## 3 Analysis of Control Strategies
### NPC Three-Level Inverter Output Stage Control Strategy
#### 3.2.1 The 120\({}^{\circ}\) SVPWM Modulation Algorithm  
1. Division of sectors  
As shown in Figure 5, for the NPC three-level inverter circuit, the switching function was defined as follows:  
\[S_{Z(Z=A,B,C)}= \begin{cases}P,(S_{X1},S_{X2}\ open,\ X=a,b,c)\\ O,(S_{X2},S_{X3}\ open,\ X=a,b,c)\\ N,(S_{X3},S_{X4}\ open,\ X=a,b,c)\end{cases}  \tag{1}\]  
The basic vectors were composed of the combination of switching functions \(S_{A}S_{B}S_{C}\), and the distribution of three-level space vectors is shown in Figure 6.  
The vector space was divided into a large sector N1, medium sector N2, and small sector N3. When moving counterclockwise from the 0\({}^{\circ}\) to the 120\({}^{\circ}\) region, N1 was 1; from the 120\({}^{\circ}\) to the 240\({}^{\circ}\) region, N1 was 2; and from the 240\({}^{\circ}\) to the 0\({}^{\circ}\) region, N1 was 3. The large sector was divided by its angular bisector into the medium sector, which was subdivided into the small sector. Red arrows indicated virtual small vectors. Blue arrows indicated virtual medium vectors. Green arrows indicated virtual large vectors. 1. Sector determination  
Given the instantaneous values of three-phase voltages \(V_{a}\), \(V_{b}\), and \(V_{c}\), the variables were defined as follows:  
\[\left\{ \begin{array}{l}X=V_{a}-V_{c}\\ Y=V_{b}-V_{a}\\ Z=V_{c}-V_{b}\end{array} \right. \tag{2}\]  
Normalization: after normalization, the length of the parallelogram formed by the first major sector was 4. \[\left\{ \begin{array}{l}x=\frac{X}{U_{d}}\times 4=\frac{4X}{U_{d}}\\ y=\frac{1}{U_{d}}\times 4=\frac{4Y}{U_{d}}\\ z=\frac{Z}{U_{d}}\times 4=\frac{4Z}{U_{d}}\end{array} \right. \tag{3}\]  
When \(x>0\) and \(z\leq 0\), N1 = 1. When \(y>0\) and \(x\leq 0\), N1 = 2. When \(z>0\) and \(y\leq 0\), N1 = 3. Then, the values \(m\) and \(n\) of the projections of the spatial voltage vectors on the 0\({}^{\circ}\) and 120\({}^{\circ}\) axes of the first major sector were calculated. According to the geometrical relations, this could be obtained using (4). \[m=\left\{ \begin{array}{l}x\ N_{1}=1\\ y\ N_{1}=2\\ z\ N_{1}=3\end{array} \right. \tag{4}\]  
Figure 6: Three-level space vector distribution map. When \(m\geq n\), N\({}_{2}\) was equal to 1; otherwise, N\({}_{2}\) was equal to 2 the values of \(m\) and \(n\) were swapped. Overmodulation occurred when the value of \(m\) or \(n\) was greater than 4. Since \(m\) needed to be greater than \(n\) after the previous processing, the following equation could be used to force \(m\) and \(n\) to be reduced to the interior of the hexagon when \(m\) was greater than 4. \[\left\{ \begin{array}{l}n=4\times\frac{n}{m}\\ m=4\end{array} \right. \tag{5}\]  
Next, the judgment of the small sector N\({}_{3}\) was carried out. When the conditions \(2\leq m\)\(<\) 2 and \(n\geq 0\) were satisfied, N\({}_{3}\) was set to 1. The judgment of other small sectors followed a similar procedure. The action time of the basic vector  
The spatial voltage vector was synthesized from the fundamental vector of the small sector in which it was located and only changed the switching state of one phase at the same moment. Table 1 below shows the small sectors where the space vectors were located and their basic vectors used when N\({}_{1}\) = 1 and N\({}_{2}\) = 1. From the distribution of basic vectors, it can be seen that V\({}_{1}\) and V\({}_{4}\) were equivalent to the same vector. The total duration of their action within one modulation cycle was defined as T\({}_{1}\), with the duration of V\({}_{2}\)'s action as T\({}_{2}\) and the duration of V\({}_{3}\)'s action as T\({}_{3}\). Below we show the calculation of the action time of each basic vector when N\({}_{3}\) = 1, and the same procedure was applied to other regions. Both the space voltage vector and the basic vector were decomposed along the 0\({}^{\circ}\) and 120\({}^{\circ}\) axes and equations were formulated based on the \"Volt-second balance\" principle. Then, the solution result was normalized, and the action time of each vector was obtained as shown in (6). \[\left\{