4.1. Main Control Strategies of the Inverter Module
Figure 6 displays voltage simulation waveforms under single-polar carrier-based modulation for PI control, PID control, PR control, and QPR control, facilitating a comparative analysis of the waveform tracking performance across these four control methods. In the waveform plot under PI control, a certain phase difference existed between the grid voltage and the reference voltage, resulting in less-than-ideal tracking performance. PID control builds upon PI control by introducing feedforward control, reducing the burden of sinusoidal regulation by the PI controller. This enhanced the control of the sinusoidal waveform output. Although the tracking performance was improved compared to PI control, achieving zero steady-state tracking error remained unattained. Under PR control, nearly zero steady-state error regulation was achieved at the zero crossing point of the voltage. However, it was evident at the waveform resonance points that the difference between the output voltage and the reference voltage was still significant. In the waveform graph under QPR control, the phase difference between the output voltage and the reference voltage was essentially eliminated, and the waveforms closely matched. This achieved precise tracking without static errors, making it the optimal waveform tracking performance.
We used the Root Mean Square Error (
RMSE) to measure the similarity between the output waveform and the reference waveform, thereby comparing the error between the output voltage waveform and the reference voltage waveform under different controls [
48,
49]. The definition of
RMSE was given by the equation:
Here, y0i and y1i represent the respective sampled values of the output voltage and the reference voltage waveforms, and z represents the total number of samples in the sinusoidal waveform. A smaller RMSE indicates a higher similarity between the output voltage waveform and the reference voltage waveform, resulting in better waveform tracking performance and improved waveform quality.
Upon calculation, the waveform zero crossing points, peak points, and overall
RMSE values under the four control modes are presented in
Table 6.
Based on the data presented in
Table 6, it is evident that of the four control modes, QPR control demonstrated the most minimal steady-state error in the output voltage, signifying superior control performance.
The waveform distortion rate [
50] was introduced to measure the magnitude of distortion occurring in the output voltage waveform. The total harmonic distortion rate (
THD) of a voltage is the percentage of the square root of the sum of the squared RMS values of all the harmonics in that voltage waveform, excluding the fundamental, to the
RMS value of the fundamental voltage of that waveform, i.e.:
where:
V1 is the fundamental voltage component;
V2,
V3,
V4,
V5, ……
VN are integer multiples of the harmonic voltage components.
Utilizing Fast Fourier Transform (FFT) analysis [
51,
52],
Figure 7 depicts the histogram of the distortion rate for each harmonic of the actual output voltage under unipolar octave modulation for the four control methods: PI control, PID control, PR control, and QPR control. It can be seen that the distortion rate of the second harmonic under PR control was large, but as the number of harmonics increased, the waveform distortion rate tended to decrease and the control effect gradually improved. At high harmonics, the harmonic distortion rates under PI and PID control were significantly higher than those under PR and QPR control.
Table 7 presents the total harmonic distortion rates for each control method. During system operation disturbances, the output voltage under PI control exhibited a total harmonic distortion rate of 0.71%, indicating a high level of harmonic content. Under PID control, the total harmonic distortion rate was reduced to 0.66%, a decrease of 0.05% compared to PI control, but still with significant harmonic content. PR control achieved a significantly lower total harmonic distortion rate of only 0.39%. Similarly, QPR control yielded a total harmonic distortion rate of 0.37%, approaching the performance of PR control and ensuring better voltage quality.
Through adjustments to operating conditions, involving variations in the DC-side voltage (
Udc) and inverter switching frequency, a comparative assessment of the four control methods was conducted. Initially, with the inverter switching frequency held constant, modifications were made to the magnitude of the DC-side voltage (
Udc). The resulting variations in the
THD of the output voltage are illustrated in
Figure 8.
As can be seen from
Figure 8, the total harmonic distortion rate of the output voltage under PR control and QPR control was smaller than that under PI control and PID control. As the voltage on the DC side increased, the total harmonic distortion rate of the output voltage decreased under PI control and PID control, resulting in improved control effectiveness. On the other hand, the performance of PR control and QPR control remained relatively consistent, but still better than that of PI control and PID control. Consequently, as the voltage on the DC side increased, the control effectiveness can be ranked as follows: QPR control > PR control > PID control > PI control.
Furthermore, keeping the input voltage constant and varying the inverter switching frequency ƒ, the change in the total harmonic distortion rate of the output voltage was obtained, as shown in
Figure 9.
Modifying the magnitude of the inverter switching frequency caused a more pronounced impact on the total harmonic distortion rate of the output voltage. At lower frequencies, the total harmonic distortion rate of the output voltage under PR control and QPR control was lower than that under PI control and PID control. At higher frequencies, the total harmonic distortion rate of the output voltage under PR and QPR control was higher than that under PI and PID control. It is apparent that the magnitude of the inverter switching frequency had a significant impact on the effectiveness of the four control methods. In high-frequency scenarios, the total harmonic distortion rate of the output voltage under QPR control was minimized, measuring below 4.2%. Conversely, in low-frequency situations, the total harmonic distortion rate under PI control was also minimized, falling below 0.47%. Compared to previous research [
29,
30,
31], the dual closed-loop control strategy with a voltage outer loop and a current inner loop used in this paper resulted in a lower distortion rate in the output voltage waveform, and it strictly conformed to the IEEE 519 standard [
53] under all operating conditions.
At lower frequencies, PR control and QPR control exhibited superior performance compared to PI control and PID control.
Figure 10 illustrates the Bode diagram for PR control and QPR control, different color lines represent the amplitude gain and phase angle of the two control methods at their respective frequencies. As the switching frequency increased, the gain of both PR and QPR control gradually decreased, resulting in a narrower bandwidth. Consequently, the control effectiveness diminished, and the stability of the system was compromised. Although PR control provided high gain at the resonant point, its bandwidth was narrower, resulting in less effective control compared to QPR control. Therefore, at lower frequencies, QPR control yields the best performance.
When operating at high frequencies, PI control and PID control outperformed PR control and QPR control. PID control, in contrast to PI control, introduced a phase lead compensation. When dealing with high-frequency situations characterized by rapid responses, system overshooting could occur, and the integrator could accumulate errors to accommodate this overshooting. Enhancing the integral action appropriately could lead to improved control performance. However, incorporating a differential component may result in high-frequency oscillations that degrade control performance [
54]. Consequently, when the switching frequency is high, PI control delivered optimal results.
4.2. Influence of the Control Method
Under the QPR control method, the total harmonic distortion rate was 0.37% after Fast Fourier Transform analysis, while the total harmonic distortion rate under the bipolar modulation method was 0.53%, 0.15% higher than that of the unipolar frequency doubling. This indicated that unipolar frequency doubling modulation had a better modulation effect and the modulated waveform was closer to the ideal sine.
Figure 11 shows the variation in the total harmonic distortion rate of the output voltage when modifying the input voltage
Udc and the inverter switching frequency
ƒ on the DC side.
Figure 11 reveals that as the voltage increased, the distortion in the waveform became more pronounced under bipolar modulation. In contrast, the
THD of the output waveform under unipolar double-frequency modulation was relatively stable, with excellent modulation performance, consistently remaining below 0.47%. As the switching frequency increased, although both bipolar modulation and unipolar double-frequency modulation exhibited degraded performance, the
THD of the output voltage under unipolar double-frequency modulation consistently remained below 4.2%, achieving a high level of modulation performance. Therefore, regardless of the operating conditions, unipolar double-frequency modulation consistently yielded the best results.