Willans Line Modeling for Powertrain Analysis and Energy Consumption of Electric Vehicles


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Virginia Tech


With electric vehicles becoming increasingly prevalent in the automotive market consumers are becoming more conscientious of total driving range. In light of this trend, reliable and accurate modeling methods are necessary to aid the development of more energy efficient vehicles with greater drivable range. Many methods exist for evaluating energy consumption of current and future vehicle designs over the US certification drive cycles. This work focuses on utilizing the well-established Willans line approximation and proposes a simplified modeling method to determine electric vehicle energy consumption and powertrain efficiency. First, a backwards physics-based model is applied to determine tractive effort at the wheel to meet US certification drive cycle demand. Second, the Willans line approximation then augments the tractive effort model and parameterizes the vehicle powertrain to establish a bi-directional power flow method. This bi-directional approach separates propel and brake phases of the vehicle over the certification City and Highway drive cycles to successfully isolate the vehicle powertrain from non-intrinsic losses, such as parasitic accessory loads. The proposed method of bi-directional modeling and parameter tuning provides significant insight to the efficiency, losses, and energy consumption of a modeled electric vehicle strictly using publicly available test data. Results are presented for eight electric vehicles with production years varying from 2016 to 2021. These electric vehicles are chosen to encapsulate the electric vehicle market as performance electric vehicles to smaller commuter electric vehicles are selected. All vehicles are modeled with an accessory load constrained between 300 and 850 W and a regenerative braking ("regen") low-speed cutoff of 5 mph with six of the eight vehicles modeled with a regenerative braking fraction of 94%. The bi-directional Willans line is then tuned to reach agreement with the net EPA energy consumption test data for each vehicle with the results presented as representative of the chosen vehicle. Lastly, a transfer function relating major model inputs to the output is derived and lends considerable insight for the sensitivity of the modeling method. Sensitivity of the proposed modeling method is conducted for a 2017 BMW i3 with the model deemed reasonably resilient to changes in input parameters. The model is most sensitive to changes in powertrain marginal efficiency with a 6% decrease of marginal efficiency leading to a 0.404 kW and 0.793 kW cycle average net battery power increase for the City and Highway drive cycles respectively. Additionally, the model is also sensitive to changes in vehicle accessory load with a direct relationship between increases of vehicle accessory load to increases of cycle average net battery power for the City and Highway cycles. The sensitivity results justify the use of the proposed model as a method for evaluating vehicle energy consumption and powertrain efficiency solely using publicly available test data.



Electric vehicles, vehicle modeling, vehicle efficiency, electric vehicle energy consumption, Willans line