In recent years, the rapid development of renewable energy has accelerated the advancement of microgrid technologies, which play a critical role in distributed generation. Microgrids leverage the characteristics of distributed energy resources to reduce the reliance on non-renewable energy sources. Beyond conventional grid-connected operation modes, microgrid systems can also operate in islanded mode without dependence on the main grid. In islanded mode, droop control methods are typically employed to achieve decentralized parallel control of multiple inverters in AC microgrids, enabling output power sharing and circulating current suppression. The essence of droop control lies in the linear relationship between the output active and reactive power and the amplitude and frequency of the output voltage. By designing droop equations, the output voltage and power are adjusted appropriately, allowing inverter controllers to autonomously distribute output power based solely on local measurements while meeting the required control quality for parallel operation.
However, traditional droop control methods exhibit poor adaptive capability when load fluctuations or system parameter variations occur. In practical applications, distributed energy systems are subject to various disturbances and noise, causing the output power, voltage amplitude, and phase to deviate from the ideal linear relationship, which poses significant challenges for controller design. In the field of power electronics, sliding mode control (SMC) methods are widely applied. Previous studies have utilized robust sliding mode controllers in DC/DC Boost converters to address instability issues caused by constant power loads in microgrids, demonstrating improved control performance. Others have employed proportional-integral (PI) controllers for voltage droop control and enhanced them with sliding mode control to improve response speed and voltage regulation capability. Research shows that sliding mode controllers can guide system outputs to desired sliding surfaces and further stabilize them near preset values, providing excellent voltage tracking and power sharing capabilities, as well as achieving frequency synchronization under various load conditions. However, practical systems face issues such as control inertia and spatiotemporal delays in power electronic devices, particularly switching devices, leading to chattering problems. Severe chattering can increase energy consumption and hardware wear, affecting control system stability. Therefore, mitigating chattering effects is crucial when applying sliding mode control methods.
Integral sliding mode (ISM) control offers discrete control action, improving control performance and robustness compared to traditional PI control, but it suffers from large chattering amplitude and slow convergence. In contrast, fast terminal sliding mode (FTSM) control provides continuous control laws with weaker chattering and faster convergence. By combining these approaches, the system state rapidly reaches the neighborhood of the sliding surface under the exponential term of terminal sliding mode, and then maintains on the sliding surface through the integral sliding mode component. Terminal sliding mode reduces convergence time and attenuates chattering, while integral sliding mode ensures better dynamic performance after reaching the sliding surface. Adaptive robust sliding mode control offers a solution for problems with bounded disturbances and has been widely used in power converters, motor control, and other fields. Thus, adaptive robust control laws can be utilized to refine the sliding surface and reduce chattering amplitude.
This paper addresses the scenario where the DC-side capacitor voltage of inverters is non-ideal in islanded mode, proposing a terminal integral sliding mode control law to achieve adaptive robust droop control. This approach mitigates the effects of system disturbances and uncertainties in DC-side voltage values, effectively suppresses external interference, and enhances system robustness. The proposed method integrates adaptive robust control with sliding mode techniques, leveraging the advantages of both to improve voltage tracking capability, steady-state performance, reliability, and robustness in three-phase inverter systems.
Mathematical Model of Three-Phase Inverter
The topology of a single LC-filtered three-phase parallel inverter is shown in the figure above. The left side of the inverter, i.e., the DC bus side, can be simplified as a voltage source with output voltage \( U_{dc} \). Due to the presence of DC-side stabilizing capacitors, it can be considered as a voltage value with bounded disturbance. S1–S6 represent six power switches, L is the filter inductance on the inverter side, \( R_L \) is the parasitic resistance of the filter inductance, and C is the filter capacitance. The inverter output currents \( i_{La}, i_{Lb}, i_{Lc} \), inverter control voltages \( U_{ia}, U_{ib}, U_{ic} \), and inverter output voltages \( U_{oa}, U_{ob}, U_{oc} \) correspond to phases a, b, and c of the three-phase inverter system. Applying Kirchhoff’s laws, the ordinary differential equations of the three-phase inverter system in the stationary coordinate system are:
$$ L \frac{di_{L}}{dt} + R_L i_{L} = U_i – U_o $$
$$ C \frac{dU_o}{dt} = i_L – i_o $$
where \( i_L \) is the inverter output current, \( U_i \) is the inverter control voltage, \( U_o \) is the inverter output voltage, and \( i_o \) is the output current. The relationship between the SPWM modulation wave \( u(t) \) and the inverter control voltage \( U_i \) is given by \( U_i = \frac{U_{dc}}{2} u(t) \).
Transforming to the dq rotating coordinate system using Park transformation, the mathematical model of the three-phase parallel inverter in the dq frame is:
$$ L \frac{di_d}{dt} = -R_L i_d + \omega L i_q + U_{id} – U_{od} $$
$$ L \frac{di_q}{dt} = -R_L i_q – \omega L i_d + U_{iq} – U_{oq} $$
$$ C \frac{dU_{od}}{dt} = i_d – i_{od} + \omega C U_{oq} $$
$$ C \frac{dU_{oq}}{dt} = i_q – i_{oq} – \omega C U_{od} $$
where \( i_d, i_q \) are the d-axis and q-axis components of the inverter output current, \( U_{id}, U_{iq} \) are the d-axis and q-axis components of the inverter control voltage, \( U_{od}, U_{oq} \) are the d-axis and q-axis components of the inverter output voltage, \( i_{od}, i_{oq} \) are the d-axis and q-axis components of the output current, and \( \omega \) is the angular frequency. This transformation converts AC quantities into DC quantities, facilitating controller design and implementation, though it introduces coupling between d-axis and q-axis variables, which must be addressed through decoupling design in the controller.
Instantaneous Power Calculation
To derive expressions for output active power P and reactive power Q, consider the Thevenin equivalent circuit for two parallel inverters. The output voltage of the i-th inverter is \( U_{oi} \), and the voltage across the load-side impedance is \( U_{load} \). The equivalent output impedance of the i-th inverter is defined as:
$$ Z_{oi} = R_{oi} + jX_{oi} $$
where the equivalent connection impedance is the sum of the equivalent output impedance and line impedance. The output power P and Q for a single inverter are:
$$ P = \frac{U_o U_{load}}{X} \sin(\delta) $$
$$ Q = \frac{U_o (U_o – U_{load} \cos(\delta))}{X} $$
where \( \delta \) is the power angle. In high-voltage networks, active power relates to frequency, and reactive power relates to voltage in a droop manner, whereas in low-voltage microgrids, this relationship is reversed. In the context of low-voltage microgrids, where output inductance values are large or distances between units are long, the line impedance and output impedance of inverters are considered predominantly inductive. Typically, distributed generation units with output power less than 10 kW are connected to low-voltage microgrids, and line impedance in low-voltage microgrids is mainly resistive, i.e., \( R \gg X \), and the power angle \( \delta \) is small, especially when similar branches are connected in parallel, so \( \sin(\delta) \approx \delta \) and \( \cos(\delta) \approx 1 \). Thus, equations (4) and (5) simplify to:
$$ P = \frac{U_o U_{load}}{X} \delta $$
$$ Q = \frac{U_o (U_o – U_{load})}{X} $$
Droop control mimics the droop characteristics of synchronous generators to control inverters. The droop control equations are:
$$ f = f_N – m_p (P – P_N) $$
$$ U = U_N – m_q (Q – Q_N) $$
where P and Q are the rated active and reactive power, f and \( f_N \) are the output voltage frequency and rated frequency of the controlled inverter, U and \( U_N \) are the output voltage amplitude and rated voltage amplitude, and \( m_p \) and \( m_q \) are the droop coefficients for active and reactive power. The droop characteristics are illustrated in the figure above.
The droop control system structure for the inverter includes Park transformation and inverse transformation modules, power calculation modules based on instantaneous power theory, low-pass filter modules, droop control modules, and voltage synthesis modules, forming the droop component of droop control.
Control Method Based on Terminal Integral Sliding Mode
To address the poor robustness of traditional droop control in parallel inverters, a droop control strategy based on adaptive robust terminal integral sliding mode control law is proposed. The convergence speed, response time, and chattering amplitude of sliding mode control are analyzed. By adding an integral component to the terminal sliding mode control law, an extended sliding mode control law is established. For system parameter disturbances and uncertainties in the DC source side voltage, adaptive robust control is employed to suppress disturbances, controlling the output voltage within an appropriate range of the reference voltage and improving system robustness.
Define the system variables as \( x_1 = e = U_o – U_{ref} \) and \( x_2 = \dot{e} \), where \( U_{ref} \) is the reference output voltage. For convenience, let:
$$ \dot{x}_1 = x_2 $$
$$ \dot{x}_2 = f(x) + g(x) u + d $$
where \( f(x) \) and \( g(x) \) are system functions, u is the control input, and d represents uncertainties and disturbances. The system equation with error and uncertainty is:
$$ \ddot{e} = f(e, \dot{e}) + g(e, \dot{e}) u + d $$
Define the integral terminal sliding mode surface functions for the d-axis and q-axis as:
$$ s_d = \dot{e}_d + \alpha_d e_d + \beta_d \int_0^t e_d(\tau) d\tau $$
$$ s_q = \dot{e}_q + \alpha_q e_q + \beta_q \int_0^t e_q(\tau) d\tau $$
where \( \alpha_d, \alpha_q, \beta_d, \beta_q > 0 \) are design parameters. The exponential reaching law is selected to define the control laws for the integral terminal sliding mode controllers on the d-axis and q-axis:
$$ \dot{s}_d = -k_d s_d – \eta_d \text{sgn}(s_d) $$
$$ \dot{s}_q = -k_q s_q – \eta_q \text{sgn}(s_q) $$
where \( k_d, k_q, \eta_d, \eta_q > 0 \). Thus, the control laws can be designed as:
$$ u_d = -g^{-1}(x) \left[ f(x) + \alpha_d \dot{e}_d + \beta_d e_d + k_d s_d + \eta_d \text{sgn}(s_d) \right] $$
$$ u_q = -g^{-1}(x) \left[ f(x) + \alpha_q \dot{e}_q + \beta_q e_q + k_q s_q + \eta_q \text{sgn}(s_q) \right] $$
Define the Lyapunov functions for the integral terminal sliding mode control systems on the d-axis and q-axis as:
$$ V_d = \frac{1}{2} s_d^2 $$
$$ V_q = \frac{1}{2} s_q^2 $$
Taking the d-axis component as an example, substitute the control law \( u_d \) into the derivative of \( V_d \):
$$ \dot{V}_d = s_d \dot{s}_d = s_d \left( -k_d s_d – \eta_d \text{sgn}(s_d) \right) = -k_d s_d^2 – \eta_d |s_d| \leq 0 $$
Similarly, for the q-axis component:
$$ \dot{V}_q = s_q \dot{s}_q = s_q \left( -k_q s_q – \eta_q \text{sgn}(s_q) \right) = -k_q s_q^2 – \eta_q |s_q| \leq 0 $$
Since \( \dot{V}_d \leq 0 \) and \( \dot{V}_q \leq 0 \), the system is stable. When \( s_d = 0 \) or \( s_q = 0 \), \( \dot{V}_d = 0 \) or \( \dot{V}_q = 0 \), and according to LaSalle’s invariance principle, the system remains on the sliding surface.
In simulations, the inverter-side power supply is often an ideal voltage source. However, in practical applications, the DC-side capacitor voltage \( U_{dc} \) of the inverter may vary. For grid-connected control scenarios, the DC-side voltage can be assumed to have small fluctuations, but in islanded mode control scenarios, DC-side fluctuations may be significant. This non-ideal condition greatly impacts inverter design, usage, and stable control, making the design of adaptive robust controllers highly valuable.
Design of Adaptive Robust Control Law
In cases of system parameter perturbations, such as aging or fluctuations in inverter output parameters, and particularly in islanded mode where the DC-side capacitor voltage value, conventionally assumed constant, may fluctuate, an adaptive robust control law is designed to suppress such fluctuations. For the system equation with bounded disturbances, assume the disturbance is a time-varying state with upper and lower bounds. The system equation can be written as:
$$ \ddot{e} = f(e, \dot{e}) + g(e, \dot{e}) u + \Delta $$
where \( \Delta \) represents the total uncertainty including disturbances and model uncertainties. Define the sliding mode surface as:
$$ s = \dot{e} + \alpha e + \beta \int_0^t e(\tau) d\tau $$
According to the exponential reaching law, the control law for the adaptive robust terminal integral sliding mode controller on the d-axis is defined as:
$$ u_d = -g^{-1}(x) \left[ f(x) + \alpha_d \dot{e}_d + \beta_d e_d + k_d s_d + \eta_d \text{sgn}(s_d) + \hat{\Delta}_d \right] $$
where \( \hat{\Delta}_d \) is the estimate of the parameter \( \Delta_d \). Similarly, for the q-axis:
$$ u_q = -g^{-1}(x) \left[ f(x) + \alpha_q \dot{e}_q + \beta_q e_q + k_q s_q + \eta_q \text{sgn}(s_q) + \hat{\Delta}_q \right] $$
Define the deviation between the parameter \( \Delta \) and its estimate \( \hat{\Delta} \) as \( \tilde{\Delta} = \Delta – \hat{\Delta} \). For decoupled computation of the adaptive law, take the estimates on the d-axis and q-axis as \( \hat{\Delta}_d \) and \( \hat{\Delta}_q \), satisfying \( \hat{\Delta} = \hat{\Delta}_d + \hat{\Delta}_q \). Define the adaptive control law on the d-axis as:
$$ \dot{\hat{\Delta}}_d = \gamma_d s_d $$
$$ \dot{\hat{\Delta}}_q = \gamma_q s_q $$
where \( \gamma_d, \gamma_q > 0 \) are adaptation gains. Substituting the control law \( u_d \) into the derivative of the Lyapunov function \( V_d = \frac{1}{2} s_d^2 + \frac{1}{2\gamma_d} \tilde{\Delta}_d^2 \):
$$ \dot{V}_d = s_d \dot{s}_d + \frac{1}{\gamma_d} \tilde{\Delta}_d \dot{\tilde{\Delta}}_d = s_d \left( -k_d s_d – \eta_d \text{sgn}(s_d) – \tilde{\Delta}_d \right) + \frac{1}{\gamma_d} \tilde{\Delta}_d ( -\dot{\hat{\Delta}}_d ) $$
Since \( \dot{\tilde{\Delta}}_d = -\dot{\hat{\Delta}}_d = -\gamma_d s_d \), then:
$$ \dot{V}_d = -k_d s_d^2 – \eta_d |s_d| – s_d \tilde{\Delta}_d + \tilde{\Delta}_d s_d = -k_d s_d^2 – \eta_d |s_d| \leq 0 $$
Similarly, for the q-axis:
$$ \dot{V}_q = -k_q s_q^2 – \eta_q |s_q| \leq 0 $$
Thus, the system is stable, and the adaptive laws ensure that the estimates converge to the true values, reducing chattering and improving robustness.
Simulation Results and Analysis
To verify the robustness of the proposed control method, a simulation model of parallel inverters with identical parameters was established, and the control performance under DC-side capacitor voltage突变 was compared between integral sliding mode controllers and conventional PI controllers. The simulation parameters are set as follows: simulation step size \( 10^{-6} \) seconds, total simulation time 1 second, constant power load power 10 kW. At 0.3 seconds, the line switch is closed, and the system response is observed when the inverter DC-side capacitor voltage drops from 800 V to 400 V. At 0.6 seconds, the line switch is opened to remove the change. The comparison of waveforms for inverter 1 is shown in the figures below.
The simulation results indicate that when the DC-side voltage fluctuates, the traditional PI controller experiences power and voltage collapse, rendering the system completely unusable. Integral sliding mode control only suffers from power loss, failing to track the load brought by the CPL, but the inverter output voltage fluctuation amplitude is better than that of traditional PI control. The sliding mode control action causes the system to have large fluctuations only when the change is removed, but the system remains out of control. In contrast, the sliding mode controller with adaptive robust control law performs excellently, maintaining system stability even when the DC-side capacitor voltage fluctuates. The exponential reaching term, integral steady-state attachment, and adaptive law in the AFTISM controller collectively suppress grid剧烈波动 and collapse, demonstrating the controller’s excellent tracking control and anti-interference ability under inverter DC-side parameter perturbations.
| Control Method | Active Power Response | Reactive Power Response | Voltage Frequency | Voltage Amplitude |
|---|---|---|---|---|
| Traditional PI | Collapse | Collapse | Unstable | Collapse |
| Integral Sliding Mode (ISM) | Poor tracking | Poor tracking | Moderate fluctuations | Large fluctuations |
| Adaptive Terminal ISM (AFTISM) | Stable | Stable | Stable | Stable |
The above table summarizes the performance of different control methods under DC-side voltage fluctuations. The AFTISM controller shows superior performance in maintaining stability and tracking accuracy.
Experimental Validation on Parallel Inverter Control Circuit
An experimental platform for parallel inverters was established to validate the performance of the proposed adaptive robust terminal integral sliding mode controller based on SPWM. The platform structure is illustrated in the figure above. To thoroughly verify the performance of the proposed control strategy, comparative experiments were designed under various conditions, including steady-state operation without disturbance, robustness under circuit parameter perturbations, and harmonic suppression capability. The experimental circuit parameters are listed in the table below.
| Parameter Name | Symbol | Value and Unit |
|---|---|---|
| Inverter output filter inductance | L | 2 mH |
| Inverter output parasitic resistance | R | 0.2 Ω |
| Inverter output filter capacitance | C | 6 μF |
| DC-side capacitor voltage | U_dc | 25 V |
| Inverter output frequency | f | 50 Hz |
| Inverter switching frequency | f_SPWM | 12.5 kHz |
| Sampling frequency | f_s | 50 kHz |
The DC side uses an NVVV 15V DC power supply, the inverter is embedded in the drive sampling inverter board, and the TMS320F28335 DSP control chip is used as the main control board with a PC as the core control part of the system. The parallel inverter experimental setup is shown in the figure above.
Steady-State Performance Analysis
Experiments on two parallel inverters were conducted in a nominal environment without additional external disturbances. The voltage waveforms under traditional PI method, ISM method, and AFTISM method are shown in the figures above. The experimental waveforms indicate that the inverter can output high-quality, distortion-free power frequency sinusoidal waveforms under traditional PI, ISM, and AFTISM methods. Data from the power tester show that the THD under the three methods are 2.21%, 1.89%, and 1.72%, respectively, consistent with simulation observations and all meeting the requirement of less than 5%, complying with international parallel quality standards. Thus, the steady-state control performance of the proposed AFTISM control method is validated.
Conclusion
This paper models the DC-side of the inverter as a voltage source with noise and disturbance terms, better aligning with the operating conditions of inverter power supplies in islanded microgrid systems. A novel control law with adaptive robustness to noise is proposed. Addressing the sliding mode control metrics—convergence speed, response time, and chattering amplitude—a terminal integral sliding mode controller with an adaptive robust control law is designed. Compared to traditional second-order linear sliding mode, the introduced integral term improves the control quality of the sliding mode, and terminal sliding mode addresses the issues of slow convergence and significant chattering in pure integral sliding mode. The adaptive robust control law adapts to system parameter changes, such as disturbances in DC-side capacitor voltage and aging of output filter inductors and capacitors, to reduce system disturbances, and also adjusts the system sliding curve to reduce chattering amplitude.
By combining sliding mode control with adaptive control methods, this paper leverages the advantages of sliding mode variable structure control, designs adaptive mechanisms for non-ideal DC bus-side capacitor voltage values in inverters, and enhances the robustness of the control system. Finally, simulation and circuit experiments on parallel three-phase inverters verify that this method significantly improves voltage tracking capability, steady-state performance, reliability, and robustness of the inverters. The proposed approach is particularly effective for three-phase inverter systems in islanded microgrids, ensuring stable operation under various disturbances and parameter uncertainties.
Future work will focus on optimizing the adaptive laws for faster convergence and extending the method to larger-scale microgrid applications with multiple three-phase inverters. Additionally, real-time implementation challenges and computational efficiency will be addressed to facilitate practical deployment in industrial settings.