**2. Vehicle Powertrain Model**

### *2.1. Multispeed Gearbox Options for Heavy Duty Vehicles*

Multispeed gearbox configurations for EVs constitute automatically shifted multispeed gearboxes, including two-speed gearboxes, and continuously variable transmissions [14]. Hydraulic automatic transmission is considered infeasible for the ECB application due to very low efficiency [15]. Additionally, gearbox designs that are still in the development stage, such as a planetary gear two-speed gearbox presented in [16], are considered out of scope.

CVT permits the electric motor to operate at the ideal point continuously. Power is transmitted by friction via a chain that runs between two axially adjustable taper discs. The diameter of the discs can be varied. Another CVT configuration is a toroidal variator. In this design the variable gear ratio is achieved by swiveling the friction gears.

The major drawback of CVT is the poor efficiency, compared to gear transmission. The traditional CVT that is used with internal combustion engine vehicles has efficiency of less than 85% but with EVs a simpler construction without a torque converter is feasible. Thus, efficiency of over 93% can be achieved [17].

Superior efficiency is achievable with gearboxes that have a discrete number of gear ratios. A greater than 95% efficiency and less than 150 millisecond gearshifts are reported for a two-speed dual clutch gearbox in [18] with 450 Nm maximum input torque and 13,000 rpm maximum speed. A single speed gearbox with the same performance specifications is reported to have two percentage points higher efficiency.

An automatic multispeed gearbox requires actuation for gear shifts. This is commonly implemented with an electrohydraulic actuator. A gear shift loss model for automatic manual transmission is presented in [19]. In this single clutch model, the losses consist of partially lost traction during clutch disengagement and engagement due to slipping, and the total loss of traction during the shifting phase. The clutch slipping is unavoidable, because engagement and disengagement must occur smoothly.

The primary power loss sources are clutch drag, gear churning, gear meshing and bearing seals [14]. Additional losses incur from energy consumption of a pumping cooling and lubricating fluid to the gearbox housing in the case of a wet clutch design. A dry clutch does not have a need for oil pumping, but can sustain lower torques.

The traction loss that occurs in shifting results in the vehicle velocity tracking loss compared to a vehicle equipped with a single speed reduction gear [19]. This loss needs to be compensated by an increased power output after the shifting event.

Based on a preliminary analysis, this work focuses on two gearbox alternatives that are estimated to have the most potential in an electric city bus. The first is a two-speed dual clutch gearbox (Figure 1) and the second is a traditional belt CVT without a torque converter. The model for the two-speed alternative includes both design (gear ratios) and operation (shifting policy) optimization levels. By contrast the CVT model simply selects, at each time step, the gear ratio that represents the highest efficiency operating point of the motor and the inverter.

**Figure 1.** Two-speed dual clutch gearbox layout.
