In modern renewable energy systems, off-grid inverters play a critical role in ensuring reliable power supply to critical loads during grid failures. Among the various types of solar inverter, such as grid-tied, off-grid, and hybrid inverters, off-grid inverters must maintain high-quality output voltage under diverse load conditions, including nonlinear and transient loads. This article explores an enhanced control strategy based on load current direct feed-forward to address these challenges, leveraging mathematical modeling and discrete analysis to optimize performance. The proposed method significantly improves dynamic response and output voltage stability, which is essential for applications involving different types of solar inverter configurations.
The conventional double-loop control structure, comprising an outer voltage loop and an inner current loop, is widely used in off-grid inverters with LC filters. However, this approach often exhibits poor dynamic response to load disturbances and limited adaptability to nonlinear loads. For instance, when dealing with various types of solar inverter setups, such as those integrated with battery storage or standalone systems, load current variations can introduce harmonics and distortions. The transfer function of a standard double-loop controlled inverter is given by:
$$ U_I = \frac{(k_{pu} s + k_{iu})(k_{pi} s + k_{ii})}{C_f s^3(L_f s + R_f) + C_f s^2(k_{pi} s + k_{ii}) + s^2 + (k_{pu} s + k_{iu})(k_{pi} s + k_{ii})} U_I^* – \frac{s^2(L_f s + R_f) + s(k_{pi} s + k_{ii})}{C_f s^3(L_f s + R_f) + C_f s^2(k_{pi} s + k_{ii}) + s^2 + (k_{pu} s + k_{iu})(k_{pi} s + k_{ii})} i_L $$
This equation shows that the output voltage $U_I$ depends not only on the reference voltage $U_I^*$ but also on the load current $i_L$, leading to potential instability. To mitigate this, a feed-forward control method is introduced, where the load current is added to the current reference. The modified transfer function becomes:
$$ U_I = \frac{(k_{pu} s + k_{iu})(k_{pi} s + k_{ii})}{C_f s^3(L_f s + R_f) + C_f s^2(k_{pi} s + k_{ii}) + s^2 + (k_{pu} s + k_{iu})(k_{pi} s + k_{ii})} U_I^* – \frac{s^2(L_f s + R_f)}{C_f s^3(L_f s + R_f) + C_f s^2(k_{pi} s + k_{ii}) + s^2 + (k_{pu} s + k_{iu})(k_{pi} s + k_{ii})} i_L $$
Although this reduces the impact of load current, it does not eliminate it entirely due to zero distribution changes. Therefore, a direct feed-forward control structure is proposed, which compensates for the voltage drop across the filter inductance by adding $i_L(L_f s + R_f)$ to the inverter output voltage. This approach decouples the load current effect, resulting in the following simplified transfer function:
$$ U_I = \frac{s^2 + (k_{pu} s + k_{iu})(k_{pi} s + k_{ii})}{C_f s^3(L_f s + R_f) + C_f s^2(k_{pi} s + k_{ii}) + s^2 + (k_{pu} s + k_{iu})(k_{pi} s + k_{ii})} U_I^* $$
For digital implementation, a discrete model is developed. The voltage and current controllers are discretized using PI controllers in the synchronous reference frame:
$$ C_u(z) = k_{pu} + \frac{4k_{iu}T z^4}{z^4 – 1}, \quad C_i(z) = k_{pi} + \frac{k_{ii}T z}{z – 1} $$
where $T$ is the sampling period. The plant’s continuous transfer function is discretized as:
$$ G(z) = \frac{1}{L_f C_f} \sqrt{\frac{1}{L_f C_f} – \frac{R_f^2}{4L_f^2}} \frac{z e^{-\frac{R_f}{2L_f} T} \sin\left( \sqrt{\frac{1}{L_f C_f} – \frac{R_f^2}{4L_f^2}} T \right)}{z^2 – 2z e^{-\frac{R_f}{2L_f} T} \cos\left( \sqrt{\frac{1}{L_f C_f} – \frac{R_f^2}{4L_f^2}} T \right) + e^{-\frac{R_f}{L_f} T}} $$
The feed-forward compensation includes a differential term to address sampling delays, modeled as:
$$ F(z) = \left( \frac{L_f}{T_1} + R_f \right) \frac{1 – z}{z} – \frac{L_f}{T_1^2} \cdot \frac{1 – z}{z – e^{-T/T_1}} $$
where $T_1$ is the time constant for high-frequency noise suppression. The overall discrete transfer function is:
$$ U_I(z) = \frac{T G(z) C_u(z) C_i(z) + T G(z)}{T z^5 + C_f (z – 1) z^3 G(z) C_i(z) + T G(z) C_u(z) C_i(z)} U_I^*(z) + \frac{T z^5 G(z) F(z)}{T z^5 + C_f (z – 1) z^3 G(z) C_i(z) + T G(z) C_u(z) C_i(z)} i_L(z) $$
Parameter design is crucial for stability. For a 15 kVA three-phase inverter with $L_f = 2$ mH, $R_f = 0.002$ Ω, $C_f = 20$ μF, and switching frequency of 80 μs, root locus analysis guides the selection of $k_{pi} = 0.1$ for the current loop and $k_{pu} = 3$, $k_{iu} = 200$ for the voltage loop. The differential coefficient in the feed-forward path is set to $1.1 L_f$ to balance compensation and noise rejection, as values beyond $2 L_f$ may lead to instability.
The performance of the proposed control method is validated through simulations and experiments. For instance, under a sudden 15 kW resistive load, the output voltage remains stable with minimal distortion. Similarly, with a nonlinear rectifier load, the voltage total harmonic distortion (THD) is maintained at 8.4% despite a current THD of 38.32%. These results highlight the robustness of the approach for various types of solar inverter applications, including those with complex load profiles.
| Parameter | Value | Description |
|---|---|---|
| $k_{pi}$ | 0.1 | Current loop proportional gain |
| $k_{pu}$ | 3 | Voltage loop proportional gain |
| $k_{iu}$ | 200 | Voltage loop integral gain |
| $L_f$ | 2 mH | Filter inductance |
| $C_f$ | 20 μF | Filter capacitance |
| THD (Voltage) | 8.4% | Under rectifier load |
| THD (Current) | 38.32% | Under rectifier load |
Furthermore, the adaptability of this control strategy to different types of solar inverter systems, such as hybrid inverters coupled with energy storage, is evident in its ability to handle rapid load changes. The direct feed-forward mechanism effectively cancels out disturbances, ensuring consistent voltage quality. This is particularly important for off-grid systems where reliability is paramount.
In conclusion, the load current direct feed-forward control method significantly enhances the performance of off-grid inverters by mitigating load-induced distortions and improving dynamic response. This approach is applicable to a wide range of types of solar inverter configurations, offering a robust solution for maintaining voltage quality in challenging environments. Future work could explore integration with advanced topologies like multi-level inverters to further optimize efficiency and scalability.