# Online Efficiency Optimization and Speed Sensorless Control of Single-Phase Induction Motors
## 6 Simulation Results
\begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{2}{c}{**Parameter**} & **Value** \\ \hline \multirow{2}{*}{PI frequency controller} & Proportional Coefficient, \(K_{p}\) & \(9\times 10^{-4}\) \\ \cline{2-3}  & Integrator Coefficient, \(K_{i}\) & 5 \\ \hline \multirow{2}{*}{PI voltage controller} & Proportional Coefficient, \(K_{p}\) & 0.108 \\ \cline{2-3}  & Integrator Coefficient, \(K_{i}\) & 0.862 \\ \hline \multirow{4}{*}{Gradient descent algorithm} & Time Delay & 5 s \\ \cline{2-3}  & convergence criteria (\(\varepsilon\)) & 0.1 \\ \cline{2-3}  & convergence coefficient (\(\lambda\)) & 0.01 \\ \cline{2-3}  & Initial value of corrective & \\ \cline{2-3}  & factor variation (\(\Delta K\)) & 0.05 \\ \hline Neural network & Sample Time & 0.01 s \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Controller parameters. Figure 5: Variations of \(K_{\text{s}}\) versus frequency. At \(t=15\) s, the reference speed is reduced to 1000 rpm. In order to track the new speed set point, the frequency is decreased to 37 Hz (see Figure 6b). Furthermore, as shown in Figure 6a,d, the STCR is changed to the new optimum value (2.5) by reducing the voltage magnitude to 0.41 pu. As shown in Figure 6e-g, as the STCR increases at \(t=15\) s, the main winding current slightly increases whereas the auxiliary winding current experiences a significant drop. Since the mechanical load is a fan, its torque is proportional to the square of speed. As such, when the speed drops from 1440 rpm to 1000 rpm, the load torque  
Figure 6: Simulation results with step change in speed set point. (**a**) rms voltage, (**b**) frequency, (**c**) voltage waveform, (**d**) stator current ratio, (**e**–g**) stator main and auxiliary winding currents, (**h**) motor and load torque, (**i**) motor speed and (**j**) speed estimation error. halves, as shown in Figure 6h. Furthermore, the electromagnetic torque dips below the load torque during transients to provide deceleration but then settles at the same value. The efficiency of the proposed drive is compared with a constant-v/f based drive in Figure 7. It can be seen that the proposed drive makes an improvement of 18% in the motor efficiency at a 50% load. Comparing the efficiency of the proposed drive with constant-v/f based drive, a 26% improvement is witnessed, which is a better improvement than the previous case, and is majorly due to the reduction in load torque. In the second scenario, the drive's performance to a load disturbance of 50% while speed set point is fixed at 1440 rpm is evaluated. Figure 8 show the simulation results of the second scenario. The load torque is initially at 50% of nominal, namely 1.2 N\(\cdot\)m. The load torque is increased from 1.2 N\(\cdot\)m to 2.4 N\(\cdot\)m at \(t\) = 20 s and then changed back to 1.2 N\(\cdot\)m at \(t\) = 35 s (see Figure 8h). Following this, the controller output voltage (shown in Figure 8a) is increased to provide the nominal torque while maintaining STCR fixed at the optimal value of 0.96. Moreover, the frequency is slightly increased to retain the motor speed at its reference value (see Figure 8b). As shown in Figure 8e-g, the main winding current increases above the auxiliary winding current during the transient acceleration period but then drops such that STCR becomes equal to 0.96. As depicted in Figure 8i, the motor speed experiences slight undershoot during the load rise but settles at it set point after a few seconds. The speed estimation error steps up to 12% at first, though diminishes to 1% after 2 s. The efficiency of constant-v/f based drive changes from 43% to 63%, while the efficiency of the proposed drive is constant and equal to 66% in both half and full load conditions, as shown in Figure 9.  
In the third scenario, in order to demonstrate the efficacy of the control scheme in tracking the optimum condition, notwithstanding the motor parameters variations, the motor parameters are to be changed. Given a specific frequency, for each set of motor parameters there exists a unique \(K_{\mathrm{s}}\).