## 4 Voltage Vector Distribution Method
### Low Switching Frequency Method  
In a SVPWM control period, there are two switch commutations of each inverter leg normally. Thus, there are six switch commutations of all three legs in a single inverter. When using the dual inverter configuration, the number of switch commutations is doubled. This can cause considerable inverter switching loss and lower the inverter efficiency. However, if one inverter is forced to output only basic voltage vectors or zero vector, there is no switch commutation during the modulation period because the desired voltage vector does not need to be synthesized by two adjacent basic vectors. Thus, the switching frequency of this clamped inverter can be lower to zero if the switch commutations at the junction of modulation periods are ignored. By clamping inverter1 to output only basic voltage vectors and zero vector, the low switching frequency method can reduce the total inverter switching frequency by nearly a half, thus the inverter switching loss is significantly reduced. The desired voltage vector of inverter \(\vec{u_{s1}^{*}}\) will be chosen among zero vector and basic voltage vectors lying in the feasible region of voltage vector distribution mentioned in Section 2.2, the one makes inverter1's output power closest to the desired value \(P_{impl}^{*}\) will be selected. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**N** & **0** & **1** & **2** & **3** & **4** & **5** & **6** \\ \hline a & 0 & z & y & \(-\)z & \(-\)x & x & \(-\)y \\ \hline b & 0 & y & \(-\)x & x & z & \(-\)y & \(-\)z \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Relation table for a, b and N.  
The specific algorithm will be introduced as follow. First, matrix \(M_{us1}\) with a size of 4 \(\times\) 7 is introduced. Each row of \(M_{us1}\) represents a candidate of voltage vectors, including six basic voltage vectors and one zero vector. The first two lines of \(M_{us1}\) are assigned as follow:  
\[\left\{ \begin{array}{l}M_{us1}(1,:)=\sqrt{\frac{2}{3}}U_{dc1}\circ[0,1,\frac{ 1}{2},-\frac{1}{2},-1,-\frac{1}{2},\frac{1}{2}]\\ M_{us1}(2,:)=\sqrt{\frac{2}{3}}U_{dc1}\circ[0,0,\frac{\sqrt{3}}{2},\frac{ \sqrt{3}}{2},0,-\frac{\sqrt{3}}{2},-\frac{\sqrt{3}}{2}]\end{array} \right. \tag{18}\]  
where \(M_{us1}(n,:)\) means the nth line of matrix \(M_{us1}\), and \"\(\circ\)\" is the operator of Hadamard product, meaning the multiplication of each corresponding element of the two participant matrixes. \(M_{us1}(1,:)\) indicates the D-axis components of each candidate vector, while \(M_{us1}(2,:)\) indicates the Q-axis components. Then the third line of \(M_{us1}\) can be calculated, which indicates inverter1's output power of the corresponding candidate vector:  
\[M_{us1}(3,:)=M_{us1}([1,2],:)^{T}\times[i_{D},i_{Q}]^{T} \tag{19}\]  
where \(i_{D}\) and \(i_{Q}\) are D-axis component and Q-axis component of \(\overset{\rightarrow}{i_{s}}\) respectively. The difference between inverter1's output power of each candidate vector and the desired value \(P^{*}_{inv1}\) can be obtained by Equation (20), assigned to the fourth line of \(M_{us1}\):  
\[M_{us1}(4,:)=|M_{us1}(3,:)-P^{*}_{dc1}| \tag{20}\]  
Then these power difference values of inverter1 are sorted in ascending sequence. Afterwards the corresponding candidate vectors are checked respectively in the sorted sequence to verify whether it is in the feasible region until we get a positive result. The specific procedure is introduced as follow. First, obtain inverter2's voltage vector of the corresponding candidate vector of inverter1 by \(\overset{\rightarrow}{u^{*}_{s2}}=\overset{\rightarrow}{u^{*}_{s1}}- \overset{\rightarrow}{u^{*}_{s}}\), which is modified from Equation (4). Then whether this voltage vector is within the modulation range of inverter2 or not is checked by voltage vector over range judgmental algorithm introduced in sector 4.1. If the result is positive, the current candidate vector of inverter1 is valid. Otherwise the next candidate vector in the sorted sequence will be checked until the voltage vector distribution is confirmed valid.