1}{A}{e}^{j{\phi }_1}$$

(1)

where, A is the amplitude of the input sinusoidal signal, Y₁ is the amplitude of the output primary harmonic component, and φ₁ is the phase of the output primary harmonic component.

The limiter shown in Figure 2 is a typical nonlinear link. When the input is a single sinusoidal signal, its description function can be expressed as follows:

$$N(A)=\frac{2K}{\pi }[\arcsin \frac{a}{A}+\frac{a}{A}\sqrt{1-{(\frac{a}{A})}^2}],A\ge a$$

(2)

When the input of the limiter is a double sinusoidal signal, its describing function can be expressed as follows:

$$N_{A_1}(A_1,A_2)=\frac{2}{\pi A_1}{\displaystyle {\int }_{-\infty }^{\infty }\{\frac{\sin u}{u^2}\cdot J_0(A_{2}u)J_1(A_{1}u)\}}du$$

(3)

$$N_{A_2}(A_1,A_2)=\frac{2}{\pi A_2}{\displaystyle {\int }_{-\infty }^{\infty }\{\frac{\sin u}{u^2}\cdot J_0(A_{1}u)J_1(A_{2}u)\}}du$$

(4)

where J₀ and J₁ are Bessel functions of the first kind. To simplify the operation, the Equation (4) can be simplified as follows [26]:

$$N_{A_2}(A_1,A_2)\approx 1-\frac{1}{\pi }({\varphi }_{\mathrm{s}}+\frac{A_1}{A_2}\sin {\varphi }_{\mathrm{s}}-\frac{{\varphi }_{\mathrm{s}}}{{\pi }^{2}n}(g_0+g_{\mathrm{n}}+2{\displaystyle \sum _{i=1}^{n-1}{g}_{\mathrm{i}}}))$$

(5)

$${\varphi }_{\mathrm{s}}={\cos }^{-1}\left[sat(\frac{1-A_1^2-A_2^2}{2A_1{A}_2})\right]$$

(6)

$$g_{\mathrm{i}}=(1+\frac{A_1}{A_2}\cos {\varphi }_{\mathrm{i}})\{-{\sin }^{-1}[sat(\frac{1}{D_{\mathrm{i}}})]+[sat(\frac{1}{D_{\mathrm{i}}})-\frac{2}{D_{\mathrm{i}}}]\cdot \sqrt{1-sat[{(\frac{1}{D_{\mathrm{i}}})}^2]}\}$$

(7)

$$D_{\mathrm{i}}=\sqrt{A_1^2+A_2^2+2A_1{A}_2\cos {\varphi }_{\mathrm{i}}}$$

(8)

$${\varphi }_{\mathrm{i}}=\frac{{\varphi }_{\mathrm{s}}}{n}$$

(9)

As for the characteristics of the limiter’s description function, when K = 1, the value of N(A) is shown in Figure 3, and the characteristics of N_A1(A₁,A₂) and N_A2(A₁,A₂) are similar to Figure 3. According to it, when A ≤ a, the gain of the limiter is 1, the limiter can be regarded as a linear link, but when A > a, the describing function of the limiter is nonlinear. Also, when the system does not trigger the limiter, the description function of the limiter is 1, and the large-signal impedance response is the same as the small-signal impedance response. When the system triggers the limiter to generate constant amplitude oscillation, the description function of the limiter becomes a nonlinear function. In this case, the impedance response of the system can only be described by the large-signal impedance model.

3.2. Small-Signal Impedance Model of Grid-Connected Inverter

When a positive sequence small disturbance voltage is injected into the connection point of the grid-connected system of the inverter shown in Figure 1, and the amplitude of the disturbance voltage does not exceed 5% of the amplitude of the grid phase voltage, the expression of the voltage and current of phase A in the time domain is as follows:

$$v_{\mathrm{a}}(t)=V_1\cos (2\pi f_{1}t)+V_{\mathrm{p}}\cos (2\pi f_{\mathrm{p}}t+{\phi }_{\mathrm{vp}})$$

(10)

$$i_{\mathrm{a}}(t)=I_1\cos (2\pi f_{1}t+{\phi }_{\mathrm{i}1})+I_{\mathrm{p}}\cos (2\pi f_{\mathrm{p}}t+{\phi }_{\mathrm{ip}})$$

(11)

where V₁ and I₁ are the amplitudes of the fundamental voltage and current of the grid, f₁ and f_p are the fundamental frequency and positive sequence disturbance frequency, φ_vp, φ_i1 and φ_ip are the initial phase angles of the positive sequence disturbance voltage, fundamental current and positive sequence disturbance current, respectively.

The voltage and current of phase A are converted to the frequency domain as follows:

$${\mathbf{V}}_{\mathrm{a}}[f]=\left\{\begin{array}{l}{\mathbf{V}}_1,f=\pm f_1\\ {\mathbf{V}}_{\mathrm{P}},f=\pm f_{\mathrm{p}}\end{array}\right.,{\mathbf{I}}_{\mathrm{a}}[f]=\left\{\begin{array}{l}{\mathbf{I}}_1,f=\pm f_1\\ {\mathbf{I}}_{\mathrm{P}},f=\pm f_p\end{array}\right.$$

(12)

where, V₁ = V₁/2, V_p = (V_p/2)e^±jφvp, I₁ = (I₁/2)e^±jφi1, I_p = (I_p/2)e^±jφip.

The positive sequence small-signal impedance of the grid-connected inverter is defined as the ratio of the positive sequence small disturbance voltage to the positive sequence small disturbance current:

$$Z_{\mathrm{p}0}(s)=-\frac{{\mathbf{V}}_{\mathrm{p}}}{{\mathbf{I}}_{\mathrm{p}}}$$

(13)

Applying Kirchhoff’s voltage law in Figure 1, the following can be obtained:

$$s{L}_{\mathrm{f}}\left[\begin{array}{l}{i}_{\mathrm{a}}\\ i_{\mathrm{b}}\\ i_{\mathrm{c}}\end{array}\right]=\left[\begin{array}{l}{m}_{\mathrm{a}}\\ m_{\mathrm{b}}\\ m_{\mathrm{c}}\end{array}\right]K_{\mathrm{m}}{V}_{\mathrm{dc}}-\left[\begin{array}{l}{v}_{\mathrm{a}}\\ v_{\mathrm{b}}\\ v_{\mathrm{c}}\end{array}\right]$$

(14)

By deriving the value of m_a and substituting it into Equation (14). The positive sequence small-signal impedance model of the grid-connected inverter can be obtained as follows [20]:

$$Z_{\mathrm{p}0}(s)=\frac{s{L}_{\mathrm{f}}+K_{\mathrm{m}}{V}_{\mathrm{dc}}[H_{\mathrm{i}}(s-j{\omega }_1)-j{K}_{\mathrm{d}}]e^{-s{T}_{\mathrm{d}}}}{1-K_{\mathrm{m}}{V}_{\mathrm{dc}}[K_{\mathrm{f}}+\frac{1}{2}{T}_{\mathrm{PLL}}(s-j{\omega }_1)\cdot H_{\mathrm{i}}(s-j{\omega }_1)\frac{I_1}{V_1}]e^{-s{T}_{\mathrm{d}}}}$$

(15)

where T_PLL(s) = [V₁H_PLL(s)/s]∙[1 + V₁H_PLL(s)/s]⁻¹.

3.3. Large-Signal Impedance Model of Grid-Connected Inverter Considering the Influence of Limiter in PLL

When only the limiter in the PLL is considered, the limiters in both the current control loop and the PWM modulation are not added, that is, only limiter₁ is retained in the limiters in Figure 1. According to the PLL control part in Figure 1, the input value v_{q_PI} of the limiter in the PLL is as follows:

$$v_{\mathrm{q}\_\mathrm{PI}}=v_{\mathrm{q}}\cdot H_{\mathrm{PLL}}(s)$$

(16)

where:

$$v_{\mathrm{q}}=\left\{\begin{array}{l}0,dc\\ [\pm j{T}_{\mathrm{PLL}1}(s)\mp j]{\mathbf{V}}_{\mathrm{p}},f=\pm (f_{\mathrm{p}}-f_1)\end{array}\right.$$

(17)

in which:

$$T_{\mathrm{PLL}1}(s)=\frac{V_1\frac{H_{\mathrm{PLL}}(s)}{\mathrm{s}}{N}_{\mathrm{PLL}}(s)}{1+V_1\frac{H_{\mathrm{PLL}}(s)}{\mathrm{s}}{N}_{\mathrm{PLL}}(s)}$$

(18)

N_PLL(s) is the description function of the limiter in PLL. According to Equation (17), the input value v_{q_PI} of the limiter in the PLL is a single sinusoidal signal, so the N_PLL(s) satisfies Equation (2).

Combining Equations (2) and (16)–(18), the value of N_PLL(s) can be solved iteratively in MATLAB, and then the value of T_PLL1(s) can be solved.

Finally, replacing T_PLL(s) in the positive sequence small-signal impedance value Z_p0(s) with T_PLL1(s), the large-signal impedance value of the inverter considering the influence of the limiter in the PLL can be obtained as follows:

$$Z_{\mathrm{p}1}(s)=\frac{s{L}_{\mathrm{f}}+K_{\mathrm{m}}{V}_{\mathrm{dc}}[H_{\mathrm{i}}(s-j{\omega }_1)-j{K}_{\mathrm{d}}]e^{-s{T}_{\mathrm{d}}}}{1-K_{\mathrm{m}}{V}_{\mathrm{dc}}[K_{\mathrm{f}}+\frac{1}{2}{T}_{\mathrm{PLL}1}(s-j{\omega }_1)\cdot H_{\mathrm{i}}(s-j{\omega }_1)\frac{I_1}{V_1}]e^{-s{T}_{\mathrm{d}}}}$$

(19)

3.4. Large-Signal Impedance Model of Grid-Connected Inverter Considering the Influence of Limiter in the Current Loop

When only the limiter in the current loop is considered, the limiter in Figure 1 only keeps limiter₂. Since the D-axis control of the current inner loop is symmetric to the Q-axis control, the description function of the current loop limiter can be obtained through the Q-axis control. The input of the limiter in the Q-axis current loop is:

$$i_{\mathrm{q}\_\mathrm{PI}}=(I_{\mathrm{qr}}-i_{\mathrm{q}})H_{\mathrm{i}}(s)=(I_{\mathrm{qr}}-i_{\mathrm{q}})(K_{\mathrm{p}\_\mathrm{I}}+K_{\mathrm{i}\_\mathrm{I}}/s)$$

(20)

in which:

$$i_{\mathrm{q}}=\frac{\pm j{I}_1}{V_1}\cos {\phi }_{\mathrm{i}1}{T}_{\mathrm{PLL}}(s){\mathbf{V}}_{\mathrm{p}}\mp j{\mathbf{I}}_{\mathrm{p}},f=\pm (f_{\mathrm{p}}-f_1)$$

(21)

By defining N_i(s) as the describing function of the limiter in the current loop and replacing H_i(s) in the positive sequence small-signal impedance Z_p0(s) with H_i(s)N_i(s), the large-signal impedance value of the inverter considering the influence of the limiter in the current loop can be obtained as follows:

$$Z_{\mathrm{p}2}(s)=-\frac{{\mathbf{V}}_{\mathrm{p}}}{{\mathbf{I}}_{\mathrm{p}}}=\frac{s{L}_{\mathrm{f}}+K_{\mathrm{m}}{V}_{\mathrm{dc}}[H_{\mathrm{i}}(s-j{\omega }_1)N_{\mathrm{i}}(s-j{\omega }_1)-j{K}_{\mathrm{d}}]e^{-s{T}_{\mathrm{d}}}}{1-K_{\mathrm{m}}{V}_{\mathrm{dc}}[K_{\mathrm{f}}+\frac{1}{2}{T}_{\mathrm{PLL}}(s-j{\omega }_1)\cdot H_{\mathrm{i}}(s-j{\omega }_1)N_{\mathrm{i}}(s-j{\omega }_1)\frac{I_1}{V_1}]e^{-s{T}_{\mathrm{d}}}}$$

(22)

Through Equation (22), the expression of disturbance current I_p with respect to disturbance voltage V_p can be obtained as follows:

$${\mathbf{I}}_{\mathrm{p}}=-{\mathbf{V}}_{\mathrm{p}}\cdot \frac{1-K_{\mathrm{m}}{V}_{\mathrm{dc}}[K_{\mathrm{f}}+\frac{1}{2}{T}_{\mathrm{PLL}}(s-j{\omega }_1)\cdot H_{\mathrm{i}}(s-j{\omega }_1)N_{\mathrm{i}}(s-j{\omega }_1)\frac{I_1}{V_1}]e^{-s{T}_{\mathrm{d}}}}{s{L}_{\mathrm{f}}+K_{\mathrm{m}}{V}_{\mathrm{dc}}[H_{\mathrm{i}}(s-j{\omega }_1)N_{\mathrm{i}}(s-j{\omega }_1)-j{K}_{\mathrm{d}}]e^{-s{T}_{\mathrm{d}}}}$$

(23)

Combining Equations (2), (20), (21) and (23), N_i(s − jω₁) is solved iteratively in MATLAB, and this value is substituted into Equation (22) to obtain the large-signal impedance value Z_p2 considering the limiter in the current loop.

3.5. Large-Signal Impedance Model of Grid-Connected Inverter Considering the Effect of Limiter in PWM

When considering the influence of the limiter in PWM, the limiter in Figure 1 only retains limiter₃. According to the principle of the PWM modulation link in Figure 1, the input of the PWM limiter can be obtained as m_a∙e^(−sTd), where m_a contains two main frequency components f₁ and f_p, and its expression is:

$$m_{\mathrm{a}}=\left\{\begin{array}{l}\frac{1}{N_{\mathrm{M}1}(M_1,M_{\mathrm{p}})\cdot K_{\mathrm{m}}{V}_{\mathrm{dc}}}\sqrt{V_1^2+{({\omega }_1{L}_{\mathrm{f}}{I}_1)}^2},f=f_1\\ {\mathbf{V}}_{\mathrm{p}}\cdot \frac{[K_{\mathrm{f}}+\frac{1}{2}{T}_{\mathrm{PLL}}(s-j{\omega }_1)H_{\mathrm{i}}(s-j{\omega }_1)\frac{I_1}{V_1}]s{L}_{\mathrm{f}}+H_{\mathrm{i}}(s-j{\omega }_1)-j{K}_{\mathrm{d}}}{s{L}_{\mathrm{f}}+K_{\mathrm{m}}{V}_{\mathrm{dc}}\cdot [H_{\mathrm{i}}(s-j{\omega }_1)-j{K}_{\mathrm{d}}]e^{-s{T}_{\mathrm{d}}}\cdot N_{\mathrm{Mp}}(M_1,M_{\mathrm{p}})},f=f_{\mathrm{p}}\end{array}\right.$$

(24)

where, N_M1(M₁,M_p) is the f₁ component of the limiter description function in PWM, and N_MP(M₁,M_p) is the f_p component of the limiter description function in PWM.

By combining Equations (5)–(9) and (24) and solving them iteratively in MATLAB, the values of N_M1(M₁,M_p) and N_MP(M₁,M_p) can be obtained.

Finally, K_m in the positive sequence small-signal impedance value Z_p0(s) is replaced by K_mN_MP(M₁,M_p) to obtain the large-signal impedance value of the inverter considering the influence of the limiter in the PWM:

$$Z_{\mathrm{p}3}(s)=\frac{s{L}_{\mathrm{f}}+K_{\mathrm{m}}{V}_{\mathrm{dc}}[H_{\mathrm{i}}(s-j{\omega }_1)-j{K}_{\mathrm{d}}]e^{-s{T}_{\mathrm{d}}}\cdot N_{\mathrm{Mp}}(M_1,M_{\mathrm{P}})}{1-K_{\mathrm{m}}{V}_{\mathrm{dc}}[K_{\mathrm{f}}+0.5T_{\mathrm{PLL}}(s-j{\omega }_1)H_{\mathrm{i}}(s-j{\omega }_1)I_1/V_1]\cdot e^{-s{T}_{\mathrm{d}}}{N}_{\mathrm{Mp}}(M_1,M_{\mathrm{p}})}$$

(25)

4. Simulation Verification

The inverter grid-connected system shown in Figure 1 is built in Simulink, and the parameter settings are shown in

Table 2. System parameters.

Parameters and Units	Value	Parameters and Units	Value
V1/V	169.7	Kp_I	0.07
f1/Hz	60	Ki_I	142.97
Lg/mH	0	Kd	0.006
Rf/Ω	3	f/Hz	10,000
Cf/μF	5	Kf	1/250
Lf/mH	4	Idr/A	15
Vdc/V	500	Iqr/A	0
Km	0.5	imax and −imin	0.2
Kp_PLL	0.13	PLLmax and −PLLmin	10
Ki_PLL	4.11	PWMmax and −PWMmin	1

. Among them, the control bandwidth of the PLL is 7.3 Hz, the control bandwidth of the current inner loop is 1190 Hz, and the active power is taken as 3818.25 W.

4.1. Large-Signal Impedance Response of Grid-Connected Inverter Considering the Influence of Limiter in PLL

Only limiter₁ in the PLL is retained in the simulation, and large-signal disturbances with amplitude of 40%V₁ and 50%V₁ are injected into the parallel points of the inverter grid-connected system, respectively.

The amplitude–frequency characteristic curves of the small-signal impedance Z_p0 of the grid-connected inverter and the large-signal impedance Z_p1 considering the limiter in the PLL under different disturbance amplitudes are shown in Figure 4 together with their measurement results. It can be seen from the figure that the theoretical values of Z_p0 and Z_p1 agree with the measured values, which verifies the correctness of the large-signal impedance model considering the limiter in the PLL in this paper. Additionally, the grid-connected inverter’s large-signal impedance amplitude increases along with the amplitude of the disturbance in the sub/super-synchronous frequency band. The phase of the impedance gradually decreases when the frequency is below 60 Hz and gradually increases when the frequency is above 60 Hz.

Figure 5 shows the large-signal impedance Z_p1 considering the limiter in the PLL for different PLL bandwidths when the disturbance amplitude is fixed at 50%V₁. The amplitude of Z_p1 is reduced and the phase also changes significantly below 300 Hz.

4.2. Large-Signal Impedance Response of Grid-Connected Inverter Considering the Influence of Limiter in the Current Loop

The theoretical amplitude–frequency characteristic curves of the small-signal impedance Z_p0 of the grid-connected inverter and the large-signal impedance Z_p2 of the inverter considering the limiter in the inner loop of the DQ-axis current are shown in Figure 6, together with their measurement results. Only limiter₂ in the current loop is retained in the simulation, and disturbances with amplitude of 40%V₁ and 50%V₁ are injected into the union points of the inverter grid-connected system, respectively. It can be seen from Figure 6 that the theoretical values of Z_p0 and Z_p2 agree with the measured values. The large-signal impedance phase and amplitude considerably fluctuate with the disturbance magnitude in the 500 Hz–2000 Hz frequency range.

Figure 7 shows the large-signal impedance of the inverter considering the limiter in the current loop for different current-loop bandwidths when the disturbance amplitude is fixed at 50% V₁. It is shown that the amplitude and phase of the large-signal impedance Z_p2 considering the limiter in the current loop change significantly in the range of 100 Hz to 3000 Hz with the increase in the bandwidth of the current loop.

4.3. Large-Signal Impedance Response of Grid-Connected Inverter Considering the Influence of Limiter in PWM

Figure 8 shows the theoretical and measured results of the small-signal impedance Z_p0 and the large-signal impedance Z_p3 considering the limiter in the PWM.

Only limiter₃ in PWM is retained in the simulation, and disturbances with amplitude of 40%V₁ and 50%V₁ are injected into the union points of the inverter grid-connected system, respectively. It can be seen from the figure that the theoretical values of Z_p0 and Z_p3 agree with the measured values, which verifies the accuracy of the large-signal impedance model of the inverter established in this paper considering the limiter in PWM. The phase of the large-signal impedance clearly increases as the disturbance amplitude does among 500 Hz–2000 Hz, and the amplitude also varies somewhat.

4.4. Large-Signal Impedance Comparison of Grid-Connected Inverter Considering the Influence of Limiter in Different Control Links

Table 3. Large-signal impedance comparison of grid-connected inverter considering the influence of limiter in different control links.

Large-Signal Impedance	Zp1	Zp2	Zp3
The frequency component of the limiter input	fp − f1	fp − f1	f1 and fp
Limiter description function	N(A)	N(A)	NA1(A1,A2) and NA2(A1,A2)
Terms that are different from Zp0	HPLL(s)NPLL(s)	Hi(s)Ni(s)	KmNMP(M1,MP)
The range of impedance characteristics affected	Around sub/super-synchronous frequency band	Around middle and high frequency band	Around middle and high frequency band

compares the large-signal impedance considering the limiter in the PLL, the limiter in the current loop, and the limiter in the PWM. Since there is only one primary frequency component (f_p − f₁) in the limiter input signal when the limiter in PLL or current loop is taken into account, the limiter description function may be directly described by the single-input limiter description function. When considering the limiter in PWM, the main frequency components of the limiter input signal are f₁ and f_p, which need to be expressed by the dual-input limiter description function. Considering the limiters in different control links, the large-signal impedance Z_p1, Z_p2 or Z_p3 of the grid-connected inverter differ from the small-signal impedance only by one term, which is H_PLL(s)N_PLL(s), H_i(s)N_i(s) and K_mN_MP(M₁,M_P), respectively. Figure 9 supplies a Bode plot of the small-signal impedance model and the large-signal impedance model that takes into account different limiters. As can be seen from the figure, the large-signal impedance Z_p1 differs from the small-signal impedance Z_p0 around the sub/super-synchronous frequency band, and the large-signal impedances Z_p2 and Z_p3 differ from the small-signal impedance Z_p0 around the middle and high frequency band.

5. System Stability Analysis Based on the Large-Signal Impedance Model

The system’s stability may be assessed based on the phase difference at the intersection of the grid impedance and the small-signal impedance of the grid-connected inverter, but it is not possible to forecast the oscillation’s magnitude. However, the system oscillation frequency and oscillation amplitude can all be predicted by analyzing the intersection of the grid impedance and the grid-connected inverter’s large-signal impedance at different disturbance amplitudes. The grid impedance value Z_g(s) in Figure 1 is calculated as follows:

$$Z_{\mathrm{g}}(s)=\frac{s{L}_g\cdot (R_f+\frac{1}{s{C}_f})}{s{L}_g+R_f+\frac{1}{s{C}_f}}$$

(26)

In this paper, the large-signal impedance model considering the limiter in the current loop is taken as an example to analyze the stability of the grid-connected system and predict the oscillation frequency and amplitude of the grid-connected system. The prediction results of two groups of parameters are given and compared with the actual system oscillation amplitude and frequency.

• Case 1

For the first set of parameters, the grid impedance is taken to be 5.7 mH and the current loop control bandwidth is 1190 Hz. Figure 10 is used to predict the constant amplitude oscillation frequency and amplitude of the system considering the limiter in the current inner loop.

The green solid line is the grid impedance Z_g(s), and the black solid line is the small-signal impedance Z_p0(s) of the grid-connected inverter. The amplitude of Z_p0(s) intersects with the grid impedance amplitude at about 582 Hz with a phase difference of 182.01°, indicating that the system is unstable. The blue solid line is the Z_p2(s) corresponding to the injection disturbance amplitude of 50%V₁. Its amplitude intersects with the grid impedance amplitude at about 580 Hz with a phase difference of 181.78°. The purple solid line is the Z_p2(s) corresponding to the injection disturbance amplitude of 54%V₁. Its amplitude intersects with the grid impedance amplitude at about 565 Hz, and the phase difference is 180.51°, indicating that the system is critically stable. The red solid line is the Z_p2(s) corresponding to the injected disturbance amplitude of 60%V₁. Its amplitude intersects with the grid impedance amplitude at about 524 Hz, and the phase difference is 167.17°, indicating that the system is stable. In conclusion, when the grid impedance is 5.7 mH, unstable oscillation will occur in the system considering a limiter in the current loop. When limiter input does not trigger clipping, the oscillation amplitude will continue to increase. When the oscillation amplitude reaches about 54%V₁, the oscillation frequency is around 565 Hz, and the divergence oscillation will eventually become a constant amplitude oscillation, indicating that the system is critically stable.

The oscillation amplitude is predicted by the Nyquist plot of the impedance ratio Z_g(s)/Z_p2(s, V_p) in Figure 11. The impedance ratio’s Nyquist curve progressively deviates from the point (−1,j0) as the injection disturbance’s magnitude rises, making the system more stable. When the amplitude of the injection disturbance is 54%V₁, the Nyquist curve just crosses the point (−1,j0), and the system is critically stable. Therefore, it is predicted that when the grid inductance is set to 5.7 mH, the grid-connected system will have a constant amplitude oscillation with an amplitude of about 54%V₁.

The grid impedance is set to 5.7 mH, and the waveform and FFT analysis results of v_a are shown in Figure 12.

As shown in Figure 12, v_a has constant amplitude oscillation. The FFT analysis of v_a shows that an oscillation with an amplitude of 83.51 V appears at 571 Hz, which is basically consistent with the oscillation amplitude and frequency predicted in Figure 10 and Figure 11.

• Case 2

The second set of parameters takes the grid impedance as 6.7 mH and the current loop control bandwidth as 1290 Hz. Figure 13 is used to predict the constant amplitude oscillation frequency and amplitude of the system considering the limiter in the current inner loop.

The green solid line represents the grid impedance Z_g(s), and the black solid line represents the small-signal impedance Z_p0(s) of the grid-connected inverter. The amplitude of Z_p0(s) intersects with the grid impedance amplitude at about 568 Hz, and the phase difference is 184.65°, indicating that the grid-connected system is unstable. The blue solid line is the Z_p2(s) corresponding to the injected disturbance amplitude of 60%V₁. Its amplitude intersects with the grid impedance amplitude at about 555 Hz, and the phase difference is 183.5°. The purple solid line is the Z_p2(s) corresponding to the injection disturbance amplitude of 64%V₁. Its amplitude intersects with the grid impedance amplitude at about 535 Hz, and the phase difference is 179.64°, indicating that the grid-connected system is critically stable. The red solid line is the Z_p2(s) corresponding to the injected disturbance amplitude of 70%V₁. Its amplitude intersects with the grid impedance amplitude at about 498 Hz, and the phase difference is 159.44°, indicating that the grid-connected system is stable. In conclusion, when considering the limiter in the current loop and the grid impedance is 5.7 mH, the grid-connected system will have unstable oscillation. When the oscillation amplitude reaches about 64%V₁, the divergence oscillation becomes constant amplitude oscillation, the system is critically stable, and the oscillation frequency is about 535 Hz.

Figure 14 uses the Nyquist plot of the impedance ratio Z_g(s)/Z_p2(s, V_p) to predict the oscillation amplitude. As the amplitude of the injection disturbance increases, the Nyquist curve of the impedance ratio gradually moves away from the point (−1,j0), and the system becomes more stable. When the amplitude of the injection disturbance is 64%V₁, the Nyquist curve just crosses the point (−1,j0), and the system is critically stable. Therefore, it is predicted that when the grid inductance is set to 6.7 mH and the current loop control bandwidth is 1290 Hz, the grid-connected system considering the limiter in the current loop will have a constant amplitude oscillation with an amplitude of about 64%V₁.

To verify whether the oscillation frequency and amplitude predicted by Figure 13 and Figure 14 are correct, the grid impedance is set to 6.7 mH, and the waveform when the grid voltage v_a is critically stable is observed. As shown in Figure 15, constant amplitude oscillations appear. FFT analysis of v_a shows an oscillation with an amplitude of 94.72 V at 550 Hz, which was basically consistent with the results predicted by Figure 13 and Figure 14.

6. Conclusions

This study establishes three large-signal impedance models that take into account the limiter in PLL, current loop, and PWM, respectively. The accuracy of each large-signal impedance model is confirmed using MATLAB/Simulink. To examine the stability of the system, the large-signal impedance model of the grid-connected inverter taking into account the limiter in the current loop is used as an example. The system’s oscillation frequency and amplitude are anticipated, and the accuracy of the analysis’s prediction findings is confirmed. In this paper, the following conclusions are obtained:

(1)

Based on the existing large-signal impedance model considering the limiter in PWM, the large-signal impedance models considering the limiter in the current control loop and the limiter in the PLL of the grid-connected inverter are developed for the first time. Their theoretical large-signal impedance characteristics are all consistent with the measured large-signal impedance characteristics;

(2)

The impedance characteristics of three large-signal impedance models and the influence of different limiters on the output impedance characteristics of grid-connected inverters are compared and analyzed. The limiter in the PLL influences the large-signal impedance characteristics of the grid-connected inverter around the sub/super-synchronous frequency band, and the limiter in the current loop and the limiter in the PWM influence those around the middle and high frequency band;

(3)

The large-signal impedance of the inverter taking into account the PLL limiter changes dramatically around the sub-super synchronous frequency range when the PLL bandwidth is increased. The amplitude and phase of the large-signal impedance of the inverter taking into account the limiter in the current loop fluctuate dramatically in the middle frequency band when the current loop’s bandwidth is increased. The PLL bandwidth has a smaller impact on Z_p1 than the current-loop bandwidth has on Z_p2;

(4)

Due to the influence of the limiter in the current loop, with the increase in the system disturbance amplitude, the phase difference at the intersection of the large-signal impedance’s amplitude and the grid impedance’s amplitude decreases gradually, and the system stability is enhanced. When the phase difference is 180°, the system is critically stable, and the oscillation frequency and amplitude of the real system are essentially compatible with the disturbance frequency and amplitude;

(5)

As the amplitude of the system disturbance increases, the Nyquist curve of Z_g(s)/Z_p2(s, V_p) considering the limiter in the current loop gradually moves away from the point (−1,j0), and the stability margin of the system increases. When the Nyquist curve just crosses the point (−1,j0), the system is critically stable, and the disturbance amplitude is basically consistent with the oscillation amplitude of the actual system.