#### Specific analysis of the influence of closed-loop control on the modulation wave

The following will take proportional-integral PI regulator, proportional-resonant PR regulator and quasi-resonant QR regulator as examples to analyze the influence of closed-loop control on the modulation wave.

(1) G_{c}(s) is the PI regulator, G_{c}(j0)→∞

Among them, k_{p}>0, k_{i}>0.

The PI regulator controls the AC signal to have a static error, so the current error i_{e}(t)=I_{e}·sin(ω_{s}t+θ_{i})-I_{o(avg)}. At this time, under the action of the PI regulator, the modulation wave V_{m}(t) can be expressed as:

From the self-balancing characteristics of 3L-NPC and formula (1.2), we can see:

① When using the Pl regulator, as time goes by, the modulating wave v_{m} will eventually produce a DC component that is opposite to I_{o(avg)} and will continue to increase;

②When V_{d}>0, that is, U1>U2, then I_{o(avg)}>0, and the final modulation wave v_{m(avg)}<0 and larger, leading to I’_{o(avg)}<0, destroying the self-balancing characteristics of 3L-NPC, Unable to achieve capacitor voltage balance;

③Assuming that the inverter is in the state of capacitor voltage equilibrium when working, that is, V_{d}=0, then I_{o(avg)}=0, and because the Pl regulator has a static difference in the control of the AC quantity, v_{m(avg)}=(k_{i}I_{e}cosθ_{i})/ω_{s} , That is, a certain DC component still exists in the modulated wave v_{m}. The magnitude of the DC component is related to the magnitude of the current error. The DC component of the modulating wave v_{m} will cause the inverter to have unequal working hours in the positive and negative half cycles, which will make the capacitor voltage unable to maintain a balanced state.

(2) G_{c}(s) is the PR regulator, G_{c}(j0) is a finite value

Among them, k_{p}>0, k_{r}>0, ω_{r} is the resonant angular frequency, and α reflects the damping of the resonant controller. In order to obtain a better current tracking effect, usually ω_{r}=ω_{s}.

Since the gain of the PR regulator to the fundamental wave tends to infinity, the fundamental wave component can be tracked without static error, so the current error i_{e}(t)=-I_{o(avg)}. At this time, under the action of the PR regulator, the DC component of the modulating wave v_{m}(t) is:

v_{m(avg)}=-I_{o(avg)}G_{P}_{R} (j0)=-k_{p}I_{o(avg)} (1.4)

From the self-balancing characteristics of 3L-NPC and formula (1.4), we can see:

① When using the PR regulator, the DC component of the modulating wave is opposite to Io(avg), and its magnitude is related to the proportional link kp and;

②When V_{d}>0, that is, U_{1}>U_{2}, then I_{o(avg)}>0, if k_{p} is small, v_{m(avg)}<0 (smaller), and satisfies u_{inv(avg)}>0, there is still I’_{o(avg)}>0, which indicates that the 3L-NPC topology still has the capability of self-balancing capacitor voltage, which can realize capacitor voltage equalization; if k_{p} is too large and v_{m(avg)}<0 (larger), u_{i}_{nv(avg)}<0, then I’_{o(avg)}<0, which will destroy the self-balancing characteristics of 3L-NPC and fail to achieve capacitor voltage equalization;

③Assuming that the inverter is in a balanced state of capacitor voltage when it is working, that is, V_{d}=0, then I_{o(avg)}=0, v_{m(avg)}=0, indicating that the inverter can maintain the balanced state.

(3) G_{c}(s) is QR regulator, G_{c}(j0)=0

Among them, k_{r}>0, in order to obtain a better current tracking effect, usually ω_{r}=ω_{s}.

Since the gain of the QR regulator to the fundamental wave is infinite, it can achieve no static error tracking of the fundamental wave component. In addition, the response of the QR regulator to the DC component is 0, and the DC component of the modulating wave v_{m(avg)}=0, which has no effect on the self-balancing characteristics of the 3L-NPC, and can realize the capacitor voltage balance control.

In order to fully illustrate the effect of closed-loop control on the voltage balance of the 3L-NPC capacitor, the 3L-NPC stand-alone inverter is simulated under the condition of symmetrical circuit parameters and stable closed-loop control system. The waveforms under PI regulator, PR regulator and QR regulator are shown in Figures 1 and 2. It can be seen from Figure 1 that when G_{c}(s) adopts a PR regulator with a larger PI or k_{p}, u_{C}_{1} keeps decreasing and u_{C}_{2} keeps increasing, and the capacitor voltage balance cannot be realized; when the k_{p} of the PR regulator is small, the capacitor voltage balance control can be realized. It can be seen from Figure 2 (a) that when the DC side capacitor has a capacitance deviation and a capacitor voltage difference, the capacitor voltage constantly tends to be balanced. Figure 2 (b) can be seen, because the QR regulator is not easy to adjust the system bandwidth, the dynamic response characteristics of the load current is poor.