# Research on Modeling and Control of Regenerative Braking for Brushless DC Machines Driven Electric Vehicles

Authors: Jian-ping Wen; Chuan-wei Zhang
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

In order to improve energy utilization rate of battery-powered electric vehicle (EV) using brushless DC machine (BLDCM), the model of braking current generated by regenerative braking and control method are discussed. On the basis of the equivalent circuit of BLDCM during the generative braking period, the mathematic model of braking current is established. By using an extended state observer (ESO) to observe actual braking current and the unknown disturbances of regenerative braking system, the autodisturbances rejection controller (ADRC) for controlling the braking current is developed. Experimental results show that the proposed method gives better recovery efficiency and is robust to disturbances.


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## Body

## 1. Introduction

Electric vehicles have been catching more attention as a means of eliminating or reducing exhaust gas emission from vehicles and dependence on oil in recent years [1]. However, the driving distance of EV, particularly battery electric vehicle (BEV), is limited by the energy in the battery [2]. One of the most important features of the EV is ability to recover braking energy when vehicle is under braking [3]. The electric motor can be controlled to produce the proper amount of braking force for recovering braking energy as much as possible which can be stored in the battery and then reused to increase the driving distance and energy efficiency.In order to improve energy efficiency and dynamic braking performance of regenerative braking, many researchers developed the control strategies for EV. Based on the modern control theory, some control strategies are applied to regenerative braking for EV [4, 5]. Reference [6] designed the mathematical models of driving and regenerative braking and researched neural network sliding mode controller to improve and recover more energy. Reference [7] proposed a cooperative regenerative braking control algorithm for a six-speed automatic transmission based parallel hybrid electric vehicle. There is a fuzzy logic control strategy to adjust the regenerative braking torque dynamically for a parallel hybrid electric vehicle and improve regenerative braking energy recovery rate of EV [8, 9]. Adaptive regenerative braking control strategy is applied to determine the braking command to guarantee the maneuvering capability [10]. Reference [11] adopts the constant current control strategies to obtain stable braking torque and improve energy recovery efficiency.BLDCM has been widely used in EV because it is more efficient and is more readily controlled than the AC motors [12]. So, regenerative braking of battery-powered EV based on BLDCM is discussed in this paper. In order to avoid the damage on batter caused by large braking current, the braking current of regenerative braking is controlled. According to power circuit topology, the model of braking current is established, on the basis of which autodisturbances rejection controller is designed to realize good control effects of regenerative braking.


## 2. Mathematical Model of BLDCM

Assuming star connected windings, no neutral line, ignoring cogging effect, homogeneous distribution windings, ignoring saturation of magnetic circuit, eddy-current loss and magnetic hysteresis loss, and no damping winding on rotor and damping effect of magnet, the voltage equation of stator windings is expressed as [13](1)uaubuc=Giaibic+pNiaibic+eaebec,where p is differential operator, ua, ub, and uc are phase voltages of three-phase stator windings, respectively, ia, ib, and ic are phase currents of three-phase stator windings, respectively, and ea, eb, and ec are back-emf three-phase stator windings, respectively. Parameters G and N are denoted as, respectively,(2)G=R000R000R,N=L-M000L-M000L-M,where R is resistance of the stator windings, L is self-inductance of the stator windings, and M is mutual inductance of the stator windings.


## 3. Modeling of Regenerative Braking

### 3.1. Regenerative Braking Process Analysis

Topological structure of three-phase driving circuit for brushless DC machine is shown in Figure1.Figure 1
Power circuit structure.With phasesa and b, for example, when EV travels in drive power modes, the power switch T1 is in PWM and the others are off. The power system is in the buck running mode, which is shown in Figure 2.Figure 2
The electrical circuit of driving of BLDCM.When EV travels in electric power generation, the BLDCM runs at generating mode and the power circuit is the same with drive power mode. The current flows in the stator windings in opposite direction. According to rotor position signals, the control system modulates the low power switches of inverter and charges the battery based on the boost chopper principle by self-inductance of windings. The kinetic energy of EV is converted to electric energy to feed the battery.With phasesa and b, for example, the power switch T2 runs under the PWM mode in the process of generative braking. The others are in stop state. The battery is charged by raising back-emf of BLDCM. The electrical circuit of regenerative braking of BLDCM is shown in Figure 3.Figure 3
The electrical circuit of regenerative braking of BLDCM.Equivalent circuit of regenerative braking is shown in Figure4. rt is equivalent series resistance of power switch, rd is equivalent series resistance of diode, La, ra, and ea are inductance, resistance, and back-emf of phase a, respectively, Lb, rb, and eb are inductance, resistance, and back-emf of phase b, respectively, ib is braking current, and im is current of stator winding.Figure 4
Equivalent circuit of regenerative braking.According to the voltage equation, the three stator windings can be regarded as the same; there areLa=Lb=Lm, ea=eb=em, and ra=rb=rm.During theT2 turning on (0≤t≤d·T), the voltage balance equation of main circuit is expressed as(3)2Lmdimdt=2em-im·2rm+rd+rt.During theT2 turning off (d·T<t≤T), the voltage balance equation of main circuit is expressed as(4)2Lmdimdt=2em-vb-im·2rm+2rd+rbat,where d is PWM signal’s duty cycle, T is PWM signal’s period, and rbat is internal resistance of battery.By averaging PWM on-time and PWM off-time ofT2, the voltage balance equation is expressed as(5)2Lmdimdt=mim+2em-1-dvb,where m=(d-2)rd-drt+(d-1)rbat-2rm.The electromagnetic torque can be expressed as(6)Jdωdt=-Kt·im-TL,where Kt is torque coefficient, TL is the load torque, J is the combined machine and load inertia coefficient, and ω is rotor angular velocity.


### 3.2. Modeling of Charging Current

The EV usually adopts battery as energy storage equipment. For protecting the battery against the damage by large charging current, the charging current is controlled. So, (5) can be rewritten as(7)i˙m=Sim+Q,where S=((d-2)rd-drt+(d-1)rbat-2rm)/2Lm and Q=[2em-(1-d)vb]/2Lm.S is decomposed into two parts and expressed as(8)S=d-2rd-drt+d-1rbat2Lm-rmLm=S1-S2.By substituting (8) in (7), the charging current im is expressed as(9)i˙m=-S2im+S1im+Q.


## 3.1. Regenerative Braking Process Analysis

Topological structure of three-phase driving circuit for brushless DC machine is shown in Figure1.Figure 1
Power circuit structure.With phasesa and b, for example, when EV travels in drive power modes, the power switch T1 is in PWM and the others are off. The power system is in the buck running mode, which is shown in Figure 2.Figure 2
The electrical circuit of driving of BLDCM.When EV travels in electric power generation, the BLDCM runs at generating mode and the power circuit is the same with drive power mode. The current flows in the stator windings in opposite direction. According to rotor position signals, the control system modulates the low power switches of inverter and charges the battery based on the boost chopper principle by self-inductance of windings. The kinetic energy of EV is converted to electric energy to feed the battery.With phasesa and b, for example, the power switch T2 runs under the PWM mode in the process of generative braking. The others are in stop state. The battery is charged by raising back-emf of BLDCM. The electrical circuit of regenerative braking of BLDCM is shown in Figure 3.Figure 3
The electrical circuit of regenerative braking of BLDCM.Equivalent circuit of regenerative braking is shown in Figure4. rt is equivalent series resistance of power switch, rd is equivalent series resistance of diode, La, ra, and ea are inductance, resistance, and back-emf of phase a, respectively, Lb, rb, and eb are inductance, resistance, and back-emf of phase b, respectively, ib is braking current, and im is current of stator winding.Figure 4
Equivalent circuit of regenerative braking.According to the voltage equation, the three stator windings can be regarded as the same; there areLa=Lb=Lm, ea=eb=em, and ra=rb=rm.During theT2 turning on (0≤t≤d·T), the voltage balance equation of main circuit is expressed as(3)2Lmdimdt=2em-im·2rm+rd+rt.During theT2 turning off (d·T<t≤T), the voltage balance equation of main circuit is expressed as(4)2Lmdimdt=2em-vb-im·2rm+2rd+rbat,where d is PWM signal’s duty cycle, T is PWM signal’s period, and rbat is internal resistance of battery.By averaging PWM on-time and PWM off-time ofT2, the voltage balance equation is expressed as(5)2Lmdimdt=mim+2em-1-dvb,where m=(d-2)rd-drt+(d-1)rbat-2rm.The electromagnetic torque can be expressed as(6)Jdωdt=-Kt·im-TL,where Kt is torque coefficient, TL is the load torque, J is the combined machine and load inertia coefficient, and ω is rotor angular velocity.


## 3.2. Modeling of Charging Current

The EV usually adopts battery as energy storage equipment. For protecting the battery against the damage by large charging current, the charging current is controlled. So, (5) can be rewritten as(7)i˙m=Sim+Q,where S=((d-2)rd-drt+(d-1)rbat-2rm)/2Lm and Q=[2em-(1-d)vb]/2Lm.S is decomposed into two parts and expressed as(8)S=d-2rd-drt+d-1rbat2Lm-rmLm=S1-S2.By substituting (8) in (7), the charging current im is expressed as(9)i˙m=-S2im+S1im+Q.


## 4. Design of Regenerative Braking ADRC Controller

### 4.1. ADRC

The typical ADRC consists of nonlinear tracking differentiator (TD), extended state observer (ESO), and nonlinear state error feedback control law (NLSEF) [14, 15]. By using extended state to observe the disturbances, nonlinear uncertain system can be approximate linearization. The second-order ADRC is shown in Figure 5.Figure 5
Flowchart of the ADRC controller.v is input signal, v1 is tracking input signal, v2 is the differential v1, u is control signal, y is system output, z1, z2, and z3 are state variables, e0 and e1 are state error, u0 is uncompensated control signal, and b is constant.Inn-order system, the TD is used to export the transient process tracking command and derivative. If the plant is one-order system, the TD becomes a filter. The disturbances observed by the ESO are entered into nonlinear state error function and the control signal is obtained.The ESO of the second-order system can be expressed as(10)z˙1=z1-e0,z˙2=z3-β01·falε1,α1,δ1+b·u,z˙3=β02·falε2,α2,δ2,where β01, β02, α1, α2, ε1, ε2, δ1, and δ2 are adjustable parameters.Nonlinear function,fal(ε,α,δ), can be denoted by(11)falε,α,δ=εαsign⁡ε,ε>δεδ1-α,ε≤δ.


### 4.2. Regenerative Braking ADRC Controller

In the braking current model, the back-emf and internal resistance and so on are difficult to detect. According to the function of ESO,(S1im+Q) in (9) is regarded as disturbances; the second-order ESO of the braking current can obtained as follows:(12)ed=zd01-im,z˙d01=-A2im+zd02-βd01faled,αd01,δd,z˙d02=-βd02faled,αd02,δd,where zd01 is used to estimate braking current, zd02 is extended state variable, and ed equals the difference between estimator and output current.In the charging process of regenerative braking, the braking currentim is first-order model. NLSEF is established as(13)Δim=zd01-i∗,im∗=kdfalΔim,αm,δm.The regenerative braking control model of BLDCM is shown in Figure6.Figure 6
Flowchart of regenerative braking of BLDCM.


## 4.1. ADRC

The typical ADRC consists of nonlinear tracking differentiator (TD), extended state observer (ESO), and nonlinear state error feedback control law (NLSEF) [14, 15]. By using extended state to observe the disturbances, nonlinear uncertain system can be approximate linearization. The second-order ADRC is shown in Figure 5.Figure 5
Flowchart of the ADRC controller.v is input signal, v1 is tracking input signal, v2 is the differential v1, u is control signal, y is system output, z1, z2, and z3 are state variables, e0 and e1 are state error, u0 is uncompensated control signal, and b is constant.Inn-order system, the TD is used to export the transient process tracking command and derivative. If the plant is one-order system, the TD becomes a filter. The disturbances observed by the ESO are entered into nonlinear state error function and the control signal is obtained.The ESO of the second-order system can be expressed as(10)z˙1=z1-e0,z˙2=z3-β01·falε1,α1,δ1+b·u,z˙3=β02·falε2,α2,δ2,where β01, β02, α1, α2, ε1, ε2, δ1, and δ2 are adjustable parameters.Nonlinear function,fal(ε,α,δ), can be denoted by(11)falε,α,δ=εαsign⁡ε,ε>δεδ1-α,ε≤δ.


## 4.2. Regenerative Braking ADRC Controller

In the braking current model, the back-emf and internal resistance and so on are difficult to detect. According to the function of ESO,(S1im+Q) in (9) is regarded as disturbances; the second-order ESO of the braking current can obtained as follows:(12)ed=zd01-im,z˙d01=-A2im+zd02-βd01faled,αd01,δd,z˙d02=-βd02faled,αd02,δd,where zd01 is used to estimate braking current, zd02 is extended state variable, and ed equals the difference between estimator and output current.In the charging process of regenerative braking, the braking currentim is first-order model. NLSEF is established as(13)Δim=zd01-i∗,im∗=kdfalΔim,αm,δm.The regenerative braking control model of BLDCM is shown in Figure6.Figure 6
Flowchart of regenerative braking of BLDCM.


## 5. Results and Analysis

### 5.1. Simulation Modeling

A simulation model of regenerative braking of the BLDCM shown in Figure6 is implemented using Matlab/Simulink to verify the braking current model and the proposed ADRC control method, as well as in comparison with standard PI controller.Initial velocity and the battery voltage in regenerative braking are principal influence factors. Under different initial velocity and the battery voltage conditions, the simulations of regenerative braking were done by using ADRC controller and PI controller, respectively. The braking current instruction value is 10 A in regenerative braking.βd01=3000, βd02=10000, αd01=αm=0.25, αd02=0.5, δd=δm=0.1, and kd=1.6. The im is obtained by detecting the current of DC bus.Figures7 and 8 show the braking current response in the simulation.Figure 7
The braking current response (a DC-link voltage of 290 V and initial velocity of 200 rad/s).Figure 8
The braking current response (a DC-link voltage of 280 V and initial velocity of 150 rad/s).It is seen that the amplitude of the actual braking current is less than that of the reference by using PI controller. When using ADRC controller, the braking current response and steady-state error are improved. Even if the initial velocity and volt DC change, the ADRC controller still maintains the good control effect.


### 5.2. Experimental Result

In order to verify the validity of the designed model in real system, the two-wheel electrical motorcycle is chosen for experiment. The battery voltage is 48 V. The constant braking current control is used. The braking current reference is 8 A. Figures9 and 10 show the braking current responses, respectively.Figure 9
The braking current response at 35 kph.
(a)
PI controller
(b)
ADRC controllerFigure 10
The braking current response at 20 kph.
(a)
PI controller
(b)
ADRC controllerThe change of the battery voltage is small in the process of one-time regenerative braking recovery. If the battery voltage is viewed as constant, the amount of energy recovery is proportional to braking current within a given period. Faster response and smaller steady-state error of braking current mean that the energy recovery increase and the recovery efficiency is improved. Simulation results show that the performance of braking current ADRC controller is superior to that of PI controller.


## 5.1. Simulation Modeling

A simulation model of regenerative braking of the BLDCM shown in Figure6 is implemented using Matlab/Simulink to verify the braking current model and the proposed ADRC control method, as well as in comparison with standard PI controller.Initial velocity and the battery voltage in regenerative braking are principal influence factors. Under different initial velocity and the battery voltage conditions, the simulations of regenerative braking were done by using ADRC controller and PI controller, respectively. The braking current instruction value is 10 A in regenerative braking.βd01=3000, βd02=10000, αd01=αm=0.25, αd02=0.5, δd=δm=0.1, and kd=1.6. The im is obtained by detecting the current of DC bus.Figures7 and 8 show the braking current response in the simulation.Figure 7
The braking current response (a DC-link voltage of 290 V and initial velocity of 200 rad/s).Figure 8
The braking current response (a DC-link voltage of 280 V and initial velocity of 150 rad/s).It is seen that the amplitude of the actual braking current is less than that of the reference by using PI controller. When using ADRC controller, the braking current response and steady-state error are improved. Even if the initial velocity and volt DC change, the ADRC controller still maintains the good control effect.


## 5.2. Experimental Result

In order to verify the validity of the designed model in real system, the two-wheel electrical motorcycle is chosen for experiment. The battery voltage is 48 V. The constant braking current control is used. The braking current reference is 8 A. Figures9 and 10 show the braking current responses, respectively.Figure 9
The braking current response at 35 kph.
(a)
PI controller
(b)
ADRC controllerFigure 10
The braking current response at 20 kph.
(a)
PI controller
(b)
ADRC controllerThe change of the battery voltage is small in the process of one-time regenerative braking recovery. If the battery voltage is viewed as constant, the amount of energy recovery is proportional to braking current within a given period. Faster response and smaller steady-state error of braking current mean that the energy recovery increase and the recovery efficiency is improved. Simulation results show that the performance of braking current ADRC controller is superior to that of PI controller.


## 6. Conclusions

In this paper, on the basis of the equivalent circuit of regenerative braking, the braking current model of BLDCM has been discussed in the battery-powered EV regenerative braking energy recovery. The ADRC control method of braking current based on ADRC and braking current model has been proposed.The performance of the ADRC controller is verified by modeling based on Matlab/Simulink and experimental result.



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*Source: 371725-2015-08-09.xml*