 #### Resonance phenomenon of filter grid-connected inverter

The three-phase inverter can be equivalent to two single-phase inverters with a coupling relationship after coordinate transformation. The research on the resonance problem can be carried out based on the single-phase grid-connected inverter model. In order to facilitate the subsequent description, Figure 1 shows a single-phase grid-connected inverter using an LCL filter. The LCL filter consists of an inverter-side inductor L1, a filter capacitor C1, and a grid-side inductor L2, and Udc is the DC side. voltage, uinv is the output voltage at the midpoint of the bridge arm of the inverter, and ug is the grid voltage. For the convenience of subsequent description, the voltage and current of each filter element have been marked in the figure. Different from the current control under the single L filter grid-connected inverter, the inverter-side inductor current iL1 and the grid-connected current ig in the LCL filter grid-connected inverter can be controlled.

Ignoring the filter element and the parasitic resistance of the line itself, the transfer functions of the inverter bridge arm output voltage uinv to the inverter side inductor current iL1 and the grid current ig are respectively

The amplitude and phase characteristics of uinv to two currents are shown in Figure 5.20, where L1=0.6mH, L2=0.36mH, and C1=7μF (the same below unless otherwise specified).
The two have higher amplitude gain at the resonant frequency [fres, see equation (5-26)].

Single-input current closed-loop control diagram

Conventional current regulators such as PI and PR do not have the s2 term, which means that there is a lack of a coefficient term in the closed-loop transfer function. According to the Rouse criterion, the closed-loop system is unstable. It should be pointed out that similar conclusions can be obtained using the logarithmic frequency stability criterion . First, the open-loop transfer function shown in Equation 5-27 has no positive real poles. It can be seen from the solid line in Figure 5.11 that the phase of the incoming current crosses -180° once near the resonant frequency. In order to ensure stability, the current regulator must ensure that the amplitude gain at the resonant peak is less than 0dB to satisfy the logarithmic frequency stability criterion. (that is, when the amplitude gain is greater than 0dB, the phase-frequency curve has no intersection at -180°); however, in order to ensure that the resonance peak is suppressed below 0dB, the open-loop cut-off frequency is severely limited, which is not conducive to improving the dynamic response of the system and suppressing current harmonics .

Taking into account the control delay caused by digital control, the frequency of the incoming current phase crossing -180° must be less than the resonant frequency. This single closed-loop control may achieve system stability, but the conditions are harsh. As shown in Figure 3, when the current regulator parameter is small (but not so small that the gain at the resonant frequency is less than 0dB), the amplitude at the -180° crossing frequency is less than 0dB, and the closed-loop control is stable; When it is larger, there is still a serious resonance phenomenon in the grid current. In addition, when the parameters are perturbed, the current control may still have instability problems.