# Inductance Estimation Based on Wavelet-GMDH for Sensorless Control of PMSM
## 5 Simulation Results  
### System Modeling  
Figure 7 represents the block diagram of the entire control system for the PMSM sensorless control, including the proposed wavelet-GMDH based magnetic saturation compensation control. Through an optimized polynomial expression-based numerical model utilizing the GMDH algorithm, real-time calculation of the inductance is achieved from eight input variables, including phase currents, voltage references, etc. In the magnetic saturation compensation block, the variation in the estimated inductance \(\Delta L_{q}\) is utilized to  
Figure 5: Comparisons of evaluation metrics (RMSE, MSE, MAE) for all cases.  
Figure 6: Wavelet-GMDH based training results. (**a**) Linear regression result; (**b**) histogram of estimation errors; (**c**) estimation errors.  
compute the extended back EMF fluctuation. Subsequently, the back EMF fluctuation \(\Delta\hat{e}_{7}\) is fed back to the sensorless position controller to compensate for the inductance variation in real time and mitigate the speed ripple.  
In this paper, simulations were conducted using PSIM v.11.1 to analyze and compare the performance of inductance estimation and speed ripple compensation using magnetic saturation compensation methods. As shown in Table 3, motor parameters equivalent to commercial motor specifications, such as resistance and the number of pole pairs/slots, were applied to simulate the real system.  
Figure 8 represents the sensorless control system modeled for the application of a magnetic saturation and compensation control system in a simulation. It is implemented as a dual motor system and consists of a sensorless motor drive and a load motor system to simulate periodic loads. The electric motor model provided by the PSIM v11.1 inherently does not account for magnetic saturation. Therefore, a detailed motor model based on mathematical models and motor parameters was implemented. Furthermore, to create an environment similar to the actual system for the wavelet-GMDH based magnetic saturation compensation algorithm, a microcontroller unit (MCU) controller was modeled through co-simulation with a DLL file based on C language programming.  
In systems exposed to severe load conditions, such as components near engines and powertrains, pumps, and compressors, periodic load torque can be compensated through a PMSM load torque compensator. However, the magnetic saturation, driven by the increasing compensation current, undermines the precision of sensorless speed control and load torque compensation. Assuming that the frequency and magnitude of the periodic load torque are predictable, the \(\delta\)-axis compensation current depending on the load is expressed as the sum of the frequency component with constant amplitude \(I_{S}\) and DC offset \(I_{DC}\)[16]:  
\[i_{\delta}=I_{S}sin\omega t+I_{DC}, \tag{22}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Unit** \\ \hline Rated power & 3 & kW \\ DC link voltage & 400 & V \\ Winding resistance & 0.3 & \(\Omega\) \\ Number of poles & 6 & - \\ Number of slots & 27 & - \\ \(d\)-axis inductance & 1.5 & mH \\ \(q\)-axis inductance & 2.0 & mH \\ \hline \hline \end{tabular}
\end{table}
Table 3: Motor parameters.  
Figure 7: Block diagram of the sensorless compensation control system based on wavelet-GMDH.  
where \(i_{\delta}\) is the \(\delta\)-axis compensation current, and \(\omega\) is the frequency component of the rotor speed. The \(\gamma\)-axis compensation current is obtained from the \(\delta\)-axis compensation current according to the maximum torque per ampere rule. With accurate rotor position estimation, even in the presence of inductance variation, the periodic load torque can be effectively compensated.