  This paper introduces a framework to systematically optimize the control and
design of an electric vehicle transmission, connecting powertrain sizing
studies to detailed gearbox design methods. To this end, we first create
analytical models of individual components: gears, shafts, bearings, clutches,
and synchronizers. Second, we construct a transmission by systematically
configuring a topology with these components. Third, we place the composed
transmission within a powertrain and vehicle model, and compute the
minimum-energy control and design, employing solving algorithms that provide
global optimality guarantees. Finally, we carry out the control and design
optimization of a fixed- and two-gear transmission for a compact family
electric vehicle, whereby we observe that a two-gear transmission can improve
the energy consumption by 0.8 %, while also achieving requirements on
gradeability and performance. We validate our framework by recreating the
transmission that is mounted in the benchmark test case vehicle and recomputing
the energy consumption over the New European Driving Cycle, where we notice an
error in energy consumption of 0.2 %, affirming that our methods are suitable
for gear ratio selection in powertrain design optimization.
