# Optimization Procedure for Computing Sampling Time for Induction Machine Parameter Estimation
## 4 Simulation and Experiment
### Testing Optimal Sampling Time on Parameter Estimation  
Using the optimal sampling time claims to improve the parameter identification procedures for the motor. To test this claim, a parameter identification procedure is tested for the measured data. The procedure used in this paper is explained in [27]. In brief, the parameter identification procedure uses constrained global optimization to compute the solution of Equation (3), with measured quantities: voltage oriented reference frame currents and rotational speed (\(\mathbf{y}[p]= \begin{bmatrix}i_{d}[p]&i_{q}[p]&\omega[p]\end{bmatrix} ^{T}\)). The model step response is computed by solving the model (Equation (13)) by Runge-Kutta method. This procedure is repeated for the original data set segmented into 25 intervals, thus returning 25 parameter vectors that represent the time varying parameters during IM line start. The parameter vector for each interval is \(\mathbf{\Theta}= \begin{bmatrix}G_{s}&G_{m}&R_{r}G_{r}&R_{r}G_{r}&\mathbf{x}_{0}^{T} \end{bmatrix} ^{T}\), where the \(\mathbf{x}_{0}\) is the initial condition of state variables (the fluxes, and the rotational speed) for each interval. Input to the system is the peak nominal voltage aligned with the \(d\)-axis, and all quantities and parameters used for identification are in per unit system where base voltage is peak voltage, base power is nominal apparent power and base speed is synchronous speed. The results for parameter identification with original sampling time of \(T_{s,min}=0.1\) ms and with optimal sampling time \(T_{s,opt}=1.4981\) ms are shown in Figures 5a,b and 6a,b. Figure 5a,b show the residuals of measured and calculated current and speed and their values calculated from the estimated model with original and optimal sampling time used in estimation. It is clear that the current residual does not show any improvement regarding its quantities or the fact that the aim is to reduce residuals to zero-mean with minimal variance. On the speed residual the change is noticeable. With the original sampling time the residual of the speed shows bias around 50 rpm in the initial moments of startup (when the actual speed is around 0), and than jumps to \(-90\) rpm after which it is starts to increase but constantly stays biased in the negative values. The optimal sampling time speed residual shows the residual with low quantities during the initial phase thus making smaller error compared to the original. After that it jumps to around 75 rpm (same as with the original sampling time) and drops to negative quantities after 0.1 s. The drop is lower than the original case, and after 0.2 s the residual becomes positive again, thus pulling the entire residual closer to the zero-mean desired shape. In the steady state, the same performance can be observed for both original and optimal sampling time, where the speed residuals consist of the eccentricity oscillations introduced by the tachometer used for measurement of the speed. To continue the investigation of optimal sampling time the identified parameters are presented. Figure 6a,b show the parameter values obtained with the original and optimal sampling time used for identification. The parameters with the original sampling time show increased steady sate stability, but the transient behaviour is improved in the optimal sampling time case. The the mutual inductance \(L_{m}\), shows the drag towards the boundaries of the parameter search set which suggest problems with the optimization procedure in three intervals. The same problem occurs only in one interval for the case of optimal sampling time parameters. The rotor resistance shows similar behaviour in both optimal and original sampling time estimates with two outlier values in the steady state for the optimal sampling time.