Cooperative Control Algorithm for Friction and Regenerative Braking Systems Considering Temperature Characteristics

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R E T R A C T E D 1 Introduction
The regenerative braking system is the most important technology for the hybrid electric vehicle (HEV); it has been mainly researched for improvement of fuel efficiency [1,2], but stability of the brake system should be considered because the function of a brake is safety.For stability, it is important to control the hydraulic and electric systems because the HEV braking system applies braking torque through the collaboration of friction and regenerative brake systems.These systems have different response times, thus the transient characteristic [3] takes place, which causes a rapid change to deceleration.But preceding researches did not take into account this dynamic characteristic.Also, the friction braking torque changes with temperature because the friction coefficient changes with temperature [4], but it is normally considered as constant, meaning the braking torque is different.Due to this property, safety of the vehicle is compromised, along with the comfort of the driver.To solve the problem of the transient region, the hydraulic and electric systems were modelled considering system dynamics.These were controlled by coincidence control for settling time which is proposed in the paper.In order to solve the problem of braking torque alternation by influence of temperature, temperature of the brake pad should be estimated.The preceding research calculated the temperature by finite element method (FEM) [5] which needs a long time to compute the temperature.But this method is not appropriate for a braking system because the environments of vehicles can be diverse.Therefore, finite difference method (FDM) was used for estimation of temperature as it calculates more rapidly.Then, the braking torque compensation algorithm, considering temperature, was developed.From coincidence control for settling time and compensation control for temperature, the integrated control algorithm was developed for the safety of HEV.In this paper, such models and control algorithms were made by MATLAB/Simulink.

Coincidence
Control for Friction and Regenerative Braking system The braking system of a HEV is usually comprised of a friction brake and a regenerative brake, this means that a total braking torque comes from the sum of friction and regenerative braking torque.The friction brake is a hydraulic system, which has a slow response time, and the regenerative braking system, which has a fast response time.This means that the hydraulic system cannot follow the response time of the electric system, and thus sudden deceleration or acceleration occurs when the regenerative braking and friction braking switch.That is the transient characteristic of regenerative braking system as shown in Fig 1.In this paper, the coincidence control for settling time of both systems was attempted to solve such a problem.For the coincidence control, the mathematical modelling was required.The hydraulic system was modelled with the flow and continuity equation.The electric system was modelled with the d-q transformation and voltage equation.The coincidence control was conducted by analysing transfer function and tuning the PI gain, which is adaptive PI control.

Friction braking system modelling
Friction braking torque is generated by hydraulic force.Therefore, the comprehension of the hydraulic system dynamics is required to control the friction braking system.Fig 2 shows the hydraulic brake circuit of HEV [6].The main valve 1 is normally open (NO), which connects the primary master cylinder to the FR and RL wheel cylinders.The main valve 2 is also normally open (NO), which connects the secondary master cylinder to the FL and RR wheel cylinders.The main valves close a few minutes after the vehicle system is active, when the door is open or ignition is turned on.The inlet valve FR, RL, FL and RR are On/Off solenoid valves which are normally open (NO), these increase the pressure of each wheel cylinder by using the pulse width modulation control (PWM control).The outlet valve FR, RL, FL and RR are On/Off solenoid valves which are normally closed (NC), decrease the pressure of each wheel cylinder, as well as using PWM control.The pump, which is driven by the electric motor, supplies flow for the hydraulic source to apply pressure inside wheel cylinders.The accumulator stores hydraulic energy when the energy is over sufficient and releases hydraulic energy when the energy is lacking.The pedal stroke simulator simulates the feeling of a driver, because the hydraulic brake circuit of the HEV is isolated with the master cylinder when braking.The hybrid control unit (HCU) calculates demanded friction braking torque by using the regenerative braking algorithm.The PWM modulator controls the inlet valve and the outlet valve to generate friction braking torque

2( )
Eq. ( 2) is the flow equation of the wheel cylinder when outlet valve is open Also the flow rate of wheel cylinder is expressed as Eq. ( 3), which is the continuity equation.

2( )
bf  is a bulk modulus of brake fluid and t is time.w Q and w V is the net flow and the volume of the wheel cylinder.w V can be obtained by Eq. ( 4).
inital w V is the initial volume of the wheel cylinder, w r is the radius of the wheel cylinder, and x is the displacement of the return spring.
Eq. ( 5) represents the relation between hydraulic force and return spring.k is the spring constant.

Friction braking system control
To control the friction braking system, the hydraulic system should be controlled.The state of

R E T R A C T E D
the inlet and outlet valve is controlled by PWM control.This generates pressure inside the wheel cylinder by repeating alternately to open each valve with a decided frequency, it is important to measure how long the valve opens in a period; this is defined as the duty ratio which is shown as Eq. ( 7).(1 )  When the reference signal is larger than the triangular wave, PWM signal is open.When the reference signal is smaller than the triangular wave, PWM signal is close.Thus, the duty ratio can be calculated as Eq. ( 9) by the similarity of the triangle.

D h H  ( 9 )
H is the amplitude of the triangular wave and h is the magnitude of the reference signal.
The friction braking system model was developed by MATLAB/Simulink, it is shown as Fig 6.The simulation was conducted on other duties with the same frequency.The result of the simulation is shown as Fig 7 .In the result, the ripple of friction braking torque ( f T ) was lessened as the PWM frequency gets higher.The relation between the duty ratio and wheel pressure in Eq. ( 8) was validated as correct.10), ( 11), (12). of time variant inductance, d-q transformation is used.Eq. ( 13) is the d-q transformation matrix [8]. ( Eq. ( 10), ( 11) and ( 12) can be converted into Eq.( 14) and (15) by using the transformation matrix which has a rotating d-q coordinate axis with the rotator angular velocity ( r  ).These Eq. ( 14) and ( 15) can be expressed as equivalent circuit, as shown in Fig 9.

Regenerative braking system control
To control regenerative braking system, the electric motor system should be controlled.The IPMSM is controlled by the flux weakening control.This control generates torque by controlling current, so the reference current should be calculated for demanded torque based on maximum torque per ampere (MPTA) control.Torque of IPMSM can be expressed as Eq. ( 16).
(L ) * ( 1) 1) In the first order system, the time constant ( ) indicates the time when the rate of response gets to 63.2% of steady-state.In Eq. (25) and Eq. ( 26), each time constant is ds pd L K and qs pq L K .
In this paper, time constant was determined depending on settling time.It is the time that reaches the rate of the steady-state between 95% and 105%.Eq. ( 25) and ( 26) are arranged into Eq.( 27) and ( 28 From Eq. ( 27) and (28), time constant must be smaller than 0.33 of demanded settling time.Time constant is empirically determined to 1/10 of demanded settling time.This way, the P gains are calculated to Eq. ( 29) and (30).Then the I gains are empirically calculated.From the result, it was confirmed for settling time to be controllable by adaptive PI control.

Coincidence control algorithm
In this paper, the coincidence control algorithm was proposed to lessen the transient characteristic.This algorithm is shown as Fig 13.The simulation was conducted during the shifts between the regenerative and friction brakes.
Regenerative braking only applied 1000 Nm in 0.8 sec, when regenerative braking was off by the regenerative braking constraint algorithm, then friction braking began to operate to compensate the total braking torque.The friction braking system whose settling time was 0.07992 sec could not meet the total braking torque because it was slower than the regenerative braking system whose settling time was 0.01151 sec.Hence the response characteristic of the regenerative braking system was controlled to coincide the response time.
Consequentially, the settling time of the regenerative braking system changed from 0.01151 sec to 0.0817 sec.It was validated that the transient region was improved by using the algorithm.

Braking torque compensation control for temperature
The friction braking torque changes with the temperature because the friction coefficient changes with the temperature.This causes the vehicle to be unstable.In this paper, the braking torque compensation control for temperature was attempted to solve the problem.For the compensation control, the braking torque estimator for temperature was needed.Accordingly the temperature estimator for disk brake system was developed.The compensation control was conducted by comparing the demanded braking torque and the estimated braking torque.

Heat transfer analysis model for disk brake system
In a braking system, various heat transfer phenomena occur, like conduction, convection and radiation.It is difficult to consider all phenomena of the brake components, and thus a simplified heat transfer analysis model is required.Fig 15 is a heat transfer analysis model.In this model, the premise is that radiation is insignificant because it is small compared to the other heat transfer phenomena.When braking system works accordingly, the pad pushes the disk and frictional heat is generated on the surface between the pad and the disk.The portion of the frictional heat that goes through the disk rises the temperature of the disk, and some of it dissipates by convection, the rest of the frictional heat travels to both pads.The frictional heat which goes to the inner pad is conducted to the piston and brake fluid in respectively, which rises the temperature of each component.The heat dissipates on the surface of the brake fluid.The frictional heat, which travels to the outer pad, is conducted from the pad to the caliper, and it dissipates on the surface of the caliper.Fig 16 shows the thermal circuit of the disk brake system [10].

Frictional heat
Frictional heat is generated by friction between the disk and the pad.It can be calculated if the braking energy is completely converted into heat energy.The braking energy per unit time is expressed by the result product of friction braking torque ( fb T ) and wheel angular velocity ( w  ).That is, the frictional heat rate ( f q ) is expressed as Eq.(31).
The heat is distributed to each disk ( d q ) and pad ( p q ).The distribution of frictional heat helps the brake system to be simply analysed because the distributed heats to the disk and pad are independent each other.Such distribution factors (  ) can be defined as Eq. ( 32) and calculated with a semi-infinite solid heat transfer.Eq. ( 33) and (34) is the pad and disk temperature equation that uses semi-infinite solid method in constant heat flux [11].Therefore, Eq. ( 32) can be derived as Eq. ( 35) [12].

Finite Difference Method
The condition of the braking system changes from moment to moment during driving.This means that the temperature solution should be capable of responding immediately to any circumstance.But it is impossible to get an exact solution because the shapes of the brake components are complex and the heat transfer coefficient changes.FDM, which is a kind of numerical analysis, is effective to solve this complicated problem.FDM approximates and solves the partial differential equation.This method makes a solution for defined discrete points.For a numerical analysis, analysed material should be divided into many sections (dashed square in Fig 17).The reference points that are placed in the middle of the sections are called nodes, this represents the section and its average temperature, and a group of such nodes is called a grid. .Eq. ( 37) is derived to obtain temperature gradient for y direction at ( , 1 / 2) i j  .Eq. ( 38) is derived to obtain temperature gradient for time ( t ) at ( , ) i j A FDM equation derivation of heat rate, conduction and convection is shown as below.
Fig 18 is the FDM grid in the disk brake system.In heat rate, energy balance method equation is written as Eq.(39).Then, the FDM equation is derived as Eq. ( 40) by arranging for 3 T in Eq. (39).
k is thermal conductivity, c is specific heat,  is density, x A is area for x direction, x  is length for x direction of cell and t  is time step for the FDM.Upper script n means previous value and upper script 1 n  is new value.
In conduction, an equation of energy balance method is derived as Eq. ( 41).Then, the FDM equation is derived as Eq. ( 42) by arranging from 2 T in Eq. ( 41).
With the FDM, the temperature estimation model for the disk brake system was developed using MATLAB/Simulink.It is shown as Fig 19.In the experiment, the temperature rose early because the thermo-coupler, attached to the pad, was in contact to a frictional surface ahead braking.The temperature became stable after 4 seconds because its capacity is small.After 4 seconds, the tendency and the value of temperature of both experiment and simulation became similar.It proves that the estimator has credibility based on the results.

Compensation control algorithm
In this paper, the algorithm of braking torque compensation was suggested to control the braking torque which changes depending on temperature.The temperature estimator for the braking system calculates the estimated temperature ( T ) by using the wheel angular velocity and the friction braking torque.The estimated temperature determines the estimated friction coefficient (  ) from the temperature-friction coefficient curve.The friction braking torque is estimated considering temperature, then the compensation controller compares total estimated braking torque     For the safety and comfort of the braking system of the HEV, the coincidence control for the friction and regenerative braking system, and the braking torque compensation algorithm for temperature were proposed in this paper.The braking system model and temperature estimator of the HEV were developed using MATLAB/Simulink, then the algorithm was applied.As a result of the compensation algorithm, it was confirmed that the algorithm compensated the demanded braking torque even if the temperature changed.Also, the coincidence control improved the transient region.With the combined algorithms, the safety of the HEV braking system could be impoved.The algorithm is expected to be used for safety of the braking system considering environmental factors.

Fig 2 :
Fig 2: Hydraulic brake circuit of HEV Fig 3 shows the inlet valve and outlet valve which control the pressure of the wheel cylinder [6].When controlling the pressure, the inlet and outlet valve complementarily operate at the same time.That is, the inlet valve is open while the outlet valve is close and the inlet valve is close while the outlet valve is open.

Fig 3 :
Fig 3: Schematic diagram of inlet and outlet valve Eq. (1) is the flow equation of the wheel cylinder when inlet valve is open.

iQ
and o Q are respectively the inlet and outlet flow rate of the wheel cylinder.v C and v A are the flow coefficient and the valve opening area of the inlet and outlet valve.p P , r P and w P are the pressure of the pump, reservoir and wheel cylinder.bf  is the density of the brake fluid.
open and closed.

D
is duty ratio, open inlet T is time of opening of the inlet valve and period T is a period.The duty ratio determines the pressure of the wheel cylinder.The relation between duty and the wheel cylinder pressure is expressed as Eq.(8) [7], where w w p P P P   .It is shown as Fig 4.

Fig 7 :Fig 8 :
Fig 7: Result of friction braking system2.3Regenerative braking system modellingRegenerative braking system uses the counter torque of the motor, which plays a role of generator in braking.In this paper, the electric respectively voltage of d and q axis.
current of d and q axis.r ds  and r qs  are linkage flux of d and q axis.

Fig 9 :
Fig 9: IPMSM equivalent circuit for d-q axis The torque is controlled by the PI control and it is possible to control the response characteristics by the adaptive PI control for settling time.The finding adequate PI gain for demanded settling time is calculated with transfer function.Fig 10 is the block diagram of the IPMSM control system with PI controller.

Fig 10 :
Fig 10: Block diagram of IPMSM control system The block diagram in Fig 10 is expressed as transfer function as Eq.(21) and (22).
) to calculate time constant to satisfy the demanded settling time ( * system model was developed by MATLAB/Simulink, it is shown as Fig 11.

Fig 11 :
Fig 11: Regenerative braking system model The result of the simulation is shown as Fig 12.

Fig 12 :
Fig 12: Result of regenerative braking system The simulation was conducted with calculated p K and i K , to control the settling time, in the same demanded torque.* m T is normalized torque of motor.The settling time was 0.0529 sec in 0.05s

Fig 13 :
Fig 13: Coincidence control algorithm for friction and regenerative braking system When braking, the HCU calculates the demanded friction braking torque ( * friction T ) and regenerative

Fig 14 :
Fig 14: Result of coincidence control

Fig 16 :
Fig 16: Thermal circuit of disk brake system 34) ierfc is integral of the complementary error function.p T , p  and p k are respectively the temperature, thermal diffusivity and thermal conductivity of the pad.d T , d  and d k are respectively the temperature, thermal diffusivity and thermal conductivity of the disk.On the frictional surface, x is 0 and p T is equal to d T .
c and p A are respectively density, specific heat and area of pad.d  , d c and d A are density, specific heat and area of disk respectively.

Fig 17 :
Fig 17: FDM gridIn FDM, a partial differential equation is transformed into the difference equation as below.Eq. (36) is an equation to obtain temperature gradient for x direction at ( 1/ 2, )i j .Eq. (37) is derived to obtain temperature gradient for y direction at ( , 1 / 2) i j  .Eq. (38) is derived to obtain temperature gradient for time ( t ) at ( , ) i j

Fig 18 :
Fig 18: FDM grid in disk brake system

Fig 19 :
Fig 19: Temperature estimator for disk brake system The brake system temperature estimator was compared with experiment result.It is shown as Fig 20.

Fig 20 :
Fig 20: Result of brake temperature estimator braking torque is added.Consequently the compensated braking torque ( ** friction T ) is transmitted to the BCU.The algorithm is shown as Fig 21.

Fig 21 :
Fig 21: Braking torque compensation control algorithm for temperature The simulation result of the compensation algorithm is of Fig 22.

Fig 22 :
Fig 22: Result of compensation algorithm The simulation was conducted in time between 5 and 13 seconds in Fig 20.The actual torque was also calculated as plotted in Fig 20.In the result, normal braking system control could not meet

Fig 23 :
Fig 23: Integrated control algorithm for coincidence and compensation control The simulation result of the integrated control algorithm was plotted in Fig 24.The simulation was conducted at 120 ℃.The shifts between regenerative and friction braking took place at 0.8 seconds, when there was the transient characteristic.Also an error occured after 0.8 seconds because the friction coefficient was 0.35 at room temperature and 0.4116 at

Fig 24 :
Fig 24: Result of integrated control algorithm 5 Conclusion