OPTIMAL CONTROL OF PARALLEL HYBRID ELECTRIC VEHICLES BASED ON THEORY OF SWITCHED SYSTEM

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1 274 Asan Journal of Control, Vol. 8, No. 3, pp , September 26 -Bref Paper- OPTIMAL CONTROL OF PARALLEL HYBRID ELECTRIC VEHICLES BASED ON THEORY OF SWITCHED SYSTEM Lcun Fang and Shyn Qn ABSTRACT The focus of ths paper s the control strategy used to control general parallel hybrd electrc vehcles (HEV). The torque splt control problem of HEV s formulated as the optmal control of a swtched system. A model-based strategy for fuel-optmal control s presented. The optmal control problem of such a swtched system s formulated as a two-stage optmzaton problem. Dynamc programmng s utlzed to determne the optmal control acton that mnmzes the cost functon. Smulated results ndcate that ths method s effectve. KeyWords: Hybrd electrc vehcles (HEV), optmal control, dynamc programmng, swtched system. I. INTRODUCTION The most challengng goals facng the automotve ndustry are ncreasng fuel economy and reducng emssons. There are a seres of constrants that are mposed not only by our socety but also by cooperatve agreements and legslatve efforts [1]. HEVs are begnnng to demonstrate ther capablty to satsfy these requrements, and they wll become a vable alternatve to conventonal vehcles n the future [2,3]. The hybrd powertran s an ntegrated system that may consst of the followng components: an nternal combuston engne (IC engne), a battery pack, and an electrc machne (EM) whch can be utlzed as a tracton motor or generator. In such a system, each sub-system s also a complex system whch has ts own functonalty and desred performance. Moreover, all of the sub-systems need to be coordnated n an optmal manner to acheve the desred objectves. Consequently, an ntegrated vehclelevel s requred to accomplsh such tasks [4]. Recent reports on HEVS have placed great emphass on the control strategy, and several approaches to t have been Manuscrpt receved July 18, 25; revsed January 25, 26, accepted May 15, 26. The authors are wth the School of Automaton Scence and Electrcal Engneerng, Behang Unversty, Bejng 183, Chna (e-mal: lcun_fang@buaa.edu.cn, qsy@buaa.edu.cn). proposed. The power management strateges can be roughly classfed nto three types: statc optmzaton methods [5,6], global optmzaton methods based on dynamc programmng [7,8], and rule-based control approaches [9,1]. HEV systems smultaneously exhbt several knds of dynamc behavor, such as contnuous-tme dynamcs, dscrete-tme dynamcs, jump phenomena, swtchng and the lke. Therefore, an HEV s a typcal hybrd dynamcal system, and many challenges exst n the desgn of a vehcle system for an HEV [4]. The focus of ths paper s the control strategy used to control general parallel HEVs. The remander of ths paper s organzed as follows. Secton 2 wll descrbe the herarchcal archtecture of the HEV system. The dfferent powertran components models used for smulaton and control are presented n secton 3. The optmal control problem s solved n secton 4. In secton 5, a case study nvestgated through smulaton s presented, and the results obtaned n optmzng the control of fuel consumpton based on the optmal strategy proposed n ths paper are gven. Conclusons are drawn n secton 6. II. CONTROL SYSTEM ARCHITECTURE An HEV s a typcal hybrd dynamcal system [4]. Most hybrd dynamcal systems have been desgned usng the concept of centralzed and dstrbuted control. A

2 L. Fang and S. Qn: Optmal Control of Parallel Hybrd Electrc Vehcles Based on Theory of Swtched Systems 275 two-level herarchcal control archtecture [11] s used n the synthess of the control law for the hybrd powertran shown n Fg. 1. The supervsory powertran (SPC) represents a hgh-level vehcle control system that can coordnate the overall powertran to satsfy certan performance ndces. Based on the drver s demand and operatng parameter feedback sgnals of the vehcle and ts components, the SPC must determne the desred propellng torque dstrbuton between the IC engne and EM, or the desred brake torque dstrbuton between the EM and frcton brake system, and the gear poston; therefore, the man functon of the SPC s to determne the desred output of varous sub-systems accordng to drver s commands, system parameter feedback sgnals and a pre-programmed control strategy. Dfferent control strateges wll result n dfferent fuel economy and emsson characterstcs. The focus of ths paper s the control strategy used to control general parallel HEVs. In order to narrow our research scope, only a double shaft parallel powertran s consdered. In ths confguraton, the engne power and electrc motor power can provde torque to the wheels separately through the transmsson. It should be ponted out that the strateges for dealng wth ths stuaton can also be appled to other powertran arrangements by makng approprate amendments. Fgure 2 shows a schematc dagram of the double shaft parallel HEV system consdered n ths paper. CAN bus Engne Drver Supervsory powertran control Motor/Battery Trans. Brake Engne Clutch Motor Trans. Vehcle The followng components are utlzed n the confguraton: spark gnton drect njecton engne: 5kW; permanent magnet motor: 1kW; NMh battery: 2kW, 2kWh; 5speed automated manual transmsson. III. SYSTEM MODEL A qualty dynamc model s paramount for the development of an effectve control strategy, especally when dealng wth a complex coordnated control problem such as a parallel HEV. An HEV powertran dynamc model descrbng key system components can be a sgnfcant ad to the development of the control law and provdes nsght nto the domnant dynamc effects n the control law synthess process. To ths end, the components of a parallel HEV are modeled. However, developng a vehcle s the key objectve of ths paper, so smplfed models are presented. Dynamc models of varous prmary HEV components, such as the ICE, motor, battery, transmsson and vehcle, are descrbed below. 3.1 Engne To avod the dependence on the avalablty of specfc effcency maps for the IC engne and the electrc motor whle optmzng and formulatng the HEV optmal control strategy, a unversal representaton, the Wllans lnes model of an IC engne and motor, s utlzed [12]. A bref summary of the parametrzaton s gven n the followng paragraphs. The scalablty of an IC engne s acheved through the use of two engne szng parameters, the dsplacement V d and the stroke S. Here, some concepts and correspondng varables, such as the mean pston speed c m, the avalable mean effectve pressure p ma and the mean effectve pressure p me, are ntroduced as follows: Sωe c m =, (1) π Battery Fg. 1. Herarchcal control archtecture of an HEV. P ma 4πQLHV m = V ω d e f, (2) Wheel Wheel Mechancal couplng Reducton gears Gear box Motor Clutch Fg. 2. Block dagram of the parallel HEV. Battery ICE P me 4πT = e, (3) V d where ω e s the ICE angular speed, T e s the ICE effectve torque, m f s the fuel mass flow rate, Q LHV s the fuel low heatng value, and e s the nner ablty to transform chemcal to mechancal energy. The engne s effcency can be approxmated by an affne relaton between the torque and fuel mass avalable n one cycle:

3 276 Asan Journal of Control, Vol. 8, No. 3, September 26 m f Te = e QLHV T ω e loss, (4) where T loss represents the nner losses. The resultng affne functon shows the domnant effect observed n all ICEs,.e., the dramatc decrease n effcency as the torque output decreases. Based on Eqs. (1) ~ (3) and some necessary substtuton n Eq. (4), a dmensonless defnton of engne effcency η can be obtaned: p me η =, (5) pma where Pme = epma Pml, (6) and p ml denotes the nner loss (see [12] for more detals). 3.2 Motor In ths secton, the same concepts that were ntroduced above for an IC engne are developed for an EM. Let p em be the avalable electrcal power. Then, for each tme nstant, the followng equaton defnes the effcency of the EM: ω T = η() p = η() V, (7) em em, e em where ω em and T em, e are the EM angular speed and effectve torque, respectvely; V and are the voltage and current, respectvely. Agan, as a frst approach, the motor s effcency can be approxmated by assumng that an affne relatonshp between the effectve torque and nput energy exsts: T = e V T em, e em em,loss ωem = e T T, (8) em em, a em,loss where e em, T em, a and T em, loss are the nner ablty to transform electrcal energy nto mechancal energy, based on the avalable mean effectve torque of the EM and nner loss (see [12] for more detals). 3.3 Battery The practcal mplementaton of ths control strategy s dependent on the avalablty of an on-lne SOC (State Of Charge) estmator. The battery performance, such as the voltage, current and effcency, from a purely electrc vewpont, s the outcome of thermally dependent electrochemcal processes that are qute complcated. A model based upon open crcut voltage s used n ths paper. Further detals about ths method can be found n [13]. 3.4 Transmsson Two man relatons descrbe the parallel double-shaft arrangement of an HEV. The frst one s the speed relaton: ω e( k) ω em( k) ω w( k) = =. (9) Rk ( ) ρ The second one s the torque relaton: Tw( k) = R( k) Te( k) η gb+ ρ Tm( k)η red, (1) where ω w s the wheel angular speed, R(k) s the gear rato, ρ s the reducton gear rato between the EM and the wheels, T w (k) s the torque of the wheels, and T e (k) and T m (k) are the output torque from the engne and the motor, respectvely. η gb denotes the effcency of the gear box, and η red s that of the reducton gears. 3.5 Vehcle The modelng of the vehcle dynamcs s derved from the basc equaton for sold-body motons (Newton s Second Law): 1 2 F = MgCr + Dar CD Af V 2 dv + M + Mgsn(θ), (11) dt where F s the force requred at the vehcle wheels to reach a certan acceleraton at speed V ; Mg, C r, D ar, C D, A f, M, and θ are the mass of the vehcle, the local acceleraton of gravty, the coeffcent of the rollng resstance between the tres and the road surface, the ar densty, the coeffcent of the aerodynamc drag for the vehcle n the travelng drecton, the frontal area of the vehcle, the nertal mass of the vehcle, ncludng the rotatonal nerta contrbuton, and the gradent of the road, respectvely. The vehcle acceleraton a and speed V are gven by 1 2 η gb RkT ( ) / r MgCr Dar CD Af V mgsn(θ) a = 2, M (12) V = adt, (13) where r s the radus of the wheels. IV. OPTIMAL CONTROL 4.1 Problem Formulaton In the operaton process of a powertran, the transmsson determnes the gear shftng sequence and selects the approprate transmsson gear based on the transmsson output speed, the poston of the acceleraton pedal and brakng pedal, the current gear poston and the clutch state. The powertran system can be formulated as a swtched system. As s well-known, a dscrete-tme swtched system can be descrbed as a set of pece-wse

4 L. Fang and S. Qn: Optmal Control of Parallel Hybrd Electrc Vehcles Based on Theory of Swtched Systems 277 dfference equatons [14]: x( k + 1) = f( x( k), u( k), m( k)) f1 ( x( k), u( k)), f m( k) = 1 = (14) fi ( xk ( ), uk ( )), f mk ( ) = I, where x(k) s a state vector of the system, u(k) s a vector of the contnuous-tme control varables, and m(k) {1, 2, I} s a fnte nteger set n whch every element corresponds to a knd of operatng mode wth event attrbutes. An HEV can be consdered as a specal case of ths model. For a vehcle control system, the contnuous-tme state vector s (V, SOC); the control varables u(k) are T e (k) and T m (k); m(k) s the gear poston, whch s a dscrete-event state varable. If the state varables and control varables are nserted nto Eq. (14), a set of pece-wse dfference equatons for an HEV can be obtaned. It should be ponted out that the motor torque becomes a dependent varable nstead of a control varable due to the equalty constrant on the drvelne torque: Twh _req = Te ( k) + Tm ( k), (15) where T wh_req s the requred torque of the vehcle. For a vehcle control system, once a drvng cycle s gven, the wheel speed profle w wh_req s known, and the torque T wh_req requred to follow the speed profle can be determned n each tme step by nversely solvng the dynamc equatons (11) ~ (13). The practcal wheel speed w wh can be detected by a speed sensor. Problem 1. The power control of an HEV can be formulated as an optmal control problem, n whch the optmzaton goal s to fnd the control nput T e (k) and swtchng sequence of gear shftng g(k) needed to mnmze a cost functon n a gven drvng cycle: N 1 J = ψ( x( N)) + L( x( k), Te ( k), m( k)) k = N 1 + M k( x, x ' ), (16) k = where N s the duraton of the drvng cycle, ψ (x (N)) s the cost functon assocated wth the error n the termnal SOC, L(x (k), T e (k), m (k)) s the nstantaneous fuel consumpton rate, and x, x represent dfferent gear postons. M k (x, x ) s the cost functon of mode swtchng between x and x ; n order to smplfy the problem, the cost functon of mode swtchng s assumed to be a constant set and to take values from ths set. In order to ensure drvablty of the vehcle and safe/smooth operaton of each subsystem, t s necessary to satsfy the followng constrants: T ( w ( k)) T T ( w ( k)), (17) e_mn e e e_max e where T e_mn (w e (k)) and T e_max (w e (k)) are the mnmal and maxmal output torque of the ICE at speed w e (k); w e_mn w e ( k) w e_max, (18) where w e_mn and w e_max are the mnmal and maxmal speeds of the ICE; SOC SOC( k) SOC, (19) mn where SOC mn and SOC max are the lower lmt and upper lmt of the SOC of the battery; m_max max Tm_mn( wm( k), SOC( k)) Tm( k) T ( w ( k),soc( k)), m (2) where T m_mn (w m (k), SOC(k)) and T m_max (w m (k), SOC(k)) are the mnmal and maxmal output torque of the EM at speed w m (k) for the battery SOC(k). 4.2 Soluton For the optmal control problem of a swtched system, the followng condtons must be satsfed [15]. Condton 1. An optmal soluton (σ *, u * ) exsts, where σ * s the optmal swtchng sequence and u * s the optmal nput control. Condton 2. For any gven swtchng sequence σ, there exsts a correspondng u * = u * (σ) such that J σ (u) = J(σ, u) s mnmzed. If the above condtons are satsfed, then, the followng equaton holds: mn J ( σ, u) = mn mn J ( σ, u). (21) σ, u U σ u U Therefore, the optmal control problem of a swtched system can be formulated as a two-stage optmzaton problem. Stage 1. (a) Fx the total number of swtches to be K, fx the order of the actve subsystem, let the mnmum value of J wth respect to u be a functon of the K swtchng nstants,.e., J 1 = J 1 (t 1, t 2,, t k ) for K, and then fnd J 1. (b) Mnmze J 1 wth respect to t 1, t 2,, t k. Stage 2. (a) Vary the order of the actve subsystem to fnd an optmal soluton under K swtches. (b) Vary the number of swtches K to fnd an optmal soluton. Dynamc programmng s a powerful tool for solvng general dynamc optmzaton problems. The man advantage s that t can easly deal wth the constrants and nonlnearty of such problems whle obtanng a globally optmal soluton. The dynamc programmng technque s based on the Bellman s Prncple of Optmalty, whch

5 278 Asan Journal of Control, Vol. 8, No. 3, September 26 states that the optmal polcy can be obtaned f we frst solve a one stage problem nvolvng only the last stage and then gradually extend t to the frst stage. As a matter of fact, dynamc programmng has been used to solve the optmal control of a swtched system [16]. By denotng Σ [k, N] as a subset of Σ [k, N] whch contans all σ Σ [k, N], startng wth subsystem at nstant k, one can compute the cost functons V (x(k), k), I by solvng Eq. (22), where, ' represent a subsystem and V (x (k), k) s the mnmum value of J f the system starts at tme k wth state x(k) and subsystem : V ( x( k), k) = mn mn J(σ, u) ' j {, } σ u U [ kn, ] ψ(( N)) f k = N mn{ Lxk ( ( ), uk ( )) = uk ( ) (22) j + mn V ( x( k + 1), k + 1)} f k < N. Consequently, the optmal soluton (σ, u) of a dscretetme swtched system can be constructed by solvng V (x(k), k) backwards and fnally fndng (x(), ). In a double-shaft parallel hybrd confguraton, condtons 1 and 2 are satsfed, and the gear rato of the transmsson only affects the torque contrbuton of the IC engne; consequently, we can dvde the development of the overall optmal control strategy nto two stages: (1) mplement a gear shftng strategy, whch selects the swtchng sequence of gears n a dscrete set Σ to optmze the operaton of the engne; (2) mplement a power splt strategy, whch defnes the best power splt between the ICE and EM. Wth the above two-stage optmzaton process, the optmal control polcy (T e (k), g(k)) can be constructed. 4.3 Problem Smplfcaton The bg challenge n utlzng the above algorthm to fnd the optmal feedback control polcy s overcomng the dffculty posed by the huge computaton burden. It s, thus, necessary to adopt some approaches to accelerate the computaton speed. In ths paper, we smplfy the above two steps as descrbed below. (1) Shftng Sequence of Gear Poston Determnng the gear shft strategy s crucal to mprovng the fuel economy of hybrd electrc vehcles [17]. The gear shftng sequence of an automatc transmsson can be modeled as a dscrete-tme dynamc system: M, mk ( ) + gk ( ) M mk ( + 1) = 1, mk ( ) + gk ( ) 1 mk ( ) + gk ( ), otherwse, (23) where M s the number of gear postons, and g(k) s constraned to take on values from among 1,, and 1, representng downshft, hold and upshft, respectvely. From the speed profle of a gven drvng cycle, the optmal gear poston can be constructed through statc optmzaton [9], the total swtchng numbers of gear poston can be fxed and the shftng sequence can be determned; consequently, the problem can be transformed nto an optmal control problem for a swtched system wth a fxed swtch number and fxed swtchng sequence. (2) The Optmal Torque Input The state and control varables are frst quantzed nto a fnte grd. For a gven drvng cycle, the vehcle model can be replaced by a fnte set of operatng ponts parameterzed by the requred torque and speed. Pre-computed look-up tables can be constructed for recordng the next states and cost functon as a functon of the quantzed states, control nputs and operatng ponts. Once these tables are bult, they can be used to solve the problem n a very effcent manner. The dynamc programmng procedure produces an * optmal, tme-varyng, state-feedback control polcy ( Te ( k ), g * ()) k for a drvng cycle. Ths control polcy can then be used as a state feedback. V. CASE STUDY To verfy the control polcy determned by the dynamc programmng algorthm, a case was studed and wll be dscussed n ths secton. The UDDS (Urban Dynamometer Drvng Schedule) drvng cycle s a standard test drvng cycle utlzed by the Envronmental Protecton Agency to certfy the fuel economy and emsson performance of vehcles whch are drven n urban areas. The optmal control polcy was found through DP (Dynamc Programmng) n a UDDS drvng cycle. In order to evaluate the fnal fuel economy, the optmal control law was appled to the full-order parallel HEV model. The termnal SOC constrant was selected as.65, and the ntal SOC n the smulaton was chosen as.65 for the purpose of calculatng the fuel economy. The dynamc trajectores of the vehcle under the optmal control polcy for the UDDS drvng cycle are shown n Fg. 3 and Fg. 4. Speed(km/h) Dfference of Tme(s) Fg. 3. Dfference between the requred and acheved speeds.

6 L. Fang and S. Qn: Optmal Control of Parallel Hybrd Electrc Vehcles Based on Theory of Swtched Systems 279 Speed(km/h) SOC Overall rato Fuel(L) B C 5.6 D.4.2 Tme(s) Fg. 4. Dynamc trajectores of the actual speed profle, battery SOC, fuel consumpton and overall rato. The largest dfference between the desred vehcle speed and the actual speed s wthn.4km/h, whch means that the control strategy can trace the objectve speed correctly. The SOC trajectory starts at.65 and ends at around.65 wth a small quantzaton error. The fuel consumpton of the control polcy s.452l, and the average fuel consumpton s 3.8 L/1Km. The results were compared wth the default control strategy n ADVISOR 22, that s, the electrcal peakng energy management strategy. The fuel consumpton of the control polcy s found to be.4695l, and the average fuel consumpton s 3.92 L/1Km, so the optmal control law s effectve n mprovng fuel economy. VI. CONCLUSIONS A The optmal torque splt problem of a parallel HEV has been formulated as the optmal control problem of a swtched system. A control strategy for general parallel HEVs has been developed n ths paper. The proposed optmal torque splt strategy for parallel HEVs, desgned wth the ad of dynamc programmng, can mprove the overall system effcency. A two-stage dynamc programmng algorthm has been used to solve the mnmum fuel optmal control problem for a parallel hybrd electrc vehcle. In order to reduce the computaton complexty, several approaches have been utlzed to accelerate the computng speed. A dynamc optmal soluton to the energy management problem over a drvng cycle has been obtaned. The smulaton results ndcate that mprovement n fuel economy s acheved by optmzng the gear shftng polcy and torque splttng strategy. In general, torque splt algorthms developed through DP are more accurate, so the approach proposed n ths paper can be utlzed to assess other control strateges. REFERENCES 1. Maclean, H.L. and L.B. Lave, Evaluaton Automoble Fuel/Propulson System Technologes, Progr. Energ. Combust. Sc., Vol. 29, pp (23). 2. Btsche, O. and G. Gutmann, System for Hybrd Cars, J. Power Sources, Vol. 127, pp (24). 3. Chau, K.T. and Y.S. Wang, Overvew of Power Management n Hybrd Electrc Vehcles, Energy Convers. Manage., Vol. 43, pp (22). 4. Phllps, A., M. Jankovc, and K. Baley, Vehcle System Controller Desgn for a Hybrd Electrc, Proc. IEEE Int. Conf. Contr. Appl., Alaska, pp (2). 5. Paganell, G., A. Brahma, G. Rzzon, and Y. Guezennec, Control Development for a Hybrd-Electrc Sport-Utlty Vehcles: Strategy, Implementaton and Feld Test Results, Proc. Amer. Contr. Conf., Vol. 6, pp (21). 6. Scarretta, A., M. Back, and L. Guzzella, Optmal Control of Parallel Hybrd Electrc Vehcles, IEEE Trans. Contr. Syst. Technol., Vol. 12, No. 3, pp (24). 7. Paganell, G., G. Ercole, A. Brahma, Y. Guezenne, and G. Rzzon, General Supervsory Control Polcy for the Energy Optmzaton of Charge-Sustanng Hybrd Electrc Vehcles, JSAE Rev., Vol. 22, No. 4, pp (21). 8. Delprat, S., J. Lauber, T.M. Guerra, and J. Rmaux, Control of a Parallel Hybrd Powertran: Optmal Control, IEEE Trans. Veh. Technol., Vol.??, No.??, pp (24). 9. Schouten, N., M. Salman, and N. Kher, Energy Management Strateges for Parallel Hybrd Vehcles Usng Fuzzy Logc, Contr. Eng. Pract., Vol.??, No.??, pp (23). 1. Pusca, R., Y. At-Amrat, A. Berthon, and J.M. Kauffmann, Fuzzy-Logc-Based Control Appled to a Hybrd Electrc Vehcle wth Four Separate Wheel Drves, IEE Proc. Contr. Theory Appl., Vol. 151, Issue 1, pp (24). 11. Ln, C.C., H. Peng, and J.W. Grzzle, Control System Development for an Advanced-Techmology Medum-Duty Hybrd Electrc Truck, SAE Paper, No (23). 12. Rzzon, G., L. Guzzella, and B.M. Baumann, Unfed Modelng of Hybrd Electrc Vehcle Drvetrans,

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