The performance of powertrain components in electric vehicles is tightly intertwined with their thermal behavior. In practical applications, their temperature must be monitored and kept below certain thresholds to avoid performance drops and failure. Sensors, however, cannot always be placed at critical locations. Instead, it is possible to use numerical models to estimate relevant magnitudes during system operation. Thermal effects in electric and electronic components can be represented in a compact way using lumped-parameter equivalent circuits. These can be combined with sensor readings from the device under study to develop digital twins and use them to monitor temperatures during test and operation. In this paper, we put forward a method to generate thermal digital twins of e-powertrain elements such as power inverters. The thermal equivalent circuit equations are obtained from a general-purpose simulation software tool and optimized to enable real-time execution. Kalman filters are then used to fuse the simulation results from this model and sensor measurements of component temperatures. The proposed method provides a way to estimate the inputs and parameters of the thermal circuit and can be used to avoid the drift of the simulation away from actual component behavior. The performance of this approach is demonstrated with a simple benchmark example and the thermal equivalent circuit of a three-phase inverter.

Index-3 augmented Lagrangian formulations with projections of velocities and accelerations represent an efficient and robust method to carry out the forward-dynamics simulation of multibody systems modeled in dependent coordinates. Existing formalisms, however, were only established for holonomic systems, for which the expression of the constraints at the position-level is known. In this work, an extension of the original algorithms for nonholonomic systems is introduced. Moreover, projections of velocities and accelerations have two side effects: they modify the kinetic energy of the system and they contribute to the constraint reaction forces. Although the effects of the projections on the energy have been studied by several authors, their role in the calculation of the reaction forces has not been described so far. In this work, expressions to determine the constraint reactions from the Lagrange multipliers of the dynamic equations and the Lagrange multipliers of the velocity and acceleration projections are introduced. Simulation results show that the proposed strategy can be used to expand the capabilities of index-3 augmented Lagrangian algorithms, making them able to deal with nonholonomic constraints and provide correct reaction efforts.