# Novel Thermal Management Strategy for Improved Inverter Reliability in Electric Vehicles
## 2 Torque Regulation of the IPMSM  
Figure 1 shows the general diagram of the implemented IPMSM torque regulation strategy including an optimum set-point generation block, a field oriented current control loop (with the required vector transformations), and the proposed TCT thermal management strategy, this last one incorporating a power loss estimator, a highest virtual junction temperature estimator and a TCT algorithm.  
From the reference torque \(T_{em}^{*}\), the current set-point generation block determines the \(i_{d}^{*}\) and \(i_{q}^{*}\) stator current references from two 2-D Look-Up Tables (LUTs). (From data obtained by Finite Element Analysis (FEM), current trajectories are determined offline and stored in the LUTs for Maximum Torque Per Ampere (MTPA), Field Weakening (FW) and Maximum Torque Per Volt (MTPV) regions. DC-link voltage dependency has been eliminated from the LUTs using the speed normalization (\(w_{norm}\)) concept proposed in [32], requiring one less dimension in the LUTs.) Unlike in conventional LUT-based current set-point generation approaches [33], the controller incorporates an additional Voltage Constraint Tracking feedback loop (VCT, Figure 1), ensuring that maximum stator voltage modulus of \(V_{DC}/\sqrt{3}\) is never exceeded during field weakening even if significant electrical machine parameter deviations exist [34].  
Once the reference currents are determined, the IPMSM torque is indirectly regulated by the current control loop. This loop follows the well-known field oriented control (FOC) approach in the synchronous reference frame (\(dq\)). However, although PI-based regulators are generally used, a second order Sliding Mode Control (SMC) with super-twisting has been implemented, as such an approach is more robust under machine non-linearities that are common in automotive IPMSMs [34; 35].  
The \(dq\) reference voltages to be synthesized by PWM have been obtained by [34]:  
\[v_{d}^{*} =v_{d,ST}+v_{d,eq}, \tag{1}\] \[v_{q}^{*} =v_{q,ST}+v_{q,eq}, \tag{2}\]  
Figure 1: General diagram of the Interior Permanent Magnet Synchronous Machine (IPMSM) controller including the proposed Temperature Constraint Tracking (TCT) algorithm.  
where \(v_{d,eq}\) (3) and \(v_{q,eq}\) (4) are the equivalent control signals, while \(v_{d,ST}\) and \(v_{q,ST}\) are computed by applying the Super-Twisting Algorithm (STA) in (5) and (6).  
\[v_{d,eq} =L_{d}\left[c_{d}e_{i_{d}}-\frac{-R_{s}i_{d}+w_{e}\Psi_{q}(i_{d},i_ {q})}{L_{d}(i_{d},i_{q})}\right], \tag{3}\] \[v_{q,eq} =L_{q}\left[c_{q}e_{i_{q}}-\frac{-R_{s}i_{q}-w_{e}\Psi_{d}(i_{d},i _{q})}{L_{q}(i_{d},i_{q})}\right],\] (4) \[v_{d,ST} =L_{d}\left[\lambda_{d}|s_{i_{d}}|^{1/2}sgn(s_{i_{d}})+\Omega_{d} \int sgn(s_{i_{d}})dt\right],\] (5) \[v_{q,ST} =L_{q}\left[\lambda_{q}|s_{i_{d}}|^{1/2}sgn(s_{i_{q}})+\Omega_{q} \int sgn(s_{i_{q}})dt\right], \tag{6}\]  
where \(R_{s}\) is the stator resistance, \(L_{d}(i_{d},i_{q})\), \(L_{q}(i_{d},i_{q})\), \(\Psi_{d}(i_{d},i_{q})\) and \(\Psi_{q}(i_{d},i_{q})\) are the current dependant stator inductances and fluxes, \(\omega_{e}\) is the electrical speed, \(e_{i_{d}}=(i_{d}^{*}-i_{d})\) and \(e_{i_{q}}=(i_{q}^{*}-i_{q})\) are the error signals, \(c_{d}\), \(c_{q}\), \(\lambda_{d}\), \(\lambda_{q}\), \(\Omega_{d}\) and \(\Omega_{q}\) are the positive gains for tuning, \(sgn(s_{i_{d}})=s_{i_{d}}/|s_{i_{d}}|\), \(sgn(s_{i_{q}})=s_{i_{q}}/|s_{i_{q}}|\), where \(s_{i_{d}}\) and \(s_{i_{q}}\) are the sliding surfaces:  
\[s_{i_{d}} =e_{i_{d}}+c_{d}\int e_{i_{d}}dt, \tag{7}\] \[s_{i_{q}} =e_{i_{q}}+c_{d}\int e_{i_{q}}dt. \tag{8}\]