  This paper presents a modeling and optimization framework to minimize the
energy consumption of a fully electric powertrain by optimizing its design and
control strategies whilst explicitly accounting for the thermal behavior of the
Electric Motor (EM). Specifically, we first derive convex models of the
powertrain components, including the battery, the EM, the transmission and a
Lumped Parameter Thermal Network (LPTN) capturing the thermal dynamics of the
EM. Second, we frame the optimal control problem in time domain, and devise a
two-step algorithm to accelerate convergence and efficiently solve the
resulting convex problem via nonlinear programming. Subsequently, we present a
case study for a compact family car, optimize its transmission design and
operation jointly with the regenerative braking and EM cooling control
strategies for a finite number of motors and transmission technologies. We
validate our proposed models using the high-fidelity simulation software
Motor-CAD, showing that the LPTN quite accurately captures the thermal dynamics
of the EM, and that the permanent magnets' temperature is the limiting factor
during extended driving. Furthermore, our results reveal that powertrains
equipped with a continuously variable transmission (CVT) result into a lower
energy consumption than with a fixed-gear transmission (FGT), as a CVT can
lower the EM losses, resulting in lower EM temperatures. Finally, our results
emphasize the significance of considering the thermal behavior when designing
an EM and the potential offered by CVTs in terms of downsizing.
