University of Huddersfield Repository · 2016. 4. 7. · a cosimulation platform is developed by using MATLAB/Simulink and CarSim sotware packages, achieving a comprehensive simulation

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Li, Guoxing, Wang, Tie, Zhang, Ruiliang, Gu, Fengshou and Shen, Jinxian

An Improved Optimal Slip Ratio Prediction considering Tyre Inflation Pressure Changes

Original Citation

Li, Guoxing, Wang, Tie, Zhang, Ruiliang, Gu, Fengshou and Shen, Jinxian (2015) An Improved

Optimal Slip Ratio Prediction considering Tyre Inflation Pressure Changes. Journal of Control

Science and Engineering, 2015. pp. 1-8. ISSN 1687-5249

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Research Article

An Improved Optimal Slip Ratio Prediction consideringTyre Inflation Pressure Changes

Guoxing Li,1,2 Tie Wang,1 Ruiliang Zhang,1 Fengshou Gu,1,2 and Jinxian Shen1

1Department of Vehicle Engineering, Taiyuan University of Technology, Taiyuan 030024, China2Centre for Eiciency and Performance Engineering, University of Huddersield, Huddersield HD1 3DH, UK

Correspondence should be addressed to Tie Wang; [email protected]

Received 7 August 2015; Revised 2 November 2015; Accepted 10 November 2015

Academic Editor: Petko Petkov

Copyright © 2015 Guoxing Li et al. his is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

he prediction of optimal slip ratio is crucial to vehicle control systems.Many studies have veriied there is a deinitive impact of tyrepressure change on the optimal slip ratio. However, the existing method of optimal slip ratio prediction has not taken into accountthe inluence of tyre pressure changes. By introducing a second-order factor, an improved optimal slip ratio prediction consideringtyre inlation pressure is proposed in this paper. In order to verify and evaluate the performance of the improved prediction,a cosimulation platform is developed by using MATLAB/Simulink and CarSim sotware packages, achieving a comprehensivesimulation study of vehicle braking performance cooperated with an ABS controller. he simulation results show that the brakingdistances and braking time under diferent tyre pressures and initial braking speeds are efectively shortened with the improvedprediction of optimal slip ratio. When the tyre pressure is slightly lower than the nominal pressure, the diference of brakingperformances between original optimal slip ratio and improved optimal slip ratio is the most obvious.

1. Introduction

he longitudinal motion of a vehicle is governed by the forcesgenerated between the tyres and the road surface. herefore,acquiring enough tyre friction is crucial to enhance vehicledynamics. According to the friction principle, the magnitudeof frictional force depends on two factors: normal pressureand friction coeicient. However, the relationship betweenthe longitudinal friction coeicient and wheel slip ratio iscomplex. In the premise of constant normal pressure, whenthe wheel slip ratio is small, longitudinal force linearlyincreaseswith slip ratio.With further increase of the slip ratio,longitudinal force increases and then decreases nonlinearly.When the longitudinal force reaches a maximum value, thecorresponding slip ratio is called optimal slip ratio.

Antilock braking system (ABS) is an automobile safetysystem that allows the vehicle wheels to maintain tractivecontact with the road surface according to driver inputsduring the braking process, preventing the wheels from bothlocking up and uncontrolled skidding. he logic thresholdcontrol method is widely applied in commercial ABS prod-ucts [1]. As an experience based control method, in the

control process, wheel slip ratio is not maintained in optimalslip ratio but luctuated near it which cannot acquire thebest braking efect. Meanwhile, if slip ratio is consideredas the control target, ABS controller can maintain thepractical slip ratio near the optimal slip ratio all the timeduring the braking process, so that vehicle controllability andstability are optimized and maximized. hat is consideredas the ideal braking method. he research on ABS control,aiming at optimal slip ratio, has been carried out for manyyears. Most researchers put the emphasis on control strategyoptimization and development of control methods based onthe control theory of self-turning PID, fuzzy PID, artiicialneural network, and so on [2–4]. Other researchers focusedon identiication of road surface and optimal slip ratio [5–7].

Up to now, most studies about optimal slip ratio controlare based on an optimal slip ratio estimation expression,proposed by Liu [8] and Bian [9], which gives a quantiicationdescription of inluence on optimal slip ratio due to changesof road adhesion coeicient, vehicle velocity, and tyre slipangle. he optimal slip ratio expression proposed by Liuet al. is mainly developed from the tyre Magic Formula,proposed by Pacejka [10], using regression analysis. In the

Hindawi Publishing CorporationJournal of Control Science and EngineeringVolume 2015, Article ID 512024, 8 pageshttp://dx.doi.org/10.1155/2015/512024

2 Journal of Control Science and Engineering

early time, the inluence of tyre inlation pressure changeswas not considered into Magic Formula based on which theinluence has not been put into existing optimal slip ratioexpression.

However, it is indicated in [11–13] that tyre inlation pres-sure changes can directly inluence the relationship betweentyre longitudinal force and slip ratio. hat is, the value ofoptimal slip ratio is inluenced by tyre inlation pressurechanges.

Based on the existing optimal slip ratio expression andan improved Magic Formula model [14], a simulation islaunched to make a study on relationship between tyreinlation pressure changes and tyre slip ratio. A second-orderfactor, representing the inluence of tyre inlation pressurechanges on the value of optimal slip ratio, is acquired. Inorder to verify and evaluate the improved optimal slip ratio,an ABS controller with optimal slip ratio as the control targetis established. Vehicle braking processes before and aterimprovement are simulated and compared.

hepaper is structured as follows. Section 2 gives a simpleintroduction of tyre dynamic principle and optimal slip ratio.Section 3 describes the proposed improved optimal slip ratioconsidering tyre inlation pressures. Section 4 describes themodelling of cosimulation system which consists of a fuzzyPID controller for ABS system and vehicle dynamics modelin MATLAB/Simulink and CarSim sotware, respectively.Section 5 veriies the improved expression through compara-tive analyses and discussions on various scenarios.he paperis concluded in Section 6.

2. Wheel Dynamics and Optimal Slip Ratio

he dynamic diferential equations for the calculation oflongitudinal motion of a vehicle are described as follows:

�� = −��, (1)

�� = �� − �� − ��, (2)

�� = �� (�) ��, (3)

where � is��,��, ��, and ��;� and � are a quarter of the vehiclemass and wheel velocity. �� is the driving resistance; � is thewheel inertia; � is the wheel rotational speed; �� and �� arethe braking torque and the rolling resistance torque; �� and�� are the friction coeicient and normal force of � wheel, asshown in Figure 1.

During the braking process, the wheel speed � can belarger than its rotation speed �� which is characterised bythe wheel longitudinal slip �:

� = � − �� . (4)

As shown in (3), the longitudinal force�� is described as afunction that depends on the longitudinal friction coeicient�(�). If the longitudinal slip � is small, the relationshipbetween the longitudinal force and slip is linear, but, with afurther increase of the slip, the longitudinal force reaches the

Tb

Fz

Fx

�

G

O

u

Figure 1: Single wheel model.

maximum at the certain value of the slip speciied by tyre-road adhesion and is saturated beyond that. When the lon-gitudinal force reaches a maximum value, the correspondingslip value is referred to as the optimal slip ratio.

In essence, the principle of ABS control is to alwaysmaintain thewheel slip ratio near the optimal slip ratio, whichensures a continuous maximum value onto tyre longitudinalforce. In order to acquire the best braking efect, ABS schemesusually take the optimal slip ratio as the control target. Upto now, researchers have made numerous studies on theinluence of road adhesion coeicient, vehicle velocity, andtyre slip angle changes on the optimal slip ratio [1–4].

Most studies about optimal slip ratio control focused onthe real-time identiication and estimation of unknown roadcondition, vehicle velocity, and tyre slip angle. Furthermore,based on a series of advanced control methods, ABS con-troller is optimized to keep the real wheel slip ratio closer tothe theoretical optimal value. In principle, previous studieshave been carried out mainly on the basis of the optimal slipratio expression proposed by Liu [8] and Bian [9]:

�op = �0 + 0.165 × log(64� ) + 0.01 × �1.5, (5)

where �op is the optimal slip ratio, �0 is the road surfacefriction coeicient, � is wheel velocity, and � is the tyre slipangle. Equation (5) was developed on the basis of the tyreMagic Formula in the period from 1989 to 1994, excludingthe inluence of tyre inlation pressure changes. However, anincreasing number of researches indicate that tyre inlationpressure change causes a direct inluence on the relationshipbetween tyre longitudinal force and slip ratio, which meansthat the value of optimal slip ratio could be inluenced by tyreinlation pressures [6–8].

3. Optimal Slip Ratio Prediction consideringthe Tyre Inflation Pressure

In order to account for the inluence of the tyre inla-tion pressure on the longitudinal friction characteristics,

Journal of Control Science and Engineering 3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

500

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3900400041004200430044004500460047004800

� (—)

Fx

(N)

pi = 0.022MPa

pi = 0.440MPa

(a)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

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0.01

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Tyre pressure (Pa)

O�

set

of

op

tim

al s

lip

rat

io�

op

(—)

−0.04

−0.03

−0.02

−0.01

×105

(b)

Figure 2: (a) ��-� curves under diferent tyre inlation pressures. (b) Relationship between optimal slip ratio and tyre inlation pressures.

Besselink et al. [14] modiied and improved the longitudinaltyre characteristic formula and related parameters. he mainefects of tyre inlation pressure changes on the longitudinaltyre characteristics are identiied as follows [10, 11]:

(i) Changes in longitudinal slip stifness and camberstifness.

(ii) Changes in longitudinal peak friction coeicient.

herefore, the Magic Formula for predicting longitudinalforce can be improved to

�� = (�� sin [��⋅ arctan {�� − �� (�� − arctan (��))}]+ ��) ��,

(6)

where �� is the overall longitudinal slip;�� is the longitudinalstifness factor; �� is shape factor for longitudinal force; ��is the peak value factor; �� is the vertical force; �� is thelongitudinal curvature factor; �

V� is vertical shit factor; and�� is comprehensive factor for combined slip, which arecalculated by

�� = ��, (7)

�� = �� , (8)

�� = ��,nom ⋅ (1 + ��1 ⋅ �� + ��1 ⋅ ��2� ) , (9)

�� = ��,nom ⋅ (1 + ��3 ⋅ �� + ��4 ⋅ ��2� ) , (10)

where �,�, ��,nom, and��,nom are longitudinal peak frictioncoeicient, longitudinal slip stifness, and their correspond-ing nominal values, respectively. �� and ��0 are measuredpressure and nominal tyre inlation pressure. And �� =(�� − ��0)/��0 is dimensionless increment of tyre inlationpressure. As can be seen in both (9) and (10), there is an

additional factor (1 + �� ⋅ �� + ��(�+1) ⋅ ��2) whichis the product of the inlation pressure increment �� andMagic Formula parameter ��, highlighting the inluencesof pressure changes.

In order to investigate the quantitative inluence of tyreinlation pressure changes on the optimal slip ratio, with tyreinlation pressure being considered as independent variable,relation curves of longitudinal force and slip ratio under dif-ferent tyre inlation pressures, ��-�, are presented in Figure 2based on the improved Magic Formula of (6). he peaklongitudinal friction point of each curve is crucially markedand the set of optimal slip ratio points corresponding to thetyre pressure input is acquired. In the igure, the results wereplaced on a standard 205/60-R15 tyre under a vertical loadof 5000N. To examine the inluences of pressure changes,the tyre pressure changes from 0.1 times to 2 times of thenominal pressure, 0.22MPa, obtaining twenty corresponding��-� curves as illustrated in Figure 2(a). As shown by themaximum longitudinal friction highlighted with the circlemarkers, the maximum longitudinal friction increases withdecreasing in tyre pressure, whereas the optimal slip ratiodecreases correspondingly. When the tyre inlation pressureis higher than the nominal value, the optimal slip ratio irstlyincreases and then decreases slightly. Fitting results show thatthe quadratic itted curve correlates to the point set of optimalslip ratio very well with a correlation coeicient of more than0.982.

In order to take quantitative efect of inlation pressureinto account, the optimal slip ratio in (5) can be modiied byincluding twomore terms as shown in the following equation:

�op = �op,nom + ��1 ⋅ �� + ��2 ⋅ ��2�= �0 + 0.165 × log(64� ) + 0.01 × �

1.5 + ��1 ⋅ ��+ ��2 ⋅ ��2� ,

(11)

where ��1 is the linear inluence coeicient of tyre inlationpressure changes and ��2 is the quadratic inluence coei-cient, both of which are decided by the characteristics of thetyre itself. he impact on the optimal slip ratio given by tyreinlation pressure changes has an extremely small coherencewith other impact factors.herefore, an independent second-order factor should be essentially added into the optimal slipratio prediction.


Table 1: Simulation parameters.

�0 0.175 Tyre type 205/60-R15 ��1/��2 −0.349/0.378V0�/(km/h) 100/80/60/40 Vehicle type Sedan/C-class ��3/��4 −0.096/0.065�/∘ 0 Engine power/kW 250 ��1 −0.213��/MPa 0.132/0.176/0.22/0.264/0.308 Max. braketorque/Nm 2000 ��2 0.179

TPMS PID controller

Fuzzy inference system

Brake torque

Vehicle dynamicsystem

Road model

+−

�op

� = (u − R�)/u

Figure 3: he block diagram of braking system based on the improved optimal slip ratio.

4. ABS Controller Based on the ImprovedOptimal Slip Ratio

he change of optimal slip ratio value can lead to a directimpact on the braking performance of ABS controller, whichin turn changes both the braking distance and total brakingtime. In order to verify the improved expression and studythe impact of tyre inlation pressure changes on brakingperformance, an ABS braking system based on the improvedoptimal slip ratio is designed, as shown in Figure 3. ABSbraking system is composed of 4 parts: optimal slip ratio�op estimator, slip ratio comparator, fuzzy PID controller,and execution system. here are 3 inputs for �op estimator:tyre inlation pressure, vehicle velocity, and tyre slip angle.he road surface friction coeicient is commonly acquiredby road type recognition algorithm. In order to simplify thecomputation, road surface friction coeicient is assigned to aconstant value in this study. Real-time signal of tyre inlationpressure is acquired by directmeasurement, such as commer-cially promoted Tyre Pressure Monitoring System (TPMS).he fuzzy PID controller is composed of a conventional PIDcontroller and a fuzzy inference system.

Taking the error between actual slip rate and the optimalslip ratio � and error rate �� as inputs, fuzzy controllerprovides proportional coeicient ��, integral coeicient��, and derivative coeicient ��, its own output linguisticvariables, as the input of PID controller. Design and selectionof self-tuning principle, fuzzy control-rule table, membershipfunctions, and universe range for the PID controller arereferred to in [12, 13].

In order to comprehensively study the braking processand performance before and ater the improvement of theoptimal slip ratio, a cosimulation platform is establishedby combining the CarSim sotware and MATLAB/Simulinkpackage based on a C-class passenger car, which then allowthe simulation studies to be performed to evaluate thebraking performance under diferent conditions. he time

step is set to 0.01 s and Runge-Kutta ode45 algorithm is used.Other key simulation parameters are shown in Table 1.

5. Results and Discussion

5.1. Inluence of Tyre Inlation Pressure on Wheel Velocityand Displacement. Based on the cosimulation system, thebraking process and performance of the vehicle are examinedfor the improved optimal slip ratios by comparing themwith those of original ratios. he curves of vehicle brakingdisplacement andwheel velocity under diferent tyre inlationpressures and optimal slip ratios are shown in Figure 4.

If �� = 0.308MPa, for example, the wheel velocityof ABS controller aims at improved slip ratio signiicantlylower than the curve of the original one, rather close to thecurve of the nominal pressure throughout the second-halfbraking process as in Figure 4(a). In order to conirm thatthe change in wheel velocity is due to the improvement onABS controller, which makes the wheel velocity continuouslyadapted to the current optimal slip ratio, it is necessary toconduct a contrastive study of real-time slip ratios controlledby ABS controllers aiming at original and improved optimalslip ratios. Next, the real-time tyre slip ratio curves under twooptimal slip ratio expressions are presented in Figure 4(b).heoretically, according to (11) and Table 1, when tyre inla-tion pressure is larger than nominal pressure, the valueof optimal slip ratio considering tyre pressure will slightlyincrease compared with the traditional slip ratio. As is shownin Figure 4(b), ABS controller aims at improved optimalslip ratio to make the absolute value of the wheel slip ratiocontinuously greater than that of a traditional one, whichis consistent with the theory. herefore, the efectiveness ofimproved expression and ABS controller is proved.

he diferences of original velocity minus improvedvelocity, �old−�improved, show that a deviation occurs from thespeed under diferent tire pressure toward the speed corre-sponding to the nominal pressure, as is shown in Figure 4(c).


2.5 3.5 4.52 3 4

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80818283848586878889

pi = 0.220

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pi = 0.308-improved

(a)

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ty d

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pi = 0.264

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pi = 0.132

(c)

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pi = 0.308

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pi = 0.264-improved

pi = 0.220

pi = 0.176

pi = 0.132

pi = 0.220-improved

pi = 0.176-improved

pi = 0.132-improved

(d)

Figure 4: (a) Wheel velocities during the braking process. (b) Real-time tyre slip ratios during the braking process. (c) Diferential valuebetween the original velocity and the improved one. (d) Braking displacements under diferent tyre pressures and �op predictions.

In particular, when tyre pressure is higher than the nominalvalue, the wheel velocity of improved slip ratio is consistentlylower than that of original one during the braking process.hat is why the velocity diferences of �� = 0.264MPa and�� = 0.308MPa are always positive. When the pressure islower than the nominal value, the wheel velocity curve showsthe almost opposite trend. In addition, as to the low pressurecasewhen�� = 0.132MPa, thewheel velocity in the inceptionphase luctuates strongly and remains higher than the originalwheel velocity during the steady phase, which proves thatABS controller has made timely amendments to the wheelspeed based on the latest improved optimal slip ratio.

Compared with the original ABS controller, brakingdistances and braking time under all nonnominal tyre pres-sures become shorter, which proves that the improved ABScontroller, aiming at the improved slip ratio, has optimized

the wheel velocity and thus achieved the beneicial result asshown in Figure 4(d).

5.2. Inluence of Tyre Inlation Pressure on Braking Perfor-mance. In order to investigate the impact of tyre inlationpressure changes on vehicle braking performance (brakingdistance and braking time), simulation studies were con-ducted under diferent initial braking speeds on the basis ofthe original slip ratio and improved optimal slip ratio.

As shown in Figure 5(a), both original braking distanceand improved braking distance increase with the initialbraking speed. In addition, there is a slight rising trend withincreasing tyre pressure. he braking distances before andater improvement are very close. It is hard to distinguish theinluence of the improved slip ratio on the braking distance.Likewise, variation trend of the braking time is similar to that


0.132 0.176 0.220 0.264 0.3085

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fB

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inu

s B

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ved

(s)

0

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(d)

Figure 5: Braking performances under diferent initial braking speeds. (a) Comparison of braking distances before and ater improvement.(b) Comparison of braking time before and ater improvement. (c) Diference of original braking distance minus improved braking distance.(d) Diference of original braking time minus improved braking time.

of the braking distance, and the diference of braking time isalso hard to be identiied.

In order to compare and analyze the degree of reduction,the diference of braking distance and braking time beforeand ater improvement under each condition is calculated,respectively. As shown in Figures 5(c) and 5(d), underall tyre inlation pressures and initial braking speeds, thebraking distance and braking time ater the improvementare all efectively shortened. According to improved optimalslip ratio expression, the value of slip ratio decreases withlowering tyre pressure, and the corresponding longitudinalforce�� increases.When tyre inlation pressure is higher thanthe nominal value, with the increasing pressure, the peak of��-� curve began to migrate to the right, accompanied bythe shape deforming.hat is, when the tyre inlation pressureis diferent from the nominal value, the correspondinglongitudinal force of the optimal slip ratio, calculated by the

original prediction which does not consider the tyre pressurechanges, is not the real maximum longitudinal force. hecorresponding longitudinal force predicted by the improvedexpression considering the tyre pressure is always morecloser to the maximum braking force under the real-timetyre condition. It is obvious that the greater braking forceis the shorter braking time will be. herefore, the brakingperformance based on the improved optimal slip ratio hasbeen enhanced. So it is in the case of tyre pressure lower thannominal pressure.

Based on the above analysis, the essence of ABS control,aiming at the improved optimal slip ratio, is by relocating theoptimal slip ratio to keep real-time wheel slip ratio near theimproved value and hence the transient longitudinal force��,improved always in a maximum value under current tyrepressure.


It can also be known fromFigures 5(c) and 5(d) that whentyre inlation pressure is larger than nominal pressure dif-ferences of braking distance and braking time continuouslyincrease with tyre pressure increasing. When tyre pressureis slightly lower than nominal pressure, optimization efectof the improved expression reaches the optimum. If tyrepressure continuously decreases, it tends to be less efective.

he relationship between performance diferences andinitial braking speed, as is shown in Figures 5(c) and 5(d),is not clearly or consistently related under the controlof improved optimal slip ratio. his illustrates that theorthogonality between tyre pressure changes, wheel velocity,and other factors is good and introducing an independentsecond-order factor to characterize the inluence of tyreinlation pressure on optimal slip ratio is reasonable andessential.

6. Conclusions

(1) Based on theABS control aiming at improved optimalslip ratio, both vehicle braking distance and brakingtime, under all nonnominal tyre inlation pressures,are efectively shortened. It proves that ABS controllerdesigned to the improved optimal slip ratio can fairlyoptimize the braking process under diferent tyrepressures.

(2) hediference of braking performances between orig-inal and improved optimal slip ratios does not changelinearly with tyre inlation pressure changes. Whentyre inlation pressure is slightly lower than the nom-inal value, the diference of braking performances isthe most obvious.

(3) he inluence of tyre inlation pressure changes onthe prediction is less afected by initial braking speedof vehicle. Introducing an independent second-orderfactor to characterize the inluence of tyre inlationpressure on optimal slip ratio is essential and rational.

Notations

�: A quarter of the vehicle mass�: Wheel velocity��: Longitudinal tyre force�: Wheel inertia�: Wheel rotational speed��: Braking torque��: Rolling resistance torque

��: Friction coeicient��: Normal force of �th wheel�: Wheel radius�: Wheel longitudinal slip ratio�op: Optimal slip ratio�: Tyre slip angle��: Overall longitudinal slip��: Longitudinal stifness factor��: Shape factor for longitudinal force��: Peak value factor��: Longitudinal curvature factor

�V�: Vertical shit factor��: Combined slip factor�: Longitudinal slip stifness��: Tyre inlation pressure��0: Nominal tyre inlation pressure��: Dimensionless increment of tyre inlation

pressure��: Magic Formula parameter

��1: Linear inluence coeicient of tyreinlation pressure changes

��2: Quadratic inluence coeicient.

Conflict of Interests

he authors declare that there is no conlict of interestsregarding the publication of this paper.

Acknowledgments

he support of the High-Tech Industrialization Projects ofShanxi Province, China (no. 2011-2368) andhe Natural Sci-ence Foundation of Shanxi Province, China (no. 2012011024-2) is gratefully acknowledged.hanks are due to Dan Song ofState Grid Skill Training Centre of SEPC for her help in ABScontroller modelling. he authors acknowledge the valuablecomments and discussions with Hangbin Zhu of TASS Inter-national and Yonghong Wang of Shanxi Dayun AutomobileManufacturing Co., Ltd.

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