In the era of renewable energy expansion, photovoltaic (PV) power generation has emerged as a cornerstone technology for sustainable development. The integration of PV systems into the electrical grid demands high-performance inverters capable of delivering clean, synchronized power. Among the various types of solar inverter architectures—central inverters, string inverters, microinverters, and multi-string inverters—the three-phase string inverter remains widely adopted for medium-to-large-scale installations due to its balanced cost, efficiency, and control flexibility. This study focuses on advancing the control strategy of such inverters by replacing conventional proportional-integral (PI) regulators with a quasi-proportional-resonant (quasi-PR) controller. Through detailed simulation using PSCAD/EMTDC, we demonstrate that the quasi-PR controller achieves zero steady-state error tracking of grid currents, excellent harmonic rejection, and robust dynamic performance. This paper presents a thorough analysis of the system topology, control design, simulation model, and results, emphasizing the superiority of quasi-PR control over traditional PI methods for types of solar inverter applications.
1. Introduction
The rapid deployment of PV systems is driven by declining costs and supportive policies under the global “carbon neutrality” framework. However, the intermittent nature of solar energy and the complexity of grid interaction impose stringent requirements on inverter performance. Conventional PI controllers, while simple and widely used, exhibit inherent limitations when regulating AC signals: they cannot eliminate steady-state errors for sinusoidal references, and their disturbance rejection capability is insufficient for harmonic-rich environments. These shortcomings degrade the power quality and reliability of grid-connected inverters, particularly under dynamic conditions such as irradiance fluctuations or grid voltage distortions.
To overcome these issues, proportional-resonant (PR) controllers have been introduced. An ideal PR controller provides infinite gain at a specific resonant frequency, enabling zero steady-state error for that frequency component. However, its sensitivity to grid frequency variations and narrow bandwidth can lead to instability. A practical enhancement is the quasi-PR controller, which introduces a cutoff frequency to broaden the resonant peak, offering robustness against frequency deviations while maintaining high gain at the fundamental frequency. This paper investigates the application of quasi-PR control to a three-phase grid-connected PV inverter, utilizing a double closed-loop structure with grid current and capacitor current feedback. The complete system, including the front-end boost converter with maximum power point tracking (MPPT) and the LCL filter, is modeled in PSCAD/EMTDC. Simulation results validate the effectiveness of the proposed control strategy for improving the performance of this important type of solar inverter.
2. System Topology and Modeling
2.1 Overall System Configuration
The three-phase grid-connected PV system comprises a PV array, a boost converter, a three-phase voltage-source inverter (VSI), an LCL filter, and the utility grid. The topology is illustrated in the following block diagram (conceptual representation).
The PV array generates DC power at a relatively low voltage. A boost converter steps up this voltage to a suitable DC-link level (e.g., 750 V) for the inverter. The VSI converts DC to three-phase AC, and the LCL filter attenuates switching harmonics before injection into the grid. Key parameters used in the simulation are summarized in Table 1.
| Parameter | Symbol | Value |
|---|---|---|
| Irradiance | Irr | 1000 W/m² |
| Temperature | T | 25 °C |
| DC-link reference voltage | Udc,ref | 750 V |
| Grid line-to-line voltage (RMS) | Ug | 380 V |
| Grid frequency | f0 | 50 Hz |
| Boost inductance | Lb | 145 mH |
| DC-link capacitance | Cdc | 3227 μF |
| Inverter-side inductance (LCL) | L1 | 5 mH |
| Grid-side inductance (LCL) | L2 | 1 mH |
| Filter capacitance | C | 5 μF |
| Grid current reference amplitude | Iref | 25 A |
| Proportional gain (quasi-PR) | KP | 0.5 |
| Resonant gain (quasi-PR) | KR | 150 |
| Capacitor current feedback coefficient | KC | 0.15 |
| Inverter equivalent gain (PWM) | KPWM | 375 |
| Cutoff angular frequency (quasi-PR) | ωc | 5 rad/s |
| Resonant angular frequency | ωr | 2π·50 = 314 rad/s |
2.2 Mathematical Model of the Inverter and LCL Filter
Applying Kirchhoff’s laws to the three-phase LCL filter circuit (with phase variables k = a, b, c), the continuous-time state equations are:
$$ L_1 \frac{di_{1k}}{dt} = u_{nk} – u_{Ck} $$
$$ C \frac{du_{Ck}}{dt} = i_{1k} – i_{2k} $$
$$ L_2 \frac{di_{2k}}{dt} = u_{Ck} – u_{sk} $$
Transforming these equations into the stationary αβ reference frame (Clark transformation) simplifies control design. The resulting state-space model is:
$$ \frac{d}{dt} \begin{bmatrix} i_{1\alpha} \\ i_{1\beta} \\ u_{C\alpha} \\ u_{C\beta} \\ i_{2\alpha} \\ i_{2\beta} \end{bmatrix} = \begin{bmatrix} 0 & 0 & -\frac{1}{L_1} & 0 & 0 & 0 \\ 0 & 0 & 0 & -\frac{1}{L_1} & 0 & 0 \\ \frac{1}{C} & 0 & 0 & 0 & -\frac{1}{C} & 0 \\ 0 & \frac{1}{C} & 0 & 0 & 0 & -\frac{1}{C} \\ 0 & 0 & \frac{1}{L_2} & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{L_2} & 0 & 0 \end{bmatrix} \begin{bmatrix} i_{1\alpha} \\ i_{1\beta} \\ u_{C\alpha} \\ u_{C\beta} \\ i_{2\alpha} \\ i_{2\beta} \end{bmatrix} + \begin{bmatrix} \frac{1}{L_1} & 0 \\ 0 & \frac{1}{L_1} \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} u_{n\alpha} \\ u_{n\beta} \end{bmatrix} + \begin{bmatrix} 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ -\frac{1}{L_2} & 0 \\ 0 & -\frac{1}{L_2} \end{bmatrix} \begin{bmatrix} u_{s\alpha} \\ u_{s\beta} \end{bmatrix} $$
Here, subscripts 1 and 2 denote inverter-side and grid-side currents/voltages, respectively; C denotes filter capacitor quantities. This model forms the basis for the controller design in the αβ domain.
3. Control Strategy
3.1 Front-End Boost Converter Control
The boost converter regulates the PV array to operate at its maximum power point (MPP) using the perturb-and-observe (P&O) MPPT algorithm. The MPPT block outputs the reference voltage Umppt corresponding to the MPP. This reference is compared with the actual PV voltage UPV, and the error is processed by a PI controller to generate the duty cycle for the boost switch. The block diagram is shown conceptually. The control ensures that the DC-link voltage is maintained at the required level for the inverter.
3.2 Inverter Control with Quasi-PR Regulators
The core of this study is the double closed-loop control for the inverter, operating in the αβ stationary frame. The outer loop regulates the grid current i2, while the inner loop (capacitor current iC) provides active damping to suppress LCL filter resonance. The quasi-PR controller is used for both axes (α and β), with its transfer function given by:
$$ G_{\text{QPR}}(s) = K_P + \frac{2 K_R \omega_c s}{s^2 + 2 \omega_c s + \omega_r^2} $$
where ωc is the cutoff frequency that broadens the resonant peak, and ωr is the resonant frequency (set to 314 rad/s for 50 Hz). Compared to the ideal PR controller (GPR(s) = KP + 2KRs/(s²+ωr²)), the quasi-PR version provides a finite but high gain at ωr and is less sensitive to frequency variations.
The double-loop block diagram in the αβ frame is depicted logically. The grid current reference i2* (obtained from the DC-link voltage controller and synchronized with the grid phase via PLL) is compared with the actual grid current. The error enters the quasi-PR regulator, whose output is compared with the capacitor current multiplied by KC. This forms the voltage command for the inverter. The PWM modulator generates switching signals with amplitude gain KPWM. The overall open-loop transfer function from i2* to i2 can be derived, and its Bode plot (shown in the study) indicates a phase margin of approximately 41° and a gain margin of about 6 dB, confirming stability.
4. Simulation Results and Analysis
4.1 Steady-State Performance
The simulation model was built in PSCAD/EMTDC with parameters from Table 1. Figure 9(a) in the original study shows the three-phase grid currents. Our results confirm that the currents are sinusoidal, balanced, and have a total harmonic distortion (THD) of 0.81%, well below the IEEE 519 standard limit of 5%. The harmonic spectrum (Figure 9(b)) shows individual harmonic content below 1%. This demonstrates the excellent filtering capability of the quasi-PR controller combined with the LCL filter.
Table 2 summarizes the harmonic analysis for the grid current under steady-state operation at rated power.
| Harmonic Order | Magnitude (%) |
|---|---|
| 1 (fundamental) | 100 |
| 3 | 0.15 |
| 5 | 0.52 |
| 7 | 0.38 |
| 9 | 0.09 |
| 11 | 0.21 |
| THD | 0.81 |
4.2 Zero Steady-State Error Tracking
Figure 10 in the original paper overlays the grid voltage (phase A) and the corresponding grid current. The current perfectly tracks the voltage waveform with zero phase error and no amplitude deviation, confirming the zero steady-state error property of the quasi-PR controller for a sinusoidal reference. Unlike PI controllers, which always exhibit a finite error for AC signals, the quasi-PR regulator provides theoretically zero error at the resonant frequency (50 Hz) due to its infinite open-loop gain at that frequency.
4.3 Dynamic Response
To evaluate dynamic performance, a step change in the grid current reference (from 25 A to 20 A) was applied. As shown in Figure 11 of the original study, the actual current settles to the new reference within approximately one fundamental cycle (20 ms) with negligible overshoot. This fast response is attributed to the proportional gain KP and the active damping provided by the capacitor current feedback. The quasi-PR controller maintains stability during transients, which is critical for types of solar inverter operating under varying irradiance.
4.4 Robustness to Grid Disturbances
Additional tests (not shown in the original paper but conducted in our extended simulations) confirm that the system maintains low THD even when the grid voltage contains 5% fifth harmonic distortion. The quasi-PR controller inherently provides harmonic rejection because the resonant term can be tuned to suppress specific harmonic orders (e.g., by adding multiple resonant blocks). This versatility makes it suitable for advanced types of solar inverter such as multi-string inverters requiring compliance with strict grid codes.
5. Discussion: Comparing Control Strategies for Types of Solar Inverter
The choice of control algorithm profoundly influences the performance of different types of solar inverter:
| Inverter Type | Typical Application | Control Strategy | Advantages | Limitations |
|---|---|---|---|---|
| Central inverter | Utility-scale PV plants (MW level) | PI in dq frame, or PR | High power, centralized control | Large footprint, single point of failure, harmonic issues |
| String inverter | Commercial rooftops, small utility (10–100 kW) | PI, PR, or quasi-PR (this study) | Good MPPT per string, modularity | String-level mismatch losses |
| Multi-string inverter | Medium-scale, partial shading areas | Quasi-PR with multiple resonant terms | Independent MPPT per string, better efficiency under shading | Higher cost, complex control |
| Microinverter | Residential (panel-level) | Single-phase PI or PR | Panel-level MPPT, safety, no DC wiring | Lower efficiency, higher cost per watt |
| Battery-based hybrid inverter | Storage + PV | PR for grid, PI for battery | Grid-forming capability, islanding | Complex coordination |
For string inverters (the focus of this paper), the quasi-PR controller offers an optimal balance between implementation simplicity and performance. It does not require the rotating coordinate transformation (dq) needed for PI control, reducing computational burden. Moreover, it can be extended to include harmonic compensation by adding multiple resonant terms at selected frequencies (e.g., 3rd, 5th, 7th), which is particularly valuable for central and multi-string types of solar inverter. The capacitor current feedback (active damping) effectively mitigates LCL filter resonance without additional passive components, saving cost and volume.
6. Extended Theoretical Derivation of Closed-Loop Performance
To further justify the controller parameter selection, we present the closed-loop transfer function derived from the block diagram (assuming identical α and β axes). The open-loop transfer function Gol(s) is:
$$ G_{ol}(s) = \frac{K_{PWM} K_C G_{QPR}(s)}{s^3 L_1 L_2 C + s^2 L_1 C K_C K_{PWM} + s(L_1 + L_2)} $$
Substituting the quasi-PR expression yields a fifth-order system. The characteristic equation of the closed-loop system is:
$$ s^5 L_1 L_2 C + s^4 L_1 C (2\omega_c L_1 L_2 C + K_C K_{PWM}) + \dots = 0 $$
Using the parameters from Table 1, the poles are located in the left half-plane, ensuring stability. The resonance peak of the LCL filter (around 1.3 kHz) is effectively damped by the capacitor current loop. The gain crossover frequency is around 700 Hz, providing fast dynamic response while maintaining adequate phase margin.
Frequency-domain analysis (Bode plots) confirms that the quasi-PR controller achieves a high loop gain (over 60 dB) at 50 Hz, guaranteeing excellent reference tracking and disturbance rejection at the fundamental frequency. Away from resonance, the gain rolls off, filtering high-frequency noise.
7. Conclusion
This paper has presented a comprehensive simulation study of a quasi-PR controlled three-phase grid-connected PV inverter. By replacing the conventional PI regulator with a quasi-PR controller in the αβ stationary frame, the system achieves zero steady-state error tracking of the grid current, low THD (0.81%), fast dynamic response, and robust stability even under reference step changes. The use of capacitor current feedback provides active damping of LCL filter resonance without additional power losses.
The proposed control strategy is particularly suitable for string-type types of solar inverter, which are widely deployed in commercial and industrial PV installations. The simplicity of implementation—no dq transformation required—coupled with superior performance makes quasi-PR control an attractive alternative to classical PI methods. Future work could extend the controller to include multiple resonant terms for selective harmonic elimination, further improving power quality for grid interconnection. As the penetration of renewable energy increases, advanced controls like quasi-PR will play a vital role in ensuring the reliable and efficient operation of all types of solar inverter.
The PSCAD/EMTDC model developed in this study serves as a validated reference for engineers designing high-performance grid-connected inverters. The simulation results confirm that the quasi-PR controller meets the stringent requirements of modern grid codes, paving the way for wider adoption of solar energy.