#### Overview of the influence of closed-loop control on capacitor voltage balance

Previously, we analyzed the self-balancing characteristics of the 3L-NPC topology from the perspective of time domain and frequency domain under open-loop operation, that is, the modulated wave is symmetrical. In practical applications, the inverter must adopt closed-loop control to control the load current i_{o} or the load voltage u_{o}. And in order to improve the stability of the system and attenuate high-frequency harmonics such as the switching frequency, usually the cut-off frequency of the system is much lower than the switching frequency. Compared with the i_{inv}(t) before the filter, the even harmonic components of the switching frequency in i_{o} and u_{o} are greatly attenuated, while the DC component is basically unchanged. Therefore, the closed-loop control basically has no effect on the self-balancing current from the even harmonic component near the switching frequency of i_{inv}(t), but has an effect on the self-balancing current from the DC component of i_{inv}(t). According to Figure 1, when Z_{eq} is a capacitive load, the 3L-NPC topology still has self-balancing characteristics under closed-loop control, but under other load conditions, closed-loop control may even destroy the capacitor voltage self-balance of 3L-NPC.

Take the single-current closed-loop control of a typical grid-connected inverter as an example, and its control block diagram is shown in Figure 2. Among them, G_{c}(s) is the regulator, K_{PW}_{M} is the inverter equivalent gain, G_{filter}(s)=u_{o}(s)/u_{inv}(s), i_{ref} is the reference current, i_{e} is the current error, and v_{m} is the modulation wave.

When the capacitor voltage is unbalanced, that is, V_{d}≠0, the load current. there is a corresponding DC component l_{o(avg)} in the load current l_{o}. At this time, record the current error i_{e}(t)=I_{e}·sin(ω_{s}t+θ_{i})-l_{o(avg)}, Among them, I_{e} and θ_{i} respectively represent the amplitude error and phase difference between the load current and the current reference, and ω_{s} is the angular frequency of the current reference signal. Take the capacitor voltage U_{1}>U_{2} as an example, Figure 3 shows the waveform of the inverter bridge arm output voltage u_{inv} when the modulation wave V_{m}(t) obtained after the current error i_{e}(t) is acted on by the regulator G_{c}(s) contains different DC components V_{m(avg)}.

It can be seen from Figure 3 that when U_{1}>U_{2,} l_{o(avg)}>0, after the closed-loop control function,

If v_{m(avg)}=0 or v_{m(avg)}>0 or v_{m(avg)}<0 (smaller), then u_{inv(avg)}>0, there is still I’_{o(avg)}>0. This shows that the 3L-NPC topology still has the ability to self-balance the capacitor voltage; if v_{m(avg) }<0 (larger), then u_{inv(avg)}<0, I’_{o(avg)}<0. This will destroy the self-balancing characteristics of the 3L-NPC topology and cannot achieve capacitor voltage balance.

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