*This is the second installment of a four-part article by: Kenneth Mendoza, Project Engineer, HiL Systems, Electronics in Applus IDIADA and Fabio Squadrani, Senior Manager, Braking Systems in Applus IDIADA on developing a control module to simulate hybrid regenerative braking on a brake dynamometer.*

Regenerative Braking Testing on Dynamometer (Part 1 of 4)

TROY, Mich. In the first part of the article, the objective of the study –i.e. the development of a control module to simulate hybrid braking on a brake dynamometer– was introduced. This second part focuses on the mathematical model and its implementation in Simulink.

#### Mathematical model definition

First of all, the moment of inertia required in a brake dynamometer is defined. This moment of inertia directly depends on the vehicle being tested in the brake dynamometer.

From kinematics, the total amount of energy of a vehicle, E_k in [J], can be described as:

Where,

m is the mass of the vehicle, in [kg],

v is the speed of the vehicle, in [m/s].

Assuming that all the kinematic energy of the vehicle can be transformed in rotational kinematic energy, E_kR in [J]:

Where,

I is the equivalent moment of inertia of the vehicle, in [kgm2],

ω is the equivalent rotational speed of the vehicle, in [rad/s].

It needs to be considered that only one side of one of the vehicles’ axles is mounted into a brake dynamometer, and that each axle’s contribution to braking is defined by the brake distribution. As a result, the equivalent kinetic rotational energy for the brake dynamometer, 〖E_kR〗_dyno in [J], is:

Where,

m is the mass of the vehicle, in [kg]

s_B is the brake split or brake distribution, per unit,

ω_d is the rotational speed of the brake dynamometer, in [rad/s]

r_r is the rolling radius speed of the wheel of the vehicle, in [m]

From the previous equations it can be deduced that the amount of moment of inertia to be mounted in a brake dynamometer for the front axle of a vehicle, I_f in [kgm2], is:

As the brake split is defined as the contribution per unit of the front axle regarding the total brake force, for the rear axle, I_r in [kgm2]:

The total torque present in a brake dynamometer during a decelerating brake application, T_t in [Nm], is:

Where,

I_t is the moment of inertia of the brake dynamometer, in [kgm2],

ω ̇ is the acceleration of the brake dynamometer, in [rad/s2].

Considering Inertia Simulation, the total torque present in the brake dynamometer can be described as:

Where,

T_brk is the torque produced by the braking system, in [Nm],

T_mot is the torque provided by the electric motor of the brake dynamometer, in [Nm].

With the previous consideration, the total moment of inertia of a brake dynamometer, I_t in [kgm2], can be described as:

Where,

I_mec is the moment of inertia of the flywheels mounted in the brake dynamometer, in [kgm2],

I_mot is the inertia provided by the electric motor of the brake dynamometer, in [kgm2].

From the equations above, the required torque to be provided by the electric motor can be described as:

If all the parameters are referred to the torque provided by the braking system:

Simplifying, the control equation for the Inertia Simulation algorithm results in:

If road losses are to be considered, the equation results in:

Where,

T_loss is summation of torque produced by all the road losses, in [Nm]

#### Mathematical model implementation with Simulink

In order to virtually validate the control equation presented in the previous section, a model has been created in Matlab Simulink.

The model implemented in Simulink consists of three blocks which are: Block 1: Computation of required torque, which implements the equation obtained in the mathematical description section

Block 2: Controller, which essentially implements a PID controller to accelerate the brake dynamometer before the brake application and the logics to switch the control mode of the motor drive motor during the brake application. Note that the “Linear load torque” gain is intended to simulate the torque when the dynamometer is accelerating freely.

Block 3: Drive and motor, which consists of the blocks that include the drive of the motor and the motor itself. It is possible to parameterize the moment of inertia at the output of the motor shaft in the “Four-Quadrant Three-Phase Rectifier DC Drive”.

Resulting graphs are shown below, where in both plots it is shown the behaviour with and without Inertia Simulation:

Case Study 1: Decreasing inertia from 50 to 35 kgm2:

In this graph it can be seen how the motor torque with Inertia Simulation activated (in blue) is negative during the brake application. This means that the motor is contributing to reduce the inertia that the braking system sees. As well as that it can be appreciated how the deceleration for the same braking torque is higher when Inertia Simulation is active (in orange). In other words as there is lesser inertia there is lesser opposition to the braking torque and therefore the resulting deceleration is higher.

Case Study 2: Increasing inertia from 50 to 80 kgm2:

This graph shows the opposite effect to the previous figure. It can be seen how the motor torque with Inertia Simulation activated (in blue) is positive during the brake application. This means that the motor is contributing to increase the inertia that the braking system sees. As well as that it can be appreciated how the deceleration for the same braking torque is lower when Inertia Simulation is active (in orange). In other words as there is more inertia there is more opposition to the braking torque and therefore the resulting deceleration is lower.

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