The integration of renewable energy sources, such as photovoltaic (PV) generation, into the power grid has accelerated significantly following the global emphasis on carbon neutrality and the establishment of modern power systems. The three-phase utility interactive inverter serves as the critical power-electronic interface between these distributed generation units and the three-phase AC grid. To mitigate current harmonics injected into the grid, an LCL filter is typically employed at the output of the inverter due to its superior high-frequency harmonic attenuation capability and smaller size compared to a simple L filter. However, the introduction of the LCL filter creates a third-order system with an inherent resonance peak that can lead to instability if not properly damped. Passive damping, involving physical resistors, is straightforward but incurs power losses. Consequently, active damping methods, realized through control algorithms, have become a preferred research focus.
Model Predictive Control (MPC) has gained considerable traction in power electronics for its intuitive concept, fast dynamic response, and flexibility in handling multiple objectives. Finite Control Set MPC (FCS-MPC) applied to utility interactive inverters often involves evaluating a cost function for all possible voltage vectors generated by the inverter and selecting the one that minimizes the function. However, conventional FCS-MPC, which applies only one voltage vector per control cycle, suffers from two main drawbacks: significant steady-state current ripple and variable switching frequency. The variable switching frequency complicates the design of the LCL filter and may inadvertently excite resonant modes. This paper proposes an enhanced Three-Vector Fixed-Frequency Predictive Current Control (TV-FFPCC) strategy for a three-phase LCL-type utility interactive inverter. The method maximizes control degrees of freedom by synthesizing an output voltage from two active vectors and one zero vector within a single control period. It incorporates active damping based on capacitor voltage feedback and employs a duty cycle calculation method derived from deadbeat control principles alongside a symmetric seven-segment pulse generation scheme to achieve constant switching frequency.
The fundamental topology of a three-phase LCL-type utility interactive inverter is shown in the schematic. The system comprises a three-phase voltage source inverter (VSI), an LCL filter, and the grid. The state-space model in the synchronous rotating (d-q) frame is essential for controller design. The continuous-time equations are:
$$
\begin{aligned}
L_1 \frac{di_{fd}}{dt} &= v_d – v_{cd} + \omega L_1 i_{fq} \\
L_1 \frac{di_{fq}}{dt} &= v_q – v_{cq} – \omega L_1 i_{fd} \\
C \frac{dv_{cd}}{dt} &= i_{fd} – i_{gd} + \omega C v_{cq} \\
C \frac{dv_{cq}}{dt} &= i_{fq} – i_{gq} – \omega C v_{cd} \\
L_2 \frac{di_{gd}}{dt} &= v_{cd} – v_{gd} + \omega L_2 i_{gq} \\
L_2 \frac{di_{gq}}{dt} &= v_{cq} – v_{gq} – \omega L_2 i_{gd}
\end{aligned}
$$
where $i_f$, $v_c$, $i_g$ represent inverter-side current, capacitor voltage, and grid-side current, respectively; $v$ is the inverter output voltage; $v_g$ is the grid voltage; $\omega$ is the grid angular frequency; $L_1$, $L_2$, and $C$ are the LCL filter parameters.
The transfer function from the inverter voltage $v$ to the grid current $i_g$ reveals the resonance issue:
$$
G_i(s) = \frac{i_g(s)}{v(s)} = \frac{1}{L_1 L_2 C s^3 + (L_1 + L_2)s}
$$
The Bode plot of this transfer function exhibits a pronounced resonance peak at frequency $f_{res} = \frac{1}{2\pi}\sqrt{\frac{L_1 + L_2}{L_1 L_2 C}}$, where the phase crosses -180° with gain > 0 dB, indicating potential instability.
A Phase-Locked Loop (PLL) based on a Second-Order Generalized Integrator (SOGI) is utilized to accurately extract the grid voltage phase angle $\theta$, which is crucial for coordinate transformations and synchronization of the utility interactive inverter.
Proposed Three-Vector Fixed-Frequency Predictive Current Control
The proposed TV-FFPCC strategy is designed to overcome the limitations of single-vector FCS-MPC. The overall control block diagram illustrates the integration of the PLL, reference generation, active damping, predictive model, and the core TV-FFPCC algorithm.
1. Discrete Predictive Model and Delay Compensation
Using forward Euler discretization with sampling period $T_s$, the discrete-time predictive model is obtained from the continuous equations. To compensate for the one-step computational delay inherent in digital implementation, the state variables are predicted two steps ahead ($k+2$). The predictions for the inverter-side current, for example, are:
$$
\begin{aligned}
i_{fd,k+2} &= i_{fd,k+1} + \frac{T_s}{L_1}(v_{d,k+1} – v_{cd,k+1} + \omega L_1 i_{fq,k+1}) \\
i_{fq,k+2} &= i_{fq,k+1} + \frac{T_s}{L_1}(v_{q,k+1} – v_{cq,k+1} – \omega L_1 i_{fd,k+1})
\end{aligned}
$$
where the states at $k+1$ are first predicted using the measured states at $k$ and the optimal voltage vector applied at $k$.
2. Capacitor Voltage Feedback Active Damping
To stabilize the system without power loss, an active damping method equivalent to connecting a virtual resistor $R_v$ in parallel with the filter capacitor is implemented. This is achieved by feeding back the high-frequency component of the capacitor voltage. The high-pass filtered capacitor voltage components, $\hat{v}_{cd}$ and $\hat{v}_{cq}$, are divided by $R_v$ to generate a virtual resistive current. This current is then subtracted from the inverter-side current reference to actively damp the resonance. The damping resistor is designed as $R_v = \frac{1}{2\zeta}\sqrt{\frac{L_2}{C}}$, where $\zeta$ is the damping ratio (typically 0.707).
3. Reference Generation and Cost Function Design
The primary control objective is to track the grid-side current reference $i_g^*$. Under steady-state conditions ($d/dt = 0$), the relationship between inverter-side and grid-side currents can be derived from the capacitor voltage equation, leading to the inverter-side current reference:
$$
\begin{aligned}
i_{fd}^* &= i_{gd}^* – \omega C v_{cq} – \frac{\hat{v}_{cd}}{R_v} \\
i_{fq}^* &= i_{gq}^* + \omega C v_{cd} – \frac{\hat{v}_{cq}}{R_v}
\end{aligned}
$$
The last term in each equation represents the active damping injection. A simple quadratic cost function $J$ is then defined based on the tracking error of the inverter-side current:
$$
J = (i_{fd}^* – i_{fd,k+2})^2 + (i_{fq}^* – i_{fq,k+2})^2
$$
4. Three-Vector Synthesis and Optimal Duty Cycle Calculation
The core innovation lies in synthesizing the output voltage over one $T_s$ using two active voltage vectors ($\mathbf{v}_1^l$, $\mathbf{v}_2^l$) and one zero vector ($\mathbf{v}_0$ or $\mathbf{v}_7$), where $l$ denotes the sector (I to VI) in which the reference voltage vector lies. The active vectors are chosen as the two adjacent vectors bounding the sector. The goal is to find the optimal duty cycles $d_1$, $d_2$, and $d_0 = 1 – d_1 – d_2$.
The duty cycles are calculated using a deadbeat principle applied to the inverter-side current. First, the current tracking errors for each candidate vector are computed using the discrete model:
$$
j_i = i_{fd}^* – i_{fd,i,k+2}, \quad k_i = i_{fq}^* – i_{fq,i,k+2}, \quad i \in \{0,1,2\}
$$
where $i=0$ corresponds to the zero vector, and $i=1,2$ correspond to the two candidate active vectors $\mathbf{v}_1^l$ and $\mathbf{v}_2^l$.
Enforcing the condition that the predicted current at $k+2$ equals its reference (deadbeat control) leads to the following system of equations:
$$
\begin{aligned}
d_1 j_1 + d_2 j_2 + d_0 j_0 &= 0 \\
d_1 k_1 + d_2 k_2 + d_0 k_0 &= 0 \\
d_0 &= 1 – d_1 – d_2
\end{aligned}
$$
Solving this system yields the optimal duty cycles:
$$
\begin{aligned}
d_1 &= \frac{k_2 j_0 – k_0 j_2}{M} \\
d_2 &= \frac{k_0 j_1 – k_1 j_0}{M} \\
d_0 &= 1 – d_1 – d_2
\end{aligned}
$$
where $M = k_0(j_1 – j_2) + k_1(j_2 – j_0) + k_2(j_0 – j_1)$.
The synthesized reference voltage vector for sector $l$ is $\mathbf{v}_s^l = d_0 \mathbf{v}_0 + d_1 \mathbf{v}_1^l + d_2 \mathbf{v}_2^l$. This calculation is performed for all six sectors. The optimal sector $l_{opt}$ and its corresponding duty cycles are those that minimize the cost function $J$.
5. Fixed-Frequency Pulse Generation
To achieve constant switching frequency, a symmetric seven-segment switching sequence is generated using the optimal sector and duty cycles. For example, if the optimal sector is I with active vectors $\mathbf{v}_1$ and $\mathbf{v}_2$, the switching sequence over one control period $T_s$ is:
$$
S = \left[ \mathbf{v}_0(\frac{d_0}{4}T_s) \rightarrow \mathbf{v}_1(\frac{d_1}{2}T_s) \rightarrow \mathbf{v}_2(\frac{d_2}{2}T_s) \rightarrow \mathbf{v}_7(\frac{d_0}{2}T_s) \rightarrow \mathbf{v}_2(\frac{d_2}{2}T_s) \rightarrow \mathbf{v}_1(\frac{d_1}{2}T_s) \rightarrow \mathbf{v}_0(\frac{d_0}{4}T_s) \right]
$$
This pattern minimizes switching transitions and ensures the average switching frequency is exactly $1/T_s$.
6. Comparative Analysis with Single-Vector FCS-MPC
The advantages of the proposed TV-FFPCC over traditional single-vector FCS-MPC for the utility interactive inverter are summarized in the table below.
| Aspect | Single-Vector FCS-MPC | Proposed TV-FFPCC |
|---|---|---|
| Vectors per Period | One fixed basic vector | Two active + one zero vector |
| Output Voltage | Limited to 8 discrete vectors | Continuously variable vector within hexagon |
| Steady-State Ripple | Relatively large | Significantly reduced |
| Switching Frequency | Variable, spectrum spread | Fixed, harmonics at $f_s$ and multiples |
| LCL Filter Design | Complex due to variable frequency | Simplified due to fixed frequency |
| Control Complexity | Often requires multi-objective cost function | Single current objective with active damping |
| Computational Burden | High sampling frequency needed for performance | Lower sampling frequency sufficient |
Simulation and Experimental Validation
The proposed TV-FFPCC strategy was validated through simulation and experiment on a three-phase LCL utility interactive inverter platform. The system parameters are listed below.
| Parameter | Symbol | Value |
|---|---|---|
| Grid Voltage (Phase, RMS) | $v_g$ | 220 V |
| Grid Frequency | $f$ | 50 Hz |
| DC-Link Voltage | $V_{dc}$ | 700 V |
| Inverter-side Inductor | $L_1$ | 2 mH |
| Grid-side Inductor | $L_2$ | 2 mH |
| Filter Capacitor | $C$ | 10 µF |
| Resonant Frequency | $f_{res}$ | ~1.6 kHz |
| Switching/Sampling Freq. | $f_s$ | 10 kHz |
| Virtual Damping Resistor | $R_v$ | 11 Ω ($\zeta=0.707$) |
Steady-State Performance
Under steady-state conditions with a grid current reference of 10 A, the proposed TV-FFPCC method demonstrated superior performance. The grid current waveform exhibited significantly lower ripple compared to the conventional single-vector method. The Total Harmonic Distortion (THD) of the grid current was markedly reduced with the proposed method. Furthermore, the harmonic spectrum confirmed the fixed switching frequency operation of TV-FFPCC, with dominant harmonics concentrated at the switching frequency $f_s$ and its integer multiples, whereas the conventional method showed a spread spectrum characteristic.
Dynamic Performance
The dynamic response was tested by stepping the grid current reference from 10 A to 20 A. Both methods provided fast tracking without exciting resonance. However, the proposed TV-FFPCC exhibited slightly lower overshoot and a smoother transient, confirming its robust dynamic capability for the utility interactive inverter.
Effectiveness of Active Damping
The critical role of the capacitor-voltage-feedback active damping was verified by disabling it ($R_v \to \infty$). Without active damping, the system became unstable, exhibiting large oscillations at the resonant frequency, and the current THD soared. With the active damping enabled, the system stabilized, and the current THD dropped to a low value, validating the effectiveness of the designed virtual resistor approach.
Performance Under Non-Ideal Grid Conditions
The robustness of the control strategy was tested under grid disturbances. During a 25% dip in one phase of the grid voltage, both controllers maintained balanced three-phase grid currents. However, the grid current under the proposed TV-FFPCC showed lower distortion. When the grid voltage contained background harmonics (3rd, 5th, 7th, 11th), the THD of the grid current increased for both methods, as expected. Nevertheless, the THD increase was substantially smaller for the proposed TV-FFPCC compared to the conventional method, demonstrating its stronger harmonic rejection capability for the utility interactive inverter operating in a polluted grid environment.
Conclusion
This paper has presented a comprehensive Three-Vector Fixed-Frequency Predictive Current Control strategy for three-phase LCL-type utility interactive inverters. The method successfully addresses the major limitations of conventional finite-control-set MPC: high steady-state ripple and variable switching frequency. By synthesizing an optimal voltage vector from two active vectors and a zero vector within each control period, the control freedom is maximized. The integration of capacitor-voltage-feedback active damping provides robust resonance suppression without additional hardware or losses. The duty cycles are calculated using a deadbeat current control principle, and a symmetric seven-segment modulation scheme ensures constant switching frequency, which simplifies filter design and improves overall performance. Extensive simulation and experimental results have validated the superior steady-state performance (lower THD, fixed frequency spectrum), excellent dynamic response, and robustness under grid disturbances of the proposed TV-FFPCC strategy, making it a highly effective solution for advanced grid-connected power conversion systems.