## 3 Control Strategy of HFL Three-Phase Four-Leg MC
### The Compensation Strategy of the N-Phase Bridge Leg  
According to the decoupling diagram of Figure 3, Figure 4 shows the topology of an ordinary three-phase four-leg inverter after decoupled a HFL three-phase four-leg MC. With respect to the fourth bridge leg of the ordinary three-phase four-leg inverter, the essence of its control is not to obtain the mathematical decoupling through complex coordinate transformation, but to separate the fourth bridge leg from the coupling control of other bridge legs. This paper uses a compensation control scheme. This solution can not only separate the fourth bridge leg control from the coupling with other bridge legs, and the fourth bridge leg structure can also completely compensate for the effects of load imbalance or nonlinearity, which gives full play to the advantages of the four-leg structure to achieve effective control of the inverters. According to the topology of the three-phase four-leg inverter, the terminal voltage of the front three bridge leg is set as follows:  
\[\left\{ \begin{array}{c}u_{\text{a}}=\text{M}_{1}(t)\sin(\omega t-\theta)\\ u_{\text{b}}=\text{M}_{2}(t)\sin(\omega t-2\pi/3-\theta)\\ u_{\text{c}}=\text{M}_{3}(t)\sin(\omega t-4\pi/3-\theta)\end{array} \right. \tag{1}\]  
If control \(u_{\text{a}}\), \(u_{\text{b}}\), \(u_{\text{c}}\), and \(u_{\text{n}}\) to make the voltage output from the inverter three-phase symmetrical, and it can satisfy zero sum of front three bridge leg terminal voltage. Then the influence of load unbalance or non-linearity can be completely eliminated by controlling the voltage \(u_{\text{n}}\) of the fourth bridge leg. According to [PERSON]'s law of voltage and current, the equation for the three-phase four-leg inverter can be obtained:  
\[\left\{ \begin{array}{c}u_{\text{a}}=L\dot{\text{d}}i_{\text{a}}/\dot{\text{ d}}t+u_{\text{AO}}+u_{\text{O}}\\ u_{\text{b}}=L\dot{\text{d}}i_{\text{b}}/\dot{\text{d}}t+u_{\text{BO}}+u_{\text {O}}\\ u_{\text{c}}=L\dot{\text{d}}i_{\text{c}}/\dot{\text{d}}t+u_{\text{CO}}+u_{\text {O}}\\ u_{\text{n}}=u_{\text{O}}-L_{\text{n}}\dot{\text{d}}i_{\text{n}}/\dot{\text{d}}t \end{array} \right. \tag{2}\]  
Figure 4: Ordinary three-phase four-leg inverter. Figure 3: The decoupled topology of HFL three-phase four-leg MC. Then:  
\[u_{\rm a}+u_{\rm b}+u_{\rm c}=L{\rm d}i_{\rm n}/{\rm d}t+(u_{\rm AO}+u_{\rm BO}+u_ {\rm CO})+3(u_{\rm n}+L_{\rm n}{\rm d}i_{\rm n}/{\rm d}t) \tag{3}\]  
In order to realize that the fourth bridge leg voltage can completely compensate the voltage drop caused by inductance. Therefore in the steady state, the voltage of the fourth bridge leg not only compensates the voltage drop of the neutral line inductance, but also compensates the voltage drop of the three-phase inductance, make \(L=L_{\rm n}\). For the given input in Equation (1), when \({\rm M}_{1}\) (\(t\)) = \({\rm M}_{2}\) (\(t\)) = \({\rm M}_{3}\) (\(t\)), we can get \(u_{\rm a}+u_{\rm b}+u_{\rm c}=0\), At this time, the following can be introduced by the Equation (3):  
\[(u_{\rm AO}+u_{\rm BO}+u_{\rm CO})+4L_{\rm n}{\rm d}i_{\rm n}/{\rm d}t+3u_{\rm n }{=~{}0} \tag{4}\]  
If the fourth bridge leg control voltage \(u_{\rm n}\) is designed as:  
\[u_{\rm n}=-\frac{4}{3}L_{\rm n}\frac{{\rm d}i_{\rm n}}{{\rm d}t} \tag{5}\]  
The following can be introduced:  
\[u_{\rm AO}+u_{\rm BO}+u_{\rm CO}=0 \tag{6}\]  
According to Equation (6), the output three-phase voltage symmetry can be obtained. Equation (5) shows the differential calculation of the current \(i_{\rm n}\) contained in the modulated wave of the fourth bridge leg, which is not feasible in practical application system, because in the actual working conditions, the differential calculation will amplify the disturbance caused by the switching action, which will easily cause instability of the system and poor quality of the output electrical parameters in the process of regulation.