In the realm of renewable energy systems, inverters play a pivotal role in converting direct current (DC) from sources like solar panels or batteries into alternating current (AC) for practical use. Among the various types of solar inverter, small off-grid inverters are particularly significant due to their portability and applicability in remote areas. These devices are essential for powering low-power appliances, but their efficiency is often compromised by significant switching losses in power electronic components. This article explores an efficiency optimization method for small off-grid inverters through carrier ratio control, focusing on reducing switching losses while maintaining output voltage quality. The types of solar inverter, including grid-tied, off-grid, and hybrid variants, each have distinct operational characteristics, but off-grid inverters face unique challenges in efficiency due to variable loads and the absence of grid support. By analyzing the relationship between carrier ratio and total harmonic distortion (THD), we propose a feedback control strategy that dynamically adjusts the carrier ratio to minimize losses without exceeding THD constraints. This approach is validated through simulations and experiments, demonstrating its practicality for improving inverter performance without additional hardware.
Small off-grid inverters typically consist of input protection circuits, DC/DC voltage regulation circuits, DC/AC inversion circuits, filter circuits, and output protection circuits. The DC/AC stage, which converts DC to AC using techniques like sinusoidal pulse width modulation (SPWM), is a major source of losses, primarily due to switching transitions in transistors. Switching losses, which depend on factors like switching frequency, current magnitude during switching, and voltage blocking, can be mitigated by reducing the carrier ratio—the ratio of carrier frequency to fundamental frequency. However, lowering the carrier ratio increases THD, compromising waveform quality. Thus, a balance must be struck to achieve efficiency gains while adhering to power quality standards. Various types of solar inverter, such as those used in residential or industrial settings, employ different control strategies, but for small off-grid systems, simplicity and cost-effectiveness are crucial. This article delves into the mathematical foundations of carrier ratio control, presents a detailed loss model, and introduces a variable-step feedback mechanism for real-time optimization.
The structure of a typical small off-grid inverter begins with an input protection circuit that safeguards against overvoltage conditions. This is followed by a DC/DC converter that adjusts the DC voltage level to suit the inversion stage. The core DC/AC circuit, often implemented as a full-bridge inverter, generates an SPWM waveform that is filtered to produce a sinusoidal output. The output protection circuit prevents damage from overloads. In this setup, the DC/AC and DC/DC stages account for the majority of losses, with switching losses in the DC/AC stage being particularly significant. For instance, in a full-bridge inverter with unipolar double-frequency modulation and an LC filter, the switching frequency directly influences both efficiency and THD. The types of solar inverter used in such configurations must handle varying loads, from linear resistive loads to non-linear rectifier loads, which affect THD differently. Understanding these dynamics is key to optimizing performance.
To analyze the inverter’s behavior, consider a full-bridge circuit with switches V1, V2, V3, and V4, an input voltage Ud, and an LC filter composed of inductor L and capacitor C, driving a load R. The modulation signals for points A and B are given by ua and ub, respectively:
$$ u_a = U_s \sin(\omega_0 t + \phi) $$
$$ u_b = U_s \sin(\omega_0 t + \phi – \pi) $$
where Us is the amplitude of the modulation wave, ω0 is the fundamental angular frequency, and φ is the initial phase. The carrier wave uc is a triangular function defined as:
$$ u_c = \begin{cases}
\frac{2}{\pi} U_C \left( \omega_c t – 2n\pi + \frac{\pi}{2} \right) – U_C, & 2n\pi – \frac{\pi}{2} \leq \omega_c t < 2n\pi + \frac{\pi}{2} \\
-\frac{2}{\pi} U_C \left( \omega_c t – 2n\pi – \frac{\pi}{2} \right) + U_C, & 2n\pi + \frac{\pi}{2} \leq \omega_c t < 2n\pi + \frac{3\pi}{2}
\end{cases} $$
where UC is the carrier amplitude, ωc is the carrier angular frequency, and n is an integer. The modulation index M and carrier ratio N are defined as:
$$ M = \frac{U_s}{U_C} $$
$$ N = \frac{\omega_c}{\omega_0} $$
The voltage at point A, uA, is determined by comparing ua and uc, leading to a series of pulses. Using double Fourier series analysis, uA can be expressed as:
$$ u_A = \frac{A_{00}}{2} + \sum_{n=1}^{\infty} \left[ A_{0n} \cos(nY) + B_{0n} \sin(nY) \right] + \sum_{m=1}^{\infty} \left[ A_{m0} \cos(mX) + B_{m0} \sin(mY) \right] + \sum_{m=1}^{\infty} \sum_{n=\pm 1}^{\pm \infty} \left[ A_{mn} \cos(mX + nY) + B_{mn} \sin(mX + nY) \right] $$
where X = ωc t and Y = ω0 t + φ. The coefficients Amn and Bmn are derived from integral expressions, and after simplification, uA becomes:
$$ u_A = \frac{M U_d}{2} \sin(\omega_0 t) + \frac{U_d}{\pi} \sum_{m=1}^{\infty} \sum_{n=\pm 1}^{\pm \infty} \left[ (-1)^m – (-1)^n \right] \frac{1}{m} J_n\left( \frac{m M \pi}{2} \right) \sin\left( (mN + n) \omega_0 t \right) $$
Here, Jn(x) is the Bessel function of the first kind. Similarly, the voltage at point B, uB, is:
$$ u_B = -\frac{M U_d}{2} \sin(\omega_0 t) + \frac{U_d}{\pi} \sum_{m=1}^{\infty} \sum_{n=\pm 1}^{\pm \infty} \left[ (-1)^{m+n} – 1 \right] \frac{1}{m} J_n\left( \frac{m M \pi}{2} \right) \sin\left( (mN + n) \omega_0 t \right) $$
The input voltage to the filter, uAB, is the difference between uA and uB:
$$ u_{AB} = u_A – u_B = M U_d \sin(\omega_0 t) + \frac{4 U_d}{\pi} \sum_{m=2,4,\ldots}^{\infty} \sum_{n=\pm 1, \pm 3, \ldots}^{\pm \infty} \frac{1}{m} J_n\left( \frac{m M \pi}{2} \right) \sin\left( (mN + n) \omega_0 t \right) $$
This expression shows that the fundamental component uf depends only on M and Ud, while the harmonic components uh are influenced by N. As N increases, harmonic frequencies shift higher, making them easier to filter out, thus reducing THD. The THD of the output voltage Uo is defined as:
$$ \text{THD} = \frac{U_{oh}}{U_{of}} $$
where Uoh is the total harmonic voltage amplitude and Uof is the fundamental voltage amplitude. The output voltage phasor Uo relates to UAB through the filter transfer function:
$$ \vec{U_o} = \frac{\vec{U_{AB}}}{1 – \omega^2 L C + j \omega \frac{L}{R}} $$
which implies that THD decreases with increasing N, as higher harmonic frequencies are attenuated by the LC filter. This relationship is crucial for understanding the trade-off between efficiency and waveform quality in various types of solar inverter.
Switching losses in the inverter are a major concern, especially for small off-grid systems where heat dissipation is limited. The total loss Psum in the switching devices includes conduction losses Pcon and switching losses PSW:
$$ P_{\text{sum}} = K (P_{\text{con}} + P_{\text{SW}}) $$
where K is the number of switches. The conduction loss Pcon is given by:
$$ P_{\text{con}} = \frac{1}{T} \int_0^T u(t) i_c(t) \tau(t) \, dt $$
where u(t) is the voltage across the device during conduction, ic(t) is the current, and τ(t) is the duty cycle. The switching loss PSW depends on the switching frequency f_sw = f0 N:
$$ P_{\text{SW}} = 0.5 U_{\text{in}} I_o (t_r + t_f) f_{\text{sw}} = 0.5 U_{\text{in}} I_o (t_r + t_f) f_0 N $$
Here, Uin is the input voltage, Io is the output current, tr and tf are the rise and fall times of the switch, and f0 is the fundamental frequency. This equation shows that PSW is proportional to N, so reducing N lowers switching losses but increases THD. Therefore, an optimal carrier ratio must be found that minimizes losses while keeping THD within acceptable limits, typically below 5% for many applications. This balance is essential in all types of solar inverter, but off-grid systems face additional challenges due to load variations.
To address this, we propose a carrier ratio control method that adjusts N based on real-time THD measurements. The control range for N is determined by analyzing THD variation under different loads. For linear loads, THD is lower, allowing for a smaller minimum N (Nmin), while for non-linear loads, THD is higher, requiring a larger maximum N (Nmax). The control algorithm starts with an initial carrier ratio N0 and continuously monitors THD. If THD exceeds a constraint THDconstraint, N is increased; if THD is below a threshold, N is decreased, provided it remains within [Nmin, Nmax]. A variable-step approach is used to enhance precision: the control range is divided into intervals with different step sizes k1, k2, k3 for fine and coarse adjustments. This method ensures that THD stays close to THDconstraint while minimizing N, thereby optimizing efficiency. The versatility of this approach makes it suitable for various types of solar inverter, especially in dynamic off-grid environments.
Simulations were conducted using PSIM to validate the proposed method. The inverter was modeled with MOSFETs FDA59N30-JSM, and parameters were set as follows:
| Simulation Parameter | Value | Control Parameter | Value |
|---|---|---|---|
| MOSFET Gate Threshold Voltage VGS(th) (V) | 3 | Nmax | 400 |
| Drain-Source On-Resistance RDS(on) (mΩ) | 30 | Nmin | 270 |
| Input Capacitance Ciss (pF) | 3538 | N1 | 300 |
| Output Capacitance Coss (pF) | 657 | N2 | 350 |
| Reverse Transfer Capacitance Crss (pF) | 280 | k1 | 3 |
| Gate Resistance RG (Ω) | 30 | k2 | 5 |
| Gate High-Level Voltage VH (V) | 12 | k3 | 10 |
| Gate Low-Level Voltage VL (V) | 0 | N0 | 350 |
| LC Filter Inductance L (μH) | 200 | THDconstraint | 5% |
| LC Filter Capacitance C (μF) | 2 | THD* | 0.3% |
| DC Input Voltage Ud (V) | 100 | Fundamental Frequency f0 (Hz) | 50 |
| Load Power P (W) | 350 | – | – |
The simulation involved switching between linear and non-linear loads at specific times. With carrier ratio control, N adjusted dynamically from 273 to 370, maintaining THD near 5% while improving efficiency. The results are summarized below:
| Operating Condition | Carrier Ratio N | THD (%) | Inverter Efficiency η (%) |
|---|---|---|---|
| Without carrier control, linear load | 400 | 3.02 | 96.80 |
| Without carrier control, non-linear load | 400 | 4.97 | 96.11 |
| With carrier control, linear load | 273 | 4.85 | 97.50 |
| With carrier control, non-linear load | 370 | 4.99 | 96.22 |
These results demonstrate that carrier ratio control increased efficiency by up to 0.70% and reduced losses by up to 21.88% compared to fixed carrier ratio operation. This highlights the method’s effectiveness in optimizing performance for small off-grid inverters, which are a key category among the types of solar inverter.
Experimental validation was performed on a full-bridge inverter using the same MOSFETs. Parameters and control settings were as follows:
| Experimental Parameter | Value | Control Parameter | Value |
|---|---|---|---|
| DC Input Voltage Ud (V) | 24 | Nmax | 400 |
| Load Power P (W) | 40 | Nmin | 250 |
| Fundamental Frequency f0 (Hz) | 50 | N1 | 300 |
| Gate High-Level Voltage VH (V) | 12 | N2 | 350 |
| Gate Low-Level Voltage VL (V) | 0 | k1 | 3 |
| LC Filter Inductance L (μH) | 200 | k2 | 5 |
| LC Filter Capacitance C (μF) | 5 | k3 | 10 |
| Gate Resistance RG (Ω) | 33 | N0 | 350 |
| – | – | THDconstraint | 30% |
| – | – | THD* | 3% |
Due to ADC limitations in THD measurement, higher THD constraints were used. The results are shown below:
| Operating Condition | Carrier Ratio N | THD (%) | Inverter Efficiency η (%) |
|---|---|---|---|
| Without carrier control, linear load | 400 | 24.7 | 95.44 |
| Without carrier control, non-linear load | 400 | 28.6 | 94.40 |
| With carrier control, linear load | 288 | 27.8 | 96.12 |
| With carrier control, non-linear load | 370 | 29.5 | 94.63 |
With carrier ratio control, efficiency improved by 0.68% for linear loads and 0.23% for non-linear loads, corresponding to loss reductions of 14.9% and 4.1%, respectively. This confirms the practical viability of the method for enhancing the performance of small off-grid inverters, which are a critical subset of the types of solar inverter used in standalone applications.
In conclusion, the carrier ratio control method presented here offers a simple yet effective way to optimize efficiency in small off-grid inverters. By dynamically adjusting the carrier ratio based on THD feedback, switching losses are minimized without compromising waveform quality. This approach is particularly beneficial for the diverse types of solar inverter deployed in remote or variable-load scenarios, as it requires no additional components and is easy to implement. Future work could explore integration with maximum power point tracking (MPPT) in solar systems or adaptation for other inverter topologies. Overall, this strategy contributes to longer component lifespan, reduced cooling needs, and improved energy efficiency, making it a valuable addition to the toolbox for inverter design and optimization.