# Research on the High Reliable Electromagnetic Stirring Power Supply
## 2 Materials and Methods
### Fault-Tolerant Control Strategy
\tag{24}\]  
where \(D_{Rx}\) is the modulating switch signal, and \(x\) = (\(a\), \(b\), \(c\)). Discretize Equation (24) to get:  
\[\left\{ \begin{array}{l}{u_{sa}(k)-{L_{s}}\frac{{{i_{sa}}(k+1)-{i_{sa}}(k)}}{ {{T_{s}}}}={u_{dc}}(k){D_{Ra}}(k)}\\ {u_{sb}(k)-{L_{s}}\frac{{{i_{sb}}(k)-{i_{sb}}(k)}}{{{T_{s}}}}={u_{dc}}(k){D_{ Rb}}(k)}\\ {u_{sc}(k)-{L_{s}}\frac{{{i_{sc}}(k+1)-{i_{sc}}(k)}}{{{T_{s}}}}={u_{dc}}(k){D_ {Rc}}(k)}\end{array} \right. \tag{25}\]  
where, \({{T_{s}}}\) is the switching period of the IGBT. Take the grid reference current as the sampling current value of the (\(k+1\))th switching cycle, namely:  
\[\left\{ \begin{array}{l}{i_{sa}(k+1)={i_{sa}^{*}}}\\ {i_{sb}(k+1)={i_{sb}^{*}}}\\ {i_{sc}(k+1)={i_{sc}^{*}}}\end{array} \right. \tag{26}\]  
According to Equation (26), we can get:  
\[\left\{ \begin{array}{l}{D_{Ra}(k)=\frac{1}{{{u_{dc}}(k)}}\Big{\{}{u_{sa}}(k )-\frac{{{L_{s}}}}{{{T_{s}}}}[{i_{sa}^{*}}(k)-{i_{sa}}(k)]}\Big{\}}\\ {D_{Rb}(k)=\frac{1}{{{u_{dc}}(k)}}\Big{\{}{u_{sb}}(k)-\frac{{{L_{s}}}}{{{T_{s} }}}[{i_{sb}^{*}}(k)-{i_{sb}}(k)]}\Big{\}}\\ {D_{Rc}(k)=\frac{1}{{{u_{dc}}(k)}}\Big{\{}{u_{sc}}(k)-\frac{{{L_{s}}}}{{{T_{s} }}}[{i_{sc}^{*}}(k)-{i_{sc}}(k)]}\Big{\}}}\end{array} \right. \tag{27}\]  
Send the three-phase modulation signal obtained by Equation (27) to the PWM modulation unit, then we can get the output of the switching drive signal of the rectifier, and the IGBT is driven to output the desired voltage and current. In this way, a composite control method of feedforward control and feedback control is formed, which effectively combines the fast response speed of feedforward control and the closed-loop tracking characteristics of feedback, and realizes the stability of the total DC side voltage. And the three-phase input current is sine wave and with high power factor, which provides high-quality DC voltage for the back-stage inverter. In the control strategy of the back-stage inverter, it is necessary to realize the tracking of the given current and the voltage equalization of the DC side capacitances. First, compare the detected voltages on the two DC capacitors with the given DC side voltages, and we can obtain:  
\[\Delta I_{1}=k_{p1}+k_{i1}\int\bigg{(}\frac{u_{dc}^{*}}{2}-u_{dc1}\bigg{)}dt \tag{28}\]  
\[\Delta I_{2}=k_{p2}+k_{i2}\int\bigg{(}\frac{u_{dc}^{*}}{2}-u_{dc2}\bigg{)}dt \tag{29}\]  
\(k_{p1}\) is the proportionality coefficient and \(k_{i1}\) is integral coefficient for the PI1 controller. \(k_{p2}\) is the proportionality coefficient and \(k_{i2}\) is integral coefficient for the PI2 controller. The amplitude of the output current of the electromagnetic stirring power supply specified by the user is \(I^{*}\), and the frequency is \(\omega^{*}\). Therefore, the corrected given current amplitude is:  
\[I_{m}=I^{*}-\Delta I_{1}-\Delta I_{2} \tag{30}\]  
The given current is:  
\[\left\{ \begin{array}{l}{i_{a}}^{*}=I_{m}\sin(\omega^{*}t)\\ \\ {i_{\beta}}^{*}=I_{m}\sin(\omega^{*}t-\frac{\pi}{2})\\ \\ {i_{\omega}}^{*}=-i_{a}^{*}-i_{\beta}^{*}\end{array} \right. \tag{31}\]  
Perform PR control on the difference between the detected output current of the electromagnetic stirrer and the given current to obtain a modulation signal, and perform SPWM modulation on the modulation signal to obtain 6 pulse signals. Finally, after resetting the fault-tolerant pulse, the final pulse signals of the 6 IGETs are obtained. The overall control process of the electromagnetic stirring system is shown in Figure 5.  
The fault-tolerant pulse reset rule in the above figure is shown in Figure 6. The red dashed box in the figure is the pulse reset required for reconstruction after primary failure, corresponding to case 1-21 in Table 2. The blue dashed box is the pulse reset required for reconstruction after secondary failure, corresponding to case 1-8 in Table 4. Take case1 of the first failure as an example: if the \(\alpha\)-phase IGBT V\({}_{1}\) fails, V\({}_{3}\) and V\({}_{4}\) will be reconstructed into the \(\alpha\)-phase, and the two capacitors on the DC side will form the \(w\)-phase.