# Research on the High Reliable Electromagnetic Stirring Power Supply
## 2 Materials and Methods
### Fault-Tolerant Control Strategy  
The control goal of the front-stage rectifier is to achieve the stability of the total DC side voltage of the two series DC capacitors. Due to the existence of the common arm in the back-stage inverter, the change of output power leads to the change of the instantaneous power of the DC side capacitor all the time. DC side voltage is not a constant value, but there are certain fluctuations that need to be suppressed. The goal of the back-stage inverter control is to achieve equal voltage on the two DC sides and make the inverter output current equal to the current reference value. And when the inverter is converted from the normal working state to the fault-tolerant working state, there is no need to change the control method. For the front-stage rectifier, the grid voltage is:  
\[\left\{ \begin{array}{l}u_{sa}=U_{sm}\sin(\omega_{0}t)\\ u_{sb}=U_{sm}\sin(\omega_{0}t-\frac{2\pi}{3})\\ u_{sc}=U_{sm}\sin(\omega_{0}t+\frac{2\pi}{3})\\ \end{array} \right. \tag{17}\]  
where \(\omega_{0}\) is an angular frequency with a frequency of 50 Hz. The measured grid current is \(i_{sa}\), \(i_{sb}\) and \(i_{sc}\). Then the apparent power \(P_{s}\) on the grid side is:  
\[P_{s}=u_{sa}i_{sa}+u_{sb}i_{sb}+u_{sc}i_{sc} \tag{18}\]  
The apparent power \(P_{s}\) on the grid side can also be expressed as:  
\[P_{s}=\frac{3U_{sm}}{2}I_{sm} \tag{19}\]  
where \(I_{sm}\) is the magnitude of the grid current, then \(I_{sm}\) is:  
\[I_{sm}=\frac{2P_{s}}{3U_{sm}} \tag{20}\]  
As a load, the operating condition of the electromagnetic stirrer is forward stirring-stop-reverse stirring. Its power is constantly changing, which causes voltage fluctuations on the DC side. Use the feedforward PI control to control the fluctuation of the DC side voltage, and we can get:  
\[\Delta I_{dc}=k_{p}(u_{dc}{}^{*}-u_{dc})+k_{i}\int_{T_{S}}(u_{dc}{}^{*}-u_{dc })dt \tag{21}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Case** & **Primary Fault** & **Connection** & **Secondary Fault** & **Secondary** \\  & & & & **Construction** \\ \hline
1 & V\({}_{1}\) & TR\({}_{\text{w}}\) \& TR\({}_{\text{13}}\) & V\({}_{4}\) & S\({}_{\text{a}}\) (V\({}_{3}\), V\({}_{2}\)) \\
2 & V\({}_{2}\) & TR\({}_{\text{w}}\) \& TR\({}_{\text{13}}\) & V\({}_{3}\) & S\({}_{\text{a}}\) (V\({}_{1}\), V\({}_{4}\)) \\
3 & V\({}_{3}\) & TR\({}_{\text{w}}\) \& TR\({}_{\text{13}}\) & V\({}_{2}\) & S\({}_{\text{a}}\) (V\({}_{1}\), V\({}_{4}\)) \\
4 & V\({}_{3}\) & TR\({}_{\text{w}}\) \& TR\({}_{\text{85}}\) & V\({}_{6}\) & S\({}_{\text{a}}\) (V\({}_{5}\), V\({}_{4}\)) \\
5 & V\({}_{4}\) & TR\({}_{\text{w}}\) \& TR\({}_{\text{13}}\) & V\({}_{1}\) & S\({}_{\text{a}}\) (V\({}_{3}\), V\({}_{2}\)) \\
6 & V\({}_{4}\) & TR\({}_{\text{w}}\) \& TR\({}_{\text{35}}\) & V\({}_{5}\) & S\({}_{\beta}\) (V\({}_{3}\), V\({}_{6}\)) \\
7 & V\({}_{5}\) & TR\({}_{\text{w}}\) \& TR\({}_{\text{35}}\) & V\({}_{4}\) & S\({}_{\beta}\) (V\({}_{3}\), V\({}_{6}\)) \\
8 & V\({}_{6}\) & TR\({}_{\text{w}}\) \& TR\({}_{\text{35}}\) & V\({}_{3}\) & S\({}_{\beta}\) (V\({}_{5}\), V\({}_{4}\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Secondary reconfigurable inverter switching status. where \(k_{p}\) is the proportionality coefficient and \(k_{i}\) is integral coefficient for the PI control. Then the magnitude of the grid reference current is:  
\[{I_{sm}}^{*}={I_{sm}}+\Delta{I_{dc}} \tag{22}\]  
The reference current of the grid is:  
\[\left\{ \begin{array}{l}{i_{sa}^{*}={I_{sm}^{*}}\sin(\omega t)}\\ {i_{sb}^{*}={I_{sm}^{*}}\sin(\omega t-\frac{2\pi}{3})}\\ {i_{sc}^{*}={I_{sm}^{*}}\sin(\omega t+\frac{2\pi}{3})}\end{array} \right. \tag{23}\]  
According to Figure 1, the output voltage of the front-stage rectifier can be expressed as:  
\[\left\{ \begin{array}{l}{u_{Ra}={u_{sa}}-{L_{s}}\frac{{{d_{is}}}}{{dt}}={u_{ dc}}{D_{Ra}}}\\ {u_{Rb}={u_{sb}}-{L_{s}}\frac{{{d_{is}}}}{{dt}}={u_{dc}}{D_{Rb}}}\\ {u_{Rc}={u_{sc}}-{L_{s}}\frac{{{d_{is}}}}{{dt}}={u_{dc}}{D_{Rc}}}\end{array} \right.