With the rapid expansion of power grid scale and increasing demands for power reliability, traditional power system technologies struggle to address operational control consistency issues. Distributed generation (DG) and microgrids have become focal points of research worldwide. In many regions, distributed generation units integrated into microgrids are gradually becoming a significant power generation method in electrical systems due to their high flexibility and controllability, enabling microgrids to play a crucial role in maintaining grid stability and reliability. When operating in islanded mode, microgrids face numerous uncertainties and external disturbances, such as intermittency in distributed generation systems, power differences between loads, and other unpredictable factors, which can cause voltage and frequency distortions, thereby affecting microgrid stability and operation.
Islanded operation is subject to various uncertainties, causing electrical quantities to fluctuate and deviate from their set references, increasing system errors. In practical design, slight deviations in filter capacitance and inductance parameters from rated values can occur, and the more random parameters there are, the greater the impact on system output. Therefore, control methods must possess anti-interference capabilities against filter parameter uncertainties. During system operation, frequent load switching by multiple users or the connection and disconnection of high-power equipment can lead to sudden changes in three-phase voltages and currents. If the system does not recover stability promptly, it can affect the normal operation of electrical equipment, and if the changes exceed certain limits, electrical faults may occur. Distributed generation equipment, due to its inherent instability, low reliability, and operational inexperience, can affect the overall system’s DC voltage input during maintenance outages or damage from natural disasters.
For islanded microgrids, the primary goal of the control system is to maintain voltage and power near their rated or reference values during normal operation or under significant load disturbances. Islanded microgrids require precise control to suppress system uncertainties and disturbances and enhance dynamic response capabilities. Model predictive control has been used to suppress disturbances in three-phase inverters, employing disturbance observers to simplify predictive models. This method eliminates the need for inner loops, offering better robustness and dynamic response. However, model predictive control is highly dependent on model accuracy, requires high precision, involves complex modeling processes, and substantial computational load, making it unsuitable for real-time control system requirements.
Another control approach involves improving the droop control loop to achieve disturbance suppression and enhanced dynamic characteristics. A novel droop control method introduces dynamic virtual impedance feedback into traditional droop inner and outer controls and incorporates integral control into the traditional power control loop, using PI controllers to adjust droop coefficients and improve system dynamic stability. A new parallel inverter load controller improves the droop equation by introducing derivative and integral components into the traditional droop control equation, enhancing dynamic response and system controllability.
Sliding mode control (SMC) is a control method known for its fast dynamic response, strong anti-external disturbance capability, and robustness, particularly suitable for situations prone to external disturbances and parameter uncertainties. However, chattering is a major drawback of sliding mode control, which can affect system stability and even damage equipment. Higher-order sliding mode control retains the advantages of traditional sliding mode control while suppressing chattering, removes relative order limitations, and improves system accuracy. Various higher-order sliding mode control methods, such as the super-twisting algorithm (STA), drift algorithm, suboptimal algorithm, and convergence algorithm, can enhance system dynamic response and harmonic suppression capabilities. Among these, STA requires less known information for computation, needing only the value of the sliding variable, and features a simple structure and fast computation, giving it strong advantages in eliminating the effects of parameter uncertainties and external disturbances.
In three-level inverters and three-phase grid-connected inverters, super-twisting control has been incorporated into voltage outer loops and power tracking loops, with switchable high-gain observers added to voltage regulators to prevent external interference from affecting control performance. This method exhibits stronger anti-interference capability and faster transient response compared to ordinary sliding mode control. A novel robust nonlinear super-twisting controller combines STA with traditional sliding mode control for distributed power sources in single-phase inverters, ensuring system stability and robustness under various load conditions in both islanded and grid-connected operations.
In the aforementioned methods, predictive control, improved droop control, and sliding mode control can all enhance system dynamic response and robustness. The super-twisting sliding mode control method demonstrates significant advantages in inverter control. Addressing the weak load disturbance suppression capability and slow voltage reference tracking of traditional PI-controlled LCL three-phase inverters, this paper leverages the excellent performance of the super-twisting sliding mode control method to design the inner and outer loops of the three-phase inverter. Based on the internal model principle, STA controllers for the system’s inner and outer loops are designed in the dq coordinate system, and system stability is proven using Lyapunov functions. The robustness of the system under different filter parameter conditions is verified, with simulation results demonstrating strong robustness to parameter uncertainties. Compared to traditional PI control, the proposed control method, validated through semi-physical experiments under different operating conditions, can suppress fluctuations within half a power frequency cycle, effectively抑制负载突变引起的干扰并提升响应速度.
Topology and Mathematical Model of Islanded Inverter
The circuit model of an LCL-type three-phase inverter in islanded operation is shown in the figure. Here, \( V_{dc} \) is the DC source voltage; \( V_1 \) to \( V_6 \) are the six IGBT switches of the three-phase bridge; \( L_1 \), \( C \), and \( L_2 \) constitute the filter structure of the three-phase LCL-type islanded inverter; \( R_1 \) and \( R_2 \) are the equivalent resistances on the inverter side and load side, respectively; \( u_{1x} \) is the unfiltered output voltage of the inverter; \( i_{1x} \) is the inverter-side current; \( u_{Cx} \) is the filter capacitor voltage; \( i_{2x} \) is the load-side current; and \( R_x \) is the three-phase load, where \( x = a, b, c \).
Assuming the main circuit topology has a three-phase balanced load and ideal switching devices, the voltage and current equations in the abc three-phase stationary coordinate system are derived from KCL and KVL:
$$ L_1 \frac{di_{1k}}{dt} = u_{1k} – R_1 i_{1k} – u_{Ck} $$
$$ C \frac{du_{Ck}}{dt} = i_{1k} – i_{2k} $$
$$ L_2 \frac{di_{2k}}{dt} = u_{Ck} – R_2 i_{2k} – u_k $$
where \( u_k \) is the load voltage, and \( k = a, b, c \). Since only one switch per arm is on at any time, denoted as 1 for on and 0 for off, the switching states of the upper IGBTs \( V_1, V_3, V_5 \) are \( \delta_{a1}, \delta_{b1}, \delta_{c1} \), and the lower IGBTs \( V_2, V_4, V_6 \) are \( \delta_{a2}, \delta_{b2}, \delta_{c2} \), satisfying:
$$ \delta_{k1} + \delta_{k2} = 1 $$
Let \( \delta_k = \delta_{k1} = 1 – \delta_{k2} \), the relationship between the inverter output line voltage and the DC voltage source \( V_{dc} \) is:
$$ U_{ab} = U_a – U_b = (\delta_a – \delta_b) V_{dc} = \delta_{ab} V_{dc} $$
$$ U_{bc} = U_b – U_c = (\delta_b – \delta_c) V_{dc} = \delta_{bc} V_{dc} $$
$$ U_{ca} = U_c – U_a = (\delta_c – \delta_a) V_{dc} = \delta_{ca} V_{dc} $$
Based on the voltage and current equations in the abc stationary coordinate system, the mathematical model of the LCL three-phase inverter is derived. Since control variables in the abc stationary coordinate system are AC quantities and lack zero-sequence components, making control complex, the system is typically transformed into a two-phase system for analysis. Considering that controllers can analyze DC quantities more easily and separately control active and reactive power, this paper uses Park transformation to convert the abc stationary coordinates to dq rotating coordinates. The 3s/2r coordinate transformation formula is:
$$ \begin{bmatrix} u_d \\ u_q \\ u_0 \end{bmatrix} = \frac{2}{3} \begin{bmatrix} \cos\theta & \cos(\theta – \frac{2\pi}{3}) & \cos(\theta + \frac{2\pi}{3}) \\ -\sin\theta & -\sin(\theta – \frac{2\pi}{3}) & -\sin(\theta + \frac{2\pi}{3}) \\ \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \end{bmatrix} \begin{bmatrix} u_a \\ u_b \\ u_c \end{bmatrix} $$
where \( \theta \) is the angle between the d-axis and the phase reference axis. Simplifying equations (1), (3), and (4) yields the voltage and current equations in the dq coordinate system:
$$ L_1 \frac{di_{1d}}{dt} = \delta_d V_{dc} – u_{Cd} – R_1 i_{1d} + L_1 \omega i_{1q} $$
$$ L_1 \frac{di_{1q}}{dt} = \delta_q V_{dc} – u_{Cq} – R_1 i_{1q} – L_1 \omega i_{1d} $$
$$ C \frac{du_{Cd}}{dt} = i_{1d} – i_{2d} + \omega C u_{Cq} $$
$$ C \frac{du_{Cq}}{dt} = i_{1q} – i_{2q} – \omega C u_{Cd} $$
$$ L_2 \frac{di_{2d}}{dt} = u_{Cd} – u_d – R_2 i_{2d} + L_2 \omega i_{2q} $$
$$ L_2 \frac{di_{2q}}{dt} = u_{Cq} – u_q – R_2 i_{2q} – L_2 \omega i_{2d} $$
where \( \omega \) is the fundamental angular frequency of the three-phase voltage. Equations (5), (6), and (7) show that the dq-axis state quantities are coupled, which affects the current tracking performance of closed-loop control. To eliminate the impact of coupling on dual-loop output, feedforward decoupling control is often introduced into dual-loop PI controllers, ensuring separate control of the dq-axes. However, this adds significant differential computations and other control parameters, increasing system complexity, and overall performance depends on the accuracy of each parameter, making it difficult to achieve theoretical results.
Power Loop Control
The power outer loop uses a droop control method to calculate voltage references based on system output voltage and current. The droop controller employs the average method in the dq rotating coordinate system, calculating three-phase active and reactive power from sampled voltages and currents on the dq-axis:
$$ P = \frac{3}{2} (u_d i_{2d} + u_q i_{2q}) $$
$$ Q = \frac{3}{2} (u_q i_{2d} – u_d i_{2q}) $$
Considering the inverter’s equivalent output impedance is more resistive than inductive, the resistive droop equation is used. The voltage and frequency references are calculated as:
$$ V = V_{\text{ref}} – k_p (P – P_{\text{ref}}) $$
$$ \omega = \omega_{\text{ref}} + k_q (Q – Q_{\text{ref}}) $$
where \( V \), \( \omega \), \( P \), and \( Q \) are the inverter output voltage amplitude reference, angular frequency reference, active power, and reactive power, respectively; \( k_p \) and \( k_q \) are the active and reactive droop coefficients; and \( V_{\text{ref}} \), \( \omega_{\text{ref}} \), \( P_{\text{ref}} \), \( Q_{\text{ref}} \) are the rated voltage amplitude, rated angular frequency, rated active power, and rated reactive power, respectively. The coordinate transformation formula in the dq coordinate system is:
$$ \frac{d\theta}{dt} = \omega $$
Using equation (4), the voltage in the dq coordinate system is obtained. The control block diagram of the resistive droop control for the three-phase inverter is shown. The three-phase active and reactive power are calculated using the dq-axis output voltages and currents, and the droop equation computes the dq-axis reference voltages for the voltage outer loop.
Dual-Loop Control Strategy
Super-Twisting Algorithm
Given a second-order system:
$$ \dot{y}_1 = y_2 $$
$$ \dot{y}_2 = \phi(t, x) + \gamma(t, x) u(t, x) $$
where \( \phi(t, x) \) and \( \gamma(t, x) \) are unknown bounded functions of time \( t \), variable \( x \), and control function \( u \), with bounds satisfying:
$$ -F < \phi(t, x) < F $$
$$ 0 < \Gamma_{\text{min}} < \gamma(t, x) < \Gamma_{\text{max}} $$
where \( F \), \( \Gamma_{\text{min}} \), and \( \Gamma_{\text{max}} \) are positive constants. Under the boundary condition, the differential of the sliding variable \( s(t, x) \) includes:
$$ \ddot{s} \in [-F, F] + [\Gamma_{\text{min}}, \Gamma_{\text{max}}] \dot{u} $$
The STA controller used in this paper is:
$$ \mu = -\lambda |s|^{1/2} \text{sign}(s) + v $$
$$ \dot{v} = -\alpha \text{sign}(s) $$
where \( s \) is the sliding variable; \( \lambda \) and \( \alpha \) are control parameters to be designed; and \( \text{sign}(s) \) is the sign function. To further reduce chattering, the discontinuous sign function \( \text{sign}(s) \) in the super-twisting controller is replaced with the continuous \( \tanh(s) \) function, serving as an approximation of the sign function:
$$ \tanh(x) = \frac{e^x – e^{-x}}{e^x + e^{-x}} $$
$$ \mu = -\lambda |s|^{1/2} \tanh(s) + v $$
$$ \dot{v} = -\alpha \tanh(s) $$
From equation (16), the sliding variable \( s \) and its derivative will converge to zero in finite time, and the two switching functions do not affect the sliding mode characteristics of the system. If \( \lambda \) and \( \alpha \) satisfy the constraints of boundary condition (12), the sufficient conditions for the sliding variable \( s \) to stabilize in finite time are:
$$ \alpha > \frac{F}{\Gamma_{\text{min}}} $$
$$ \lambda^2 \geq \frac{4F}{\Gamma_{\text{min}}} \frac{\Gamma_{\text{max}}}{\Gamma_{\text{min}}} \frac{\alpha + F}{\alpha – F} $$
Dual-Loop Design
Capacitor Voltage Outer Loop
Since the control bandwidth of the inductor current loop is much larger than that of the capacitor voltage loop, the dual-loop control structure can be designed separately. First, the sliding variables for the capacitor voltage outer loop are defined as:
$$ s_{vd} = u_d^* – u_d $$
$$ s_{vq} = u_q^* – u_q $$
The control objective of the capacitor voltage outer loop is to track the reference voltage \( u_d^* \) and \( u_q^* \) generated by the droop controller and ultimately generate the current reference signals \( i_{1d}^* \) and \( i_{1q}^* \). Thus, equation (6) can be rewritten as:
$$ C \frac{du_{Cd}}{dt} = i_{1d}^* – i_{2d} + \omega C u_{Cq} $$
$$ C \frac{du_{Cq}}{dt} = i_{1q}^* – i_{2q} – \omega C u_{Cd} $$
Differentiating equation (18) once and combining with equation (19) yields:
$$ C \frac{ds_{vd}}{dt} = C \frac{du_d^*}{dt} – (i_{1d}^* – i_{2d} + \omega C u_{Cq}) $$
$$ C \frac{ds_{vq}}{dt} = C \frac{du_q^*}{dt} – (i_{1q}^* – i_{2q} – \omega C u_{Cd}) $$
Observing equation (20), the sliding variables \( s_{vd} \) and \( s_{vq} \) contain the current reference signals \( i_{1d}^* \) and \( i_{1q}^* \) with order zero, so the first-order sliding variable equation (18) can be applied to the control design of this LCL three-phase inverter system. To avoid chattering and keep the control signals \( i_{1d}^* \) and \( i_{1q}^* \) continuous, the super-twisting algorithm is introduced:
$$ i_{1d}^* = i_{2d} – \omega C u_{Cq} + C_0 \frac{du_d^*}{dt} + \mu_v(s_{vd}) $$
$$ i_{1q}^* = i_{2q} + \omega C u_{Cd} + C_0 \frac{du_q^*}{dt} + \mu_v(s_{vq}) $$
where \( \mu_v(s_{vd}) \) and \( \mu_v(s_{vq}) \) are the voltage sliding mode controllers, and \( C_0 \) is the rated value of the filter capacitor \( C \).
According to equation (16), \( \mu_v(s_{vd}) \) and \( \mu_v(s_{vq}) \) can be written as:
$$ \mu_v(s_{vd}) = \lambda_{vd} |s_{vd}|^{1/2} \tanh(s_{vd}) + \alpha_{vd} \int_0^t \tanh(s_{vd}) d\tau $$
$$ \mu_v(s_{vq}) = \lambda_{vq} |s_{vq}|^{1/2} \tanh(s_{vq}) + \alpha_{vq} \int_0^t \tanh(s_{vq}) d\tau $$
where \( \lambda_{vd} \), \( \lambda_{vq} \), \( \alpha_{vd} \), and \( \alpha_{vq} \) are positive proportional-integral coefficients. From equations (20) and (21), the dynamic equations of the sliding variables \( s_{vd} \) and \( s_{vq} \) are:
$$ C \frac{ds_{vd}}{dt} = -\mu_v(s_{vd}) + \Delta C \frac{du_d^*}{dt} $$
$$ C \frac{ds_{vq}}{dt} = -\mu_v(s_{vq}) + \Delta C \frac{du_q^*}{dt} $$
where \( \Delta C \) is the uncertainty of the filter capacitor, \( \Delta C = C – C_0 \). Since \( \Delta C \) is bounded, there exist two positive constants \( C_{vn1} \) and \( C_{vn2} \) satisfying the constraints:
$$ C_{vn1} \geq \left| \frac{d}{dt} \left( \Delta C \frac{du_d}{dt} \right) \right| $$
$$ C_{vn2} \geq \left| \frac{d}{dt} \left( \Delta C \frac{du_q}{dt} \right) \right| $$
According to equation (17), the sufficient conditions for the sliding variables \( s_{vd} \) and \( s_{vq} \) to converge to the sliding surfaces \( s_{vd} = \dot{s}_{vd} = 0 \) and \( s_{vq} = \dot{s}_{vq} = 0 \) in finite time are:
$$ \alpha_{vd}^2 > 4C^2 C_{vn1} \frac{\lambda_{vd} + C_{vn1}}{\lambda_{vd} – C_{vn1}}, \quad \lambda_{vd} > C C_{vn1} $$
$$ \alpha_{vq}^2 > 4C^2 C_{vn2} \frac{\lambda_{vq} + C_{vn2}}{\lambda_{vq} – C_{vn2}}, \quad \lambda_{vq} > C C_{vn2} $$
Inductor Current Inner Loop
The sliding variables for the inductor current inner loop are defined as:
$$ s_{id} = i_{1d}^* – i_{1d} $$
$$ s_{iq} = i_{1q}^* – i_{1q} $$
The inductor current inner loop aims to track \( i_{1d}^* \) and \( i_{1q}^* \) generated by the voltage outer loop. Differentiating the current loop sliding variables (26) and combining with equation (5) ultimately generates the modulation signals \( \delta_d \) and \( \delta_q \):
$$ \delta_d = \frac{1}{V_{dc}} \left[ L_{10} \frac{di_{1d}^*}{dt} + \mu_i(s_{id}) + u_{Cd} + R_1 i_{1d} – L_1 \omega i_{1q} \right] $$
$$ \delta_q = \frac{1}{V_{dc}} \left[ L_{10} \frac{di_{1q}^*}{dt} + \mu_i(s_{iq}) + u_{Cq} + R_1 i_{1q} + L_1 \omega i_{1d} \right] $$
where \( \mu_i(s_{id}) \) and \( \mu_i(s_{iq}) \) are the current sliding mode controllers, and \( L_{10} \) is the rated value of the filter inductor \( L_1 \).
According to equation (16), \( \mu_i(s_{id}) \) and \( \mu_i(s_{iq}) \) can be written as:
$$ \mu_i(s_{id}) = \lambda_{id} |s_{id}|^{1/2} \tanh(s_{id}) + \alpha_{id} \int_0^t \tanh(s_{id}) d\tau $$
$$ \mu_i(s_{iq}) = \lambda_{iq} |s_{iq}|^{1/2} \tanh(s_{iq}) + \alpha_{iq} \int_0^t \tanh(s_{iq}) d\tau $$
where \( \lambda_{id} \), \( \lambda_{iq} \), \( \alpha_{id} \), and \( \alpha_{iq} \) are positive proportional-integral coefficients. From equations (5), (26), and (27), the dynamic equations of the sliding variables \( s_{id} \) and \( s_{iq} \) are:
$$ L_1 \frac{ds_{id}}{dt} = \Delta L_1 \frac{di_{1d}^*}{dt} – \mu_i(s_{id}) – \Delta L_1 \omega i_{1q} $$
$$ L_1 \frac{ds_{iq}}{dt} = \Delta L_1 \frac{di_{1q}^*}{dt} – \mu_i(s_{iq}) + \Delta L_1 \omega i_{1d} $$
where \( \Delta L_1 \) is the uncertainty of the filter inductor, \( \Delta L_1 = L_1 – L_{10} \). Since \( \Delta L_1 \) is bounded, there exist positive constants \( L_{1n1} \) and \( L_{1n2} \) satisfying the constraints:
$$ L_{1n1} \geq \left| \frac{d}{dt} \left( \Delta L_1 \frac{di_{1d}}{dt} \right) \right| $$
$$ L_{1n2} \geq \left| \frac{d}{dt} \left( \Delta L_1 \frac{di_{1q}}{dt} \right) \right| $$
According to equation (30), the sufficient conditions for the sliding variables \( s_{id} \) and \( s_{iq} \) to converge to the sliding surfaces \( s_{id} = \dot{s}_{id} = 0 \) and \( s_{iq} = \dot{s}_{iq} = 0 \) in finite time are:
$$ \alpha_{id}^2 > 4L_1^2 L_{1n1} \frac{\lambda_{id} + L_{1n1}}{\lambda_{id} – L_{1n1}}, \quad \lambda_{id} > L_1 L_{1n1} $$
$$ \alpha_{iq}^2 > 4L_1^2 L_{1n2} \frac{\lambda_{iq} + L_{1n2}}{\lambda_{iq} – L_{1n2}}, \quad \lambda_{iq} > L_1 L_{1n2} $$
The overall control structure of the dual-loop control based on STA is shown in the figure. First, power is calculated from the output voltage and current, then the droop equation computes the voltage references \( u_d^* \) and \( u_q^* \) in the dq coordinate system. The capacitor voltage outer loop calculates the current references \( i_{1d}^* \) and \( i_{1q}^* \) for the inductor current inner loop using equation (21). Then, the current references are used to compute the modulation signals \( \delta_d \) and \( \delta_q \) via equation (27). Finally, through inverse Park transformation, the signals are converted to the abc stationary coordinate system, generating SPWM modulation signals to control the switching of \( V_1 \) to \( V_6 \) in the three-phase inverter.
Observing the super-twisting dual-loop control equations (21) and (27), derived from the essence of the super-twisting control method, the voltage loop design includes the output current \( i_{2d} \), and the current loop design includes \( u_{Cd} \) as feedforward terms, enhancing the system’s anti-disturbance performance and response to reference changes. Compared to traditional sliding mode control, the introduction of “feedforward” quantities reduces the gains \( \mu_v(s_v) \) and \( \mu_i(s_i) \) in the super-twisting control, acting as fine adjustments in the control system. Due to the finite-time convergence特性 of the super-twisting algorithm, it does not affect the system’s response speed to steady state.
Stability Analysis
To prove the stability of the dual-loop control system based on STA, equation (32) is chosen as the Lyapunov candidate function for the voltage loop:
$$ R_V(s_v) = \frac{1}{2} C s_v^2 \geq 0 $$
Differentiating equation (32) and combining with equations (18) to (22) yields:
$$ \frac{dR_{vd}}{dt} = -s_{vd} \left[ \lambda_{vd} |s_{vd}|^{1/2} \tanh(s_{vd}) + \alpha_{vd} \int_0^t \tanh(s_{vd}) d\tau \right] $$
$$ \frac{dR_{vq}}{dt} = -s_{vq} \left[ \lambda_{vq} |s_{vq}|^{1/2} \tanh(s_{vq}) + \alpha_{vq} \int_0^t \tanh(s_{vq}) d\tau \right] $$
When the sliding variable \( s_{vd} > 0 \), the hyperbolic tangent function \( \tanh(s) > 0 \), so \( dR_{vd}/dt < 0 \); conversely, when \( s_{vd} < 0 \), \( \tanh(s) < 0 \), so \( dR_{vd}/dt > 0 \). Therefore, the Lyapunov function \( R_V(s_v) \) is positive definite and its first derivative with respect to time is negative definite, proving that the sliding variable \( s_{vd} \) can asymptotically converge to zero, i.e., the proposed STA-based voltage loop controller is asymptotically stable. Similarly, the stability of the current loop can be proven.
Robustness Analysis
In distributed generation systems, unpredictable measurement noise, electromagnetic radiation induction internally, coupling induction from distributed capacitance and inductance between device wires, and errors between actual filter element parameters and rated parameters can affect overall system stability. Considering various uncertainties, the sliding variables of the voltage outer loop (23) are modified as follows:
$$ C \frac{ds_{vd}}{dt} = -\mu_v(s_{vd}) + \xi_d $$
$$ C \frac{ds_{vq}}{dt} = -\mu_v(s_{vq}) + \xi_q $$
where \( \xi \) is the sum of internal and external disturbances of the system. Then equation (33) can be modified as:
$$ \frac{dR_{vd}}{dt} = -s_{vd} \left[ \lambda_{vd} |s_{vd}|^{1/2} \tanh(s_{vd}) + \alpha_{vd} \int_0^t \tanh(s_{vd}) d\tau – \xi_d \right] $$
$$ \frac{dR_{vq}}{dt} = -s_{vq} \left[ \lambda_{vq} |s_{vq}|^{1/2} \tanh(s_{vq}) + \alpha_{vq} \int_0^t \tanh(s_{vq}) d\tau – \xi_q \right] $$
According to the Lyapunov stability condition, as long as the parameters \( \lambda_v \) and \( \alpha_v \) are reasonably selected, the first derivative of the function \( R_V(s_v) \) with respect to time can be made negative definite. The proposed STA-based dual-loop controller possesses strong robustness.
Simulation and Experimental Results
Robustness Performance Simulation
The simulation is based on the PLECS platform to verify that the proposed STA control method is robust to uncertainties in system filter parameters. The system parameters for simulation and experimental platform design are shown in Table 1.
| Parameter Name | Value |
|---|---|
| DC Bus Voltage \( V_{dc} \) | 700 V |
| Rated Output Voltage Amplitude | 311 V |
| Rated Output Voltage Angular Frequency | 100π rad/s |
| System Sampling Frequency \( T_s \) | 10 kHz |
| System Switching Frequency \( f_s \) | 10 kHz |
| Rated Inductance \( L_{f1} \) | 2 mH |
| Rated Inductance \( L_{f2} \) | 0.03 mH |
| Rated Capacitance \( C_f \) | 8 mF |
Figure 6 shows the robustness simulation results under three different operating conditions with the same control parameters, at a rated power of 3 kW. In Figure 6(a), Condition I corresponds to the Phase A output voltage waveform with rated filter parameters; Condition II with 1.5 times the rated filter parameters; Condition III with 2 times the rated filter parameters. The output RMS value serves as a reference for control accuracy. From Figure 6(a), it can be seen that the Phase A voltage and current waveforms under all three conditions show no fluctuations, and the system is unaffected by disturbances from filter parameter variations. The output current quickly tracks load switching changes. Observing Figure 6(b), as filter parameters change, the system output voltage RMS shows small deviations of approximately 0.22 V (Condition II). When filter parameters are twice the rated value, a voltage fluctuation of about 1.2 V occurs, stabilizing after about 1.2 rated cycles (24 ms). Simulation results indicate that even with twice the filter parameter disturbance, the impact on system output voltage is minimal. The above simulation results demonstrate that the proposed three-phase islanded inverter system based on STA has good robustness to filter parameter disturbances.
Load Step Change Experiment Comparison
To further verify the anti-interference capability and response speed of the STA control method, an experimental prototype of a three-phase LCL inverter system was built, as shown in Figure 7. The DC power source uses Chroma 62150H-1000S; voltage and current signals are sampled by LEM LV25-P voltage Hall sensors and HLSR 32-P current Hall sensors, respectively; verification is performed using RT Box developed by Plexim GmbH, PLECS simulation software, and a PC.
Figures 8 and 9 show the experimental waveforms of Phase A voltage, current, active power, and voltage RMS for the two control methods during load switching. Observing Figure 8(a), when the load is connected at time \( t_1 \), the ST-controlled output voltage and current quickly reach the steady-state value of 6.19 A, but the PI-controlled voltage and current waveforms exhibit fluctuations (peak voltage drops from 311.5 V to 298.9 V), stabilizing after about one cycle (approximately 20 ms). In Figure 8(b), the ST-controlled output voltage RMS has a maximum fluctuation of only 0.3 V, with almost no voltage or power fluctuations, while the PI control method has a voltage RMS deviation of about 10 V, requiring about one cycle to transition to steady state.
Comparing Figure 9(a), when the load is disconnected, the PI-controlled output voltage and current exhibit fluctuations for about one cycle (20 ms), while the ST-controlled output voltage and current quickly transition to steady state with no significant voltage or current overshoot. In Figure 9(b), the PI-controlled voltage RMS has a fluctuation of about 10.3 V after load disconnection, while the ST-controlled voltage RMS fluctuation is only 0.3 V. Through load switching comparison figures, the proposed ST-controlled three-phase inverter system under islanded conditions has a good dynamic response process during load connection and disconnection and can suppress changes caused by system load disturbances.
Reference Step Change Comparison
Figures 10 and 11 show the experimental comparison of Phase A for reference step changes under the two control methods. Observing the output voltage and current waveforms in Figure 10(a), when the reference voltage suddenly drops by 30 V at time \( t_1 \), the proposed ST-controlled voltage and current quickly track the voltage reference, while the traditional PI-controlled voltage and current waveforms, though able to track the voltage reference change, take about 2 cycles (40 ms) to reach steady state, with a long dynamic process and tracking error. Comparing the output voltage RMS and active power in Figure 10(b), PI control takes about 2 cycles (40 ms) to track the voltage reference change, while ST control has minimal active power fluctuation and can quickly track the voltage reference change within one cycle (20 ms).
From Figure 11(a), it can be seen that when the reference voltage suddenly returns to the rated voltage reference, the PI-controlled output voltage and current waveforms take about 2 cycles (40 ms) to track the rated voltage, while the proposed ST-controlled output voltage and current achieve zero-error tracking of the reference voltage. The experimental results in Figures 10 and 11 show that reference voltage changes have almost no impact on the proposed ST-controlled three-phase inverter system, virtually eliminating the process of tracking voltage references. Compared to the traditional PI control method, in practical applications, it can reduce system voltage, current, and power fluctuations, enhancing system stability.
Conclusion
Starting from the basic principles of internal model control, a mathematical model of super-twisting sliding mode control is designed in the dq coordinate system. Since the controller introduces output voltage and current as feedforward variables, it can effectively improve system response speed and anti-interference capability. Compared with traditional PI control, the proposed disturbance suppression method has been proven to have good robustness to filter parameter uncertainties; in addition, load step change and voltage reference step change experiments validate the effectiveness of the proposed islanded disturbance suppression strategy based on super-twisting sliding mode control. The dynamic changes of voltage and current can quickly track changes in load or voltage reference, and effectively suppress system fluctuations caused by various disturbances.