In the context of escalating energy scarcity and growing global support for renewable energy, distributed generation technologies have experienced vigorous development. Among these, photovoltaic (PV) power generation stands out as one of the most significant and promising forms of utilization. The electrical energy generated by a PV array is direct current (DC). To integrate this power into the alternating current (AC) grid, a grid-connected inverter is essential. Modern on grid inverter control predominantly employs Pulse Width Modulation (PWM) techniques. While effective, PWM switching introduces high-frequency harmonics into the grid current. Therefore, installing a filter on the AC side of the on grid inverter is a standard practice to attenuate these harmonics and ensure power quality.
Compared to a simple L filter, an LCL filter offers superior high-frequency attenuation, allowing for a significant reduction in the overall inductance value and, consequently, the filter’s physical size and cost. This makes the LCL filter highly attractive and widely adopted in PV on grid inverter applications. However, the LCL filter is a third-order system inherently prone to resonance, which can lead to instability and threaten the safe operation of the PV system if not properly addressed. The primary method to mitigate this resonant peak is to increase system damping. Traditionally, passive damping methods involve adding physical resistors in series or parallel with the filter components. While simple, this approach increases power losses, reducing the overall system efficiency. Active damping methods, on the other hand, achieve resonance suppression algorithmically within the controller, eliminating the need for power-dissipating components and thereby maintaining high efficiency. This article focuses on a double-stage three-phase LCL-type PV on grid inverter, investigating its control strategy with an emphasis on active damping to suppress the LCL filter resonance.
The system topology, as shown in the figure, consists of two main power stages. The first stage is a PV array connected to a DC-DC Boost converter. This stage serves two critical functions: it elevates the variable PV voltage to a stable, higher DC bus voltage required by the subsequent inverter, and it provides a convenient point for implementing Maximum Power Point Tracking (MPPT) control. The second stage is a three-phase DC-AC inverter equipped with an LCL filter. This inverter’s core function is to convert the stabilized DC power into high-quality AC power synchronized with the grid. The comprehensive control of this on grid inverter involves stabilizing the DC-link voltage, achieving unity power factor operation, and actively damping the LCL filter resonance.
Mathematical Modeling and Parameter Design of the LCL Filter
Under balanced three-phase conditions, the state-space equations of the LCL filter in the abc stationary reference frame can be derived from the circuit topology. The equations for the inverter-side inductors (L₁), the filter capacitors (C), and the grid-side inductors (L₂) are as follows:
For the inverter-side inductor currents \(i_{1a}, i_{1b}, i_{1c}\):
$$L_1 \frac{di_{1a}}{dt} = u_a – u_{ca}$$
$$L_1 \frac{di_{1b}}{dt} = u_b – u_{cb}$$
$$L_1 \frac{di_{1c}}{dt} = u_c – u_{cc}$$
For the capacitor voltages \(u_{ca}, u_{cb}, u_{cc}\):
$$C \frac{du_{ca}}{dt} = i_{1a} – i_{2a}$$
$$C \frac{du_{cb}}{dt} = i_{1b} – i_{2b}$$
$$C \frac{du_{cc}}{dt} = i_{1c} – i_{2c}$$
For the grid-side inductor currents \(i_{2a}, i_{2b}, i_{2c}\):
$$L_2 \frac{di_{2a}}{dt} = u_{ca} – e_a$$
$$L_2 \frac{di_{2b}}{dt} = u_{cb} – e_b$$
$$L_2 \frac{di_{2c}}{dt} = u_{cc} – e_c$$
Where \(u_{a,b,c}\) are the inverter output voltages, \(u_{ca,cb,cc}\) are the capacitor voltages, \(e_{a,b,c}\) are the grid voltages, and \(i_{2a,2b,2c}\) are the grid currents. Applying the Clarke transformation to these equations simplifies the model by converting it to the αβ stationary reference frame, where the α and β components are orthogonal and decoupled. The transformed equations for one axis (e.g., α) are:
$$L_1 \frac{di_{1\alpha}}{dt} = u_{\alpha} – u_{c\alpha}$$
$$C \frac{du_{c\alpha}}{dt} = i_{1\alpha} – i_{2\alpha}$$
$$L_2 \frac{di_{2\alpha}}{dt} = u_{c\alpha} – e_{\alpha}$$
The transfer function from the inverter output voltage \(u(s)\) to the grid current \(i_2(s)\) is fundamental for stability analysis and controller design. From the state equations, this transfer function for the LCL filter is derived as:
$$G_{LCL}(s) = \frac{i_2(s)}{u(s)} = \frac{1}{L_1 L_2 C s^3 + (L_1 + L_2)s}$$
This clearly shows the third-order nature of the system. The denominator reveals a resonant pole pair. The resonant frequency \(f_{res}\) is a critical parameter given by:
$$f_{res} = \frac{1}{2\pi} \sqrt{\frac{L_1 + L_2}{L_1 L_2 C}}$$
Proper design of the LCL filter parameters is crucial. The goal is to achieve sufficient harmonic attenuation above the switching frequency while keeping the resonant frequency within a manageable range for the controller to dampen. The total inductance \(L_{total} = L_1 + L_2\) limits the current ripple. The capacitor value is limited by the reactive power it generates at the fundamental frequency. Furthermore, the resonant frequency should be placed well above the grid frequency to avoid low-frequency oscillations and well below half the switching frequency to allow effective active damping. Common design guidelines are summarized in the table below:
| Parameter | Design Constraint / Guideline | Rationale |
|---|---|---|
| Total Inductance \(L_{total}\) | $$\frac{U_{dc}}{0.2 \times 8 f_{sw} I_n} \leq L_{total} \leq \frac{\sqrt{U_{dc}^2/2 – E^2}}{\omega_0 I_n}$$ | Upper bound limits current ripple; lower bound ensures current control bandwidth and limits voltage drop. |
| Inductance Ratio \(L_1 / L_2\) | Typically between 4 and 6 | Optimizes the distribution of harmonic attenuation between the two inductors and influences the resonant frequency. |
| Filter Capacitance \(C\) | $$C \leq \frac{0.05 P_0}{3 \omega_0 E^2}$$ | Limits the reactive power drawn by the capacitor at the fundamental frequency to less than 5% of rated power. |
| Resonant Frequency \(f_{res}\) | $$10 f_0 \leq f_{res} \leq 0.5 f_{sw}$$ | Avoids interference with grid frequency and ensures active damping is feasible within the controller bandwidth. |
Where \(U_{dc}\) is the DC-link voltage, \(f_{sw}\) is the on grid inverter switching frequency, \(I_n\) and \(P_0\) are the rated current and power, \(E\) is the grid phase voltage (RMS), and \(\omega_0 = 2\pi f_0\) is the grid angular frequency.
System Control Strategy
The control architecture for the double-stage on grid inverter is hierarchically structured, addressing the objectives of each power stage and the interaction between them.
Maximum Power Point Tracking (MPPT) Control
In the double-stage topology, the Boost converter provides a natural and effective interface for MPPT. The relationship between its input voltage \(U_{pv}\) and output voltage \(U_{dc}\) is \(U_{pv} = (1-D)U_{dc}\), where \(D\) is the duty cycle. Since the subsequent on grid inverter control maintains \(U_{dc}\) at a constant reference value, adjusting the duty cycle \(D\) directly controls the operating voltage of the PV array. By dynamically adjusting \(D\) according to an MPPT algorithm, the PV array can be forced to operate at its maximum power point (MPP).
The Incremental Conductance (INC) algorithm is a highly effective and widely used MPPT method. It is based on the characteristic that at the MPP, the instantaneous conductance (\(I/U_{pv}\)) equals the negative of the incremental conductance (\(dI/dU_{pv}\)). The power-voltage (P-V) curve of a PV array has a unique maximum. The slope of this curve, \(dP/dU_{pv}\), is positive to the left of the MPP, zero at the MPP, and negative to the right. Since \(P = U_{pv}I\), we have:
$$\frac{dP}{dU_{pv}} = I + U_{pv} \frac{dI}{dU_{pv}}$$
Setting \(dP/dU_{pv} = 0\) for the MPP condition yields:
$$\frac{dI}{dU_{pv}} = -\frac{I}{U_{pv}}$$
The INC algorithm uses this as the decision criterion:
- If \( \frac{\Delta I}{\Delta U_{pv}} > -\frac{I}{U_{pv}} \), the operating point is to the left of the MPP; increase \(U_{pv}\) (decrease \(D\)).
- If \( \frac{\Delta I}{\Delta U_{pv}} \approx -\frac{I}{U_{pv}} \), the operating point is at the MPP; maintain \(D\).
- If \( \frac{\Delta I}{\Delta U_{pv}} < -\frac{I}{U_{pv}} \), the operating point is to the right of the MPP; decrease \(U_{pv}\) (increase \(D\)).
This logic is implemented in a digital controller to continuously track the MPP under varying irradiation and temperature conditions.
Grid-Side Inverter Control
The three-phase on grid inverter employs a cascaded control structure: an outer voltage loop and an inner current loop. The outer loop regulates the DC-link voltage \(U_{dc}\). Its error signal is processed by a Proportional-Integral (PI) controller, which generates the reference for the active component of the grid current \(i_{2d}^*\). For unity power factor operation, the reference for the reactive current component \(i_{2q}^*\) is set to zero. These references in the synchronous rotating (dq) frame are then transformed into the stationary (αβ) frame as \(i_{2\alpha}^*\) and \(i_{2\beta}^*\).
The inner current loop is responsible for fast, accurate tracking of these current references and for providing active damping of the LCL resonance. A dual-loop current control strategy is adopted, consisting of a grid-current outer loop and a capacitor-current inner loop. This structure effectively stabilizes the system.
Proportional-Resonant (PR) Control for Grid Current: Traditional PI controllers in the dq frame are excellent for tracking DC references but require complex decoupling networks. When transformed to the αβ frame, the current references become sinusoidal signals at the fundamental grid frequency. A PI controller has finite gain at this frequency, leading to steady-state tracking error. A Proportional-Resonant (PR) controller, however, provides theoretically infinite gain at a specific resonant frequency (tuned to the grid frequency \(\omega_0\)), enabling zero steady-state error for sinusoidal signals. The transfer function of an ideal PR controller is:
$$G_{PR}(s) = K_P + \frac{K_R s}{s^2 + \omega_0^2}$$
where \(K_P\) is the proportional gain and \(K_R\) is the resonant gain. The PR controller is applied to the grid current error in the αβ frame, ensuring precise tracking of the sinusoidal reference.
Capacitor Current Feedback for Active Damping: The capacitor current inner loop is the key to active damping. By feeding back the capacitor current \(i_c\) (or a scaled version of it, \(K_e \cdot i_c\)) and subtracting it from the output of the PR controller, an equivalent virtual resistor in parallel with the filter capacitor is emulated. This increases the damping of the LCL resonant poles without any physical power loss. The inner loop typically uses a simple proportional gain \(K_e\).
The complete block diagram of the dual-current-loop control in the αβ frame is shown conceptually below, leading to the overall closed-loop transfer function.
The open-loop plant from the modulator output to the grid current, including the feedback of the capacitor current (which equals \(i_1 – i_2\)), can be derived. Combining the PR controller \(G_{PR}(s)\), the proportional capacitor current feedback gain \(K_e\), and the PWM gain \(K_{PWM}\), the closed-loop transfer function from the grid current reference \(i_2^*(s)\) to the actual grid current \(i_2(s)\) is:
$$\frac{i_2(s)}{i_2^*(s)} = \frac{A_2 s^2 + A_1 s + A_0}{B_5 s^5 + B_4 s^4 + B_3 s^3 + B_2 s^2 + B_1 s + B_0}$$
where
$$A_2 = K_P K_e K_{PWM}, \quad A_1 = K_R K_e K_{PWM}, \quad A_0 = K_P K_e K_{PWM} \omega_0^2$$
$$B_5 = L_1 L_2 C, \quad B_4 = K_e K_{PWM} L_2 C, \quad B_3 = L_1 + L_2 + L_1 L_2 C \omega_0^2$$
$$B_2 = K_P K_e K_{PWM} + K_e K_{PWM} L_2 C \omega_0^2, \quad B_1 = K_R K_e K_{PWM} + (L_1 + L_2)\omega_0^2, \quad B_0 = K_P K_e K_{PWM} \omega_0^2$$
Proper tuning of the controller parameters (\(K_P, K_R, K_e\)) is essential. For the PR controller, to maintain robust performance against grid frequency variations (e.g., ±0.5 Hz) and ensure low current tracking error, specific gain and phase margins are targeted:
- Gain at fundamental frequency \(T(f_0) > 75 \text{ dB}\) for less than 1% amplitude error.
- Phase Margin \(PM > 45^\circ\) for good dynamic response.
- Gain Margin \(GM > 3 \text{ dB}\) for sufficient robustness.
These criteria guide the selection of \(K_P\) and \(K_R\), while \(K_e\) is chosen to adequately damp the resonance peak in the closed-loop system, often based on root locus or frequency response analysis.
Simulation Verification and Analysis
A detailed simulation model of the complete double-stage three-phase LCL on grid inverter system was built in MATLAB/Simulink to validate the proposed control strategy. The key system and controller parameters used in the simulation are listed in the table below.
| Component / Parameter | Value |
|---|---|
| PV Array & Boost Stage | |
| PV Array Rated Power | 4000 W |
| MPPT Algorithm | Incremental Conductance (INC) |
| DC-AC Inverter Stage | |
| DC-Link Voltage Reference \(U_{dc}^*\) | 600 V |
| Grid Phase Voltage (RMS) | 220 V |
| Grid Frequency \(f_0\) | 50 Hz |
| Switching Frequency \(f_{sw}\) | 20 kHz |
| Inverter-side Inductor \(L_1\) | 5 mH |
| Grid-side Inductor \(L_2\) | 1 mH |
| Filter Capacitor \(C\) | 10 µF |
| Calculated Resonant Freq. \(f_{res}\) | ~918 Hz |
| Controller Parameters | |
| DC Voltage PI: \(K_{P\_v}, K_{I\_v}\) | 2, 43 |
| Grid Current PR: \(K_P, K_R\) | 1, 47 |
| Cap. Current Feedback Gain \(K_e\) | 10 |
| PWM Gain \(K_{PWM}\) | 1 |
The simulation results demonstrate the effectiveness of the integrated control strategy. The INC-based MPPT controller successfully drives the PV array to operate at its maximum power point, with the output power stabilizing around the rated 4000 W after a short transient.
The DC-link voltage control loop performs robustly. Despite initial transients during grid connection, the voltage is regulated and maintained tightly at its 600 V reference, providing a stable input for the on grid inverter.
The performance of the grid-side on grid inverter is critical. The waveform of the grid voltage (scaled down) and the Phase-A grid current shows perfect synchronization and phase alignment, confirming the achievement of unity power factor operation. A Fast Fourier Transform (FFT) analysis of the grid current quantifies its quality. The fundamental current amplitude is approximately 7.954 A, corresponding to an output power of about 3.7 kW, which aligns with the power from the PV array minus system losses. More importantly, the Total Harmonic Distortion (THD) of the grid current is measured at 1.35%, which is well below the standard limit of 5%. The harmonic spectrum shows effective attenuation of switching-frequency harmonics and, crucially, no pronounced peak at the LCL resonant frequency, validating the success of the capacitor-current-feedback active damping strategy implemented in the dual-loop current controller.
Conclusion
This research has presented a comprehensive analysis and control strategy for a double-stage three-phase LCL photovoltaic on grid inverter. The system’s mathematical model was established in the stationary reference frame, forming the basis for controller design. The proposed strategy effectively addresses all key control objectives: MPPT is reliably achieved via the INC algorithm in the Boost converter stage; DC-link voltage stability is ensured by a PI-based outer voltage loop in the inverter; and high-quality grid connection with active resonance damping is accomplished through a sophisticated dual-current-loop control scheme.
The inner current loop combines a Proportional-Resonant (PR) controller for the grid current to achieve zero steady-state error and unity power factor, with a proportional capacitor current feedback loop to actively damp the inherent LCL filter resonance. This active damping method enhances system stability without the efficiency penalty associated with passive damping. Simulation results under standardized conditions have verified the strategy’s effectiveness, showing stable MPPT operation, excellent DC voltage regulation, low grid current THD, and the absence of resonant peaks. The control framework for the LCL-type on grid inverter established here provides a solid theoretical foundation. Future work will focus on optimizing this strategy for robustness against wider environmental variations and grid disturbances, and exploring advanced adaptive or nonlinear control techniques to further enhance performance under non-ideal conditions.