602 31,,.,,,,. HEV,.,, HEV,,,.,,. 2 (Basic assumptions) HEV, : 1) P ice (t), (continuously variable transmission, CVT) ω ice (t), T ice (t), 10 ; 2) (

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1 DOI: /CTA Control Theory & Applications Vol. 31 No. 5 May 2014, (, ) :.,. 5,,. : ; ; : U463 : A Real-time optimization control for hybrid electric vehicles based on quadratic performance index XIA Chao-ying, ZHANG Cong (School of Electrical Engineering and Automation, Tianjin University, Tianjin , China) Abstract: In order to solve the real-time optimization control problem for hybrid electric vehicles (HEVs), this paper proposes an optimal power split scheme based on the quadratic performance index optimization. Under several assumptions and approximations, a linear model of HEV is established. The control problem of HEVs is transformed to a quadratic optimal regulating problem, from which a real-time optimization control with simple construction is obtained. Simulation results under five driving cycles show that the proposed real-time method can achieve the optimization control for HEVs with the future driving conditions unknown. Furthermore, the fuel economy of this real-time method is found to be very close to that of the global optimal control calculated off-line aiming to minimize fuel consumption. Key words: quadratic optimal control; real-time control; hybrid electric vehicle 1 (Introduction),, (hybrid electric vehicle, HEV). HEV (state of charge, SOC),, HEV., HEV 1. SOC,., HEV 2.., (equivalent consumption minimization strategy, ECMS) 3 4. ECMS,., ECMS 5,., 6 7,., 8 9, (dynamic programming, DP) HEV,., (Pontryagin s minimum principle, PMP) DP. SOC,, PMP 10. DP PMP : ; : xiachaoying@126.com.

2 602 31,,.,,,,. HEV,.,, HEV,,,.,,. 2 (Basic assumptions) HEV, : 1) P ice (t), (continuously variable transmission, CVT) ω ice (t), T ice (t), 10 ; 2) ( ),,,, ; 3) SOC, 5 ; 4),,, HEV. 1 Fig. 1 Structure diagram of parallel hybrid electric vehicle,. SOC SOC Pice(t) Pess(t),, Tice(t), ωice(t), Tice(t) ωice(t) = Pice(t),, ωice(t), Pess(t). 3 (Acquisition of object parameters and the linear model) ADVISOR HEV, 1., FC SI41 emis, MAP 2. 1 HEV Table 1 Parameters of HEV 1287 kg 41 kw / 12 Ah/320 V 49 kw 2.0 m Fig. 2 Optimal operating line of the engine 1),, ṁ(p ice (t))( 3), ṁ(p ice (t)). HEV, HEV., M HEV, v F M dv dt + F r = F, (1) F r. (1) v, 1 2 M dv2 dt + F rv = P ice + P ess, (2)

3 5 : 603 : P ice, P ess. ( (2) F r v) 4. MATLAB cftool fv 2,, 1 f = 17.23, 1 2 M dv2 dt + fv2 = P ice + P ess. (3) 3 FC SI41 emis Fig. 3 Optimal fuel consumption line of FC SI41 emis 4 Fig. 4 Relation between vehicle velocity and the power to overcome resistance, 6 Ah/320 V, SOC V E P bat (t) dsoc(t) E = V (V V 2 4RP bat (t)). (4) dt P bat (t) 30 kw, 0 0, 30 kw SOC 0.6, 0.8, P bat (t)/( V (V V 2 4RP bat (t)) ) , 1 1, (P bat (t) 0, P bat (t) < 0 ), (4) { , P bat (t) 0, η bat = , P bat (t) < 0. 2), η m, P bat (t) = P ess (t)/η m (5) { , P bat (t) 0 η m = , P bat (t) < 0, dsoc(t) E = P ess(t). (6) dt η bat η m x 1 (t)=v 2 (t), x 2 (t)=soc(t) E, u 1 (t) = P ice (t), u 2 (t) = P ess (t), (3) (6) HEV ẋ(t) = Ax(t) + Bu(t), (7) 2 2 : A = 2f M 0 M M, B = , x = η bat η m x 1 u 1, u =,,, x 2 u 2 : 0 P ice (t) P ice max, t t 0, t f, (8) P ess min P ess (t) P ess max, t t 0, t f, (9) SOC = SOC(t f ) SOC(t 0 ) 0, (10) : P ice max, t 0 t f, P ess min P ess max. 4 (Real-time optimization control based on the linear quadratic regulator (LQR)),,,.,., 5,. 5 Fig. 5 Approximated step line of a driving cycle

4 HEV SOC α 1 0 α 3 0 Q =, R =,, HEV (7) 0 α 2 0 α 4 tf J LQR = 1 α 1 (x 1 (t) r 2 ) 2 + α 2 (x 2 (t) 2 t 0 x 2e ) 2 + α 3 (u 1 (t) k 1 r 2 f ) 2 + α 4 (u 2 (t) k 2 r 2 f ) 2 dt, (11) : r, x 2e, k 1 k 2. α 1, α 2, α 3, α 4,, ( 1 ), ( 2 ), ( 3 ) ( 4 ),,.,,,,., 3 : 1), P req (t) P 0, k 1 = 1, k 2 = 0 ( ), SOC α 1, α 2, α 3, α 4, ; 2), P ice (t) < 0, α 3., α 3, P ice (t) 0, ; 3) P req (t) P 0, k 1 = 0, k 2 = 1 ( ), α 2 = 0,, SOC. 1, HEV. ũ 1 u 1 (t) r 2 f =, u 2 (t) 0 ũ 2 x 1 = x 2 x 1 (t) x 2 (t) r 2 x 2e. (12), (7), x = A x + Bũ. (13) (11)( k 1 = 1, k 2 = 0) J LQR = 1 2 tf t 0 xq x T + ũrũ T dt, (14) Q 0, R > 0., (13), t f, (14), ũ = R 1 B T K x, (15), K T = K > 0 KA + A T K KBR 1 B T K + Q = 0 (16)., (11) (14). (15),,,. 5 (Simulation results and analysis) 5.1 LQR (Real-time optimization control with LQR) SOC(t 0 ) = 0.7, P 0 = 5 kw. 3, : 1), α 1 = 300, α 2 = , α 3 = 0.004, α 4 = 0.001,, K = ; e 4 2), α 1 = 300, α 2 = , α 3 = 0.05, α 4 = 0.001,, K = ; e 4 3), α 1 = 200, α 2 = 0, α 3 = 0.1, α 4 = ,, K =. 0 0 (15) (2)(4) (5), 5 6 ; 6.

5 5 : LQR Fig. 6 Results of LQR and global optimal control 5.2 (Global optimal control) LQR, J FC = tf, t 0 ṁ(p ice (t))dt (17) SOC(t f ) = SOC(t 0 ), (18)., (2) P req (t), P ice (t), P ice (t) = P req (t) P ess (t), (4) (5) SȮC E = V (V V 2 4RP ess (t)/η m ). (19) P ess (t) SOC (17). H = ṁ(p req (t) P ess (t)) λ(t) V (V V 2 4RP ess (t)/η m ), (20) λ(t) λ = H SOC. (21) 3), SOC V R, H SOC, λ = 0, λ. 12, MATLAB fminbnd (8) (9). (19) SOC(t f ) (18), Table 2 Fuel consumptions 7 UDDS(urban dynamometer driving schedule), SOC λ, 6. 1),, 3. 7 SOC λ Fig. 7 Relationship between SOC and λ 5.3 (Simulation analysis of results) 2 5,,,, 10%., 5, LQR,, λ( 3), 7, SOC = SOC(t f ) SOC(t 0 ) λ,. (SOC(t 0 ) = 0.7) (SOC(t 0 ) = 0.7) /(L (100km) 1 ) /(L (100km) 1 ) SOC(t f ) /(L (100km) 1 ) SOC(t f ) /% UDDS CSHVR UDDSHDV INDIA URBAN WVUSUB

6 λ Table 3 Values of λ in global optimal algorithm UDDS CSHVR UDDSHDV INDIA URBAN WVUSUB 6.166e e e e e 5, 6 SOC, SOC., 20 kw, 26 kw, 7C,,,. 6 (Conclusions) HEV HEV SOC,. 5,, 10%. (References): 1 LIN C C, PENG H E, GRIZZIE J W. Power management strategies for a parallel hybrid electric truck J. IEEE Transactions on Control Systems Technology, 2003, 38(11): ISG D. :, (ZHANG Hu. Study on the multi-energy management strategy for ISG(Integrated starter/generator) parallel diesel Hybrid electric vehicle D. Shanghai: Shanghai Jiao Tong University, 2007.) 3 SCIARRETTA A, BACK M, GUZZELLA L. Optimal control of parallel hybrid electric vehicles J. IEEE Transactions on Control Systems Technology, 2004, 12(3): PAGANELLI G, DELPRAT S, GUERRA M T, et al. Equivalent consumption minimization strategy for parallel hybrid powertrains C //Proceedings of IEEE Vehicular Technology Conference. Piscataway, NJ, USA: IEEE, 2002: SERRAO L, ONORI S, RIZZONI G. ECMS as a realization of Pontryagin s minimum principle for HEV control C //Proceedings of American Control Conference. Piscataway, NJ, USA: IEEE, 2009: KIM N, CHA W S, PENG H. Optimal equivalent fuel consumption for hybrid electric vehicles J. IEEE Transactions on Control Systems Technology, 2012, 20(3): MUSARDO C, RIZZONI G, STACCIA B. A-ECMS: an adaptive algorithm for hybrid electric vehicles energy management C //Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference. Piscataway, NJ, USA: IEEE, 2005: PEREZ V L, BOSSIO R G, MOITRE D, et al. Optimization of power management in an hybrid electric vehicle using dynamic programming J. Mathematics and Computers in Simulation, 2006, 73(1): ,. J., 2011, 28(3): (SHEN Caiying, XIA Chaoying. Control strategy of series hybrid electric vehicle based on improved dynamic programming J. Control Theory & Applications, 2011, 28(3): ) 10 KIM N, CHA S, PENG H. Optimal control of hybrid electric vehicles based on pontryagin s minimum principle J. IEEE Transactions on Control Systems Technology, 2011, 19(5): PISU P, RIZZONI G. A comparative study of supervisory control strategies for hybrid electric vehicles J. IEEE Transactions on Control Systems Technology, 2007, 15(3): DELPRAT S, LAUBER J, GUERRA M T, et al. Control strategy optimization for an hybrid parallel powertrain C //Proceedings of American Control Conference. Piscataway, NJ, USA: IEEE, 2001: : (1958 ),,,, xiachaoying@126.com; (1988 ),,,, zhangcong@tju.edu.cn.

428 28,,,, 1. 1 Fig. 1 Scheme of a series HEV 2, P ft (t) t /, P bat (t) t, P el (t) t,, P req (t) t.,,,,.,, P ft (t) + P bat (t) = P req (t) + P el (

428 28,,,, 1. 1 Fig. 1 Scheme of a series HEV 2, P ft (t) t /, P bat (t) t, P el (t) t,, P req (t) t.,,,,.,, P ft (t) + P bat (t) = P req (t) + P el ( 28 3 2011 3 : 1000 8152(2011)03 0427 06 Control Theory & Applications Vol. 28 No. 3 Mar. 2011, (, 300072) :,,., PSATv6.1.,, ;, 28.47%. : ; ; : U463 : A Control strategy of series hybrid electric vehicle