In recent years, the rapid development of distributed photovoltaic systems has significantly increased the application of micro solar inverters. Ensuring the stable operation of these micro solar inverters is a critical requirement for their safe and reliable integration into the power grid. Traditional analysis methods, such as the state-space averaging (SSA) model, have been widely used to study the stability of solar inverters. However, these conventional models often neglect the switching process of power devices within the circuit, making it difficult to accurately describe the high-frequency dynamic behavior and stability characteristics of the solar inverter. To address this issue, I propose a novel z-s domain modeling method that explicitly accounts for the switching transitions. This approach combines the discrete-time mapping of switching states with continuous frequency-domain analysis, offering a more accurate representation of the solar inverter’s stability.
First, I analyze each operating mode of the micro solar inverter. The topology under study is a single-phase flyback-type micro solar inverter, where the primary stage is a flyback converter for boosting the voltage, and the secondary stage is a line-frequency bridge inverter for polarity reversal. The inverter operates in discontinuous conduction mode (DCM) to simplify the analysis. Within one line-frequency cycle, there are six operating modes, but due to the symmetry of the positive and negative half-cycles, I focus on the positive half-cycle for modeling. The key state variables selected are the magnetizing current \( i_m \), the output current \( i_g \), and the filter capacitor voltage \( u_c \). The state-space equations for each mode are derived as follows:
Mode 1: The primary switch \( V_p \) is turned on. The input voltage charges the magnetizing inductance, and the magnetizing current increases linearly. The output filter capacitor supplies the load. The state equation is:
\[ \dot{x} = A_1 x + B_1 u_{PV} \]
Mode 2: The primary switch is turned off. The energy stored in the transformer is transferred to the secondary side through the diode. The magnetizing current decreases linearly, and the output current flows to the grid and charges the filter capacitor. The state equation is:
\[ \dot{x} = A_2 x + B_2 u_{PV} \]
Mode 3: The magnetizing current drops to zero, and the energy transfer is complete. The load is solely supplied by the filter capacitor. The state equation is:
\[ \dot{x} = A_3 x + B_3 u_{PV} \]
where the matrices \( A_i \) and \( B_i \) are defined based on the circuit parameters, including the magnetizing inductance \( L_m \), filter inductance \( L_f \), filter capacitance \( C_f \), transformer turns ratio \( N \), and various parasitic resistances. The specific matrices are summarized in the table below.
| Mode | Matrix A | Matrix B |
|---|---|---|
| 1 | \[ A_1 = \begin{bmatrix} -(r_m+r_s)/L_m & 0 & 0 \\ 0 & -(r_c+r_L+R_g)/L_f & 1/L_f \\ 0 & -1/C_f & 0 \end{bmatrix} \] | \[ B_1 = [1/L_m \quad 0 \quad 0]^T \] |
| 2 | \[ A_2 = \begin{bmatrix} -(N^2 r_m + r_{VD} + r_c)/(N^2 L_m) & r_c/(N L_m) & -1/(N L_m) \\ r_c/(N L_f) & -(r_c+r_L+R_g)/L_f & 1/L_f \\ 1/(N C_f) & -1/C_f & 0 \end{bmatrix} \] | \[ B_2 = [0 \quad 0 \quad 0]^T \] |
| 3 | \[ A_3 = \begin{bmatrix} 0 & 0 & 0 \\ 0 & -(r_c+r_L+R_g)/L_f & 1/L_f \\ 0 & -1/C_f & 0 \end{bmatrix} \] | \[ B_3 = [0 \quad 0 \quad 0]^T \] |
Next, I derive the discrete-time iterative equations for the solar inverter. By solving the state-space equations, the relationship between the state variables at the beginning and end of each mode can be expressed using matrix exponentials. For a small switching period \( T \), I approximate the matrix exponential using a Taylor series expansion: \( e^{A T} \approx I + A T \). This linearization is valid because the switching frequency is high relative to the system dynamics. The discrete iterative equations for the three modes within the \( n \)-th switching cycle are:
\[ x_{n1} = e^{A_1 d_{n1} T} x_n + A_1^{-1}[e^{A_1 d_{n1} T} – I] B_1 \]
\[ x_{n2} = e^{A_2 d_{n2} T} x_{n1} \]
\[ x_{n3} = e^{A_3 d_{n3} T} x_{n2} \]
where \( d_{n1} \), \( d_{n2} \), and \( d_{n3} \) are the duty ratios of each mode, and they satisfy \( d_{n1} + d_{n2} + d_{n3} = 1 \). The condition of DCM operation provides an additional relationship between the duty ratios, which allows me to solve for \( d_{n2} \) and \( d_{n3} \) in terms of \( d_{n1} \). After linearizing around a steady-state operating point, I obtain the small-signal discrete-time model for the solar inverter.
To transform this model into the frequency domain, I introduce small perturbations around the steady-state values. Let \( x = X + \hat{x} \) and \( d = D + \hat{d} \). Substituting these into the discrete equations and neglecting higher-order terms, I derive the linearized perturbation equations. The resulting state-space representation in discrete-time is:
\[ \hat{x}_{n+1} = A_{total} \hat{x}_n + B_{total} \hat{d}_n \]
where \( A_{total} \) and \( B_{total} \) are functions of the steady-state operating point and the system matrices. The detailed expressions for these matrices are complex but can be computed numerically. The open-loop transfer function from the duty ratio \( \hat{d} \) to the output current \( \hat{i}_g \) in the z-domain is then given by:
\[ G_{id}(z) = \frac{\hat{i}_g(z)}{\hat{d}(z)} = F [zI – (A_{n3a}A_{n2}A_{n1} + A_{n3b}A_{n1})]^{-1} (A_{n3a}A_{n2}B_{n1} + A_{n3b}B_{n1} + A_{n3a}B_{n2} + B_{n3}) \]
where \( F = [0 \quad 1 \quad 0] \) extracts the output current from the state vector. The matrices \( A_{n1}, A_{n2}, A_{n3a}, A_{n3b}, B_{n1}, B_{n2}, B_{n3} \) are derived from the linearization process and depend on the steady-state values and the duty ratios. This z-domain model captures the switching dynamics of the solar inverter.
To enable analysis in the s-domain, which is more familiar for control system design, I apply the bilinear transformation (also known as Tustin’s method):
\[ z = \frac{1 + 0.5 s T}{1 – 0.5 s T} \]
Substituting this into the z-domain transfer function yields the s-domain transfer function \( G_{id}(s) \) for the micro solar inverter. This allows me to use Bode plots, Nyquist diagrams, and other continuous-time frequency-domain tools to analyze the stability of the solar inverter. The resulting model is termed the z-s domain model of the solar inverter.
I now apply the z-s domain model to analyze the stability of the micro solar inverter under current-mode control with a proportional-resonant (PR) controller. The controller transfer function is \( G_c(s) \), and the overall open-loop transfer function of the solar inverter system is:
\[ G_o(s) = G_c(s) G_{id}(s) \]
The simulation parameters for the solar inverter are: input voltage \( U_{PV} = 35 V \), output current RMS \( I_g = 0.707 A \), transformer turns ratio \( N = 6 \), filter inductance \( L_f = 10 mH \), magnetizing inductance \( L_m = 3 \mu H \), filter capacitance \( C_f = 1 \mu F \), switching frequency of the flyback stage \( f_s = 10 kHz \), and grid frequency \( 50 Hz \).
First, I consider the case without the PR controller, i.e., direct current-mode control. The Bode plot of the open-loop transfer function is shown in the analysis below. The gain margin is 3.67 dB, the phase margin is 26.7°, and the crossover frequency is 12.2 kHz. The system is stable, and the solar inverter can achieve good grid-connection performance.
Next, I add a PR controller and vary its parameters to study the stability boundaries. The PR controller is designed to track the grid-frequency reference. I compare the stability predictions from the z-s domain model with those from the traditional SSA model. The key results are summarized in the table below.
| PR Parameters | Model | Gain Margin (dB) | Phase Margin (deg) | Stability Prediction | Simulation Result |
|---|---|---|---|---|---|
| \( K_p = 0.8, K_r = 50 \) | z-s Domain | 5.58 | 42.9 | Stable | Stable |
| SSA | Infinite | 77.1 | Stable | Stable | |
| \( K_p = 5, K_r = 500 \) | z-s Domain | -10.4 | -62.6 | Unstable | Unstable |
| SSA | Infinite | 27 | Stable | Unstable |
For \( K_p = 0.8 \) and \( K_r = 50 \), both the z-s domain model and the SSA model predict stability. The simulated output current waveform confirms that the solar inverter operates stably with a smooth sinusoidal current. For \( K_p = 5 \) and \( K_r = 500 \), the z-s domain model correctly predicts instability, as indicated by negative gain and phase margins. In contrast, the SSA model still predicts stability with a positive phase margin of 27°. The simulation results for these parameters clearly show an oscillating output current, confirming the instability. This demonstrates the superior accuracy of the proposed z-s domain model for the solar inverter stability analysis.
I further validate the model through experimental tests on a hardware prototype of the micro solar inverter. The experimental setup includes the inverter, controller, DC power supply, and auxiliary circuits. The electrical parameters are consistent with the simulations. The experimental waveforms are captured for the three cases. Without the PR controller, the output current is stable and sinusoidal, matching the z-s domain model prediction. With the PR controller set to \( K_p = 0.8 \) and \( K_r = 50 \), the solar inverter remains stable. With \( K_p = 5 \) and \( K_r = 500 \), the output current exhibits clear oscillations, validating the instability predicted by the z-s domain model. The experimental results are in full agreement with the simulation and the z-s domain analysis.
In conclusion, I have proposed a z-s domain modeling method for micro solar inverters that captures the discrete switching dynamics and maps them to the continuous s-domain for stability analysis. Compared to the conventional state-space averaging model, the z-s domain model provides significantly higher accuracy in predicting instability, especially when the system operates near stability boundaries. The model is derived through a systematic process: analyzing operating modes, deriving discrete iterative equations, linearizing around the steady-state, converting to the z-domain, and finally applying the bilinear transform to obtain the s-domain transfer function. This approach allows engineers to use familiar frequency-domain tools like Bode plots to design stable controllers for solar inverters. The simulation and experimental results confirm the correctness and effectiveness of the proposed modeling method, making it a valuable tool for the design and analysis of micro solar inverters in distributed photovoltaic systems.