In the context of global environmental degradation and energy supply constraints, renewable energy generation, particularly photovoltaic and wind power, has become a crucial component of power systems. However, the integration of these distributed renewable energy sources and numerous power electronic devices poses new challenges to the stable operation of traditional power grids. The stability issues of grid-connected inverter systems are especially pronounced in weak grid environments. To address this problem, researchers have extensively studied stability control strategies for grid-connected inverters. For instance, one study proposed a stability control strategy based on q-axis impedance reshaping for grid-connected inverters, introducing an impedance controller that reshapes the q-axis impedance into a positive resistance in the low-frequency band, effectively mitigating system instability caused by phase-locked loops and grid voltage feedforward control. Another study focused on modeling, stability analysis, and control of inverter grid-connected systems based on power electronic transformers, developing a complex frequency-domain model for distributed generation grid-connected systems and utilizing advanced stability assessment methods to analyze system stability, while exploring the impact of different control parameters on system stability boundaries. Based on this, two stability control methods were proposed: virtual impedance adjustment and digital filter application, which effectively enhanced system stability. Although these methods have achieved certain results in stability control of grid-connected inverters, they rely on specific hardware and control strategies, lacking generality and flexibility. Therefore, I propose a stability control strategy for grid-connected inverters based on virtual synchronous generators (VSGs). By emulating the operational mechanisms of synchronous generators, this strategy enhances system stability and improves response speed.
To begin, I establish a comprehensive model for grid-connected inverters to accurately simulate their working process. Given that grid-connected inverters are responsible for efficiently and stably converting DC power into AC power integrated into the grid, the model delves into the typical system architecture of inverters, integrating multiple core modules to ensure high efficiency and stability in power conversion. The model structure is illustrated below.
The core of the model is based on a transformation mechanism that converts three-phase AC signals into a rotating dq coordinate system, thereby transforming them into DC components, facilitating subsequent signal processing and control. The PWM control module receives these transformed signals and generates pulse sequences to drive the switching devices of the inverter, enabling precise regulation of the output voltage. The inverter itself adopts an LCL filter structure to suppress harmonic components and improve power quality. At the output of the model, I define the filter capacitor voltage and the corresponding line inductor current, while fully considering the effects of line inductance and line resistance on power transmission. To more accurately describe the operating state of the inverter, the model also embeds a power calculation module that monitors the output power of the inverter in real-time and provides necessary feedback for control algorithm reference. During the model construction process, I also thoroughly consider the control principles of grid-connected inverters, equating the grid to an ideal voltage source in series with a purely inductive impedance model, thereby simplifying the analysis process. The grid-connected inverter model constructed through these means accurately reflects the dynamic characteristics of the actual system.
Next, I introduce virtual synchronous generator (VSG) technology to precisely emulate the core behavior of traditional synchronous generators. The incorporation of VSG technology enables the inverter to accurately replicate key characteristics of synchronous generators, including excitation control, the relationship between reactive power and voltage, the dynamic behavior of power-frequency controllers, and the relationship between inverter output voltage and current. The excitation controller, as a key component of VSG technology, precisely adjusts based on the deviation between the voltage reference value and the actual output voltage. This process can be quantified as:
$$E_f^*(t) = E_0 + K_v (V_{ref} – V_{out}(t)) + \int_0^t K_i (V_{ref} – V_{out}(\tau)) d\tau$$
where \(E_f^*(t)\) is the excitation electromotive force reference value of the VSG, \(E_0\) is the baseline value of the excitation electromotive force, \(K_v\) is the proportional gain of the voltage controller, \(V_{ref}\) is the voltage reference value, \(V_{out}(t)\) is the actual output voltage, and \(K_i\) is the integral gain of the voltage controller. This equation ensures stable voltage control. Additionally, VSG technology reveals the intrinsic relationship between reactive power and voltage:
$$Q(t) = \frac{E_f^*(t) \cdot V_{out}(t) \cdot \sin \delta}{X}$$
where \(Q(t)\) is the reactive power output by the VSG, \(X\) is the impedance to reactive power, and \(\delta\) is the phase difference between the VSG output current and voltage. This relationship is influenced by factors such as the excitation electromotive force reference value, actual output voltage, and power angle. The dynamic behavior of the power-frequency controller is also a key aspect of VSG technology implementation. It maintains the output frequency near the grid frequency by adjusting the angular velocity. This process can be expressed as:
$$\omega(t) = \omega_0 + \frac{P_{ref} – P(t)}{M} – D(\omega(t) – \omega_{grid})$$
where \(\omega(t)\) is the angular velocity of the VSG, \(\omega_0\) is the baseline angular velocity, \(M\) is the virtual inertia, \(P_{ref}\) is the active power reference value, \(P(t)\) is the actual output active power, \(D\) is the damping effect of the VSG on frequency deviation, and \(\omega_{grid}\) is the angular velocity value of the grid frequency. Based on Kirchhoff’s voltage law, the relationship between inverter output voltage and current can be described as:
$$V_{out,a,b,c}(t) = e_{a,b,c}(t) – R \cdot i_{a,b,c}(t) – L \cdot \frac{d i_{a,b,c}(t)}{d t}$$
where \(V_{out,a,b,c}(t)\) are the three-phase components of the inverter output voltage, \(R\) is the resistive loss at the inverter output, \(i_{a,b,c}(t)\) are the three-phase components of the inverter output current, \(L\) is the inductance at the inverter output, and \(\frac{d i_{a,b,c}(t)}{d t}\) is the rate of change of the three-phase components of the inverter output current. Finally, the equivalent circuit equation of the virtual synchronous generator can be expressed as:
$$E_f^*(t) = \sqrt{3} \cdot V_{line}(t) \cdot \cos(\theta(t) – \phi)$$
where \(V_{line}(t)\) is the effective value of the inverter output line voltage, \(\theta(t)\) is the angle of internal mechanical rotation of the VSG, and \(\phi\) is the phase relationship between the VSG output current and the line voltage. This equation embodies the core idea of VSG technology: equating the output characteristics of the inverter to the excitation-induced electromotive force of a synchronous generator. This implementation allows grid-connected inverters to closely mimic the behavior of synchronous generators, thereby achieving high stability in frequency and voltage output.
To better control the stability of grid-connected inverters, I delve into strategies for reasonable power allocation. First, I clarify the principle of active power allocation, which is based on the relationship between the actual output active power of the inverter and its rated capacity. This can be expressed as:
$$f_{P_i S_i} = \frac{\sum_{i=1}^n P_i}{\sum_{i=1}^n S_i}$$
where \(P_i\) is the actual output active power of inverter \(i\), and \(S_i\) is the rated capacity of inverter \(i\). To achieve this allocation principle, I adopt an active droop control strategy. This strategy adjusts the output active power of the inverter to approach its rated value and sets a droop coefficient. This coefficient maintains a positive correlation with the rated active power of the inverter, which can be expressed as:
$$m_i = k_p \cdot P_{r,i}$$
where \(m_i\) is the droop coefficient of inverter \(i\), \(k_p\) is the proportionality coefficient, and \(P_{r,i}\) is the rated active power of inverter \(i\). Under the droop control strategy, the relationship between the output active power \(P_i\) and its angular frequency \(\omega_i\) of inverter \(i\) can be expressed as:
$$\omega_i = \omega_{ref} – \frac{1}{m_i} (P_i – P_{r,i})$$
where \(\omega_{ref}\) is the reference angular frequency. For the issue of reactive power allocation, since inverter output voltages are equal, differences in line impedance can lead to uneven reactive power distribution. To compensate for this difference, I not only employ a reactive droop control strategy but also combine it with adaptive virtual impedance control to achieve balanced reactive power distribution. Under the adaptive virtual impedance control strategy, the virtual impedance value \(R_{v,i}\) is dynamically adjusted based on the reactive power \(Q_i\) of inverter \(i\) and the load power factor. The dynamic adjustment process can be expressed as:
$$R_{v,i} = k_q \cdot |Q_i – Q_{ref,i}| + k_i \int_0^t (Q_i(\tau) – Q_{ref,i}) d\tau$$
where \(k_q\) is the adjustment coefficient, \(k_i\) is the integral coefficient, and \(Q_{ref,i}\) is the reactive power reference value of inverter \(i\). In summary, combining the active droop control strategy with the adaptive virtual impedance control strategy ensures appropriate allocation of inverter output power, thereby effectively improving the stability of grid-connected inverters based on virtual synchronous generators.
To validate the stability control method for grid-connected inverters, I use MATLAB/Simulink as the experimental platform to build a simulation model. To ensure the accuracy and reliability of the simulation results, I set a series of key simulation parameters, as detailed in Table 1.
| Parameter Name | Parameter Value |
|---|---|
| DC-side voltage \(U_{dc}\) (V) | 500 |
| AC-side voltage effective value \(U_{ac}\) (V) | 230 |
| AC-side frequency \(f_{ac}\) (Hz) | 50 |
| Filter inductor \(L_f\) (mH) | 5 |
| Filter capacitor \(C_f\) (μF) | 10 |
| Switching frequency \(f_{sw}\) (kHz) | 10 |
During the experimental preparation phase, I also inspect and debug all experimental equipment to ensure proper operation, while establishing a comprehensive data acquisition system to ensure the accuracy and completeness of experimental data.
To comprehensively evaluate the stability control method for grid-connected inverters, I define the following two key experimental evaluation metrics.
(1) Output Voltage Fluctuation Rate. The output voltage fluctuation rate is an important metric for measuring the stability of inverter output voltage. It can be quantified by calculating the percentage of the standard deviation of the output voltage relative to its average value. The specific expression is:
$$V_f = \frac{\sqrt{\frac{1}{N} \sum_{i=1}^N (V_i – \bar{V})^2}}{\bar{V}} \times 100\%$$
where \(V_i\) is the output voltage value at the \(i\)-th sampling moment, \(\bar{V}\) is the average value of the output voltage, and \(N\) is the total number of sampling points. A smaller output voltage fluctuation rate indicates more stable output voltage, reflecting better stability of the grid-connected inverter.
(2) Stability Margin. The stability margin is a key metric for measuring the system’s ability to resist disturbances and its stability. In frequency domain analysis, the stability margin typically includes phase margin and gain margin. To more intuitively estimate the stability margin in time domain analysis, I employ a simplified time-domain stability margin expression based on the ratio of the maximum change in system state variables to the initial state. This can be specifically expressed as:
$$SM = \frac{1}{\max_t \left( \frac{\| x(t) – x_0 \|}{x_0} \right)}$$
where \(x(t)\) is the value of the system state variable at time \(t\), \(x_0\) is the initial state of the system, and \(\| \cdot \|\) is the Euclidean norm. A larger value of this metric indicates stronger system resistance to disturbances and better stability. This metric effectively evaluates the stability performance of grid-connected inverters when facing various disturbances.
Through the definition and calculation of these two experimental evaluation metrics, the performance of the stability control method for grid-connected inverters can be comprehensively and objectively assessed.
To demonstrate the superiority of my method, I compare it with two existing methods through comparative experiments. Comparison Method 1 is the strategy proposed in literature [1], and Comparison Method 2 is the method proposed in literature [2]. To comprehensively evaluate the performance of each method, I design multiple experimental scenarios and record the output voltage fluctuation rates for each method. The results are shown in Table 2.
| Experimental Scenario | Proposed Method (%) | Comparison Method 1 (%) | Comparison Method 2 (%) |
|---|---|---|---|
| Steady-state operation | 0.15 | 0.45 | 0.60 |
| Load step change | 0.25 | 0.80 | 1.00 |
| Grid voltage fluctuation | 0.35 | 0.95 | 1.20 |
| Non-linear load | 0.40 | 1.10 | 1.50 |
| Grid frequency fluctuation | 0.81 | 1.50 | 2.00 |
From Table 2, it can be observed that my method demonstrates significant advantages in output voltage fluctuation rate for stability control of grid-connected inverters. In the steady-state operation scenario, the output voltage fluctuation rate achieved by my method is extremely low, far below that of the comparison methods, fully showcasing its exceptional ability to maintain system steady-state performance. When the load undergoes sudden changes, my method also performs excellently, with its output voltage fluctuation rate much lower than that of the comparison methods. This proves that my method can quickly adjust the system state and effectively maintain output voltage stability when facing load mutations. Moreover, in scenarios of grid voltage fluctuation and non-linear loads, my method also demonstrates strong anti-interference ability, with relatively low output voltage fluctuation rates. Especially under complex conditions of grid frequency fluctuation, the output voltage fluctuation rate of my method is as low as 0.81%, further validating its superior stability and robustness in complex grid environments.
Next, I further examine the stability margins of each method at different time points to more comprehensively evaluate their performance. The comparison of stability margins for each method at different time points is shown in Table 3.
| Time Point (s) | Proposed Method | Comparison Method 1 | Comparison Method 2 |
|---|---|---|---|
| 0.5 | 0.95 | 0.75 | 0.65 |
| 1.0 | 0.92 | 0.70 | 0.60 |
| 1.5 | 0.90 | 0.65 | 0.55 |
| 2.0 | 0.88 | 0.60 | 0.50 |
| 2.5 | 0.85 | 0.55 | 0.45 |
| 3.0 | 0.82 | 0.50 | 0.40 |
From Table 3, it can be seen that the stability margins of my method at different time points are generally higher than those of the comparison methods, fully demonstrating the significant advantage of my method in maintaining system stability. Over the entire time range, the stability margin of my method fluctuates slightly and remains at a high level, indicating that it can maintain stable performance when facing different time points of grid environments and disturbances. In contrast, the stability margins of the comparison methods fluctuate more significantly and noticeably decrease at some time points, indicating that their system stability may be greatly affected in complex grid environments and disturbances. In summary, my method exhibits significant superiority over existing methods in stability control of grid-connected inverters, ensuring stable system operation in complex grid environments.
To further elaborate on the mathematical foundations, I present additional formulas and analyses. The dynamic response of the grid-connected inverter under VSG control can be modeled using state-space equations. Consider the state vector \(x = [\omega, E_f, V_{out}, i_{out}]^T\), where \(\omega\) is the angular frequency, \(E_f\) is the excitation electromotive force, \(V_{out}\) is the output voltage, and \(i_{out}\) is the output current. The state-space model can be derived from the previously mentioned equations. For instance, the differential equation for angular frequency is:
$$\frac{d\omega}{dt} = \frac{P_{ref} – P}{M} – D(\omega – \omega_{grid})$$
Similarly, the excitation control can be expressed in differential form:
$$\frac{dE_f}{dt} = K_v (V_{ref} – V_{out}) + K_i \int (V_{ref} – V_{out}) dt$$
These equations form the basis for the stability analysis of the grid-connected inverter system. The stability of the system can be assessed using eigenvalue analysis or Lyapunov methods. For example, the Jacobian matrix of the linearized system around an equilibrium point can be computed to determine the eigenvalues. If all eigenvalues have negative real parts, the system is locally stable. The incorporation of VSG technology inherently introduces damping and inertia, which improves the stability margins.
Moreover, the adaptive virtual impedance control enhances the reactive power sharing among multiple grid-connected inverters. The virtual impedance \(R_{v,i}\) is adjusted based on the reactive power error, as shown in the formula above. This adaptive mechanism ensures that even in the presence of parameter variations or disturbances, the reactive power distribution remains balanced, contributing to overall system stability. The effectiveness of this approach can be validated through simulation studies, as demonstrated in the experimental results.
In practice, the implementation of VSG-based control for grid-connected inverters requires careful tuning of parameters such as \(M\), \(D\), \(K_v\), \(K_i\), \(k_p\), \(k_q\), and \(k_i\). These parameters influence the dynamic response and stability of the system. For instance, a larger virtual inertia \(M\) provides more damping but may slow down the response, while a smaller \(M\) leads to faster response but reduced stability. Therefore, a trade-off must be considered. Table 4 summarizes the recommended parameter ranges based on my analysis and simulations.
| Parameter | Symbol | Recommended Range |
|---|---|---|
| Virtual inertia | \(M\) | 0.1 – 1.0 kg·m² |
| Damping coefficient | \(D\) | 5 – 20 N·m·s/rad |
| Voltage proportional gain | \(K_v\) | 0.5 – 2.0 |
| Voltage integral gain | \(K_i\) | 10 – 100 |
| Active droop coefficient factor | \(k_p\) | 0.01 – 0.1 |
| Reactive adjustment coefficient | \(k_q\) | 0.1 – 1.0 |
| Reactive integral coefficient | \(k_i\) | 1 – 10 |
These parameter ranges are derived from extensive simulations and are intended to guide the practical deployment of VSG-based grid-connected inverters. It is important to note that the optimal parameters may vary depending on specific system configurations and grid conditions. Therefore, online adaptation or optimization algorithms can be further explored to enhance performance.
The integration of renewable energy sources via grid-connected inverters is crucial for modern power systems. The proposed stability control strategy addresses key challenges such as low inertia, parameter sensitivity, and weak grid conditions. By emulating synchronous generator behavior, the grid-connected inverter can provide frequency and voltage support, improving grid stability. This is particularly important in microgrids or isolated systems where conventional generators are absent. The VSG technology enables grid-connected inverters to participate in grid regulation services, such as primary frequency control and voltage regulation, thereby enhancing the overall reliability and resilience of the power system.
In conclusion, through the introduction of the virtual synchronous generator concept, I have successfully emulated the dynamic characteristics of synchronous generators, providing necessary inertia and damping for grid-connected inverters, thereby significantly enhancing system stability and response speed. By combining the principles and control strategies of virtual synchronous generators, I have designed a novel control strategy for grid-connected inverters. This strategy not only effectively copes with grid voltage fluctuations and interference but also provides necessary support and protection during grid faults, ensuring stable operation of the power system. In future research, I will continue to deepen the study of virtual synchronous generator technology, optimize its parameter design and control strategies to improve overall system performance. Simultaneously, I will explore more advanced control methods and technologies to address future challenges faced by power systems, providing more reliable solutions for the integration of renewable energy and stable operation of power systems.