2.1 - Regenerative Braking Calculations
2.1 - Regenerative Braking Calculations
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Written 28/08/22 by Tom Johnson
Oh boy, this is going to be a long one. Today I'm discussing the plausibility of a regenerative braking system for UH27. This article will detail all theoretical calculations, for such a system, and a follow-up article on the physical implementation for the system will be available at a later date.
The basic operating principles for regenerative braking involves reducing the frequency of current supplied to the stator of an AC (alternating current) motor, which in turn makes the flux wave travel slower than the rotor, effectively turning the motor into a generator. This is unique to AC motors, and, for the purposes of this article, we are assuming that UH27 will use the same Fischer Ti-085 AC motors that have been used by UH Racing on the last three generations of UH vehicles. This is not only out of cost, as purchasing new motors is a multi-thousand pound investment that our university is not keen to make, but also as this setup is arguably better than the equivalent AMK motor-inverter solution, due to the higher torque characteristics of the Fischer motors.
Vehicle Dynamics:
Based on previous figures for both UH25 and UH26, we can estimate the braking forces that UH27 will experience.
UH27 will have an estimated mass of 284kg, and will ideally be split 50/50 between the front and rear axle. When static, both axles will experience 1393N of load, but this will change dramatically when the vehicle is experiencing ~1g of deceleration.
UH27's wheelbase is estimated at 1.64m and has a barycentre height of 0.339m. Given our target deceleration of 1g, we can calculate that 70.6% of the vehicles load will be on the front axle, while 29.4% of the vehicles load will be on the rear axle. Now is probably a good time to mention that aerodynamic drag is not being accounted for, as it is on the order of hundreds of Newtons, whereas the hydraulic braking force is on the order of tens of thousands of Newtons. Therefore, for simplicity, I have not decided to include it in this article.
Using the equations above, and an estimated friction coefficient between the tyre and the track surface of 1.3, we can therefore calculate that the braking force on the front axle is 2363N, while the braking force on the rear axle is 981N.
Multiplying the braking force by our tyre radius (0.2271m), gives us the braking torque on each axle, which will be 537Nm on the front axle and 223Nm on the rear axle. However, UH27 has a FDR (Final Drive Ratio) of 8.5, which means that for every 8.5 rotations made by the motor output shaft, the wheel spins once. In order to calculate the effective motor regenerative torque, we must divide the braking torque by the FDR. This leaves us with a combined motor torque of 63Nm on the front axle, and 26Nm on the rear axle.
Regeneration Power:
The following section uses a maximum velocity of 27.7m/s (62mph), as that was the maximum velocity recorded around a lap of the FSG (Formula Student Germany) track. This was used as UH Racing has not yet run an electric vehicle in the dynamic events at FSUK.
Based on the equation above, this gives the front axle a maximum regenerative power of 65.32kW, and the rear axle a maximum regenerative power of 26.96kW.
Current Limitations:
As discussed in the 'Cell Type' article, one major drawback of the cylindrical cells used in the Energus Li1x4pVTC6T modules is the fact that the maximum burst charge current is 64A. For safety reasons, we must assume a worst case scenario, which for us is all 144 of these modules to be operating at 2.5V, giving us an minimum voltage of 360V.
We must also consider internal impedance, which can be calculated like so:
As the EPA070170SP cell has an estimated internal impedance of 0.015 Ohms, and we are using 144 cells in series, our overall resistance is 0.216 Ohms.
We can now form a quadratic equation which allows us to calculate the estimated charging current for any given regenerative power.
The resultant charging currents for the front and rear axle are 164.8A and 64.5A respectively. As mentioned previously, our cells can sustain a maximum burst charge current of 120A. Reworking the equation to ensure the current does not exceed 120, we are able to use a regenerative power of around 48kW, using 46Nm of braking torque. Based on running data provided by the University of Sapienza, I believe this configuration could give us around 850Wh of energy across the endurance event, which increases the theoretical capacity of our accumulator to 7.24kWh, which I believe should be enough to finish the endurance event, putting UH27 in the 59th percentile for accumulator capacity (according to the FSG 2021 team provided data)
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