## 3 The Proposed PI-Tuned Speed Controller
Figure 3.7  
Here, Equation (14) can be re-written in terms of power by multiplying both sides with instantaneous mechanical speed input \(\omega_{\mathrm{m}}\), as in Equation (16):  
Figure 3.8  
Different strategies have been used to achieve the relevant results [10; 18]. The hysteresis-based current controller (CC) operates by relating a current error variation between the calculated and required 3-phase currents, versus a constant hysteresis band. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Sr. No** & **Parameter Name** & **Impedance Parameters** & **Value** \\ \hline
1 & Stator winding resistance & \(\mathrm{R_{s}}\) & 87 m\(\Omega\) \\ \hline
2 & Stator leakage inductance & \(\mathrm{L_{Is}}\) & 0.8 mH \\ \hline
3 & Excitation inductance & \(\mathrm{L_{m}}\) & 34.7 mH \\ \hline
4 & Rotor resistance & \(\mathrm{R_{r}}\) & 227 m\(\Omega\) \\ \hline
5 & Rotor leakage inductance & \(\mathrm{L_{tr}}\) & 0.8 mH \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Induction machine electrical parameters. An inner hysteresis band is employed to sense the changing moments via digital logic for use in Equation (17). M, in Equation (18), is the angular momentum of the rotor at a synchronous speed. Here, \(\mathrm{P_{m}}\), \(\mathrm{P_{g}}\), and \(\mathrm{P_{D}}\) are the mechanical, electrical, and damping power in MW (megawatts), respectively. In various swing equations, the parameter H is the merged inertia constant of the prime mover-generator-exciter system, expressed in seconds. It is generally accepted to have a time range of (1-10) seconds for all types of machines under test; the calculated value is based upon the three-phase MWA ratings of the IM under examination, as given in Equation (19). \[\frac{\mathrm{M}\cdot\mathrm{d}^{2}\delta_{\mathrm{m}}}{\mathrm{dt}^{2}}= \mathrm{P_{net}}=(\mathrm{P_{m}}\ -\mathrm{P_{g}}-\mathrm{P_{D}}) \tag{17}\]  
\[\mathrm{M}=\mathrm{J}\cdot\omega_{\mathrm{m}} \tag{18}\]  
\[\mathrm{H}=\frac{\mathrm{J}\omega_{\mathrm{s}}^{2}}{2\cdot\mathrm{S_{rated}}} \tag{19}\]  
After substituting M (Equation (18)) into Equation (17), Equation (20) is obtained:  
\[2\mathrm{H}\cdot\left(\frac{\mathrm{S_{rated}}}{\omega_{\mathrm{s}}^{2}} \right)\omega_{\mathrm{m}}\cdot\left(\frac{\mathrm{d}^{2}\delta_{\mathrm{m}}}{ \mathrm{dt}^{2}}\right)=\mathrm{P_{net}}=(\mathrm{P_{m}}-\mathrm{P_{g}}- \mathrm{P_{D}}). \tag{20}\]  
Finally, the present net power \(\mathrm{P_{net}}\) is calculated in Equation (21):  
\[\left(\frac{2\mathrm{H}}{2\cdot\omega_{s}}\right)\cdot\left(\frac{\mathrm{d}^{ 2}\delta_{\mathrm{m}}}{\mathrm{dt}^{2}}\right)=\mathrm{P_{net}}=(\mathrm{P_{m} }-\mathrm{P_{g}}-\mathrm{P_{D}}). \tag{21}\]  
Figure 4: Simulink implementation of the proposed scheme. * Indicates number of multiport switches. The dataset for the proposed model is derived from Figure 5. The input parameter voltage (V\({}_{\text{ab}}\)), phase current (I), torque (T), and output parameter speed (\(\omega_{\text{r}}\)) are recorded for 1 million time samples. The dataset of both the input and output parameters is loaded into a multiple regression algorithm for predictive analysis. Then, the multiple linear regression algorithm predicts the speed from the input parameters of voltage (V\({}_{\text{ab}}\)), phase current (I), and torque (T), respectively. The mathematical equations of the regression algorithm are presented as Equations (22) and (23):  
\[y_{\text{i}}=\sum_{i=1}^{n}\text{f}(x_{\text{i}}\,\ \beta)+\ \varepsilon_{ \text{i}} \tag{22}\]  
\[y_{\text{i}}=\text{f}(x_{\text{i}},\ \beta) \tag{23}\]  
where \(\varepsilon_{\text{i}}\) is the difference between actual and estimated value, \(\text{f}(x_{\text{i}},\ \beta)\) is a polynomial equation, \(x_{\text{i}}\) is the number of independent variables, and \(\beta\) is the coefficient of the polynomial equation.