## 4 Voltage Vector Distribution Method
### Accurate Power Following Method  
Because low switching frequency method only uses basic voltage vectors and zero vector, the difference of inverter1's output power between actual and desired value, known as verter1's power following deviation, is not minimized. The accurate power following method is proposed to solve this problem when this deviation of low switching frequency method is beyond the tolerance range. The principle of this method is simple: to generate inverter1's desired voltage vector in phase with \(\overset{\rightarrow}{i_{s}}\), and its amplitude is determined by the desired output power of inverter1, making inverter1's power following deviation fully eliminated. The collinearity of \(\overset{\rightarrow}{u_{s1}}\) and \(\overset{\rightarrow}{i_{s}}\) could also make inverter1 free of bearing any reactive power.  
The specific algorithm will be introduced as follow. First, the amplitude of \(\overset{\rightarrow}{i_{s}}\) is obtained using its D-axis and Q-axis components by Equation (21):  
\[\overset{\rightarrow}{|\overset{\rightarrow}{i_{s}}|}=\sqrt{i_{D}^{2}+i_{Q}^ {2}} \tag{21}\]  
Then the amplitude of inverter1's desired voltage vector can be determined as follow, which makes the corresponding output power of inverter1 equal to the desired value \(P_{inv1}^{*}\).  
\[|\overset{\rightarrow}{u_{s1}^{*}}|=\frac{P_{inv1}^{*}}{|\overset{\rightarrow }{i_{s}}|} \tag{22}\]  
Therefore, \(\overset{\rightarrow}{u_{s1}^{*}}\) can be obtained by Equation (23) to be in phase with \(\overset{\rightarrow}{i_{s}}\):  
\[\overset{\rightarrow}{u_{s1}^{*}}=\frac{\overset{\rightarrow}{i_{s}}}{| \overset{\rightarrow}{i_{s}}|}|\overset{\rightarrow}{u_{s1}^{*}}| \tag{23}\]  
Afterwards the voltage vector over range judgmental algorithm is executed to check whether the current calculated \(\overset{\rightarrow}{u_{s1}^{*}}\) is within the modulation range of inverter1. If the result is positive, the current calculated \(\overset{\rightarrow}{u_{s1}^{*}}\) will be assigned to inverter1's desired voltage vector \(u_{s1\_AF}^{*}\), where the subscript AF means Accurate Following. Otherwise, the current calculated \(\overset{\rightarrow}{u_{s1}^{*}}\) need to be shortened to the boundary of inverter1's modulation range by the following procedure.  
We define the proportions of adjacent basic vectors for synthesizing the current calculated \(\overset{\rightarrow}{u_{s1}^{*}}\) as \(a_{1}^{\prime}\) and \(b_{1}^{\prime}\), which are the results from the voltage vector over range judgmental algorithm, and we  
Figure 6: Voltage vector distribution examples of low switching frequency method. (**a**) First situation; (**b**) second situation.  
have \(a_{1}^{\prime}+b_{1}^{\prime}>1\) so the current calculated \(\overset{\rightharpoonup}{u_{s1}^{*}}\) is beyond the modulation range. To make \(a_{1}+b_{1}=1\), we have:  
\[\left\{\begin{array}{l}a_{1}=\frac{a_{1}^{\prime}}{a_{1}^{\prime}+b_{1}^{ \prime}}\\ b_{1}=\frac{b_{1}^{\prime}}{a_{1}^{\prime}+b_{1}^{\prime}}\end{array}\right. \tag{24}\]  
where \(a_{1}\) and \(b_{1}\) are the proportions of adjacent basic vectors for the shortened \(\overset{\rightharpoonup}{u_{s1}^{*}}\), which is on the boundary of inverter1's modulation range. Then the amplitude of this shortened \(\overset{\rightharpoonup}{u_{s1}^{*}}\) can be obtained by Equation (25):  
\[|u_{s1}^{*}|=\sqrt{\frac{2}{3}}U_{dc1}\sqrt{1-a_{1}b_{1}} \tag{25}\]  
Afterwards, the shortened \(\overset{\rightharpoonup}{u_{s1}^{*}}\) can be obtained by using Equation (23) again. Then the shortened \(\overset{\rightharpoonup}{u_{s1}^{*}}\) can be assigned to \(\overset{\rightharpoonup}{u_{s1}^{*}\_{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{ \_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{\_}{ \_}{\_}{\_}{