The optimization of controllers and the order reduction of high-order systems remain focal points of research interest among scholars both domestically and internationally. The introduction of complex factors into controllers has been demonstrated to significantly enhance control performance. In the context of modeling, incorporating the concept of complex numbers facilitates not only the transformation of a high-order multi-input multi-output (MIMO) system into a lower-order single-input single-output (SISO) system but also improves the system’s output performance. For instance, prior research has shown that introducing complex factors enables zero-error tracking of AC signals by PI controllers in orthogonal coordinates for three-phase grid-tied inverters. Further work proposed a generalized complex controller model, broadening the applicability of such controllers. Another study utilized complex factors to simplify the control structure under asymmetric current conditions. These investigations collectively indicate that integrating the concept of complex numbers with traditional control methods can effectively enhance control performance.
However, implementing complex-domain control based on a real-domain model often increases system complexity. This is particularly challenging for systems like the three-phase LCL-type grid tied inverter, whose real-domain model in the synchronous rotating (dq) frame constitutes a high-order MIMO system, posing difficulties for practical engineering applications. Given that a MIMO system in the real domain can be equivalent to a SISO system in the complex domain, with the order of the complex-domain model being half that of the real-domain model, researchers have proposed modeling methods based on the complex domain to reduce model order and simplify controller analysis and design. For example, the concept of complex transfer functions has been applied to three-phase induction motors, a typical high-order system, resulting in a more concise system structure. Similarly, transforming the real-domain model of a doubly-fed induction generator into an equivalent, reduced-order complex-domain model has simplified the analysis and design of feedback control loops, enabling direct pole placement. Studies on three-phase power systems with symmetrical properties have applied complex coefficients to convert real-domain MIMO systems into complex-domain SISO systems, thereby extending the applicability of root locus analysis. These studies underscore the significant value of complex-domain modeling methods in simplifying high-order system models.
In control system design, stability criteria form the foundation for controller development. For the analysis and design of complex-domain models, however, the applicability of traditional control theory tools is somewhat limited. The Hurwitz criterion is an effective stability criterion applicable to complex-domain models. It determines system stability based on the coefficients of the characteristic equation without requiring the solution of complex differential equations. Nonetheless, when the order of the system’s characteristic equation is high, using the Hurwitz criterion for stability analysis involves substantial computational effort, limiting its application in high-order systems. Complex-domain modeling can simplify the system model and reduce its order, effectively overcoming this limitation. Research has combined complex-domain modeling with the Hurwitz criterion, proposing and proving a complex Hurwitz criterion. A key feature of this criterion is that for a system modeled in the complex domain, the order of its characteristic polynomial is halved, dramatically reducing the computational complexity of the stability analysis. This criterion has been successfully validated and applied in the stability analysis of complex-domain models for self-excited induction motors and doubly-fed induction generators. Similar to three-phase motors, a three-phase grid tied inverter is also a high-order system. Employing traditional real-domain stability criteria for its analysis similarly encounters computational complexity. Existing literature presents limited research on the application of the complex Hurwitz stability criterion to three-phase grid tied inverters. Therefore, this study applies the complex Hurwitz stability criterion to the controller design and stability analysis of a three-phase grid-tied inverter.
This paper focuses on the three-phase LCL-type grid tied inverter as the research subject. Initially, a high-order real-domain model of the inverter in the synchronous rotating (dq) frame is established. Leveraging the symmetry of the state matrix, an equivalent, reduced-order complex-domain model is derived, yielding the system’s state equation in the complex domain. Based on this low-order complex-domain model, active damping and grid current control strategies are designed directly. By incorporating capacitor current feedback, internal coupling within the filter is eliminated, simplifying the design of the grid current controller. Furthermore, stability analysis is conducted using the complex Hurwitz criterion, deriving the necessary and sufficient conditions for system stability and obtaining the permissible range for controller parameters. Finally, simulation and experimental results from a prototype verify the feasibility and effectiveness of the proposed method for stability analysis and controller design of the grid tied inverter.
Mathematical Model of the Three-Phase LCL-Type Grid-Tied Inverter
The main circuit topology of a three-phase LCL-type grid tied inverter is illustrated below. In this topology, \(L_1\) represents the inverter-side inductance, \(L_2\) represents the grid-side inductance, and \(C_f\) is the filter capacitor, collectively forming the LCL filter.
Based on Kirchhoff’s laws and the Park transformation, the differential equations of the three-phase LCL-type grid tied inverter in the rotating (dq) reference frame are established as follows:
$$
\begin{aligned}
L_1 \dot{i}_{1d} &= v_d + \omega_0 L_1 i_{1q} – v_{cd} – R_1 i_{1d} \\
L_2 \dot{i}_{2d} &= v_{cd} + \omega_0 L_2 i_{2q} – v_{gd} – R_2 i_{2d} \\
C \dot{v}_{cd} &= i_{1d} – i_{2d} + \omega_0 C v_{cq} \\
L_1 \dot{i}_{1q} &= v_q – \omega_0 L_1 i_{1d} – v_{cq} – R_1 i_{1q} \\
L_2 \dot{i}_{2q} &= v_{cq} – \omega_0 L_2 i_{2d} – v_{gq} – R_2 i_{2q} \\
C \dot{v}_{cq} &= i_{1q} – i_{2q} – \omega_0 C v_{cd}
\end{aligned}
$$
where \(i_{1d}\), \(i_{1q}\), \(i_{2d}\), \(i_{2q}\), \(v_{cd}\), \(v_{cq}\), \(R_1\), \(R_2\) denote the d- and q-axis components of the inverter-side inductor current, grid-side current, capacitor voltage, inverter-side inductor parasitic resistance, and grid-side inductor parasitic resistance, respectively. \(\omega_0\) is the grid angular frequency, i.e., the rotation frequency of the coordinate transformation matrix. The system is a three-phase three-wire configuration without a zero-sequence path; therefore, the 0-axis is omitted in the modeling. For analytical convenience, Equation (1) is transformed into the s-domain via the Laplace transform.
Defining the Laplace-transformed state variables, input variables, and output variables as:
$$
\begin{aligned}
\mathbf{X}^T &= (I_{1d}, I_{2d}, V_{cd}, I_{1q}, I_{2q}, V_{cq}) \\
\mathbf{U}^T &= (V_d, V_{gd}, V_q, V_{gq}) \\
\mathbf{Y}^T &= (I_{2d}, I_{2q})
\end{aligned}
$$
the state-space equation in the s-domain is obtained:
$$
\begin{cases}
\mathbf{A} \mathbf{X}(s) = \mathbf{B} \mathbf{U}(s) \\
\mathbf{Y}(s) = \mathbf{C} \mathbf{X}(s)
\end{cases}
$$
where \(\mathbf{A}\) is the system characteristic matrix, \(\mathbf{B}\) is the input matrix, and \(\mathbf{C}\) is the output matrix, given by:
$$
\mathbf{A} = \begin{bmatrix}
L_1 s + R_1 & 0 & 1 & -\omega_0 L_1 & 0 & 0 \\
0 & L_2 s + R_2 & -1 & 0 & -\omega_0 L_2 & 0 \\
-1 & 1 & C s & 0 & 0 & -\omega_0 C \\
\omega_0 L_1 & 0 & 0 & L_1 s + R_1 & 0 & 1 \\
0 & \omega_0 L_2 & 0 & 0 & L_2 s + R_2 & -1 \\
0 & 0 & \omega_0 C & -1 & 1 & C s
\end{bmatrix}, \quad
\mathbf{B} = \begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & -1 \\
0 & 0 & 0 & 0
\end{bmatrix}, \quad
\mathbf{C} = \begin{bmatrix}
0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0
\end{bmatrix}
$$
From Equation (6), it is evident that the three-phase LCL grid tied inverter in the dq frame is a 6th-order MIMO high-order system.
Complex-Coefficient Hurwitz Stability Criterion
Conventional Routh or Hurwitz stability criteria are applicable when the coefficients of the system’s characteristic equation are real constants. This study is based on a complex-domain system design, where the resulting characteristic equation has complex coefficients. Therefore, stability criteria for constant-coefficient characteristic equations are not directly applicable here. Prior research has addressed stability determination for polynomials with complex coefficients. Herein, the necessary and sufficient conditions for a complex-coefficient polynomial to have zeros with negative real parts are outlined.
Consider a complex-coefficient polynomial:
$$
P(s) = s^n + \alpha_1 s^{n-1} + \alpha_2 s^{n-2} + \ldots + \alpha_n, \quad \alpha_k = p_k + j q_k, \quad k = 1, 2, \ldots, n
$$
The corresponding complex-coefficient Hurwitz stability criterion states that a necessary and sufficient condition for system stability (all characteristic roots having negative real parts) is that the determinants formed from the real and imaginary parts of the characteristic polynomial’s coefficients are all positive:
$$
\Delta_i > 0, \quad i = 1, 2, \ldots, n
$$
where:
$$
\Delta_1 = p_1
$$
$$
\Delta_k = (-1)^{k(k-1)/2} \det(\mathbf{D}_k), \quad k=2,3,\ldots,n
$$
with matrix \(\mathbf{D}_k\) constructed as follows (setting \(p_r = q_r = 0\) for \(r > n\)):
$$
\mathbf{D}_k = \begin{bmatrix}
p_1 & p_3 & p_5 & \ldots & p_{2k-1} & -q_2 & -q_4 & \ldots & -q_{2k-2} \\
1 & p_2 & p_4 & \ldots & p_{2k-2} & -q_1 & -q_3 & \ldots & -q_{2k-3} \\
0 & p_1 & p_3 & \ldots & p_{2k-3} & 0 & -q_2 & \ldots & -q_{2k-4} \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \ldots & p_k & 0 & 0 & \ldots & -q_{k-1} \\
0 & q_2 & q_4 & \ldots & q_{2k-2} & p_1 & p_3 & \ldots & p_{2k-3} \\
0 & q_1 & q_3 & \ldots & q_{2k-3} & 1 & p_2 & \ldots & p_{2k-4} \\
0 & 0 & q_2 & \ldots & q_{2k-4} & 0 & p_1 & \ldots & p_{2k-5} \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \ldots & q_k & 0 & 0 & \ldots & p_{k-1}
\end{bmatrix}
$$
In practical engineering, systems described by complex-coefficient matrices are uncommon, and thus characteristic polynomials with complex coefficients are not typical. However, for certain symmetric systems, the system matrix can be represented using complex coefficients to reduce the system order, simplify analysis, and decrease computational load.
For the autonomous system in the s-domain described by Equation (8), stability conditions can be obtained by computing its characteristic polynomial \(\det(\mathbf{A}(s))\). If the matrix \(\mathbf{A}(s)\) possesses a specific symmetrical structure, the system can be reduced, halving its order and significantly reducing computational complexity.
Specifically, if the matrix has the following symmetric structure:
$$
\mathbf{A}(s) = \begin{bmatrix}
\mathbf{A}_{11}(s) & -\mathbf{A}_{12}(s) \\
\mathbf{A}_{12}(s) & \mathbf{A}_{11}(s)
\end{bmatrix}
$$
i.e., a structure where the main diagonal blocks are equal and the off-diagonal blocks are negatives of each other, then the original matrix can be represented in complex form as:
$$
\mathbf{A}_c(s) = \mathbf{A}_{11}(s) + j\mathbf{A}_{12}(s)
$$
where the order of \(\mathbf{A}_c(s)\) is half that of \(\mathbf{A}(s)\), and the eigenvalues of \(\mathbf{A}_c(s)\) are equivalent to those of \(\mathbf{A}(s)\). Once the characteristic equation of \(\mathbf{A}_c(s)\) is found, the necessary and sufficient conditions for system stability can be determined using the complex-coefficient Hurwitz stability criterion. Applying this complex-coefficient matrix approach to assess the stability of systems like the grid tied inverter directly reduces a high-order system to a lower-order one for solution, significantly simplifying the computational process compared to using the real-coefficient matrix.
Complex-Domain Model and Controller Design for the Three-Phase LCL Grid-Tied Inverter
The characteristic polynomial matrix from Equation (5) is:
$$
\mathbf{A} = \begin{bmatrix}
L_1 s + R_1 & 0 & 1 & -\omega_0 L_1 & 0 & 0 \\
0 & L_2 s + R_2 & -1 & 0 & -\omega_0 L_2 & 0 \\
-1 & 1 & C s & 0 & 0 & -\omega_0 C \\
\omega_0 L_1 & 0 & 0 & L_1 s + R_1 & 0 & 1 \\
0 & \omega_0 L_2 & 0 & 0 & L_2 s + R_2 & -1 \\
0 & 0 & \omega_0 C & -1 & 1 & C s
\end{bmatrix}
$$
Matrix \(\mathbf{A}(s)\) satisfies the symmetry requirement. We can define:
$$
\mathbf{A}_{11}(s) = \begin{bmatrix}
L_1 s + R_1 & 0 & 1 \\
0 & L_2 s + R_2 & -1 \\
-1 & 1 & C s
\end{bmatrix}, \quad
\mathbf{A}_{12}(s) = \begin{bmatrix}
\omega_0 L_1 & 0 & 0 \\
0 & \omega_0 L_2 & 0 \\
0 & 0 & \omega_0 C
\end{bmatrix}
$$
Therefore, the corresponding complex-coefficient matrix is:
$$
\mathbf{A}_c(s) = \mathbf{A}_{11}(s) + j\mathbf{A}_{12}(s) = \begin{bmatrix}
L_1 s + R_1 + j\omega_0 L_1 & 0 & 1 \\
0 & L_2 s + R_2 + j\omega_0 L_2 & -1 \\
-1 & 1 & C s + j\omega_0 C
\end{bmatrix}
$$
The corresponding complex state variables are:
$$
\mathbf{X}_c^T = (I_1, I_2, V_c), \quad \text{where} \quad I_1 = I_{1d} + jI_{1q}, \quad I_2 = I_{2d} + jI_{2q}, \quad V_c = V_{cd} + jV_{cq}
$$
Defining the complex input variables as \(V = V_d + jV_q\) and \(V_g = V_{gd} + jV_{gq}\), the state-space equation of the three-phase LCL grid tied inverter in the complex domain is obtained:
$$
\begin{bmatrix}
L_1 s + R_1 + j\omega_0 L_1 & 0 & 1 \\
0 & L_2 s + R_2 + j\omega_0 L_2 & -1 \\
-1 & 1 & C s + j\omega_0 C
\end{bmatrix}
\begin{bmatrix}
I_1 \\ I_2 \\ V_c
\end{bmatrix}
=
\begin{bmatrix}
V \\ -V_g \\ 0
\end{bmatrix}
$$
From Equation (20), the structural block diagram of the three-phase LCL grid tied inverter in the dq frame within the complex domain can be derived.
Based on the system’s complex-domain structure, the grid current expression is:
$$
I_2 = Y_m V + Y_o V_g
$$
where:
$$
Y_m = \frac{Z_c}{Z_c Z_1 + Z_c Z_2 + Z_1 Z_2}, \quad Y_o = -\frac{Z_1 + Z_c}{Z_c Z_1 + Z_c Z_2 + Z_1 Z_2}
$$
and \(Z_1, Z_2, Z_c\) represent the complex-domain impedances of the inverter-side inductor \(L_1\), grid-side inductor \(L_2\), and filter capacitor \(C\) in the dq frame, respectively:
$$
Z_1 = sL_1 + j\omega_0 L_1 + R_1, \quad Z_2 = sL_2 + j\omega_0 L_2 + R_2, \quad Z_c = \frac{1}{sC + j\omega_0 C}
$$
If the grid voltage is treated as a disturbance, considering \(V\) as the input and \(I_2\) as the output, the Bode plot of the system’s open-loop transfer function reveals a resonant peak at the resonant frequency accompanied by a -180° phase jump, which can easily lead to instability. Therefore, this resonant peak must be suppressed. This paper employs capacitor current feedback active damping to inhibit the resonant peak. The addition of active damping significantly suppresses the peak and smoothens the phase transition.
Accordingly, the system’s structural block diagram in the complex domain is designed as follows. A PI controller is adopted in the complex domain for grid current control. Capacitor voltage feedforward is also added to eliminate the internal feedback relationship within the filter, simplifying the design of the current loop.
Neglecting grid voltage disturbance, the closed-loop transfer function in the complex domain is:
$$
G_{out\_c} = \frac{G_{pi} G_{active} G_{L2C}}{1 + G_{pi} G_{active} G_{L2C}}
$$
where \(G_{pi}, G_{active}, G_{L2C}\) are the transfer functions of each loop:
$$
\begin{aligned}
G_{pi} &= K_p + \frac{K_i}{s} \\
G_{L2C} &= \frac{1}{(sL_2 + j\omega_0 L_2 + R_2)(sC + j\omega_0 C) + 1} \\
G_{active} &= \frac{K_{pwm}}{sL_1 + j\omega_0 L_1 + R_1 + K_{pwm} K_c}
\end{aligned}
$$
Based on Equation (24), the closed-loop characteristic equation is:
$$
1 + K_{pwm} G_{pi} G_{active} G_{L2C} = 0
$$
This equation is expanded into a complex-coefficient polynomial:
$$
s^4 + (a_1 + jb_1)s^3 + (a_2 + jb_2)s^2 + (a_3 + jb_3)s + (a_4 + jb_4) = 0
$$
The coefficients \(a_i\) and \(b_i\) are functions of the system parameters \((L_1, L_2, C, R_1, R_2, \omega_0)\), the pulse-width modulation (PWM) gain \(K_{pwm}\), the active damping coefficient \(K_c\), and the PI controller parameters \(K_p\) and \(K_i\). Their detailed expressions are derived from algebraic manipulation of the characteristic equation.
According to the complex Hurwitz criterion, the necessary and sufficient condition for system stability is that all characteristic roots have negative real parts, which is equivalent to the determinants \(\Delta_i\) formed from the polynomial’s real and imaginary part coefficients all being positive:
$$
\Delta_i > 0, \quad i = 1, 2, 3, 4
$$
Given that all physical and controller parameters are positive, \(\Delta_1 = a_1 > 0\) generally holds. Therefore, the stability condition reduces to requiring \(\Delta_2 > 0\), \(\Delta_3 > 0\), and \(\Delta_4 > 0\) simultaneously. The expressions for \(\Delta_2\), \(\Delta_3\), and \(\Delta_4\) are determinants constructed as specified in the complex Hurwitz criterion, incorporating the coefficients \(a_i\) and \(b_i\). Substituting the expressions for \(a_i\) and \(b_i\) into these determinant inequalities yields the permissible range for the PI controller parameters \(K_p\) and \(K_i\) that ensure stable operation of the grid tied inverter.
Simulation and Experimental Verification
A control model based on the three-phase LCL-type grid tied inverter was established using MATLAB/Simulink simulation software to validate the effectiveness of the proposed method. Both the simulation and the experimental prototype employ a three-phase three-wire connection with a rated power of 3.3 kW. The main parameters are listed in the table below.
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| \(L_1\) | 3.7 mH | Switching Frequency \(f_s\) | 20 kHz |
| \(L_2\) | 0.6 mH | Rated Grid Current \(I_g\) | 15 A |
| \(C_f\) | 4.7 μF | Rated Grid Voltage \(V_g\) | 220 V |
Based on these parameters and the proposed method, the range of stabilizing controller parameters \(K_p\) and \(K_i\) was calculated. For the given system, the stabilizing range was found to be approximately:
$$
0.715 < K_p < 4.382; \quad 10 < K_i < 43
$$
A further analysis was conducted: setting \(K_i = 20\) and varying \(K_p\) from 0.5 to 4.382 and then jumping to 15, the zero-pole maps of the system’s closed-loop transfer function were plotted. The results show that initially, poles reside in the right-half plane. As \(K_p\) increases to about 0.715, poles migrate to the left-half plane, indicating stability. When \(K_p\) exceeds 4.382, poles start moving back to the right-half plane, leading to instability. This observation aligns with the theoretically derived range.
The simulation waveforms for the phase-A voltage and three-phase grid currents of the grid tied inverter using the proposed complex-domain control method are shown. The steady-state grid currents are sinusoidal and in phase with the corresponding grid voltages, accurately tracking their reference values. This demonstrates that the PI control method based on the complex-domain model enables the three-phase grid tied inverter to achieve excellent steady-state performance.
An experimental prototype was built based on the simulation parameters. The experimental platform for the three-phase grid tied inverter mainly consists of the main power circuit, the control circuit, and the sampling circuit. The three-phase 220 V AC grid voltage is supplied via a three-phase grid-connected transformer and a voltage regulator. The DC-side voltage is provided by a DC regulated power supply.
The steady-state experimental waveforms of the grid currents are shown. Using the controller design strategy based on the complex-domain model, the experimental results show that the grid currents are sinusoidal and in phase with the grid voltages, and their amplitudes accurately track the reference values. The experimental results are consistent with the simulation results, validating the proposed approach.
Conclusion
This paper employed a complex-domain averaging modeling method for the three-phase LCL-type grid tied inverter. By exploiting the symmetry of the system matrix, the high-order real matrix was reduced to a lower-order complex matrix, thereby transforming the MIMO system in the real domain into a SISO system in the complex domain. This order reduction simplifies the control object, facilitating controller design. Subsequently, the complex Hurwitz stability criterion was applied to analyze the stability of the three-phase grid tied inverter, determining the range of control parameters for stable system operation. Both simulation and experimental results verify the effectiveness and feasibility of the proposed complex-domain modeling method and controller design strategy for the grid tied inverter.