Modeling, Dynamics, and Control of Electrified Vehicles Edited by Hui Zhang, Dongpu Cao, and Haiping Du WP WOODHEAD PUBLISHING MODELING, DYNAMICS, AND CONTROL OF ELECTRIFIED VEHICLES Related titles Electric and Hybrid Vehicles: Power Sources, Models, Sustainability, Infrastructure and the Market (ISBN 978-0-444-53565-8) Scrosati, Garche and Tillmetz, Advances in Battery Technologies for Electric Vehicles (ISBN 978-1-78242-377-5) MODELING, DYNAMICS, AND CONTROL OF ELECTRIFIED VEHICLES Edited by HUI ZHANG Beihang University, Beijing, China DONGPU CAO Cranfield University, Bedford, United Kingdom HAIPING DU University of Wollongong, Wollongong, NSW, Australia Woodhead Publishing is an imprint of Elsevier The Officers’ Mess Business Centre, Royston Road, Duxford, CB22 4QH, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, OX5 1GB, United Kingdom Copyright r 2018 Elsevier Inc All rights reserved No part of this 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Publisher: Joe Hayton Acquisition Editor: Sonnini R Yura Editorial Project Manager: Ana Claudia Garcia Production Project Manager: Omer Mukthar Cover Designer: Victoria Pearson Typeset by MPS Limited, Chennai, India CONTENTS List of Contributors ix Modeling, Evaluation, and State Estimation for Batteries Hao Mu and Rui Xiong 1.1 Introduction 1.2 Battery Modeling 1.3 Evaluation of Model Accuracy 1.4 State Estimation 1.5 Conclusions References High-Power Energy Storage: Ultracapacitors Lei Zhang 2.1 Introduction 2.2 Modeling 2.3 UC State Estimation 2.4 Conclusions Further Reading HESS and Its Application in Series Hybrid Electric Vehicles Shuo Zhang and Rui Xiong 3.1 Introduction 3.2 Modeling and Application of HESS 3.3 Conclusion References 25 34 35 39 39 45 66 69 70 77 77 80 115 117 Transmission Architecture and Topology Design of EVs and HEVs 121 Jibin Hu, Jun Ni and Zengxiong Peng 4.1 Introduction 4.2 EV and HEV Architecture Representation 4.3 Topology Design of Power-Split HEV 4.4 Topology Design of Transmission for Parallel Hybrid EVs 4.5 Conclusion Reference 121 125 129 143 157 157 v vi Contents Energy Management of Hybrid Electric Vehicles Hong Wang, Yanjun Huang, Hongwen He, Chen Lv, Wei Liu and Amir Khajepour 5.1 Introduction 5.2 Energy Management of HEVs 5.3 Case Study 5.4 Model Predictive Control Strategy 5.5 Results 5.6 Conclusions References Structure Optimization and Generalized Dynamics Control of Hybrid Electric Vehicles Liang Li, Sixiong You, Xiangyu Wang and Chao Yang 6.1 Introduction 6.2 Generalized Dynamics Models 6.3 Extended High-Efficiency Area Model 6.4 Typicals Applications 6.5 Conclusions References 159 159 161 182 192 195 198 198 207 207 208 212 215 241 243 Transmission Design and Control of EVs 245 Xiaoyuan Zhu and Fei Meng 7.1 Introduction 7.2 EVs Equipped with IMT Powertrain System 7.3 Problem Formulation 7.4 Oscillation Damping Controller Design 7.5 Simulation Results 7.6 Conclusion Funding References Further Reading 245 248 253 259 265 271 272 272 274 Brake-Blending Control of EVs Chen Lv, Hong Wang and Dongpu Cao 8.1 Introduction 8.2 Brake-Blending System Modeling 8.3 Regenerative Braking Energy-Management Strategy 8.4 Dynamic Brake-Blending Control Algorithm 8.5 Conclusion 275 275 278 283 292 306 Contents References Further Reading Dynamics Control for EVs Yafei Wang and Hiroshi Fujimoto 9.1 Introduction 9.2 Modeling and Control of EVs 9.3 Sensing and Estimation 9.4 Active Safety Control 9.5 Riding and Energy Efficiency Control 9.6 Conclusions References 10 Robust Gain-Scheduling Control of Vehicle Lateral Dynamics Through AFS/DYC Hui Zhang and Junmin Wang 10.1 Introduction 10.2 Development of Uncertain Vehicle Dynamics Model 10.3 Main Results 10.4 Simulation Results 10.5 Conclusions Acknowledgments References 11 State and Parameter Estimation of EVs Brett McAulay, Boyuan Li, Philip Commins and Haiping Du 11.1 Introduction 11.2 Velocity Estimation (Longitudinal, and Total, Preferred Method and Alternatives) 11.3 Slip-Angle Estimation 11.4 Tire-Force and TireRoad Friction Coefficient Estimation 11.5 Vehicle Mass- and Road Slope-Estimation Method 11.6 Conclusions References Further Reading vii 306 308 309 309 315 321 326 332 336 336 339 339 342 355 359 364 365 365 369 369 372 374 381 395 405 406 407 12 Modeling and Fault-Tolerant-Control of Four-Wheel-IndependentDrive EVs 409 Rongrong Wang and Junmin Wang 12.1 Introduction 409 viii Contents 12.2 System Modeling and Problem Formulation 12.3 Fault-Tolerant Tracking Controller Design 12.4 Simulation Investigations 12.5 Conclusions References 13 Integrated System Design and Energy Management of Plug-In Hybrid Electric Vehicles Xiaosong Hu 13.1 Introduction 13.2 Powertrain Modeling 13.3 Heuristic Scenarios 13.4 Emission Mitigation via Renewable Energy Integration 13.5 Optimal Scenario With Integrated System Design and Energy Management 13.6 Battery-Health Implication 13.7 Conclusions References Appendix 14 Integration of EVs With a Smart Grid Xiaosong Hu 14.1 Introduction 14.2 Powertrain Modeling 14.3 Formulation of Cost-Optimal Control Problem 14.4 Results and Discussion 14.5 Conclusions References Index 411 418 437 448 448 451 451 453 455 463 465 468 471 473 474 475 475 477 483 485 494 495 497 LIST OF CONTRIBUTORS Dongpu Cao Cranfield University, Bedford, United Kingdom Philip Commins University of Wollongong, Wollongong, NSW, Australia Haiping Du University of Wollongong, Wollongong, NSW, Australia Hiroshi Fujimoto The University of Tokyo, Tokyo, Japan Hongwen He Beijing Institute of Technology, Beijing, China Jibin Hu Beijing Institute of Technology, Beijing, China Xiaosong Hu Chongqing University, Chongqing, China Yanjun Huang University of Waterloo, ON, Canada Amir Khajepour University of Waterloo, ON, Canada Boyuan Li University of Wollongong, Wollongong, NSW, Australia Liang Li Tsinghua University, Beijing, China Wei Liu Beijing Institute of Technology, Beijing, China Chen Lv Cranfield University, Cranfield, United Kingdom; Cranfield University, Bedford, United Kingdom Brett McAulay University of Wollongong, Wollongong, NSW, Australia Fei Meng Shanghai Maritime University, Shanghai, China Hao Mu Beijing Institute of Technology, Beijing, China Jun Ni Beijing Institute of Technology, Beijing, China ix Integration of EVs With a Smart Grid 479 with β j $ 0; jAf0; 2g The near-optimal values of engine on/off control signal e can be obtained by heuristics that turn the ICE on if the vehicle power demand exceeds a threshold Pon , i.e., 1; Tv ðkÞωðkÞ $ Pon eðkÞ (14.6) 0; otherwise: The optimal Pon is found by iteratively solving the CP problem over a grid covering the allowable vehicular power range (Hu et al., 2013b, 2015a, 2016) 14.2.3 Battery 14.2.3.1 Electrical Model A lithium-ion battery pack is comprised of strings in parallel, with each string containing the same number of cells in series Each battery cell is modeled by an open-circuit voltage (OCV), uc ; in series with a resistor, Rc For simplicity, the impact of current direction on Rc is neglected, since it has already been addressed in Hu et al (2013b, 2015a, 2016) The terminal battery power is calculated by: Pbt ðkÞ uc ðkÞIc ðkÞnb Rc Ic2 ðkÞnb (14.7) where Ic is the cell current and nb is the total number of cells corresponding to the rating energy capacity of kWh An approximate affine OCVSoC model is adopted as follows, which ensures a good fit within the typically allowed battery SoC range in PHEVs (Hu et al., 2013b, 2015a, 2016): uc ðkÞ Qc SoCc ðkÞ uc0 Cc (14.8) In Eq (14.8), Qc represents the rated cell capacity (in As), and Cc and uc0 are coefficients to be fitted (in F and V), respectively The associated battery energy E b can be thereby expressed as: ! Ð SoCc ðkÞ Ð SoCc ðkÞ Qc uc ðτÞdτ nb Qc τ uc0 dτ E b ðkÞ nb Qc Cc ! Qc nb Cc 2 n b Qc SoCc ðkÞ uc0 SoCc ðkÞ uc ðkÞ u2c0 : 2Cc (14.9) 480 Modeling, Dynamics, and Control of Electrified Vehicles Then, in light of Eqs (14.7) and (14.9), the terminal battery power can be yielded by: Pbt ðkÞ P b ðkÞ Rc Cc P 2b ðkÞ 2Eb ðkÞ u2c0 Cc nb (14.10) where P b ðkÞ uc ðkÞIc ðkÞnb is the internal battery power Similar to Eq (14.2), Eq (14.10) can be relaxed as an inequality for preserving the convexity in the cost-optimal control problem in Section 14.3 The convex battery constraints applied to both charging during parking and power management during on-road driving are summarized as: Eb ðk 1Þ Eb ðkÞ ΔtP b ðkÞ; kAMc , Md > > < Rc Cc P 2b ðkÞ P bt ðkÞ P b ðkÞ # 0; kAMc , Md (14.11) 2Eb ðkÞ u2c0 Cc nb > > : Eb;min # Eb ðkÞ # Eb;max ; kAMc , Md where Mc and Md are the sets of discrete time-steps for charging and driving, respectively The time interval is represented by Δt, which could be different in charging and driving The battery energy limits Eb;min and Eb;max are evaluated at the SoC limits SoCb;min and SoCb;max by Eq (14.9), respectively Different current/power limits are applied to charging and driving, as illustrated by: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !ffi !ffi u u > > u u 2E b ðkÞ 2E b ðkÞ < Ic;min tnb 1u2c0 nb #P b ðkÞ #Ic;max tnb 1u2c0 nb ; kAMd C C c c > > : P bt;min #P bt ðkÞ #0; kAMc (14.12) where Ic;min and Ic;max are the cell current limits for power management during on-road driving, and P bt;min is the battery-charging power limit during parking As treated in Patil et al (2013), net-zero battery energy transfer over a 24-hour work period is enforced, so that today’s control does not interfere with performance tomorrow by, for instance, depleting the battery; namely: Eb ð0Þ Eb ðN Þ (14.13) with N corresponding to the final time-step 14.2.3.2 Health Model Battery aging unavoidably happens during practical PHEV operations, and its fading rate depends on a range of factors In Wang et al (2011), Integration of EVs With a Smart Grid 481 aging tests accounting for diverse current rates (C-rates), temperatures, and depths-of-discharge were carried out for A123’s lithium-ironphosphate battery cells (ANR26650m1) This research showed that the capacity loss of this type of cell greatly depends on the current rate and temperature, whereas the depth-of-discharge factor is negligible The capacity loss is mimicked by the following semiempirical model: Ea ðcÞ ΔQc HðcÞexp (14.14) BðcÞz RTc where ΔQc is the percentage of capacity loss (in %), c is the C-rate, and H is a preexponential factor as a function of the C-rate, as shown in Table 14.1 The ideal gas constant is denoted by R (i.e., 8.31 J/mol K), Tc is the lumped cell temperature (in K), and B is the discharged amperehour (Ah) throughput The activation energy Ea (in J/mol) and the power-law factor z are calibrated as: Ea ðcÞ 31; 700 370:3c; (14.15) z 0:55: The capacity loss of 20% (ΔQc 20%) is symptomatic of end-of-life (EOL) of an automotive battery, and the associated total discharged Ah throughput Btol is calculated by: 31=z 20 : Btol ðcÞ (14.16) 2Ea ðcÞ HðcÞexp RTc Then, the total Ah throughput including both the charge and discharge processes equals to 2Atol, and the number of cycles until the battery EOL, Neol, is achieved by 2Btol ðcÞ 3600Btol ðcÞ Neol ðcÞ Qc 2Qc =3600 (14.17) Table 14.1 Preexponential factor as a function of the C-rate C-rate ca H a 0.5 31,630 21,681 C-rate corresponds to 2A (Wang et al., 2011) 12,934 10 15,512 482 Modeling, Dynamics, and Control of Electrified Vehicles where one cycle corresponds to an ampere-second throughput of 2Qc An energy throughput-based battery SoH model can be established as follows (Johannesson et al., 2013): ðt ðt 1 Pnc ðτÞdτ SoHðtÞ Ic ðτÞuc dτ 2Neol ðcÞQc uc 2Neol ðcÞEnc (14.18) where uc is the nominal voltage of the cell (i.e., 3.3 V), Pnc Ic uc is the nominal internal power, and Enc Qc uc is the nominal energy capacity jPnc jQc (in Ws) Given c , the discrete-time SoH dynamics can be char2Enc acterized by: Pnc ðkÞ Δt: (14.19) SoHðk 1Þ SoHðkÞ 2Neol ðjPnc ðkÞjÞEnc It is reasonable to assume that the lumped cell temperature Tc can be kept constant by an advanced thermal-management system (Hu et al., 2015a) In this study, Tc 298:15 K (room temperature, 25˚C) is considered, as recommended by most battery manufacturers The SoH model, Eq (14.19), is precisely approximated by four piecewise quadratic functions: SoHðkÞ ðd0;1 Pnc2 ðkÞ d1;1 ÞΔt; Pnc ðkÞ# 3:68 > > < SoHðkÞ ðd P ðkÞ d ÞΔt; 3:68 # P ðkÞ # 40:31 0;2 nc 1;2 nc SoHðk 1Þ Pnc ðkÞ # 57:14 > SoHðkÞ ðd P ðkÞ d ÞΔt; 40:31 , 0;3 nc 1;3 > : SoHðkÞ ðd0;4 Pnc2 ðkÞ d1;4 ÞΔt; Pnc ðkÞ 57:14 (14.20) with d0;j ; jAf1; 2; 3; 4g being all negative The original and approximate SoH models are compared in Fig 14.2 Because the battery cell has a very flat OCV curve in the operating SoC window, P b ðkÞ nb P nc ðkÞ As a result, Eq (14.20) can be rewritten as: Pb2 ðkÞ SoHðk 1Þ SoHðkÞ d0;j d1;j Δt 0; jAf1; 2; 3; 4g: nb (14.21) As manipulated for Eqs (14.2) and (14.10), Eq (14.21) can be relaxed as inequalities to preserve the convexity in the next cost-optimal control problem, i.e., Pb2 ðkÞ SoHðk11Þ2SoHðkÞ2 d0;j 1d1;j Δt #0;jAf1;2;3;4g;kAMc ,Md : nb (14.22) Integration of EVs With a Smart Grid 483 Cycles (A) 10,000 8000 6000 4000 2000 SoH change rate (1/s) (B) 20 40 60 80 x10–7 Original model Approximation 0 20 40 60 Nominal internal power | Pnc| (W) 80 Figure 14.2 Original and approximate SoH models: (A) Number of cycles until EOL, Neol, and (B) SoH change rate 14.3 FORMULATION OF COST-OPTIMAL CONTROL PROBLEM The daily PHEV operation herein comprises two identical driving routes (starting at a.m in the morning and at p.m in the afternoon) and parking, which is typical of operational patterns on work days The Federal Test Procedure (FTP-75) was chosen to emulate urban driving The FTP-75 drive cycle and the timing of the routes over the 24-hour horizon are shown in Fig 14.3 We can readily consider discrepant trip knowledge, in terms of velocity profile, trip length, and trip timing However, the impact of trip/traffic conditions on PHEV energy consumption is beyond the concentration of this work (see, e.g., Sun et al., 2015b) We used the US Midwest ISO data on the cost of purchasing electric power from the utilities (Johannesson et al., 2013), which are within the range of 0.33 b/kWh, depending on grid-power demand Additional costs include transmission and distribution costs, other operational costs, and profit margins of the utilities Consequently, the electricity price charged by the grid in the US Midwest (e.g., Michigan) at different times of the day (again based on grid demand) ranges from to 25 b/kWh The resultant electricity price trace on a certain day is plotted in Fig 14.4 The objective function of the optimal control problem is the total daily cost of PHEV, Gtol , which is caused by on-road gasoline 484 Modeling, Dynamics, and Control of Electrified Vehicles Velocity (m/s) (A) 20 10 0 500 1000 (B) Velocity (m/s) 1500 2000 Time (s) 30 20 10 0 12 Time (h) 16 20 24 Figure 14.3 FTP-75 driving cycle and timing of the daily PHEV operation: (A) FTP-75 cycle and (B) timing of the 24-hour PHEV usage (The 24-hour period in the figure starts at a.m., and thus the first hour corresponds to a.m.) Electricity price ( /kWh) 16 14 12 10 6 12 Time (h) 18 24 Figure 14.4 Hourly electricity price of the grid Data from Patil, R.M., 2012 Combined design and control optimization: application to optimal PHEV design and control for multiple objectives Ph.D Dissertation, University of Michigan, Ann Arbor, MI, USA (Patil, 2012) consumption, electricity charged from the grid during parking, and concomitant battery aging: Gtol Ggas Ggrid Gbat (14.23) Integration of EVs With a Smart Grid where cg Δtd X > > Pf ðkÞ Ggas > > Lg ρg kAMd > < Δtc X > G cc ðkÞP bt ðkÞ grid > > ηc kAMc > > : Gbat 5cbat ðSoHð0Þ SoHðNÞÞ 485 (14.24) where cg is the gasoline price (in USD/L), Lg is the lower heating value of gasoline (in J/g), and ρg is the gasoline density (in g/L) The grid electricity price (in USD/kWh) is represented by cc, and ηc denotes the average charger efficiency Note that P bt is negative during charging, and we use Δtd second when driving and Δtc minute when charging (Patil et al., 2013; Patil, 2012) Furthermore, Gbat is an equivalent operational cost due to daily battery aging, with cbat being the battery pack price (in USD/kWh) covering battery management circuitry and packaging The optimization variables include T, Tbrk , Eb , P egu , P b , P bt , and SoH Clearly, the objective function Gtol is convex with respect to these optimization variables The constraints in the CP framework (convex inequality and affine equality constraints) consist of the power/torque balance (14.1) and (14.4) when kAMd , battery electrical constraints (14.11)(14.13), battery SoH constraint (14.22), as well as the following operating bounds of the EGU, EM, mechanical braking, and battery SoH: Tmin ðωðkÞÞ # T ðkÞ # Tmax ðωðkÞÞ; kAMd > > > > < # P egu ðkÞ # P egu;max ; kAMd Tbrk ðkÞ # 0; kAMd (14.25) > > > # SoHðkÞ # 1; kAMc , Md > : SoHð0Þ where Tmin and Tmax are the torque limits of EM and P egu;max is the maximal EGU output power The CP framework mentioned above can be quickly and efficiently solved by commercially available tools, e.g., CVX (Boyd and Vandenberghe, 2004), resulting in proven global optimality 14.4 RESULTS AND DISCUSSION The key parameters of the PHEV are provided in Table 14.2, while the main specifications of its onboard power sources are listed in Table 14.3 486 Modeling, Dynamics, and Control of Electrified Vehicles Table 14.2 Main vehicle parameters (Hu et al., 2016) Parameter Value Parameter Value Frontal area (m2) 2.000 1.155 Aerodynamic drag coefficient Air density (kg/m3) 0.300 Vehicle mass excluding battery pack (ton) EM inertia (kgm2) Inertia of final drive and wheels (kgm2) Vehicular auxiliary power Pau (kW) Final gear 1.086 1.184 Rolling-resistance coefficient Wheel radius (m) 0.010 0.308 Table 14.3 Main specifications of the onboard power sources Parameter Value Parameter a Gasoline price cg (USD/L) Gasoline lower heating value Lg (J/g) Gasoline density ρg (g/L) Maximum EGU power Pegu;max (W) Nominal battery capacity Qc (As) Nominal battery voltage (V) Battery-cell resistance Rc (Ω) 0.01 Battery-cell mass (kg) 0.07 a 0.100 0.800 4.000 Value 1.99 42,600 Maximum discharge Ic,max (A) Maximum charge Ic,min (A) 70 235 749 35,000 Initial battery SoC (%) Maximum battery SoC (%) 70 90 8280 Minimum battery SoC (%) 30 3.3 Maximum charging power P bt;min (W) Average charger efficiency ηc (%) Battery-pack price cbat (USD/ kWh)b 22500 98 990 Designated according to the current price in the US Midwest Adopted from Hu et al (2015a) b The additional mass caused by packaging and circuitry is assumed to account for 12.3% of the total mass of the battery pack (Hu et al., 2015a) 14.4.1 Optimization Results The optimized battery-state evolutions are shown in Fig 14.5 It is clear that the optimal charging appears in the proximity of the lowest electricity price The SoH decreases when the SoC rises during charging or drops during on-road driving The optimized on-road power management in the first trip is illustrated in Fig 14.6 The PHEV during driving operates in a blended paradigm, where the battery energy is mainly used, and the EGU Integration of EVs With a Smart Grid 487 Electricity ( /kWh) (A) 15 10 12 18 24 (B) SoC (%) 90 60 30 12 18 24 12 Time (h) 18 24 (C) SoH (%) 100 99.995 99.99 Figure 14.5 Optimized battery-state evolutions: (A) Grid electricity price, (B) SoC, and (C) SoH (A) EGU Power (kW) 20 10 –10 –20 7.1 7.2 (B) Efficiency (%) Battery 7.3 Time (h) 7.4 7.5 40 OOL 20 0 10 15 Operating point 20 25 30 35 Power (kW) Figure 14.6 Optimized on-road power management in the first trip: (A) power split between EGU and battery and (B) EGU efficiency 488 Modeling, Dynamics, and Control of Electrified Vehicles exhibits high operating efficiency in spite of being used less frequently Fig 14.7 shows the outcome of cumulative costs over the 24-hour period As can be seen, the gasoline cost is much smaller than for charging and battery degradation In other words, the CP-based optimal control policy uses less gasoline The daily cost is minimized at around $0.85 In order to further show the cost-saving capability of the optimized scenario, we conduct a comparison with three heuristic PHEV scenarios as given in Table 14.4 The comparative result is presented in Fig 14.8 The optimal and PHEV2 scenarios have larger SoC swing, provoking added SoH degradation The moderately increased SoH decay, however, is successfully offset by substantial savings of charging and gasoline costs in the optimal scenario, which in turn give rise to the minimal total cost (see the subplot (C)) The optimality losses in the PHEV1 and PHEV2 scenarios are mainly due to the expensive charging timing The gasoline cost in the PHEV3 scenario is considerably higher than those in other (A) 0.1 0.05 0 12 18 24 12 18 24 (B) 0.4 Cost (USD) 0.2 (C) 0.6 0.3 0 12 18 24 12 Time (h) 18 24 (D) 0.5 0 Figure 14.7 Cumulative costs: (A) gasoline cost, (B) charging cost, (C) battery-aging cost, and (D) total cost 489 Integration of EVs With a Smart Grid Table 14.4 Three heuristic PHEV scenarios Type Portrayal PHEV1 The PHEV has a heuristic charging protocol/timing: P bt ðkÞ 0:1kW ; kAMc1 is applied, where Mc1 contains all the time points during parking The PHEV has a heuristic charging protocol/timing: P bt ðkÞ 1kW ; kAMc2 is applied, where Mc2 only contains the h just before each trip The PHEV has a heuristic on-road power management strategy: the EGU is always on during driving PHEV2 PHEV3 SoC (%) (A) 90 60 30 Optimum 12 PHEV1 18 12 18 24 12 Time (h) 18 24 PHEV2 24 PHEV3 (B) SoH (%) 100 99.995 99.99 (C) Cost (USD) 1.5 0.5 Figure 14.8 Comparison of the three heuristic PHEV scenarios: (A) SoC, (B) SoH, and (C) total cost scenarios, because of low EGU efficiency The cost reductions in the optimal scenario are 28.33%, 40.32%, and 33.08% with regard to the PHEV1, PHEV2, and PHEV3 scenarios, respectively We also contrast our CP framework with DP (a commonly-used benchmark) in terms of accuracy and computational speed, as indicated 490 Modeling, Dynamics, and Control of Electrified Vehicles Table 14.5 Comparison between CP and DP Algorithm Total cost (USD) Timea (s) CP DP 37.65 7600 a 0.85 0.83 A 2.3 GHz microprocessor with GB RAM was used in Table 14.5 The CP ensures very close accuracy to DP while running approximately 200 times faster (the slight accuracy penalty arises from the convex modeling approximations and engine on/off control) 14.4.2 Sensitivity to Price Changes Despite relatively steady electricity rate for transportation applications, the increasingly severe oil shortage and supply instability may lead to growing gasoline price At the same time, battery-price reduction most probably results from economies of scale, progressive fabricating and management technologies, as well as increasingly mature chemistries In this sense, it is interesting to look at how the optimal PHEV scenario is sensitive (or resilient) to such price changes Since accurate forecasting of gasoline and battery prices is far beyond the scope of this chapter (and is the job of economics researchers), a simple but nontrivially useful sensitivity analysis is carried out, where we consider a 100% increase of gasoline price and a 50% reduction of battery price, relative to the baseline case listed in Table 14.2 The battery-state trajectories are compared in Fig 14.9 The PHEV in the changed-price case relies on more electricity and battery utilization, as exposed by about 10% larger SoC depletion during driving The consequent SoH decay is increased Furthermore, the cumulative costs are contrasted in Fig 14.10 It can be seen that the PHEV evolves as a pure electric vehicle in the changed-price case Notwithstanding enlarged charging cost and SoH decay, the battery-price decrease helps increase cost effectiveness 14.4.3 V2G Implication PHEVs are technically viable to sell electric energy back to the grid Such V2G functionality often acts as an ancillary service (e.g., spinning reserve and regulation reserve) to stabilize the grid But from the PHEV side, does it lower vehicle economy? To answer this question, we only Integration of EVs With a Smart Grid (A) SoC (%) 90 Baseline Changed price 60 72 62 52 30 (B) SoH (%) 491 7.5 6 12 18 24 18 24 100 99.995 99.99 99.985 12 Time (h) Figure 14.9 Battery-state trajectories with baseline and changed gasoline/battery prices: (A) SoC and (B) SoH (A) 0.1 Baseline Changed price 0.05 0 12 18 24 (C) 12 18 24 (B) 0.4 Cost (USD) 0.2 0.6 0.3 0 12 18 24 12 Time (h) 18 24 (D) 0.5 Figure 14.10 Cumulative costs with baseline and changed gasoline/battery prices: (A) gasoline cost, (B) charging cost, (C) battery-aging cost, and (D) total cost 492 Modeling, Dynamics, and Control of Electrified Vehicles need to the following minor changes to the CP framework in Section 14.3: ! X > P bt ðkÞ < G Δt cc ðkÞmin ; P bt ðkÞηc grid c η (14.26) c kAM c > : P bt;min # P bt ðkÞ # P bt;max ; kAMc where P bt;max is the V2G power limit The related optimization result is compared to that without the V2G setting (see Fig 14.11) Interestingly, the optimization does not allow V2G activities and thus produces an identical result as the no V2G case The underlying reason is that the V2G-induced battery-aging expense outweighs the V2G-added revenue The same conclusion is even attained in the changed-price case with 50% reduced battery price Further comparisons are made with a heuristic solution with V2G activities around high electricity price in both the baseline and changed-price cases (see Figs 14.12 and 14.13) It is found that the imposed V2G activities in both cases incur larger SoC swing and concomitantly augmented SoH decay, thus noticeably reducing the vehicle economy The losses are 57.2% and 53.4% in the baseline and SoC (%) (A) 90 60 30 12 Optimum-no V2G 12 18 24 12 18 24 SoH (%) (B) 18 Optimum-V2G 24 100 99.99 99.98 Cost (USD) (C) 0.5 0 Time (h) Figure 14.11 Optimization results with/without V2G addition: (A) SoC, (B) SoH, and (C) total cost ... states of batteries, like the state of charge (SoC), state of health (SoH), and state of function (SoF) directly due to the complicated electrochemical process Modeling, Dynamics, and Control of Electrified. .. Dynamics, and Control of Electrified Vehicles DOI: http://dx.doi.org/10.1016/B978-0-12-812786-5.00002-1 Copyright © 2018 Elsevier Inc All rights reserved 39 40 Modeling, Dynamics, and Controlof Electrified. .. Systematic chart of P2D model 6 Modeling, Dynamics, and Control of Electrified Vehicles comprised of three regions that imply four distinct boundaries The specific descriptions of this model can