 #### 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) Gc(s) is the PI regulator, Gc(j0)→∞

Among them, kp>0, ki>0.

The PI regulator controls the AC signal to have a static error, so the current error ie(t)=Ie·sin(ωst+θi)-Io(avg). At this time, under the action of the PI regulator, the modulation wave Vm(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 vm will eventually produce a DC component that is opposite to Io(avg) and will continue to increase;

②When Vd>0, that is, U1>U2, then Io(avg)>0, and the final modulation wave vm(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, Vd=0, then Io(avg)=0, and because the Pl regulator has a static difference in the control of the AC quantity, vm(avg)=(kiIecosθi)/ωs , That is, a certain DC component still exists in the modulated wave vm. The magnitude of the DC component is related to the magnitude of the current error. The DC component of the modulating wave vm 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) Gc(s) is the PR regulator, Gc(j0) is a finite value

Among them, kp>0, kr>0, ωr is the resonant angular frequency, and α reflects the damping of the resonant controller. In order to obtain a better current tracking effect, usually ωrs.

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 ie(t)=-Io(avg). At this time, under the action of the PR regulator, the DC component of the modulating wave vm(t) is:

vm(avg)=-Io(avg)GPR (j0)=-kpIo(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 Vd>0, that is, U1>U2, then Io(avg)>0, if kp is small, vm(avg)<0 (smaller), and satisfies uinv(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 kp is too large and vm(avg)<0 (larger), uinv(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, Vd=0, then Io(avg)=0, vm(avg)=0, indicating that the inverter can maintain the balanced state.

(3) Gc(s) is QR regulator, Gc(j0)=0