## 4 Voltage Vector Distribution Method
### Linear Partition Method  
From Equation (4) we know, when the output voltage vectors of two inverters are 180\({}^{\circ}\) out of phase, meaning \(\overset{\rightarrow}{u_{s1}}\) and \(\overset{\rightarrow}{u_{s2}}\) are collinear but in the opposite direction, the synthesized \(\overset{\rightarrow}{u_{s}}\) has the maximum amplitude. Thus, if we assign \(\overset{\rightarrow}{u_{s1}}\) and \(\overset{\rightarrow}{u_{s2}}\) by linearly partition \(\overset{\rightarrow}{u_{s}}\) in the proper position, the possibility of \(\overset{\rightarrow}{u_{s1}}\) and \(\overset{\rightarrow}{u_{s2}}\) in the feasible region can be maximized. This is the strategy of the linear partition method. If both the two methods above have failed to distribute the voltage vectors efficiently, the linear partition method is the last chance. Even if it fails to generate the desired stator voltage vector \(\overset{\rightarrow}{u_{s}}\) integrally with the limitation of two inverters' DC-bus voltages, in which situation the two hexagons centered on \(O_{1}\) and \(O_{2}\) have no overlapping region, the linear partition method can provide a \(\overset{\rightarrow}{u_{s}}\) having the same direction and the nearest amplitude with \(\overset{\rightarrow}{u_{s}}\). The specific algorithm will be introduced as follow. First, the amplitude of \(\overset{\rightarrow}{u_{s}}\) is calculated using its D-axis and Q-axis components as follow:  
\[|\overset{\rightarrow}{u_{s}^{*}}|=\sqrt{u_{D}^{*2}+u_{Q}^{*2}} \tag{27}\]  
Then the desired amplitude of \(\overset{\rightarrow}{u_{s1}^{*}}\) can be obtained by Equation (28), making the corresponding output power of inverter1 follows \(P_{inv1}^{*}\) accurately. \[|\overset{\rightarrow}{u_{s1}^{*}}|=\frac{P_{inv1}^{*}}{u_{s}^{*}}|\overset{ \rightarrow}{i_{s}}| \tag{28}\]  
Similar to Equation (23), \(\overset{\rightarrow}{u_{s1}^{*}}\) can be obtained as follow to be in phase with \(\overset{\rightarrow}{u_{s}^{*}}\):  
\[\overset{\rightarrow}{u_{s1}^{*}}=\frac{\overset{\rightarrow}{u_{s}^{*}}}{| \overset{\rightarrow}{u_{s}^{*}}|}|\overset{\rightarrow}{u_{s1}^{*}}| \tag{29}\]  
Then 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 answer is negative, the current calculated \(\overset{\rightarrow}{u_{s1}^{*}}\) need to be shortened to the boundary of inverter1's modulation range by the procedure introduced in sector 4.3, that is using Equations (24), (25) and (29) in turn. Afterwards, obtain \(\overset{\rightarrow}{u_{s2}^{*}}\) by \(\overset{\rightarrow}{u_{s2}^{*}}=\overset{\rightarrow}{u_{s1}^{*}}- \overset{\rightarrow}{u_{s}^{*}}\), which is modified from Equation (4). Then check whether \(\overset{\rightarrow}{u_{s2}^{*}}\) is within the modulation range of inverter2. If positive, the latest calculated \(\overset{\rightarrow}{u_{s1}^{*}}\) will be assigned to inverter1's desired voltage vector \(\overset{\rightarrow}{u_{s1\_LP}^{*}}\), where the subscript LP means Linear Partition. Otherwise, the current calculated \(\overset{\rightarrow}{u_{s2}^{*}}\) need to be shortened to the boundary of inverter2's modulation range by the same way with \(\overset{\rightarrow}{u_{s1}^{*}}\). After \(\overset{\rightarrow}{u_{s2}^{*}}\) is shortened, we need to recalculate the corresponding \(\overset{\rightarrow}{u_{s1}^{*}}\) by \(\overset{\rightarrow}{u_{s1}^{*}}=\overset{\rightarrow}{u_{s}^{*}}+ \overset{\rightarrow}{u_{s2}^{*}}\), and check whether it is within the modulation range again. If not, shorten it to the boundary of its modulation range. Finally the recalculated \(\overset{\rightarrow}{u_{s1}^{*}}\) can be assigned to \(\overset{\rightarrow}{u_{s1\_LP}^{*}}\). Similarly, the difference between inverter1's output power corresponding to \(\overset{\rightarrow}{u_{s1\_LP}^{*}}\) and the desired value \(P_{inv1}^{*}\), denoted by \(D_{Pinv1\_LP}\), needs to be calculated for voltage vector distribution method selection strategy.