In recent years, the demand for efficient and reliable solar inverters has grown significantly due to the widespread adoption of photovoltaic systems in residential, industrial, and commercial applications. Among various inverter topologies, LC-type solar inverters are particularly favored for their superior output voltage quality and electromagnetic compatibility. However, controlling these inverters to achieve high performance remains challenging, especially under parameter uncertainties and dynamic operating conditions. Traditional model-based predictive control methods, such as Model Predictive Voltage Control (MPVC), rely heavily on accurate system parameters, and their performance degrades when parameter mismatches occur. To address this, researchers have explored parameter identification techniques, but these are often susceptible to system disturbances. As an alternative, Model-Free Predictive Voltage Control (MFPVC) methods have been proposed, which eliminate the dependency on precise parameters by leveraging voltage and current differentials. However, conventional MFPVC methods utilize only the basic voltage vectors of the inverter for prediction, leading to significant output voltage harmonics and wide harmonic frequency ranges. In this work, we introduce a novel Model-Free Predictive Harmonic Optimal Control (MFP HOC) method for LC-type solar inverters. Our approach synthesizes virtual vectors from basic vectors, computes their operation times based on voltage differentials, and updates differential information in real-time to minimize harmonics and improve output voltage performance. Through extensive experimentation, we validate the effectiveness of our method in reducing voltage harmonics and enhancing the overall performance of solar inverters.
The topology of a three-phase LC-type solar inverter consists of three half-bridge circuits, a three-phase LC filter circuit, and a resistive load circuit. Key components include the filter inductance $L$, filter capacitance $C$, load resistance $R$, DC input voltage $U_{dc}$, inverter-side current $i_i$, capacitor voltage $u_c$, load current $i_o$, and inverter output voltage $u_i$. The mathematical model of the LC-type solar inverter can be described by the following differential equations:
$$ \begin{cases} L \frac{di_i}{dt} = u_i – u_c \\ C \frac{du_c}{dt} = i_i – i_o \end{cases} $$
In traditional MPVC, the predicted current and voltage at the next sampling instant $(k+1)$ are derived from the discrete-time model, which depends on inductance and capacitance parameters. This dependency can lead to performance issues when parameters are inaccurate. To overcome this, MFPVC methods use voltage and current differentials instead of the exact model. The differentials $\Delta u_c$ and $\Delta i_i$ are defined as:
$$ \Delta u_c(k-1) = u_c(k) – u_c(k-1) $$
$$ \Delta i_i(k-1) = i_i(k) – i_i(k-1) $$
These differentials are stored in a lookup table corresponding to the basic voltage vectors, as shown in Table 1. The predicted values are then computed as:
$$ i_i(k+1) = i_i(k) + \Delta i_i(k) $$
$$ u_c(k+1) = u_c(k) + \Delta u_c(k) $$
To achieve accurate output voltage tracking and suppress resonance caused by the LC filter, a cost function is defined:
$$ g = \lambda_i [i_i(k+1) – i_{i,ref}(k+1)]^2 + [u_c(k+1) – u_{c,ref}(k+1)]^2 $$
where $\lambda_i$ is a weighting factor for the inverter-side current. Although this approach improves robustness, it still results in high voltage harmonics due to the limited set of basic vectors.
| Vector | $\Delta u_c$ | $\Delta i_i$ |
|---|---|---|
| $U_0$ | $\Delta u_{c0}$ | $\Delta i_{i0}$ |
| $U_1$ | $\Delta u_{c1}$ | $\Delta i_{i1}$ |
| $U_2$ | $\Delta u_{c2}$ | $\Delta i_{i2}$ |
| $U_3$ | $\Delta u_{c3}$ | $\Delta i_{i3}$ |
| $U_4$ | $\Delta u_{c4}$ | $\Delta i_{i4}$ |
| $U_5$ | $\Delta u_{c5}$ | $\Delta i_{i5}$ |
| $U_6$ | $\Delta u_{c6}$ | $\Delta i_{i6}$ |
| $U_7$ | $\Delta u_{c7}$ | $\Delta i_{i7}$ |
To address the limitations of conventional MFPVC, we propose the MFP HOC method, which focuses on harmonic optimization through virtual vector synthesis and real-time differential updates. The core idea is to synthesize virtual vectors from the basic vectors of the solar inverter, reducing the harmonic distortion in the output voltage. Specifically, we use two zero vectors and two adjacent active vectors to form six virtual vectors, as detailed in Table 2. The operation times for the first active vector $U_x$, second active vector $U_y$, and zero vector $U_z$ are calculated based on their impact on the cost function. The times $t_x$, $t_y$, and $t_z$ are given by:
$$ t_x = \left[ \frac{G_y G_z}{G_x G_y + G_y G_z + G_z G_x} \right] T_s $$
$$ t_y = \left[ \frac{G_z G_x}{G_x G_y + G_y G_z + G_z G_x} \right] T_s $$
$$ t_z = \left[ \frac{G_x G_y}{G_x G_y + G_y G_z + G_z G_x} \right] T_s $$
where $G_x$, $G_y$, and $G_z$ are the cost function values corresponding to vectors $U_x$, $U_y$, and $U_z$, respectively, and $T_s$ is the control period. The cost function for capacitor voltage tracking is $G = [u_{c,ref} – u_c(k+1)]^2$, with $u_c(k+1) = u_c(k) + \Delta u_c(k)$.
| Virtual Vector | Basic Vector Sequence |
|---|---|
| $U_{s1}$ | {$U_0$, $U_1$, $U_2$, $U_7$, $U_7$, $U_2$, $U_1$, $U_0$} |
| $U_{s2}$ | {$U_0$, $U_3$, $U_2$, $U_7$, $U_7$, $U_2$, $U_3$, $U_0$} |
| $U_{s3}$ | {$U_0$, $U_3$, $U_4$, $U_7$, $U_7$, $U_4$, $U_3$, $U_0$} |
| $U_{s4}$ | {$U_0$, $U_5$, $U_4$, $U_7$, $U_7$, $U_4$, $U_5$, $U_0$} |
| $U_{s5}$ | {$U_0$, $U_5$, $U_6$, $U_7$, $U_7$, $U_6$, $U_5$, $U_0$} |
| $U_{s6}$ | {$U_0$, $U_1$, $U_6$, $U_7$, $U_7$, $U_6$, $U_1$, $U_0$} |
A critical aspect of our MFP HOC method is the real-time update of voltage and current differentials. When a virtual vector is applied, the differentials $\Delta u_c$ and $\Delta i_i$ are combinations of those from the constituent basic vectors. For a virtual vector $U_s$ composed of vectors $U_x$, $U_y$, and $U_z$ with times $t_x$, $t_y$, and $t_z$, the differentials are:
$$ \Delta u_c(k-1) = \frac{t_x \Delta u_{cx}(k-1) + t_y \Delta u_{cy}(k-1) + t_z \Delta u_{cz}(k-1)}{T_s} $$
$$ \Delta i_i(k-1) = \frac{t_x \Delta i_{ix}(k-1) + t_y \Delta i_{iy}(k-1) + t_z \Delta i_{iz}(k-1)}{T_s} $$
To update the differentials for the basic vectors, we analyze the relationship between the virtual vector and its components. By substituting the virtual vector $U_s$ into the differential equations, we derive expressions that allow us to compute the differentials for zero and active vectors. For instance, if the virtual vector and zero vector have equal coordinate components, the differentials are directly updated as:
$$ \Delta u_{cz}(k-1) = \Delta u_c(k-1) $$
$$ \Delta i_{iz}(k-1) = \Delta i_i(k-1) $$
If the components are not equal, we use historical differential data for adjustment:
$$ \Delta u_{cz}(k-1) = [\Delta u_{cz}(k-2) – \Delta u_c(k-2)] + \Delta u_c(k-1) $$
$$ \Delta i_{iz}(k-1) = [\Delta i_{iz}(k-2) – \Delta i_i(k-2)] + \Delta i_i(k-1) $$
Similarly, the differentials for the active vectors are updated based on the zero vector differentials. This real-time update mechanism ensures that the differential information remains accurate, further improving the output voltage quality of the solar inverter.
The proposed MFP HOC method involves several steps in each control cycle. First, we sample the capacitor voltage and inverter-side current at time $k$. Then, we compute the differentials for the virtual vector and update the differentials for all basic vectors using the relationships described above. Next, we calculate the operation times for each virtual vector based on the voltage differentials. Using these times and the updated differentials, we predict the capacitor voltage and inverter-side current at time $k+1$:
$$ u_c(k+1) = u_c(k) + \frac{t_x \Delta u_{cx}(k) + t_y \Delta u_{cy}(k) + t_z \Delta u_{cz}(k)}{T_s} $$
$$ i_i(k+1) = i_i(k) + \frac{t_x \Delta i_{ix}(k) + t_y \Delta i_{iy}(k) + t_z \Delta i_{iz}(k)}{T_s} $$
We evaluate the cost function for each virtual vector and select the one with the minimum value as the optimal vector. The duty cycles for the three phases are then computed as:
$$ d_a = \frac{S_{aux} t_x + S_{auy} t_y + S_{auz} t_z}{T_s} $$
$$ d_b = \frac{S_{bux} t_x + S_{buy} t_y + S_{buz} t_z}{T_s} $$
$$ d_c = \frac{S_{cux} t_x + S_{cuy} t_y + S_{cuz} t_z}{T_s} $$
where $S_{a}$, $S_{b}$, and $S_c$ are switching functions for the three phases. This optimal vector is applied in the next control cycle to achieve harmonic-optimized performance.
To validate the proposed MFP HOC method, we conducted experiments on a hardware platform equipped with a DSP controller, current and voltage sensors, an LC filter, and resistive loads. The system parameters included a DC input voltage of 200 V, output voltage reference of 100 V, filter inductance of 1.7 mH, filter capacitance of 15 μF, load resistance of 10 Ω, and a control frequency of 20 kHz. We compared the performance of our method with traditional MFPVC under both steady-state and dynamic conditions.
In steady-state operation, the traditional MFPVC method exhibited significant output voltage errors due to the use of only basic vectors, resulting in a Total Harmonic Distortion (THD) of 3.36%. In contrast, our MFP HOC method achieved a much lower THD of 1.29%, demonstrating its effectiveness in reducing voltage harmonics. The virtual vector synthesis and differential updates contributed to a smoother output voltage waveform with minimized ripple. Under dynamic conditions, such as a step change in the output voltage reference from 100 V to 50 V, both methods showed fast tracking capabilities. However, the MFP HOC method maintained superior voltage quality with comparable dynamic performance, confirming its robustness and practical applicability for solar inverters.
In conclusion, we have developed a novel Model-Free Predictive Harmonic Optimal Control method for LC-type solar inverters. By synthesizing virtual vectors from basic vectors and updating differential information in real-time, our approach significantly reduces output voltage harmonics and improves overall performance. Experimental results validate the method’s effectiveness in both steady-state and dynamic scenarios, highlighting its potential for enhancing the reliability and efficiency of solar power systems. Future work will focus on extending this method to grid-connected solar inverters and exploring its applicability under varying environmental conditions.