The transition towards renewable energy systems has positioned photovoltaic (PV) technology at the forefront of sustainable power generation. A critical component in harnessing solar energy is the grid-connected solar inverter, which serves as the essential interface between the PV modules and the utility grid. Its primary control objective is to inject sinusoidal current that is synchronized with the grid voltage at a specified power factor, typically unity, while maintaining a Total Harmonic Distortion (THD) well below 5% to comply with stringent grid codes. The quality of this injected power is directly influenced by the inverter’s control strategy and the design of its output filter. This article presents a comprehensive analysis of an inductor current feedback control strategy for single-phase solar inverters utilizing an LC filter, focusing on achieving high-quality grid current with unity power factor operation.
The core challenge in solar inverter control lies in accurately tracking a sinusoidal reference despite disturbances from the grid and nonlinearities within the power stage. Various current control techniques have been explored in the literature, each with distinct advantages and limitations. Hysteresis (or bang-bang) control offers excellent dynamic response and inherent stability but results in variable switching frequency, complicating filter design and increasing losses. Classical Proportional-Integral-Derivative (PID) controllers, while ubiquitous in industrial control, struggle with zero steady-state error for sinusoidal references in the stationary frame. Repetitive control can effectively suppress periodic harmonics but requires significant memory and careful filter design to ensure stability. Model Predictive Control (MPC) offers fast dynamics but its performance is highly sensitive to the accuracy of the system model. In recent years, the Proportional-Resonant (PR) controller has gained prominence for AC current regulation due to its ability to provide theoretically infinite gain at a specific resonant frequency (e.g., the grid fundamental frequency), thereby achieving zero steady-state error for sinusoidal signals. This makes it particularly suitable for grid-following solar inverters.
A common inverter topology for residential and commercial applications is the two-stage configuration, consisting of a front-end DC-DC boost converter for Maximum Power Point Tracking (MPPT) and voltage adaptation, followed by a full-bridge inverter. The output of the inverter bridge is connected to the grid through a low-pass filter. The choice of filter is crucial: a simple L filter is a first-order system with limited high-frequency attenuation; an LCL filter is a third-order system offering superior harmonic attenuation but introducing potential resonance issues and control complexity; an LC filter represents a balanced compromise, providing better filtering than an L filter with less complexity than an LCL filter. This analysis focuses on the LC filter-based solar inverter.
The schematic of a typical single-phase grid-connected solar inverter system is shown conceptually above. The DC link capacitor, C, stabilizes the bus voltage provided by the boost stage (comprising L, D, T). The full-bridge inverter (switches T1-T4) generates a Pulse Width Modulated (PWM) voltage at nodes A and B. This voltage is filtered by the inductor L_f and capacitor C_f before being connected to the grid voltage u_g. The control system typically includes a Phase-Locked Loop (PLL) for grid synchronization, an MPPT algorithm, and the critical current control loop. Upon grid synchronization, the controller initially matches the inverter output voltage to the grid voltage before closing the relay. Subsequently, the current control loop operates to inject power at unity power factor.
Design and Modeling of the LC Output Filter
The LC filter is a second-order low-pass filter. Its transfer function, relating the inverter bridge output voltage U_i(s) to the filter output voltage U_o(s) (which is essentially the grid voltage in steady state), is given by:
$$ \frac{U_o(s)}{U_i(s)} = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2} $$
where the natural resonant angular frequency $\omega_n$ and the damping factor $\zeta$ are defined as:
$$ \omega_n = \frac{1}{\sqrt{L_f C_f}}, \quad \zeta = \frac{R_d}{2} \sqrt{\frac{C_f}{L_f}} $$
Here, $R_d$ represents a small parasitic resistance or an intentionally added damping resistor. The filter’s cutoff frequency $f_c$ must be strategically chosen: it must be higher than the grid fundamental frequency (50/60 Hz) to allow the fundamental current to pass, but significantly lower than the lowest significant harmonic frequency present in the PWM voltage (typically the switching frequency sidebands) to provide adequate attenuation. The characteristic impedance $Z_0$ of the filter is:
$$ Z_0 = \sqrt{\frac{L_f}{C_f}} $$
The values of $L_f$ and $C_f$ can then be derived from the chosen $f_c$ and $Z_0$:
$$ L_f = \frac{Z_0}{2\pi f_c}, \quad C_f = \frac{1}{2\pi f_c Z_0} $$
Typically, $Z_0$ is selected between 0.5 to 0.8 times the base impedance of the system, and $f_c$ is set between 100 Hz to 500 Hz for a 50 Hz grid. The capacitor value is also constrained by the maximum allowable reactive power consumption at full voltage.
Vector Analysis and Control Objectives
To understand the control requirements, a phasor diagram analysis is essential. Under ideal unity power factor operation, the grid current $i_g$ is in phase with the grid voltage $u_g$. The capacitor $C_f$ draws a current $i_C$ that leads $u_g$ by 90°. Therefore, the inductor current $i_L$, which is the sum of $i_g$ and $i_C$, necessarily leads $u_g$ by an angle $\theta$. The inductor voltage $u_L$ leads $i_L$ by 90°. The fundamental component of the inverter’s PWM output voltage $u_{pwm}$ is the vector sum of $u_g$ and $u_L$.
The key relationships are:
$$ I_C = 2\pi f C_f U_g $$
$$ \theta = \arctan\left(\frac{I_C}{I_g}\right) $$
$$ U_L = 2\pi f L_f I_L $$
where $U_g$, $I_g$, $I_C$, $I_L$ are RMS values, and $f$ is the grid frequency. This analysis reveals a critical insight: for the solar inverter to deliver active power at unity power factor from the grid perspective, the *inductor current* must lead the grid voltage. The required lead angle $\theta$ depends on the output power level ($I_g$) and the fixed capacitive VARs ($I_C$).
System Model and Proposed Control Strategy
The control block diagram for the inductor current feedback strategy is developed as follows. The plant includes the PWM inverter, modeled as a gain $K_{PWM}$ and a small delay, and the LC filter impedance. The controller $G_c(s)$ processes the error between the reference inductor current $i_L^*$ and the measured inductor current $i_L$. A key enhancement is the inclusion of a grid voltage feedforward path $G_{ff}(s)$.
Starting from the circuit equations:
$$ u_{pwm} = L_f \frac{di_L}{dt} + u_C $$
$$ i_L = C_f \frac{du_C}{dt} + i_g $$
$$ u_C = u_g + L_g \frac{di_g}{dt} $$
Assuming the grid impedance $L_g$ is small or accounted for, and focusing on the inductor current dynamics, we can derive the open-loop transfer function. The output of the current controller is a voltage command $v_c$. The modulator produces $u_{pwm} = K_{PWM} v_c$. Therefore, the relationship is:
$$ K_{PWM} v_c = L_f \frac{di_L}{dt} + u_C $$
To achieve decoupling and improve disturbance rejection, the grid voltage (or more precisely, the capacitor voltage $u_C$) is fed forward. The modified command becomes:
$$ v_c’ = v_c + \frac{1}{K_{PWM}} u_C $$
Substituting this into the equation yields:
$$ K_{PWM} v_c + u_C = L_f \frac{di_L}{dt} + u_C \Rightarrow K_{PWM} v_c = L_f \frac{di_L}{dt} $$
This ideal feedforward action effectively cancels the grid voltage disturbance, linearizing the plant to a simple integrator ($i_L(s) / v_c(s) = K_{PWM} / (L_f s)$). This significantly simplifies controller design and enhances the solar inverter’s robustness against grid voltage distortions.
Current Controller Design: P vs. PR Regulators
The performance of the solar inverter hinges on the design of $G_c(s)$. We analyze two primary candidates: the Proportional (P) and the Proportional-Resonant (PR) controller.
1. Proportional (P) Controller: Here, $G_c(s) = K_p$. While simple, a P controller introduces a steady-state phase and amplitude error when tracking a sinusoidal reference. Referring to the phasor diagram, the controller output $v_c$ is in phase with the current error $\Delta i = i_L^* – i_L$. For a nonzero $v_c$ to exist in steady state (to generate the required $u_L$), $\Delta i$ cannot be zero. This results in a phase lag of the actual $i_L$ relative to the reference $i_L^*$. To compensate and force the final grid current $i_g$ to be in phase with $u_g$, the reference current $i_L^*$ must be commanded with a phase advance $\alpha$ greater than the theoretical $\theta$. This advance $\alpha$ must compensate for both the capacitor phase shift and the phase lag introduced by the P controller’s steady-state error.
2. Proportional-Resonant (PR) Controller: The ideal PR controller is expressed as:
$$ G_c(s) = K_p + \frac{K_r s}{s^2 + \omega_0^2} $$
where $\omega_0 = 2\pi f_{grid}$. The resonant term provides theoretically infinite gain at $\omega_0$, enabling zero steady-state error in both magnitude and phase for the sinusoidal signal at that frequency. In practice, a “non-ideal” or “quasi” PR controller is used for numerical stability and bandwidth control:
$$ G_c(s) = K_p + \frac{2K_r \omega_c s}{s^2 + 2\omega_c s + \omega_0^2} $$
where $\omega_c$ is the cutoff bandwidth around the resonant frequency. With a well-tuned PR controller, the steady-state error $\Delta i$ approaches zero. Consequently, the required phase advance for the reference current $i_L^*$ is precisely the angle $\theta$ derived from the LC filter vector analysis, i.e., $i_L^*$ should lead $u_g$ by $\theta = \arctan(2\pi f C_f U_g / I_g)$. This makes the controller design more intuitive and the system performance more predictable across different power levels.
| Control Technique | Key Principle | Advantages | Disadvantages | Suitability for Solar Inverter |
|---|---|---|---|---|
| Hysteresis | Maintains current within a fixed band around reference. | Very fast dynamic response, simple implementation, robust. | Variable switching frequency, high current ripple, difficult filter design. | Moderate. Good for simplicity but not for optimal filter design. |
| PI in dq-frame | Transforms AC quantities to DC for PI control. | Zero steady-state error, constant switching frequency. | Requires complex transformation (PLL), performance depends on transformation accuracy. | High. Industry standard for three-phase systems. |
| Proportional (P) | Uses simple proportional gain. | Extremely simple, stable. | Steady-state amplitude and phase error for AC signals. | Low. Requires compensation (phase advance) and yields poorer THD. |
| Proportional-Resonant (PR) | Provides high gain at a specific resonant frequency. | Zero steady-state error for AC signals in stationary frame, no transformation needed. | Design of resonant term parameters, potential stability issues with multiple resonators. | Very High. Excellent for single-phase solar inverters. |
| Repetitive Control | Learns and cancels periodic errors over one fundamental period. | Excellent for rejecting periodic harmonics, high steady-state accuracy. | Slow response, high memory requirement, complex design for stability. | Moderate/High. Often used as an add-on to improve THD. |
| Model Predictive Control (MPC) | Uses a model to predict future behavior and optimize switch states. | Very fast dynamics, multi-variable constraint handling. | Computationally intensive, performance highly sensitive to model accuracy. | Growing interest. Suited for high-performance applications with powerful processors. |
Performance Analysis and Parameter Tuning
The system’s closed-loop performance can be analyzed using the linearized model after feedforward decoupling. The open-loop transfer function $G_{ol}(s)$ with a PR controller is approximately:
$$ G_{ol}(s) = G_c(s) \cdot \frac{K_{PWM}}{L_f s} $$
The controller parameters $K_p$, $K_r$, and $\omega_c$ are tuned for stability, bandwidth, and harmonic rejection. $K_p$ primarily affects the dynamic response and the gain at frequencies away from resonance. $K_r$ determines the magnitude of the resonant peak at $\omega_0$, directly influencing the steady-state tracking accuracy. $\omega_c$ sets the width of the resonant band, affecting robustness against grid frequency variations. A standard tuning approach involves:
- Setting $\omega_c$ to a small value (e.g., 5-15 rad/s) for good frequency selectivity.
- Choosing $K_r$ to achieve the desired high gain at $\omega_0$ without causing instability.
- Adjusting $K_p$ to shape the gain crossover frequency and phase margin for satisfactory transient response and stability.
The required reference current phase advance $\phi_{adv}$ for unity power factor operation is a function of the controller type and operating point:
| Controller Type | Phase Advance Formula ($\phi_{adv}$) | Notes |
|---|---|---|
| Proportional (P) | $\phi_{adv} = \arctan\left(\frac{2\pi f C_f U_g}{I_g}\right) + \arctan\left(\frac{2\pi f L_f I_L}{K_p K_{PWM}}\right)$ | Includes compensation for controller-induced lag. Varies with $I_g$. |
| Proportional-Resonant (PR) | $\phi_{adv} = \arctan\left(\frac{2\pi f C_f U_g}{I_g}\right) = \theta$ | Only depends on filter parameters and operating power. More consistent. |
Simulation and Experimental Validation
The proposed strategy was validated on a 3 kW single-phase solar inverter prototype with the following parameters: DC link voltage = 400 V, $L_f$ = 6 mH, $C_f$ = 16 µF, switching frequency = 18 kHz. A programmable DC source simulated the PV array. The control algorithm was implemented on a digital signal processor (DSP).
The experimental results confirmed the theoretical analysis. With the PR controller and the correctly phased reference current, the solar inverter achieved unity power factor operation across its power range. The grid current THD was measured below 3% at rated power, comfortably meeting standard requirements (e.g., IEEE 1547, EN 61000-3-2). The table below summarizes key performance metrics under different control approaches for the solar inverter.
| Output Power (kW) | Controller Type | Applied Phase Advance | Grid Current THD (%) | Power Factor | Remarks |
|---|---|---|---|---|---|
| 1.5 | P | 8.5° | 4.2 | 0.998 | Requires precise advance tuning per operating point. |
| 1.5 | PR | $\theta \approx 6.2°$ | 2.5 | 1.000 | Stable, low THD, advance follows theoretical $\theta$. |
| 2.5 | PR | $\theta \approx 3.7°$ | 2.8 | 1.000 | Consistent high performance at higher power. |
| 3.0 | P | 4.8° | 3.9 | 0.999 | THD higher than PR, advance angle changes with power. |
| 3.0 | PR | $\theta \approx 3.1°$ | 2.9 | 1.000 | Optimal performance at rated power for the solar inverter. |
The relationship between the required phase advance and output power was plotted, clearly showing that the advance angle decreases with increasing power for both controllers, as predicted by the $\arctan(I_C/I_g)$ relationship. However, the P controller consistently required a larger advance due to its inherent phase lag. Furthermore, it was demonstrated that by actively controlling the reference current phase advance $\phi_{adv}$, the solar inverter’s displacement power factor can be directly and precisely regulated, enabling both unity and non-unity power factor operation as required by certain grid support functions.
Conclusion
This detailed analysis presents a robust and high-performance control strategy for single-phase grid-connected solar inverters based on inductor current feedback and an LC filter. The integration of grid voltage feedforward decoupling effectively linearizes the plant and suppresses grid disturbance effects. The use of a quasi-Proportional-Resonant current controller provides superior sinusoidal tracking with zero steady-state error in the stationary frame, eliminating the need for complex coordinate transformations. A critical insight is the necessity to command the inductor current reference with a specific phase advance relative to the grid voltage to achieve unity power factor at the point of common coupling; this advance is precisely defined by the filter capacitor’s reactive current and is simpler to implement with a PR controller than with a P controller. The theoretical principles were validated on a 3 kW solar inverter platform, confirming that the strategy yields grid currents with low harmonic distortion (THD < 3%) and precise unity power factor across the operational range. This control approach provides a effective solution for modern, high-efficiency solar inverters, ensuring compliance with grid codes and high-quality power injection.