L. Guzzella
`
`A. Sciarretta
`
`Vehicle Propulsion
`Systems
`
`Introduction to Modeling
`and Optimization
`
`Second Edition
`
`@ Springer
`
`BMW v. Paice, IPR2020-01386
`BMW1108
`Page 1 of 29
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`

`

`Lino Guzzella · Antonio Sciarretta
`
`Vehicle Propulsion Systems
`
`BMW v. Paice, IPR2020-01386
`BMW1108
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`

`

`Lino Guzzella · Antonio Sciarretta
`
`Vehicle Propulsion Systems
`
`Introduction to Modeling and Optimization
`
`Second Edition
`
`With 202 Figures and 30 Tables
`
`123
`
`BMW v. Paice, IPR2020-01386
`BMW1108
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`

`

`Prof. Dr. Lino Guzzella
`Dr. Antonio Sciarretta
`ETH Zürich
`Inst. Mess- und Regeltechnik
`Sonneggstr. 3
`8092 Zürich
`Switzerland
`lguzzella@ethz.ch
`Antonio.Sciarretta@ifp.fr
`
`Library of Congress Control Number: 2007934932
`
`ISBN 978-3-540-74691-1 2nd Edition Springer Berlin Heidelberg New York
`ISBN 978-3-540-25195-8 1st Edition Springer Berlin Heidelberg New York
`
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`© Springer-Verlag Berlin Heidelberg 2005, 2007
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`

`Contents
`
`1
`
`1
`Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
`1
`1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
`2
`1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
`5
`1.3 Upstream Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
`1.4 Energy Density of On-Board Energy Carriers . . . . . . . . . . . . . . . 10
`1.5 Pathways to Better Fuel Economy . . . . . . . . . . . . . . . . . . . . . . . . . 12
`
`2 Vehicle Energy and Fuel Consumption – Basic Concepts . . . 13
`2.1 Vehicle Energy Losses and Performance Analysis . . . . . . . . . . . . 13
`2.1.1 Energy Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
`2.1.2 Performance and Drivability . . . . . . . . . . . . . . . . . . . . . . . . 18
`2.1.3 Vehicle Operating Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
`2.2 Mechanical Energy Demand in Driving Cycles. . . . . . . . . . . . . . . 21
`2.2.1 Test Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
`2.2.2 Mechanical Energy Demand . . . . . . . . . . . . . . . . . . . . . . . . 23
`2.2.3 Some Remarks on the Energy Consumption . . . . . . . . . . 27
`2.3 Methods and Tools for the Prediction of Fuel Consumption . . . 32
`2.3.1 Average Operating Point Approach . . . . . . . . . . . . . . . . . . 32
`2.3.2 Quasistatic Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
`2.3.3 Dynamic Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
`2.3.4 Optimization Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
`2.3.5 Software Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
`
`3
`
`IC-Engine-Based Propulsion Systems . . . . . . . . . . . . . . . . . . . . . . 43
`3.1 IC Engine Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
`3.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
`3.1.2 Normalized Engine Variables . . . . . . . . . . . . . . . . . . . . . . . . 44
`3.1.3 Engine Efficiency Representation . . . . . . . . . . . . . . . . . . . . 45
`3.2 Gear-Box Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
`3.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
`3.2.2 Selection of Gear Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
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`3.2.3 Gear-Box Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
`3.2.4 Losses in Friction Clutches and Torque Converters . . . . . 51
`3.3 Fuel Consumption of IC Engine Powertrains . . . . . . . . . . . . . . . . 54
`3.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
`3.3.2 Average Operating Point Method . . . . . . . . . . . . . . . . . . . . 54
`3.3.3 Quasistatic Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
`
`4 Electric and Hybrid-Electric Propulsion Systems . . . . . . . . . . 59
`4.1 Electric Propulsion Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
`4.2 Hybrid-Electric Propulsion Systems . . . . . . . . . . . . . . . . . . . . . . . . 60
`4.2.1 System Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
`4.2.2 Power Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
`4.2.3 Concepts Realized . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
`4.2.4 Modeling of Hybrid Vehicles . . . . . . . . . . . . . . . . . . . . . . . . 69
`4.3 Electric Motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
`4.3.1 Quasistatic Modeling of Electric Motors . . . . . . . . . . . . . . 74
`4.3.2 Dynamic Modeling of Electric Motors . . . . . . . . . . . . . . . . 89
`4.3.3 Causality Representation of Generators . . . . . . . . . . . . . . 90
`4.4 Batteries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
`4.4.1 Quasistatic Modeling of Batteries . . . . . . . . . . . . . . . . . . . 95
`4.4.2 Dynamic Modeling of Batteries . . . . . . . . . . . . . . . . . . . . . 103
`4.5 Supercapacitors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
`4.5.1 Quasistatic Modeling of Supercapacitors . . . . . . . . . . . . . . 111
`4.5.2 Dynamic Modeling of Supercapacitors. . . . . . . . . . . . . . . . 115
`4.6 Electric Power Links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
`4.6.1 Quasistatic Modeling of Electric Power Links . . . . . . . . . 117
`4.6.2 Dynamic Modeling of Electric Power Links . . . . . . . . . . . 117
`4.7 Torque Couplers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
`4.7.1 Quasistatic Modeling of Torque Couplers . . . . . . . . . . . . . 119
`4.7.2 Dynamic Modeling of Torque Couplers . . . . . . . . . . . . . . . 120
`4.8 Power Split Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
`4.8.1 Quasistatic Modeling of Power Split Devices . . . . . . . . . . 121
`4.8.2 Dynamic Modeling of Power Split Devices . . . . . . . . . . . . 126
`
`5 Non-electric Hybrid Propulsion Systems . . . . . . . . . . . . . . . . . . . 131
`5.1 Short-Term Storage Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
`5.2 Flywheels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
`5.2.1 Quasistatic Modeling of Flywheel Accumulators . . . . . . . 137
`5.2.2 Dynamic Modeling of Flywheel Accumulators . . . . . . . . . 138
`5.3 Continuously Variable Transmissions . . . . . . . . . . . . . . . . . . . . . . . 140
`5.3.1 Quasistatic Modeling of CVTs . . . . . . . . . . . . . . . . . . . . . . 141
`5.3.2 Dynamic Modeling of CVTs . . . . . . . . . . . . . . . . . . . . . . . . 144
`5.4 Hydraulic Accumulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
`5.4.1 Quasistatic Modeling of Hydraulic Accumulators . . . . . . 146
`5.4.2 Dynamic Modeling of Hydraulic Accumulators . . . . . . . . 152
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`XI
`
`5.5 Hydraulic Pumps/Motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
`5.5.1 Quasistatic Modeling of Hydraulic Pumps/Motors . . . . . 154
`5.5.2 Dynamic Modeling of Hydraulic Pumps/Motors . . . . . . . 156
`5.6 Pneumatic Hybrid Engine Systems . . . . . . . . . . . . . . . . . . . . . . . . 157
`5.6.1 Modeling of Operation Modes . . . . . . . . . . . . . . . . . . . . . . . 158
`
`6 Fuel-Cell Propulsion Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
`6.1 Fuel-Cell Electric Vehicles and Fuel-Cell Hybrid Vehicles . . . . . 165
`6.1.1 Concepts Realized . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
`6.2 Fuel Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
`6.2.1 Quasistatic Modeling of Fuel Cells . . . . . . . . . . . . . . . . . . . 179
`6.2.2 Dynamic Modeling of Fuel Cells . . . . . . . . . . . . . . . . . . . . . 193
`6.3 Reformers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
`6.3.1 Quasistatic Modeling of Fuel Reformers . . . . . . . . . . . . . . 200
`6.3.2 Dynamic Modeling of Fuel Reformers . . . . . . . . . . . . . . . . 204
`
`7
`
`Supervisory Control Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
`7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
`7.2 Heuristic Control Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
`7.3 Optimal Control Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
`7.3.1 Optimal Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
`7.3.2 Optimization Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
`7.3.3 Real-time Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 219
`
`8 Appendix I – Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
`8.1 Case Study 1: Gear Ratio Optimization . . . . . . . . . . . . . . . . . . . . 227
`8.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
`8.1.2 Software Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
`8.1.3 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
`8.2 Case Study 2: Dual-Clutch System - Gear Shifting . . . . . . . . . . . 231
`8.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
`8.2.2 Model Description and Problem Formulation . . . . . . . . . . 231
`8.2.3 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
`8.3 Case Study 3: IC Engine and Flywheel Powertrain . . . . . . . . . . . 234
`8.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
`8.3.2 Modeling and Experimental Validation . . . . . . . . . . . . . . . 236
`8.3.3 Numerical Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
`8.3.4 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
`8.4 Case Study 4: Supervisory Control for a Parallel HEV. . . . . . . . 241
`8.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
`8.4.2 Modeling and Experimental Validation . . . . . . . . . . . . . . . 241
`8.4.3 Control Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
`8.4.4 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
`8.5 Case Study 5: Optimal Rendez-Vous Maneuvers . . . . . . . . . . . . . 251
`8.5.1 Modeling and Problem Formulation . . . . . . . . . . . . . . . . . . 251
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`8.5.2 Optimal Control for a Specified Final Distance . . . . . . . . 253
`8.5.3 Optimal Control for an Unspecified Final Distance . . . . 257
`8.6 Case Study 6: Fuel Optimal Trajectories of a Racing FCEV . . . 261
`8.6.1 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
`8.6.2 Optimal Control
`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
`8.6.3 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
`8.7 Case Study 7: Optimal Control of a Series Hybrid Bus . . . . . . . 270
`8.7.1 Modeling and Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
`8.7.2 Optimal Control
`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
`8.7.3 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
`8.8 Case Study 8: Hybrid Pneumatic Engine . . . . . . . . . . . . . . . . . . . 280
`8.8.1 HPE Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
`8.8.2 Driveline Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
`8.8.3 Air Tank Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
`8.8.4 Optimal Control Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . 284
`8.8.5 Optimal Control Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
`
`9 Appendix II – Optimal Control Theory . . . . . . . . . . . . . . . . . . . . 289
`9.1 Parameter Optimization Problems . . . . . . . . . . . . . . . . . . . . . . . . . 289
`9.1.1 Problems Without Constraints . . . . . . . . . . . . . . . . . . . . . . 289
`9.1.2 Numerical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
`9.1.3 Minimization with Equality Constraints . . . . . . . . . . . . . . 293
`9.1.4 Minimization with Inequality Constraints . . . . . . . . . . . . . 296
`9.2 Optimal Control
`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
`9.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
`9.2.2 Optimal Control for the Basic Problem . . . . . . . . . . . . . . . 298
`9.2.3 First Integral of the Hamiltonian . . . . . . . . . . . . . . . . . . . . 302
`9.2.4 Optimal Control with Specified Final State . . . . . . . . . . . 304
`9.2.5 Optimal Control with Unspecified Final Time . . . . . . . . . 305
`9.2.6 Optimal Control with Bounded Inputs . . . . . . . . . . . . . . . 306
`
`10 Appendix III – Dynamic Programming . . . . . . . . . . . . . . . . . . . . 311
`10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
`10.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
`10.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
`10.2.2 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
`10.3 Implementation Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
`10.3.1 Grid Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
`10.3.2 Nearest Neighbor or Interpolation . . . . . . . . . . . . . . . . . . . 316
`10.3.3 Scalar or Set Implementation . . . . . . . . . . . . . . . . . . . . . . . 318
`
`References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
`
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`7 S
`
`upervisory Control Algorithms
`
`In all types of hybrid vehicles, a supervisory controller must determine how
`to operate the single power paths, in order to satisfy the power demand of
`the drive line in the most convenient way. The main objective of that opti-
`mization is the reduction of the overall energy use, usually in the presence of
`various constraints due to driveability requirements and the characteristics of
`the components.
`Based on the review article [225], this chapter describes the theoretical
`concepts of various types of control strategies for parallel hybrid-electric vehi-
`cles. Appendix I contains examples of the application of these ideas. Similar
`approaches that have been investigated also for series hybrids [272, 19, 40],
`combined hybrids [209, 51], and fuel-cell hybrids [184, 213] are not treated in
`this book.
`
`7.1 Introduction
`
`A parallel hybrid-electric vehicle can be operated in any of the modes sum-
`marized in Table 7.1. Besides the power split ratio u (see Chap. 4 for its
`definition), additional control variables are the clutch status and the engine
`status. Both are Boolean, clutch engaged (Bc = 1) or disengaged (Bc = 0),
`engine on (Be = 1) or off (Be = 0). Both zero-emission (ZEV) and regen-
`erative braking modes (u = 1) can be operated in principle either with the
`engine shut down and disengaged or shut down but still engaged. The other
`modes (ICE, power assist, battery recharge) all require the engine to be on
`and engaged. In the trivial case of no power demand, these values are always
`zero, of course (u = 0, Be = 0, Bc = 0).
`In relation to the torque and speed values required at the drive line the
`supervisory controller determines at each instant the operating mode to be
`adopted and the value of the ratio u(t). In all practical control strategies, the
`engine is shut down when the torque at the wheels is negative or zero, i.e.,
`
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`206
`
`7 Supervisory Control Algorithms
`
`Table 7.1. Control parameters for different parallel hybrid operating modes.
`
`Mode
`ICE
`ZEV
`ZEV
`Regenerative braking
`Regenerative braking
`Power assist
`Recharge
`
`u
`0
`1
`1
`1
`1
`∈ (0, 1)
`< 0
`
`Be
`1
`0
`0
`0
`0
`1
`1
`
`Bc
`1
`0
`1
`0
`1
`1
`1
`
`1
`2a
`2b
`5a
`5b
`3
`4
`
`when the vehicle is coasting or braking. The control strategies in the literature
`differ for the choice of u(t) and of Be(t) when the power required is positive.
`Control strategies may be classified according to their dependency on the
`knowledge of future situations. Non-causal controllers require the detailed
`knowledge of the future driving conditions. This knowledge is available when
`the vehicle is operated along regulatory drive cycles, or for public transporta-
`tion vehicles that have prescribed driving profiles. In all other cases, driving
`profiles are not predictable in advance, at least not in the sense that the exact
`speed and altitude profiles as a function of time would be known a priori. In
`these cases, causal controllers must be used.
`A second classification can be made among heuristic, optimal, and sub-
`optimal controllers. The first class of controllers represents the state of the
`art in most prototypes and mass-production hybrids. Optimal controllers are
`inherently non-causal, although the next sections will show how to substan-
`tially reduce the amount of information required. Sub-optimal controllers are
`often causal.
`
`7.2 Heuristic Control Strategies
`
`Heuristic controllers are based on Boolean or fuzzy rules involving various
`vehicular variables. A typical heuristic approach, sometimes called “electric
`assist” strategy [272, 273, 150, 207], is based on the torque demand and on
`the vehicle speed:
`• below a certain vehicle speed the motor is used alone (u = 1);
`• above this speed threshold and below the maximum engine torque at the
`current engine speed, the engine alone is used (u = 0);
`• however, if the battery state of charge is too low, the engine is forced to
`deliver excess torque to recharge the battery (u < 0);
`• if the state of charge is too high, the motor is used alone (u = 1); and
`• above the engine maximum torque at the current engine speed, the motor
`is used to assist the engine (0 < u < 1).
`
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`7.2 Heuristic Control Strategies
`
`207
`
`In the baseline “electric assist” strategy the key control parameter is the
`speed threshold at which the choice is made between motor or engine oper-
`ation (see Fig. 7.1). Other strategies may define different thresholds based
`on different combinations of vehicular variables. One possibility is to oper-
`ate the engine only above a specified fraction of the maximum torque of the
`engine at the current engine speed [272]. Another strategy consists of using
`an acceleration threshold at which the choice is made between the quasi-
`stationary, ICE-based (u = 0) mode, and the transient, electrical (u = 1)
`mode [35]. Also power thresholds are often used. They offer the advantage
`of being immediately comparable with the power limits of the prime movers
`[151, 33, 269, 29, 80]. More complex combinations of power demand, speed,
`and possibly other variables may also be used [14].
`
`Fig. 7.1. Typical heuristic power management control for parallel hybrids, in terms
`of motor torque and vehicle speed (a) or motor power and state of charge of the
`battery (b).
`
`T
`
`P
`
`0 < u < 1
`
`u = 0
`
`u = 1
`
`u = 1
`
`active braking
`
`(a)
`
`0 < u < 1
`
`u < 0
`
`u = 0
`
`u = 1
`active braking
`(b)
`
`v
`
`u = 1
`
`q
`
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`208
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`7 Supervisory Control Algorithms
`
`When both prime movers are on, the value of u is mainly determined by
`the battery state of charge. In the baseline “electric assist” strategy, the engine
`delivers excess torque only when the state of charge reaches a specified lower
`bound. In contrast, the strategy sometimes referred to as “balanced electric
`assist” [272] continuously modulates the power split ratio to keep the state of
`charge at a constant level.
`In other control systems, the value of u varies continuously as a function
`of two to four vehicular parameters, based on a “fuzzy” set of rules. Vehicular
`variable sets may include vehicle speed and engine speed [142], state of charge,
`power demand and motor speed [223, 218], temperature, and vehicle accel-
`eration and speed, state of charge [88], torque demand and state of charge
`[11, 24], vehicle acceleration and engine speed [148].
`The main advantage of heuristic controllers is that they are intuitive to
`conceive and rather easy to implement. If properly tuned, they can provide
`good results in terms of fuel consumption reduction and charge sustainability.
`Unfortunately, the behavior of heuristic controllers strongly depends upon the
`choice of the thresholds involved, which actually can vary substantially with
`the driving conditions [227]. The resulting limited robustness of heuristic con-
`trollers, in addition to the tuning effort required, motivates the development
`of model-based controllers that optimize the power flows.
`
`7.3 Optimal Control Strategies
`
`7.3.1 Optimal Behavior
`
`The main objective of the energy-management controller is to minimize fuel
`consumption along a route. Obviously, it is not necessary to minimize the fuel
`mass-flow rate at each instant of time, but rather the total fuel consumed
`during a driving mission.
`Possible missions are single or multiple repetitions of the governmental
`test-drive cycles, see Chap. 2. Alternatively, missions can be driving patterns
`recorded on typical routes. During operation, an HEV energy-management
`controller can explicitly use all of the available information about the mission.
`The mission information is either provided by the driver or identified implicitly
`by the control algorithm.
`The energy-management controller must respect various hard and soft
`constraints. For instance, the battery must never be depleted below a specified
`threshold, while the torque provided by the engine is limited.
`
`Performance Index
`
`As illustrated in Figure 7.2, the simplest performance index J = mf (tf ) is
`the fuel mass mf consumed over a mission of duration tf . Hence, J can be
`written as [62, 20, 172, 236, 143, 245, 268, 276, 140, 282]
`
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`7.3 Optimal Control Strategies
`
`209
`
`(7.1)
`
`∗m
`
`f (t, u(t)) dt.
`
`Z tf
`
`0
`
`J =
`
`Pollutant emissions can also be included in the performance index J by
`considering the more general expression
`
`Z tf
`
`J =
`
`L(t, u(t)) dt,
`
`(7.2)
`
`0
`
`where L(·) is the cost function. The emission rates of the regulated pollutants
`can be included in the performance index (7.1) by introducing a weighting
`factor for each pollutant species [125, 31, 153]. However, if the ICE is a spark-
`ignited engine operated with stoichiometric air/fuel ratios, its pollutant emis-
`sions can usually be reduced to negligible levels using a three-way catalytic
`converter. Accordingly, the pollutant emission is not considered as part of the
`optimization problem, although in practice “duty-cycle” (on/off operation) or
`engine shutoff at idle can cause problems due to excessive pollutant emissions
`caused by engine or catalyst cooling.
`Drivability issues are sometimes included in the optimality criterion. For
`example, the cost function in [284] includes an anti-jerk term, which consists of
`the engine acceleration squared, multiplied by an arbitrary weighting factor.
`Smoothness and driver-acceptance considerations are included in [268] among
`the local constraints discussed below.
`
`Fig. 7.2. Typical trajectories of the state variable q(t) and consumed fuel mass
`mf (t) (bold) along a mission.
`
`mf
`
`q
`
`J
`
`q(t )f
`q(0)
`
`qmin
`
`net storage
`net depletion
`
`0
`
`Mission
`
`t
`
`ft
`
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`210
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`7 Supervisory Control Algorithms
`
`Integral Constraints
`
`Obviously, the drive mode that minimizes the performance index (7.1) cor-
`responds to a purely electrical strategy in which all of the traction power
`is provided by the battery. However, if the energy recovered by regenerative
`braking is not sufficient to sustain the battery charge, this choice can leave
`the battery completely discharged at the end of the mission.
`The sustenance of the energy-storage system is required for the vehicle
`certification process. Since only small deviations from the nominal value of
`the state of charge (SoC) are permitted at the end of tests to assess vehicle
`energy consumption, energy-management controllers must ensure small SoC
`variations over drive cycles.
`In principle, the sustenance constraint can be taken into account in two
`different ways, namely, as a soft constraint, that is, by penalizing deviations
`from the initial value of the energy stored at the end of the mission, or as a
`hard constraint, by requiring that the energy stored at the end of the mission
`equal the value at the start of the mission.
`To represent constraints on the final SoC q(tf ), a penalty function φ(q(tf ))
`is added to the performance index (7.2) to obtain a charge-sustaining perfor-
`mance index of the form
`
`J = φ(q(tf )) +
`
`L(t, u(t)) dt.
`
`(7.3)
`
`Z tf
`
`0
`A hard constraint, in which q(tf ) must exactly match the initial value q(0),
`is often explicitly assumed [62, 20, 172, 236, 143, 245, 268, 140, 282].
`Soft constraints can be added as functions of the difference q(tf ) − q(0).
`In [152] the quadratic penalty function φ(q(tf )) = α · (q(tf ) − q(0))2 is used,
`where α is a positive weighting factor. In [153], the term α · (q(t) − q(0))2 is
`included in the cost function.
`A quadratic penalty function tends to penalize deviations from the target
`SoC, regardless of the sign of the deviation. In contrast, a linear penalty
`function of the type
`
`φ(q(tf )) = w · (q(0) − q(tf )) ,
`where w is a positive constant, penalizes battery use, while favoring the energy
`stored in the battery as a means for saving fuel in the future. Since the penalty
`function (7.4) can be expressed as
`
`(7.4)
`
`Z tf
`
`φ(q(tf )) = w ·
`˙q dt,
`(7.5)
`the variable w · ˙q(t) can be added to the cost function of (7.3) to yield the
`performance index
`
`0
`
`Z tf
`
`0
`
`J =
`
`(L(t, u(t)) + w · ˙q(t)) dt.
`
`(7.6)
`
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`The parameter w is often chosen arbitrarily [125, 284, 276], while in the reg-
`ulatory standard SAE J1711 [216], w = 38 kWh per gallon of gasoline. Other
`physically meaningful definitions of w are discussed below.
`The piecewise-linear penalty function
`wdis · (q(0) − q(tf )), q(tf ) > q(0),
`wchg · (q(0) − q(tf )), q(tf ) < q(0),
`
`(
`
`φ(q(tf )) =
`
`7.3 Optimal Control Strategies
`
`211
`
`(7.7)
`
`is at the core of equivalent-consumption minimization strategies (ECMS)
`[183], which represent real-time implementations of optimal control algo-
`rithms. Their formulation is presented in the next section.
`
`Local Constraints
`
`Local constraints can also be imposed on the state and control variables. These
`constraints mostly concern physical operation limits, notably the maximum
`engine torque and speed, the motor power, or the battery state of charge. Con-
`straints on the control variables are imposed in [268] to enhance smoothness
`and driver acceptance.
`
`7.3.2 Optimization Methods
`
`This section presents various approaches to evaluating optimal control laws.
`These approaches are grouped into three subclasses, namely, static optimiza-
`tion methods, numerical dynamic optimization methods, and closed-form dy-
`namic optimization methods.
`
`Static Optimization
`
`Since a mission usually lasts hundreds to thousands of seconds, while, at each
`time t, multiple values of u(t) must be evaluated, finding the optimal con-
`trol law by inspecting all possible solutions requires excessive computational
`and memory resources. The simplified approach described in [20] for a series
`HEV can easily be extended to parallel HEVs. This approach does not re-
`quire detailed knowledge of the actual power demand at the wheels Pm(t),
`but only its average and root mean square values. The charge sustenance is
`guaranteed only when duty-cycle operations are performed. For continuous
`operation of the primary energy source, charge sustenance must be achieved
`using an additional slow, integrative controller, which makes the controller
`inherently suboptimal. Dynamic optimization techniques, as presented in the
`next section, avoid this drawback.
`
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`7 Supervisory Control Algorithms
`
`Numerical Optimization Methods
`
`Dynamic programming is commonly used for optimization over a given time
`period [172, 284, 152, 153, 13]. This method can be used to minimize the
`performance index (7.1) in the presence of a hard or a soft constraint on the
`terminal value of the SoC.
`Dynamic programming requires gridding of the state and time variables,
`and thus the optimal trajectory is calculated only for discretized values of
`time and SoC. Consequently, the integral (7.3) and the state dynamics
`
`˙q(t) = f(t, q(t), u(t))
`
`(7.8)
`
`are replaced by discrete counterparts. A useful property of all dynamic pro-
`gramming algorithms is that their computational burden increases linearly
`with the final time tf . Since the computational burden increases exponen-
`tially with the number of state variables, however, reasonably long missions
`can be analyzed only if the number of state variables is small. Conveniently,
`(7.8) is a scalar equation.
`The cost-to-go function Γ (t, q) is the cost over the optimal trajectory pass-
`ing through the point (t, q) in the time-state space, up to the terminal time
`tf , as shown in Fig. 7.3. Based on this definition, the value of J = Γ (0, q(0)).
`To evaluate Γ (t, q), the computation proceeds with a time-discretization step
`∆t backward from time tf − ∆t to time t = 0 [30, 27]. The cost-to-go function
`is then evaluated from the recursion
`{Γ (t + ∆t, q(t) + ˙q(t, q(t), u(t)) · ∆t) + L(t, u(t)) · ∆t} .
`(7.9)
`
`Γ (t, q(t)) = min
`u
`
`The initial condition for (7.9) is imposed at time tf by
`
`Γ (tf , q(tf )) = φ(q(tf )).
`
`(7.10)
`
`The feedback function
`{Γ (t + ∆t, q(t) + ˙q(t, q(t), u(t)) · ∆t) + L(t, u(t)) · ∆t}
`(7.11)
`
`U(t, q(t)) = arg min
`u
`
`represents the control strategy to be stored for real-time operation.
`Due to the discretization of the state, the values of q are either inter-
`polated or approximated by the nearest available values on the grid. In the
`latter case, rounding can artificially increase or decrease the battery energy
`calculated over the optimal trajectory. The energy artificially introduced or
`deleted by rounding determines whether the adopted state-space discretiza-
`tion is acceptable or the number of grid points must be increased.
`Improved algorithms with reduced computational time are available. They
`will be discussed in detail in Appendix III. For example, the iterative dynamic
`programming algorithm in [13] is based on the adaptation of the state space.
`
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