In modern photovoltaic (PV) grid-connected systems, the solar inverter plays a critical role in converting DC power from solar panels into AC power suitable for the grid. Among various topologies, the two-stage single-phase solar inverter, consisting of a front-end DC-DC converter (e.g., Boost) and a rear-end full-bridge inverter, is widely adopted due to its flexibility in handling wide input voltage ranges and simplified control. However, a significant challenge in such systems is the presence of a double-line frequency ripple (DLFR) in the input voltage, which stems from the inherent power pulsation in single-phase AC output. This ripple severely degrades the efficiency of maximum power point tracking (MPPT), reducing the overall energy harvest from PV panels. In this paper, I delve into a feedforward control strategy based on DC-link voltage to suppress the input voltage DLFR, aiming to enhance the performance of solar inverters. I will analyze the generation mechanism, develop a small-signal model, derive the feedforward controller from the perspective of closed-loop audio susceptibility, and validate the approach through experimental results. Throughout this discussion, the term “solar inverter” will be emphasized to highlight its centrality in renewable energy systems.
The two-stage single-phase solar inverter typically operates with the front-end Boost converter performing MPPT control to extract maximum power from the PV array, while the rear-end inverter regulates the DC-link voltage and injects synchronized current into the grid. In ideal conditions, the instantaneous output power of a single-phase solar inverter contains a constant component and a sinusoidal component at twice the grid frequency, as given by:
$$ p_g(t) = i_g(t) u_g(t) = I_g U_g \cos \phi – I_g U_g \cos(2\omega t + \phi) $$
where \( I_g \) and \( U_g \) are the peak grid current and voltage, respectively, \( \omega \) is the grid angular frequency, and \( \phi \) is the power factor angle. For unity power factor operation (\(\phi = 0\)), this simplifies to:
$$ p_g(t) = P_g – P_g \cos(2\omega t) $$
with \( P_g = I_g U_g / 2 \) being the average power. This pulsating power causes the DC-link capacitor to charge and discharge at 100 Hz (for a 50 Hz grid), inducing a DLFR in the DC-link voltage. Consequently, through the input-output dynamics of the Boost converter, this ripple propagates to the input voltage, perturbing the PV operating point and reducing MPPT accuracy. Studies indicate that to maintain MPPT efficiency above 99%, the input voltage ripple must be kept below 6%, underscoring the need for effective suppression techniques in solar inverter design.
To understand the DLFR generation in detail, consider the power balance in a two-stage solar inverter. The front-end Boost converter, under MPPT control, ideally delivers a constant power \( P_{pv} \) from the PV array. However, due to losses and dynamics, the instantaneous input power \( p_{in}(t) \) may vary. The DC-link capacitor \( C_{dc} \) absorbs the difference between the Boost output power and the inverter input power. From energy conservation:
$$ C_{dc} \frac{d u_{dc}(t)}{dt} u_{dc}(t) = p_{boost}(t) – p_g(t) $$
Assuming the Boost converter operates with high efficiency, \( p_{boost}(t) \approx P_{pv} \), and substituting \( p_g(t) \) from above, the DC-link voltage can be expressed as:
$$ u_{dc}(t) = U_{dc} + \Delta u_{dc} \sin(2\omega t + \theta) $$
where \( U_{dc} \) is the average DC-link voltage, and \( \Delta u_{dc} \) is the ripple amplitude. This ripple then couples to the input voltage \( u_{in} \) through the Boost converter’s transfer functions, as analyzed in the small-signal model. The impact on MPPT is significant because PV cells exhibit a nonlinear voltage-power characteristic, and even small voltage deviations can lead to substantial power losses. Therefore, mitigating the DLFR is crucial for optimizing solar inverter performance.
Various methods have been proposed to address this issue in solar inverters, including increasing the DC-link capacitance, using active power decoupling circuits, and implementing advanced control algorithms. However, these approaches often add cost, complexity, or reduce power density. Feedforward control offers a promising alternative by directly compensating for disturbances. In this work, I focus on a DC-link voltage feedforward strategy that injects a corrective signal into the Boost converter’s duty cycle to cancel the DLFR effect. This method leverages the audio susceptibility of the system, which describes how input voltage responds to output voltage perturbations. By designing the feedforward controller to nullify this susceptibility at the ripple frequency, the input voltage ripple can be significantly reduced, enhancing the solar inverter’s MPPT efficiency.
To design the feedforward controller, I first develop a small-signal model of the Boost converter in the solar inverter. The converter is typically operated in continuous conduction mode (CCM), and its power stage includes an input inductor \( L_1 \), input capacitor \( C_{in} \), and output capacitor \( C_{dc} \), along with parasitic resistances. The PV array is modeled as a current source \( i_{ph} \) in parallel with a diode, but for small-signal analysis around an operating point, it can be represented as a Norton equivalent with a current source \( i_{ph} \) and a dynamic resistance \( R_{pv} \). However, for input voltage control, the focus is on the transfer functions from duty cycle \( d \) and DC-link voltage \( u_{dc} \) to input voltage \( u_{in} \). The small-signal model is derived using state-space averaging, considering the switch network’s average behavior. The key parameters include the duty cycle \( D \), average input voltage \( U_{in} \), average inductor current \( I_L \), and average DC-link voltage \( U_{dc} \). From the model, the control-to-input-voltage transfer function \( G_{uin,d}(s) \) and the audio susceptibility \( A_u(s) \) are obtained as:
$$ G_{uin,d}(s) = \frac{\tilde{u}_{in}(s)}{\tilde{d}(s)} \bigg|_{\tilde{i}_{ph}(s)=0, \tilde{u}_{dc}(s)=0} = \frac{-U’_{dc} (1 + s C_{in} R_{C_{in}})}{s^2 L_1 C_{in} + s C_{in} R_1 + 1} $$
$$ A_u(s) = \frac{\tilde{u}_{in}(s)}{\tilde{u}_{dc}(s)} \bigg|_{\tilde{i}_{ph}(s)=0, \tilde{d}(s)=0} = \frac{(1-D) (1 + s C_{in} R_{C_{in}})}{s^2 L_1 C_{in} + s C_{in} R_1 + 1} $$
where \( U’_{dc} = U_{dc} – I_L R_{on} + U_F \) (with \( R_{on} \) as the MOSFET on-resistance and \( U_F \) as the diode forward voltage), \( R_{C_{in}} \) is the input capacitor ESR, and \( R_1 = R_{L_1} + D R_{on} + (1-D) R_F + R_{C_{in}} \) represents the total parasitic resistance. These transfer functions form the basis for analyzing the solar inverter’s dynamic behavior.
The input voltage control loop in a typical solar inverter uses a PI controller \( G_c(s) = k_p + k_i/s \) to regulate \( u_{in} \) to the MPPT reference. The modulation gain \( F_m \) relates the control signal to the duty cycle, and the input voltage sensor gain is \( K_{uin} \). Without feedforward, the closed-loop audio susceptibility \( A_{u,c}(s) \) is given by:
$$ A_{u,c}(s) = \frac{A_u(s)}{1 + K_{uin} F_m G_c(s) G_{uin,d}(s)} $$
This function determines how much DC-link voltage ripple appears at the input. At 100 Hz, \( A_{u,c}(s) \) typically has a non-zero magnitude, leading to significant DLFR. To suppress it, I introduce a feedforward path from the measured DC-link voltage \( u_{dc} \) to the duty cycle, with a transfer function \( G_{ff}(s) \). The modified control block diagram includes this path, and the new closed-loop audio susceptibility becomes:
$$ A_{u,c}(s) = \frac{A_u(s) – G_{ff}(s) F_m G_{uin,d}(s)}{1 + K_{uin} F_m G_c(s) G_{uin,d}(s)} $$
The goal is to make \( A_{u,c}(s) \) as small as possible at the ripple frequency. Ideally, if \( A_u(s) – G_{ff}(s) F_m G_{uin,d}(s) = 0 \), then \( A_{u,c}(s) = 0 \), meaning perfect rejection. Solving for \( G_{ff}(s) \) yields:
$$ G_{ff}(s) = \frac{A_u(s)}{F_m G_{uin,d}(s)} = -\frac{1-D}{F_m U’_{dc}} $$
This result shows that the feedforward controller is a constant gain dependent on the operating point (duty cycle \( D \) and adjusted DC-link voltage \( U’_{dc} \)). For practical implementation in a solar inverter, this gain can be updated in real-time based on the current MPPT point to maintain effectiveness across varying PV conditions.
The design of the feedforward controller highlights its simplicity and effectiveness in solar inverters. However, the performance depends on accurate parameter knowledge. To assess the impact, I evaluate the frequency response of \( A_{u,c}(s) \) with and without feedforward. Using typical solar inverter parameters: \( L_1 = 0.15 \text{ mH} \), \( C_{in} = 10 \mu\text{F} \), \( C_{dc} = 470 \mu\text{F} \), \( R_{on} = 20 \text{ m}\Omega \), \( U_F = 0.7 \text{ V} \), \( F_m = -1/2.4 \), and MPPT point at \( U_{in} = 31.3 \text{ V} \), \( I_{in} = 8.96 \text{ A} \), \( U_{dc} = 48 \text{ V} \), I calculate \( D = 1 – U_{in}/U_{dc} \approx 0.348 \), \( I_L = I_{in} / (1-D) \approx 13.7 \text{ A} \), and \( U’_{dc} \approx 48.7 \text{ V} \). Then, \( G_{ff} = – (1-0.348) / (-1/2.4 \times 48.7) \approx 0.032 \). The PI controller parameters are chosen as \( k_p = 0.5 \) and \( k_i = 400 \) based on stability criteria. The table below summarizes these key parameters for the solar inverter.
| Parameter | Symbol | Value |
|---|---|---|
| Input Inductance | \( L_1 \) | 0.15 mH |
| Input Capacitance | \( C_{in} \) | 10 μF |
| DC-Link Capacitance | \( C_{dc} \) | 470 μF |
| MOSFET On-Resistance | \( R_{on} \) | 20 mΩ |
| Diode Forward Voltage | \( U_F \) | 0.7 V |
| Modulation Gain | \( F_m \) | -1/2.4 |
| Duty Cycle at MPPT | \( D \) | 0.348 |
| Adjusted DC-Link Voltage | \( U’_{dc} \) | 48.7 V |
| Feedforward Gain | \( G_{ff} \) | 0.032 |
| PI Controller Proportional Gain | \( k_p \) | 0.5 |
| PI Controller Integral Gain | \( k_i \) | 400 |
With these values, the magnitude of \( A_{u,c}(s) \) at 100 Hz drops from approximately -10 dB without feedforward to -60 dB with feedforward, indicating a 50 dB improvement in ripple rejection. This dramatic reduction demonstrates the potential of the feedforward strategy in solar inverters. To further illustrate, the following equation shows the closed-loop susceptibility magnitude at frequency \( \omega_r = 2\pi \times 100 \text{ rad/s} \):
$$ |A_{u,c}(j\omega_r)| = \left| \frac{A_u(j\omega_r) – G_{ff} F_m G_{uin,d}(j\omega_r)}{1 + K_{uin} F_m G_c(j\omega_r) G_{uin,d}(j\omega_r)} \right| $$
With feedforward, the numerator approaches zero, minimizing the magnitude. This analysis confirms that the solar inverter’s input voltage ripple can be effectively suppressed without altering the main control loop.
For experimental validation, I constructed a 280 VA prototype of a two-stage single-phase solar inverter. The system includes a Boost converter with the parameters listed above and a full-bridge inverter with an output filter inductor \( L_2 = 0.38 \text{ mH} \) and parasitic resistance \( R_{L_2} = 0.2 \Omega \). The grid voltage is 220 V RMS at 50 Hz, and the switching frequency is 20 kHz for both stages. The MPPT algorithm uses a perturb-and-observe method, but for consistency, the operating point is fixed near the standard test conditions. The control is implemented digitally using a microcontroller, with the feedforward gain calculated online based on the measured duty cycle. Below is a table of the full-bridge inverter parameters.
| Parameter | Symbol | Value |
|---|---|---|
| Rated Power | \( S_N \) | 280 VA |
| Switching Frequency | \( f_s \) | 20 kHz |
| Grid Voltage (RMS) | \( U_g \) | 220 V |
| Grid Frequency | \( f \) | 50 Hz |
| Output Filter Inductance | \( L_2 \) | 0.38 mH |
| Output Filter Parasitic Resistance | \( R_{L_2} \) | 0.2 Ω |
The experimental setup aims to verify the feedforward control’s efficacy in a real solar inverter environment. The key measurements include input voltage \( u_{in} \), DC-link voltage \( u_{dc} \), grid voltage \( u_g \), and grid current \( i_g \). Without feedforward, the input voltage exhibits a DLFR with a ripple rate (peak-to-peak over average) of 7.03%, while the DC-link voltage ripple is 22.9%. After enabling the feedforward control, the input voltage ripple reduces to 3.19%, meeting the 6% requirement for high MPPT efficiency. The DC-link voltage ripple remains unchanged, as expected, since the feedforward does not affect its regulation. The grid current maintains low distortion, confirming that the solar inverter’s overall performance is preserved. The image below shows the prototype solar inverter system used in the experiments, highlighting its compact design and integration.
The results demonstrate that the DC-link voltage feedforward strategy effectively suppresses input voltage DLFR in two-stage single-phase solar inverters. This improvement directly enhances MPPT accuracy, leading to higher energy yield from PV systems. Compared to other methods, such as increasing capacitance or adding passive components, this approach minimizes cost and complexity, making it attractive for commercial solar inverters. However, the feedforward gain’s dependence on operating conditions requires careful calibration. In practice, adaptive tuning based on real-time measurements can be employed to maintain optimal performance across varying solar irradiance and temperature. Future work could explore integration with more advanced MPPT algorithms or extend the strategy to three-phase solar inverters, where DLFR is absent but other challenges exist.
In conclusion, I have presented a comprehensive analysis of input voltage double-line frequency ripple suppression in two-stage single-phase solar inverters. By deriving a small-signal model and designing a feedforward controller from closed-loop audio susceptibility, I achieved significant ripple reduction, as validated experimentally. The proposed method balances performance and simplicity, contributing to the ongoing development of efficient and reliable solar inverters for renewable energy integration. As the demand for clean energy grows, such advancements in solar inverter technology will play a pivotal role in maximizing the potential of photovoltaic systems worldwide.