Systematic Optimization and Control Design for Downsized Boosted Engines with Advanced Turbochargers

Home , Engine displacement

THESIS

Presented in Partial Fulﬁllment of the Requirements for the Degree

Master of Science

in the Graduate School of The Ohio State University

Yuxing Liu, B.E.

Graduate Program in Mechanical Engineering

The Ohio State University

2014

Master’s Examination Committee:

Professor Marcello Canova, Advisor

Professor Giorgio Rizzoni c Copyright by

Yuxing Liu

2014 Abstract

Downsized boosted engines are a promising solution to increase fuel economy and reduce

CO2 emissions. With the arrival of several advanced charging technologies, the complexity of matching an engine to charging systems to achieve high levels of boost pressure over a wide range of engine speed increases considerably. Besides, the air-path control of advanced boosting systems is more challenging due to the presence of multiple actuators. This research aims at developing a methodology for systematic optimization and control design for downsized boosted engines with advanced boosting systems.

The objective motivates the development of parametric models for families of compressors and turbines, considering the effects of geometric parameters on the flow and efficiency characteristics. The parametric modeling approach is used to replace and existing map-based model of the two-stage turbocharger of a Diesel engine simulator, to provide a validation against full engine test data.

Next, a study is conducted to assess the opportunity of controlling the air-path of

Diesel engine with two-stage turbochargers to increase the exhaust gas energy upstream of the aftertreatment system. The problem is formulated as model predictive control tracking problem, and the controller and observer are designed on a piecewise affine system. The effects of different weights in cost function are compared, and a Pareto front is obtained.

Finally, a heuristic control is developed based on MPC results.

As main contributions, the parametric model of turbocharger and model predictive control of air-path will serve as ground work for future research on systematic optimization and control design of engines with advanced boosting systems.

ii This document is dedicated to my parents.

I would not be who I am today without them.

iii Acknowledgments

I would like to express my sincere gratitude to my advisor, Dr. Marcello Canova, for his patience, motivation, knowledge, and providing me with an excellent atmosphere for doing research. I would like to thank the other committee member, Dr. Giorgio Rizzoni, who oﬀers me two edifying courses and leads me to the ﬁeld of automotive engineering.

I would also like to thank Dr. Lisa Fiorentini for her insightful comments and generous helps on the part of control design, and Dr. Stephanie Stockar for her detailed literature review on combined design and control optimization. Special thank goes to Junqiang Zhou, who as a good colleague and friend, is always willing to help and give his best suggestions.

Many thanks to Dr. Fabio Chiara for his original work in previous project, to Massimo

Naddeo for his helps on parametric turbocharger models, to Paolo Di Martino for supplying turbochargers data utilized in this study.

I am grateful to General Motors Corporation for providing engine experimental results and ﬁnancial support for this study, and in particular to Dr. Y.Y. Wang for supporting the research activity that led to the content of this thesis.

Finally, I would like to thank my parents Zhicheng Liu and Fang Yang, for supporting me spiritually throughout my life.

iv Vita

August 22, 1989 ...... Born in Lanzhou, China

2008 ...... Lanzhou No.1 Senior High School, Lanzhou, China

2012 ...... B.E. in Mechanical Engineering, Tongji University,

Shanghai, China

2012 to present ...... Graduate Research Associate, Center for

Automotive Research, The Ohio State University

Fields of Study

Major Field: Mechanical Engineering

Specialization: Modeling, control, estimation, and diagnosis for automotive systems

v Table of Contents

Abstract ...... ii

Dedication ...... iii

Acknowledgments ...... iv

Vita ...... v

Table of Contents ...... vi

List of Figures ...... ix

List of Tables ...... xiii

1 Introduction ...... 1 1.1 Background...... 1 1.2 Motivations...... 3 1.3 Objectives...... 5 1.4 Outline...... 6

2 Literature Review ...... 7 2.1 Advanced Boosting Systems...... 7 2.1.1 Single-Stage Turbocharging Systems...... 7 2.1.2 Two-Stage Turbocharging Systems...... 8 2.1.3 Electric Boosting Systems...... 11 2.2 Analytical Model for Turbocharger...... 13 2.2.1 Compressor Model...... 14 2.2.2 Turbine Model...... 14 2.3 Diesel Engine Air-Path Control...... 15 2.3.1 EGR-VGT Control of Diesel Air-Path System...... 15 2.3.2 Model Predictive Control for Diesel Air-Path System...... 16 2.3.3 Integrated Control of Engine and Aftertreatment Systems...... 17 2.4 Combined Design and Control Optimization...... 18

vi 3 Parametric Model of Turbocharger ...... 19 3.1 Basic Turbomachinery Theory...... 19 3.1.1 Dimensionless Representation...... 19 3.1.2 Similitude Theory...... 20 3.1.3 Turbocharger Geometric Parameters...... 21 3.2 Compressor Model...... 22 3.2.1 Assumptions and Compressor Selection...... 23 3.2.2 Compressor Mass Flow...... 25 3.2.3 Compressor Eﬃciency...... 33 3.2.4 Surge and Choke Model...... 37 3.3 Turbine Model...... 39 3.3.1 Turbine Selection...... 41 3.3.2 Turbine Mass Flow...... 42 3.3.3 Turbine Eﬃciency...... 50 3.4 Mean-Value Model of Diesel Engine...... 53 3.4.1 Valves and Flow Restrictions...... 55 3.4.2 Receivers...... 55 3.4.3 Mass Flow through Engine...... 56 3.4.4 Engine-out Temperature and Torque Production...... 57 3.4.5 Turbocharger Speed...... 58 3.5 Integration and Validation of Two-Stage Turbocharged Engine Model... 58 3.5.1 Steady-State Validation...... 59 3.5.2 Simulation of FTP Cycle...... 61

4 Air-Path Control of Diesel Engine for Rapid Warm-up ...... 63 4.1 Thermal Model of Exhaust System...... 65 4.1.1 Bypass Valve Thermal Model...... 65 4.1.2 Catalyst Thermal Model...... 69 4.2 Preliminary Analysis on FTP Cycle...... 69 4.2.1 Baseline Scenario...... 69 4.2.2 Influence of HP Bypass Valve...... 69 4.2.3 Influence of LP Waste-Gate Valve...... 70 4.3 Piecewise Affine System Development...... 71 4.3.1 Model Order Reduction...... 71 4.3.2 Linearization and Discretization...... 73 4.3.3 Piecewise Affine System Validation...... 77 4.4 Model Predictive Controller Design...... 78 4.4.1 Introduction of Model Predictive Control...... 78 4.4.2 Model Predictive Control for the Engine Air-Path...... 80 4.4.3 MPC Implementation Details...... 84 4.5 Observer Design...... 87 4.5.1 Reduced Order Observer...... 88 4.5.2 Observer Design Considering Model Uncertainties...... 90

vii 4.6 Simulation Results and Discussion...... 93 4.6.1 Feedforward Control...... 93 4.6.2 MPC and Pareto Analysis...... 95 4.6.3 Heuristic Control...... 98

5 Conclusions and Future Work ...... 101 5.1 Conclusions...... 101 5.1.1 Parametric Model of Turbocharger...... 101 5.1.2 Model Predictive Control of Diesel Engine Air-Path...... 103 5.2 Future Work...... 105

References ...... 107

A Nomenclature ...... 120 A.1 Abbreviations...... 120 A.2 Subscripts...... 121 A.3 Variables and Constants...... 122

viii List of Figures

2.1 Variable Geometry Turbine [24]...... 7

2.2 Twin Turbo Parallel Operation (Left) versus Sequential Operation (Right) [32]9

2.3 Regulated 2-Stage Charging System: Commercial Application [39] (Left) and

Passenger Car Application [40] (Right)...... 10

2.4 Electrically Assisted Turbocharger (Left) and eBooster Charging System

(Right) [54]...... 13

3.1 Deﬁnition of the Main Geometric Parameters...... 22

3.2 Example of Compressor Maps: Corrected Dimensional Maps (Top) and Di-

mensionless Maps (Bottom)...... 24

3.3 Compressor Maps of the Original Two-Stage Turbocharger and Similar Mod-

els of the GT Series...... 24

3.4 Identiﬁcation of the JK Flow Model...... 26

3.5 Identiﬁcation of the SE Flow Model...... 26

3.6 Extrapolation of the JK and SE Models...... 28

3.7 Prediction Error for JK Model (Type 1: Pressure Ratio as Input)...... 30

3.8 Prediction Error for SE Model (Type 1: Pressure Ratio as Input)...... 30

3.9 Prediction Error for JK Model (Type 2: Mass Flow as Input)...... 31

3.10 Prediction Error for SE Model (Type 2: Mass Flow as Input)...... 31

3.11 Compressor Flow Parametric Model Validation (JK Model, Type 2).... 32

3.12 Schematic Diagram for Momentum Equation...... 32

3.13 Identiﬁcation of the Eriksson Eﬃciency Model...... 33

3.14 Identiﬁcation of the Shaver Eﬃciency Model...... 34

ix 3.15 Identiﬁcation of the Martin Eﬃciency Model...... 34

3.16 Extrapolation of 3 Eﬃciency Models...... 35

3.17 Prediction Error for 3 Eﬃciency Models...... 36

3.18 Compressor Eﬃciency Parametric Model Validation (Eriksson Model)... 37

3.19 Identiﬁcation of the Surge and Choke Model...... 39

3.20 Integrated Compressor Flow Model and Surge-choke Model...... 39

3.21 Turbine Maps of the Original Two-Stage Turbocharger and Similar Models

of the GT Series...... 42

3.22 Interpretation of the Turbine Flow Model Parameters...... 43

3.23 Identiﬁcation of the Turbine Flow Model with Constraints on the Parameter

m ...... 44

3.24 Turbine Flow Parametric Model Validation...... 45

3.25 Comparison of Constrained Fitting and Global Fitting...... 46

3.26 Variable Geometry Turbine Flow Model Validation (HPT)...... 47

3.27 Variable Geometry Turbine Flow Parametric Validation (GT32 and GT17) 47

3.28 Turbine Flow Model Parameter as Function of VGT Position: 3 VGTs (left)

and Normalization (Right)...... 48

3.29 VGT Flow Parametric Model Validation (Extrapolated VGT Maps from

Fixed Geometry Turbine Data)...... 49

3.30 BSR-based Turbine Eﬃciency Model Validation (HPT)...... 50

3.31 Power-based Turbine Eﬃciency Model Validation (HPT)...... 51

3.32 Modiﬁed Power-based Turbine Eﬃciency Model Validation (HPT)..... 52

3.33 Variable Geometry Turbine Eﬃciency Model...... 52

3.34 Air-Path Conﬁguration of Diesel Engine with Two-Stage Turbocharger... 54

3.35 Functional Scheme of the System Model...... 55

3.36 Engine Operating Conditions and Control Inputs for Model Calibration.. 59

3.37 Block Diagram of Engine Model with Open-loop Control Strategy..... 59

x 3.38 Steady-State Validation of the Engine Model...... 60

3.39 Comparison of Compressor Operating Points: Original Look-up Table Model

(Left) and Parametric Model (Right)...... 61

3.40 Engine Speed and Torque Proﬁle during FTP Cycle...... 62

3.41 Commands during FTP Cycle: Fueling Rate (Left) and Actuators Positions

(Right)...... 62

3.42 Simulation Results of the Engine Model during FTP Cycle...... 62

4.1 Air-Path Conﬁguration of Two-stage Turbocharged Engine with DOC... 64

4.2 Sketch of Bypass Valve Thermal Model: Converging-Diverging Nozzle (Left)

and Turbine with Bypass (Right)...... 65

4.3 The Bypass Valve Outlet Temperature...... 68

4.4 FTP Cycle Analysis: Baseline...... 70

4.5 FTP Cycle Analysis: Inﬂuence of Bypass...... 70

4.6 FTP Cycle Analysis: Inﬂuence of Wastegate...... 71

4.7 Validation of Reduced Order Model at Steady-State Conditions...... 72

4.8 Validation of Reduced Order Model during FTP Cycle...... 73

4.9 Partition of the Engine Map Based on FTP Cycle (0-200s)...... 74

4.10 Validation of Linearized Model at Region 6: Change of Actuator Position. 74

4.11 Validation of Linearized Model at Region 6: Change of Operating Point.. 75

4.12 Validation of Linearized Model at Region 18: Change of Actuator Position. 76

4.13 Validation of Linearized Model at Region 18: Change of Operating Point. 76

4.14 Validation of Piecewise Aﬃne System during FTP Cycle...... 77

4.15 A Discrete MPC Scheme...... 78

4.16 Diagram of Engine Air-Path Control System...... 81

4.17 References Signal Maps (Left) and Comparison with FTP Baseline Results

(Right)...... 84

xi 4.18 Validation of Observer on PWA System...... 89

4.19 Validation of Observer on Full Order Engine Model...... 89

4.20 Steady-State Estimation Error at Region 18: eig(Ai − LiCi) = 0.9 · eig(Ai). 91

4.21 Steady-State Estimation Error at Region 18: eig(Ai − LiCi) = eig(Ai)... 91

4.22 Steady-State Estimation Error at Region 18: Li Obtained by Pole Placement 92 4.23 Validation of Redesigned Observer on Full Order Engine Model...... 92

4.24 Comparison of Feedforward Control with Diﬀerent Bypass Opening.... 94

4.25 MPC for Rapid Warm-up on FTP Cycle (wT = 0, wp = 50, we = 50)... 95

4.26 MPC for Rapid Warm-up on FTP Cycle (wT = 2000, wp = 10, we = 50). 95 4.27 Comparison of Feedforward and Feedback Portions of Actuators Positions. 96

4.28 State Estimation of Closed-loop System...... 96

4.29 Comparison of MPC with Diﬀerent Weights wT = 2000, we = 50...... 97 4.30 Comparison of Feedforward Control and MPC...... 98

4.31 Identiﬁcation of Heuristic Control of Bypass Valve Opening...... 99

4.32 Heuristic Control for Rapid Warm-up on FTP Cycle...... 99

4.33 Comparison of Heuristic Control and MPC...... 100

xii List of Tables

3.1 Pseudo-dimensionless Representation...... 20

3.2 Dimensionless Representation...... 20

3.3 GT Compressors Speciﬁcation...... 23

3.4 Comparison of the Two Compressor Flow Models...... 29

3.5 Comparison of the Three Compressor Eﬃciency Models...... 35

3.6 GT Turbines Speciﬁcation...... 41

3.7 GM Duramax Engine Characteristics...... 54

4.1 Comparison of 3 Output Injection Matrices...... 90

4.2 Performance Indices of Feedforward Control...... 93

4.3 Performance Indices of MPC...... 97

4.4 Comparison of Heuristic Control and MPC...... 100

xiii Chapter 1

Introduction

1.1 Background

HE automotive industry is currently facing the challenge of meeting the conﬂicting T goals of improving vehicle fuel economy, complying to stringent emission standards, satisfying the customer appeal for performance and keeping aﬀordable costs. In the US, the

Corporate Average Fuel Economy (CAFE) standards have been increased over time to 26.0 mpg by 1985, to 27.5 mpg by 1989, and to 37.8 mpg by 2016 in the passenger car ﬂeet [1,2].

National Highway Traﬃc Safety Administration has recently developed new footprint-based

CAFE standards for both passenger car and light-duty truck fleets [3]. These standards propose to double the required fleet fuel economy by 2025. As electric vehicles and fuel cell technologies present some deficiencies and are immature to be largely spread in mass production, internal combustion engines will remain the mainstay of growing mobility over the next 20 years. As a result, automotive companies will heavily rely on engine technology improvements to meet fuel economy improvement and emissions reduction goals.

These technology improvements have been realized in different engine fields. First, important progress has occurred in engine control. With the advance of electronic devices, the quantity and quality of sensors and actuators has progressively grown, and control strategies have become considerably sophisticated [4]. The injection and combustion processes have also been largely modified with the arrival of common rail systems [5]. These systems are able to realize multiple injections per engine stroke and present high flexibility to optimize the injection timings.

1 Aftertreatment systems have become necessary for Diesel engines [6]. Diesel oxidation catalyst (DOC) is implemented to reduce the emissions of CO, total hydrocarbon emissions, and volatile organic fractions [7]. Diesel particulate ﬁlter (DPF) is introduced to achieve extremely low particulate matters emissions [8]. Lean NOx traps (LNT) have proved to be

a promising technique for reducing the NOx emissions from light-duty Diesel engines, while selective catalytic reduction (SCR) systems, especially urea-SCR systems, have become

the leading devices to reduce tailpipe NOx emissions from medium- to heavy-duty Diesel engines [9]. Meanwhile, exhaust gas recirculation (EGR) has become common practice to

meet the mandatory NOx emissions reductions [10]. EGR works by recirculating a portion of exhaust gas back to the intake manifold. Diluting the fresh air charge with exhaust inert

gas lowers the peak combustion temperatures and the generation of NOx. Air-path also has considerable improvements with the use of ﬂexible valve actuation

and turbocharging. Variable valve timing (VVT) is able to independently control the in-

take and exhaust valve opening to optimize the cylinder charge and residual gas fraction [11].

Similarly, variable valve actuation (VVA) provides more ﬂexibility for engine performance

optimization, due to the ability of controlling valve timing and lift [12]. Exhaust gas tur-

bochargers increases the engine speciﬁc power by forcing even more air to enter into the

cylinders. With higher air density in the combustion chamber, engine performance and spe-

ciﬁc fuel consumption have been improved. Variable geometry turbine (VGT) has gradually

replaced ﬁxed geometry turbine with a waste-gate. By varying the swallowing capacity of

a turbine as the engine operating, VGT could increase useful engine operating speed range

and reduce turbo-lag [13].

All these technological developments have been equipped in modern Diesel or SI engine

to meet the strict emissions regulations requirements. However, their potential improve-

ments to achieve the next standards (especially NOx and CO2) are insuﬃcient and new technologies have to be developed.

Two technologies will be applied to reduce NOx emissions. First, more sophisticated

2 alternative combustion modes such as low temperature combustion (LTC), homogeneous charge compression ignition (HCCI), will be employed at low speed and low load regions

[14, 15]. Second, low pressure (LP) EGR systems, extracting exhaust gases after the DPF to recirculate them at the compressor inlet, will arrive in complement or substitution of the conventional high pressure (HP) EGR systems [16].

To reduce CO2 emissions, downsized boosted engines are the most promising solution currently followed by the major automakers. By reducing the engine displacement, lower fuel consumption is achieved while still making use of conventional aftertreatment systems,

At the same time, engine peak performance and driveability are retained through the use of boosting solutions. Several studies have shown that 20% fuel economy improvement can be achieved with a 40% downsizing and turbocharging. The addition of advanced fuel management strategies, such as fuel cut oﬀ during idling and coasting, increases the fuel economy to about 30%, matching the performance of full hybrid vehicles without the added weight and cost penalties [17–19].

An important requirement for boosting system is to keep high pressure level over a wide range of engine speed. The high boost pressure required for the rated power can only be produced by a larger device, while the turbocharger design needs to be small to achieve a high boost pressure at low speeds. To solve this conﬂict, the state of art is the well-known method of designing the turbocharger with VGT. However, these systems are unable to provide the higher boost and wider operating range required by the downsized engine. So, advanced charging systems will be introduced for further engine development.

1.2 Motivations

In single-stage arrangements, turbochargers have been the dominant technology for engine charging system, taking almost 100% of the market share, because of its high eﬃciency and ability to recover waste energy. With the arrival of multistage systems, the chargers

3 can be arranged in series or parallel, working together or sequentially. Even though the main charger is generally a turbocharger, the second one can be turbocharged, mechan- ically assisted (supercharger) or electrically assisted. Variable geometry elements and an additional intercooler between stages can also be considered to enhance the performance.

Depending on the control requirements, multiple actuators could be used to regulate the boost pressure. All these possibilities represent a large number of conﬁgurations and no standard solutions exist for all the diﬀerent applications. In order to optimize the performance of the downsized engine, various boosting architectures should be analyzed and compared, including the component designs (sizing).

The system complexity and interactions between subsystems make the control of advanced boosting systems challenging. For instance, in a series/sequential arrangement, mode-switching control is required to improve the robustness and smoothness of response in transitions between one charger and the other. For the case of electrically assisted chargers, system-level energy optimization needs to be designed to optimize the energy utilization of the system. Therefore, there is a need to develop a systematic methodology for system optimization and control design for downsized engines with advanced boosting systems.

For typical automotive applications, the sizing and optimization of the engine system conﬁguration and the control system optimization are usually performed in sequence. Al- though the approach of decoupling the engine hardware design from the control software design phase substantially simpliﬁes the overall engineering process, a sequential procedure may not provide favorable performances, as the optimality for one single problem could be disrupted by another. Although more challenging, a combined optimization method, where the engine air-path system design and the control algorithm are simultaneously optimized, is more appealing. Such an approach would allow one to integrate individual problems

(design and control) together, through multi-objective optimization.

To enable a coupled and systematic model-based optimization, parametric models for the boosting device components, such as compressor and turbine, are of critical importance.

4 These models will allow one to systematically evaluate the inﬂuence of design parameters on a set of performance objectives, particularly in cases where experimental data are not easily accessible. Moreover, considering the complexity of the optimization, the parametric models should be of low computational load, easy for the calibration and accurate enough at least within the range of interest for the geometric parameters.

1.3 Objectives

The content of this thesis is developed as part of a larger research project with General

Motors Corporation oriented to explore model-based system optimization and control design for advanced engine charging systems, with a research approach that combines thermo-

ﬂuid modeling, engine/powertrain system simulation, and the use of estimation and control design methods for linear and nonlinear systems.

In this project, the model-based system-level optimization and control design methodology will be applied to a series/sequential two-stage turbocharger (2ST) system. The speciﬁc objectives which are pursued in the current period are: a. Development of a parametric model that can represent the performance of a family of

turbochargers based on key geometric parameters. b. Control design of 2ST engine system for rapid warm-up of the aftertreatment system.

c. Co-optimization of system design and control for 2ST conﬁgurations to improve the

system performance in steady-state and transient conditions. d. Design of robust mode-switching control strategies between two stages to guarantee

smooth boost response during transitions.

According to the progress of this ongoing project, this Master’s thesis will focus on the objectivesa andb. The remaining tasks will be part of the PhD research program of the author.

5 1.4 Outline

This thesis is organized as follows. Chapter 2 contains an exhaustive literature review on several relative topics. First, the state of the art of advanced boosting systems is explored. Then, several analytical modeling approaches for turbochargers are evaluated.

Next, various control approaches for engine air-path systems as well as integrated engine and aftertreatment systems are summarized. Finally, all existing works on combined design and control optimization are brieﬂy introduced.

In Chapter 3, the development of parametric model for the compressor and turbine maps is described. First, a brief overview of basic turbomachinery theory is given, where the dimensionless parameters and relevant geometric parameters for the characterization of flow and efficiency maps are defined. Then, a formal derivation of the parametric maps for families of compressors and turbines is discussed together with the model results. The parametric modeling approach is applied to an engine model with 2ST, which is validated against steady-state and transient experimental data.

Chapter 4 presents the design and application of an air-path control strategy of the

2ST Diesel engine for rapid catalyst warm-up, based on model predictive control (MPC).

First, thermal models for waste-gate and catalyst are introduced. Preliminary study on the FTP city driving cycle is conducted to analysis the inﬂuence of bypass opening on catalyst warm-up. Then, model order reduction and linearization are applied, resulting in a discrete-time piecewise aﬃne (PWA) system. The rapid warm-up problem is formulated as

MPC tracking control, and controller and state observer are designed on the PWA system.

Finally, MPC and observer are implemented on the full order nonlinear engine model, and a heuristic control for bypass opening is extracted from the solution of MPC.

The thesis ends with conclusions and recommendations for future research.

6 Chapter 2

Literature Review

2.1 Advanced Boosting Systems

2.1.1 Single-Stage Turbocharging Systems

URBOCHARGER technology history can be presented through the evolution of

compressor and turbine over the last decades. The exhaust-gas turbocharger with TChapter 2 Section 2.1 variable geometry turbine (VGT) has been widely adopted for Diesel engines. VGT is used tocomponents control a desired of the boost flow velocity pressure and over a the high engine enthalpy operating gradient range, lead which to a is high achieved turbine work output and thus a high boost pressure. Whereas in the fully open by guide vanes arranged upstream of the turbine rotor, as shown in Figure 2.1. Many position of the guide vanes, maximum turbine flow capacity is obtained with researcha high papers centripetal are available share on of the the velocity topic of vector the variable and a relatively geometry lower turbines turbine for Diesel work output. The advantage of this power output control over a wastegate enginescontrol [13,20– relates22]. Even to the though fact thatthe spark the full ignition exhaust-gas (SI) engine mass benefits flow is from always this routed technology, the highvia exhaust the turbine gas temperature and can be haveconverted prevented to output. the variable The blades geometry are turbine actuated from by being means of levers which are controlled by an adjusting sleeve accommodated widelyin used the [ turbine23]. casing. This adjusting sleeve can be actuated by a number of different pneumatic or electrical actuators.

FigureFigure 2.1: 2.2: Variable Variable Geometry geometry Turbine turbine. [24]

A typical engine characteristic curve of a VGT application is plotted on a compressor characteristic map in figure7 2.3. The conflict of goals of wishing to achieve ever-higher starting torques with lower engine speeds and, at the same time, a higher rated power output can be clearly illustrated by the compressor map. The engine characteristic curve is very near to the surge limit and the compressor efficiencies are not at the optimum at these points. In turn the rated output point is very near the compressor’s maximum mass flow (choke) limit where low efficiencies lead to high boost-air temperatures. So, an increase in the starting torque is restricted by the surge limit whereas an increase in rated engine output is restricted by wheel speed limits and altitude reserve.

The requirements applicable to turbochargers with variable turbine geometry have become constantly more stringent in recent years. Just a while ago, turbocharged engines with a per-liter output of 35-40 kW/l was suﬃcient, but now the current state of art demands 50-58 kW/l and new advanced developments push already the turbocharger technology limits up to 65 kW/l.

19 Wide compressor operating ranges are a challenge for a single-stage compressor with high pressure ratio, since it is limited by surge at low ﬂow rates and choke at high ﬂow rates.

More recently, variable geometry compressor (VGC) systems have been considered as a possibility to improve the compressor performance and stability. Centrifugal compressors can be equipped with either variable inlet guide vanes [25–29] or with a variable geometry diffuser [30,31]. Experimental and simulation results have shown improvements in efficiency and pressure ratio over fixed geometry compressor at low speed conditions, in conjunction with the opportunity to expand the operating range in terms of surge and choke limits.

2.1.2 Two-Stage Turbocharging Systems

Two-stage turbocharging is a generic term used to describe the installation of two turbines and compressors to boost a single engine installation. This technology has excellent transient response and it is capable of meeting the needs of current and future downsizing options. The variations of such installations are numerous and there is an issue with the nomenclature used by various OEMs which are explained below.

Parallel Boosting Systems

In parallel boosting systems, also called twin turbo, identically-sized turbochargers are used, each fed by a separate set of exhaust streams from the engine. The pressurized air from both compressors is added up in a common manifold and passed through an intercooler and then fed into intake manifold [32, 33]. A typical layout of a twin turbo ﬁtted on a V6 engine is shown in the left of Figure 2.2. The main advantage of twin turbochargers systems when compared with single turbocharger systems is the turbo-lag phenomenon diminishes considerably. But it does not improve the surge line and the torque at low speed operations.

One solution is to operate only with one turbo at low engine speed and operate both in parallel connection in high speed operation, which is called sequential parallel turbocharging as shown in Figure 2.2 on the right side. Since the sizes of two turbochargers are not

8 Section 2.2 Chapter 2

FigureFigure 2.5: 2.2: Twin Twin turbo Turbo Parallelparallel Operation operation (Left) (left) versus versus Sequential sequential Operation turbo (Right) operation [32] (right) [63].

necessarily the same in parallel sequential case, this kind of system is able to improve Diesel manyengine other transient speciﬁc response equipment since low inertia but the turbocharger base function is used remains at low speed. a pure parallel one. The main advantage of twin turbochargers systems when compared with Galindo et al. [34–36] applied the parallel sequential boosting system on a 2.2L modern single turbocharger systems is the turbo-lag phenomenon diminishes considerably.passenger-car Theoretically, Diesel engine the to reach turbo-lag similar could performance be reduced in terms by of acceleration 40% according capability to turbo-machinesand low end torque rules as a [63]. 2.7L However,single-stage classicalturbo equivalent parallel engine. connection Turbocharger of the match- two turbos does not improve the surge line and the torque at low speed opera- ing for the parallel sequential turbocharging system is discussed in [34]. A control system tions. To compensate this, a solution is to operate only with one turbo at low enginedecides speeds when to and operate parallel in each connection one of the modes, of compressors and when and in how the to high perform speed the oper- tran- ationsition area. based This on engine solution steady-state represents operation the as second well as case transient called conditions sequential (acceleration, parallel turbocharging as shown in ﬁgure 2.5 on the right side. braking, gearbox shifts), is proposed in [35]. To smooth the torque oscillations during mode

transition,Parallel fuel sequential correction turbocharging strategy and control systems valve are pre-lift able strategy to improve are analyzed diesel in engine [36]. transient response since low inertia turbochargers allow faster accelerations andCompared have advantages to serial arrangement, in terms of there packaging is an advantage and cost in carbon because monoxide of the (CO) lower and turbohydrocarbon sizes. emissions When compared because of a to reduced serial gas sequential contact surface architectures, and only one there expansion is an advantagestage. This in allows carbon lower monoxide heat losses in (CO) the turbochargers and hydrocarbon and a higher emissions temperature because at cata- of a reduced gas contact surface and only one expansion stage. This allows lower heat losses in the turbochargers and a higher temperature at catalyst entrance during the start up phase. But9 it has some disadvantages on rated power due to the small two turbochargers which oﬀers a lower global eﬃciency

22 Chapter 2 Section 2.2

In this arrangement, flow characteristicsSection are 2.2 similar to that of a single turbine, Chapter 2 but with significant changes in the split of expansion ratio between the stages. Once the expansion ratio has increasedVGT to on the the point LP stage, where not the for turbines transient are operation, but to optimize multiple nearly chocked, the HP turbine thenobjectives runs at largely with higher constant control conditions. flexibility. If a further increase in expansion ratio is required, for example when operating at altitude, additional expansion will onlyFor occur passenger across the car LP applications, turbine as Saulnier shown et al. [288] tried at first to use a in figure 2.9 (right side). regulated 2-stage system on a highly downsized 1.5-liter diesel engine. The objective was to achieve the same performance as a similar 2.0-liter engine For heavy- and light duty applications, flexibility of operation rather than reaching a specific power of 70 kW/l with boost pressures higher than 3 bar. absolute gain in performance is generally preferred. In 1998, Pflüger [269] was One difference with the charging system proposed by Pflüger[269] was the the first to introduce a regulated 2-stage turbocharging system for commercial addition of a particulate filter between the two turbos in order to get enough diesel engines. These objectives were not only to increase the engine rated enthalpy flux for regeneration purposes. Defining optimal turbochargers com- power but also make available a very high maximum torque at very low engine speeds and over a wide speed range.bination, High boost they pressures show the at steady low engine state speeds performance could be reached by several improve the engine accelerating behavior,turbochargers reduce matchings, exhaust smoke whereas and transient allow performance needs a real opti- relatively early high mean effective pressure.mization This considering allows, the in connection full boost system with a as a whole and not stage by stage. suitably matched drive train, to reduceAfter engine optimization, speeds (downspeeding) despite an optimal without turbochargers balance in terms of tur- compromising the drivability criteriabine and flow thus capacity, to improve transient fuel response consumption targets could not be reached and “take and noise levels. On the downside,off” the problems disadvantages due to low of regulated engine displacement 2-stage at low RPM were observed. The turbocharging system are additionalHP cost, turbine complexity, was already packaging quite and small poor with low low efficiency and VGT technology load performance. In fact, the lowerdoes pressure not yet ratioexist for per very stage small compared turbines. to So at that time, it was not possible the single-stage system combined toto the decrease low flow even rate more results the turbine in less diameterefficient maintaining tolerable efficiency to operations at partial loads. improve the transient response.

28 Single-stage Two-stage Potential: peak cylinder pressure = 107% 24 ] r Charge Air Regulating Charge Air Regulating a b

[ 2000Cooler Nm Valve Cooler Valve 20 P E

HP stage M Compressor HP stage HPC-BP HPT-BP Turbocharger B Bypass Turbocharger327 kW 16 Ext. BP LP stage LP stage Turbocharger Turbocharger298 kW with Bypass 12 40 60 80 100 Related engine speed [ % ] (a) (b) Figure 2.3: Regulated 2-Stage Charging System: CommercialFigure 2.12: Application Left: diagram [39 of] a(Left) regulated and 2-stagePassenger charging system for passenger Figure 2.10:Figure Left: diagram 2.3: DeﬁnitionCar of a Application regulated of the Main 2-stage [40 Geometric] (Right) chargingapplication Parameters system for [315]- commercial Right: FEV2 concept [352]. application - Right: BMEP curve comparison [269].

In 2004, Schmitt et al. [290, 315] modified the regulated 2-stage system for lyst entrance during the start up phase. A major27 disadvantage of the parallel arrangement is that, since the differently sized compressorspassenger share the car same application pressurepaying ratio, the special boost attention level to the transient response. is that,is limited since by the the differently maximum sized pressure compressors ratioThe of the share system, smaller the schematized same turbocharger pressure in [figure ratio,37]. Nevertheless,2.12, the boost has been level extended with a waste gate is limited by the maximum pressure ratio of the smaller turbocharger [37]. Nevertheless, the parallel arrangement can cover a higher flow rate range. This system is suitable30 for SI theengine parallel which arrangement require wider can flow cover range a higher but lower flow boost ratepressure range. This [38]. system is suitable for SI engine which require wider flow range but lower boost pressure [38]. Series Boosting Systems

SeriesSerial Boosting turbocharging Systems systems were primary designed to

Serial turbocharging systems were primary designed to increase the absolute boost pressure in power production and marine applications. For automotive applications, flexibility of operation rather than absolute gain in performance is generally preferred. In 1998, Pflüger was the first to introduce a regulated 2-stage turbocharging system for commercial Diesel engines [39]. These objectives were not only to increase the engine rated power but also make available a very high maximum torque at very low engine speeds and over a wide speed range.

The basic set-up of series charging system, schematized in Figure 2.3, consists of two turbochargers connected in series with a bypass valve placed across the high pressure tur- 11 bine. The larger low pressure (LP) turbine is matched to set the maximum power, while the

10 small high pressure (HP) turbine is matched to fulfill the low end torque requirements and to reduce the turbo-lag. At low engine speeds, the bypass is closed and the low exhaust gas energy is mainly used in the high pressure stage to compress fresh air. As the air flow rate increases, the capacity of the HP stage will be approached and the flow will consequently become choked. Therefore, in order to avoid excessive back pressures and overspeed of the turbocharger shaft, the HP stage is normally bypassed at high flow rates [39, 41].

Variable geometry turbine can enhance even more the regulated 2-stage charging system performance. Lee et al. [41] replaced the HP fixed geometry turbine with its waste-gate by a VGT. They claims the system with VGT can improve both steady-state and transient performance. Canova et al. [42] compared two 2-stage turbocharger configurations, with different VGT locations. The results show that the proposed new configuration, which has a HP stage with bypass and LP stage with VGT, can achieve higher control flexibility.

For passenger car applications, Schmitt et al. [40,43,44] modiﬁed the regulated 2-stage system to improve the transient response. The system, schematized in the right of Fig- ure 2.3, has been extended with a waste-gate across the LP turbocharger and an adequately large cross-section bypass valve across the HP compressor. The advantage of this arrangement is the ﬂow can be fully diverted around the HP stage at high engine speeds, leaving the rotating parts to achieve just a basic speed level necessary for quick changes in load.

Therefore these elements make possible the use of turbines with low absorption capacity and smaller compressors, allowing a high boost pressure to be achieved at a very early point and improving the transient response. More applications on passenger car can be seen in [45–48].

2.1.3 Electric Boosting Systems

Electric boosting systems have been used in two ways, either by having the electric motor attached directly on the turbocharger shaft, or by having an extra electrically driven compressor supporting the main turbocharger at low loads and during transients.

11 Electrically Assisted Turbochargers

The electrically assisted turbocharger, or EAT, is a turbocharger ﬁtted with an electric motor/generator around the turbocharger shaft and between the compressor and turbine housings, as depicted in the left side of Figure 2.4. The impact of a EAT system has previously been shown to improve acceleration performance, reduced turbo-lag and therefore improve regulation of boost demand, reduce soot emissions, and improve fuel economy

[49–53].

Using electrical energy, additional torque can be applied to the turbocharger shaft when there is not much exhaust gas energy available. This results in faster acceleration of the turbocharger during transient operations and increased boost pressure at low speed operation. When the available turbine power is higher than the power necessary to drive the compressor, the EAT system can be operated in generator mode using part of the turbine work to recover electric energy. This technique is called electric turbocompound.

Kolmanovsky et al. [49–51] applied optimal control techniques to a medium-size passenger car with a EAT system capable of bi-directional energy transfer between the turbocharger shaft and energy storage. The main objective is to optimize the transient response during full load accelerations. In [50], it is found that the system with 1.5kW EAT can achieve same performance and similar fuel consumption as with 10.3kW hybrid version.

Similar conclusion is reported in [53].

Electric Boosters

The other way of using electric assistance is to have an extra centrifugal compressor driven by an electric motor in series with the main turbocharger. This electric supercharger, given in the left of Figure 2.4, is also called electric booster or eBooster. Since the eBooster is completely independent from the turbocharger, there is no way to utilize electric turbocompound. Depending on the amount of electrical energy available from the

12 Section 2.4 Chapter 2

2.4 Electric Boosting Systems

Electric boosting systems have been used in two ways, either by having the electric motor attached directly on the turbocharger shaft or by having an extra electrically driven compressor installed in series with the turbocharger.Section 2.4 Chapter 2 Both arrangements are described in the following subsections. forced oil cooling in the stator and forced air cooling in the rotor [252], see figure 2.35. Rated power was also slightly reduced at 2 kW. Results obtained 2.4.1 Electrically Assisted Turbochargers on a 1.7l turbocharged engine showed engine torque could be increased by approximately 17% at 1000 and 1200 rpm with turbo spool-up time cut down by 33% [164, 375]. The electrically assisted turbocharger, generally called eu-ATL or EAT, consists of a standard wastegate or variable geometry turbine turbocharger with an additional high speed electric motor. Using electrical energy, ad-2.4.2 Electric Boosters ditional torque can be applied to the turbocharger shaft when there is not much exhaust gas energy available. This results in either faster acceleration Another way of using electric assistance is to have an extra radial flow compressor driven by an electric motor in series with the turbocharger. This of the turbocharger during transient operations or increased boost pressureelectric supercharger is called eBooster. It can be placed in the intake air path under low end operating conditions. Other functions such as maintaining tur-before or after the turbocharger, but placing it before provides more flexibility bocharger speed during gearshift to reduce emissions [193], or compressing airin terms of packaging and some minor advantages in terms of power. To keep before cranking (intake air over 100¥C) to improve cold starting can also bethe pressure losses as low as possible, a regulated bypass valve is added around the eBooster to completely divert the air mass flow when it is switched off. realized with the electrical assistance. Valve actuation is supervised by appropriate control strategies to avoid flow recirculation during valve opening [217]. A diagram of the serial configuration with a view of an eBooster is given in figure 2.36.

Intercooler Intercooler Engine Compres

-sor

s s

Engine a

p y e-Motor B

s ECU

s Turbine

a p

y M

B e-Motor

a p

y M

B ECU Turbine Compressor Compressor

(a) Figure(b) 2.36: eBooster charging system [247]. Figure 2.32: Left: the TurbostarterFigure by Turbodyne 2.4: Electrically System Assisted Inc. [253] Turbocharger - Right: diagram (Left) and eBooster Charging System (Right) [54] The eBooster is completely independent from the turbocharger and the of the electrically driven turbocharger [150]. thermal energy of the exhaust gases. Thermomechanical loads are relatively low and there is no way to utilize an energy recovery system due to the sep- The ﬁrst EAT system calledsystem, Turbostarter eBooster wascan be presented used at low in engine1995 by speeds Tur-aration as a regulated of the turbine two-stage and the charging electric motor.system Depending on the amount of bodyne System Inc. [253]. The system was equipped with an external brush- to increase the steady-state torque, or only in transient to reduce turbo-lag eﬀects.60

Lefebvre54 et al. [55] compared a 3-cylinder 1L turbocharged SI engine with eBooster to a baseline 1.8L naturally aspirated engine. With a 1.6kW eBooster, the transient response

and low end torque are largely improved. Moveover, the eBooster can make the main

compressor operate in areas of higher eﬃciency and higher surge margin.

Tavcar et al. [56] compared diﬀerent electric boosting systems, including EAT, eBooster,

and electrically split turbocharger. Results show that the engine utilizing an EAT can

improve transient engine performance, and also possesses the potential to increase high-

speed torque for a limited period of time. While eBooster system has highest low-speed

torque and the fastest transient response during acceleration.

2.2 Analytical Model for Turbocharger

In control-oriented engine models, a turbocharger is usually modeled with a data-driven

(black-box) approach. However, only a limited fraction of the compressor and turbine

operating domain is typically available from a turbocharger supplier. Besides, standard

table interpolation routines are not continuously diﬀerentiable, extrapolation is unreliable

13 and the table representation is not compact [57]. Several modeling approaches have been proposed, ranging from curve-ﬁtting algorithms to more complex approaches in part based on physical considerations. Physics-based models have been also developed to predict the performance of turbocharger, but they are not well suited to control-oriented engine models.

2.2.1 Compressor Model

For compressor mass flow and efficiency models, dimensionless representation is often used for the performance characterization. Jensen-Kristensen method [57], which is widely adopted, used a hyperbolic function to fit the head parameter and flow parameter. This model provide high accuracy in interpolating characteristic maps, but the ability to extrapolate at surge and choke regions is limited. Eriksson et al. proposed an ellipsis model [58], which can be extended to describe reversed flow, surge, normal operation, choke and restriction [59,60]. In contrast, when the turbocharger is mainly operate at a particular range rather than the entire map, a simplified third-order polynomial model is developed in [61].

The compressor isentropic efficiency is typically fitted as a quadratic function of the dimensionless flow parameter. In [57], the coefficients of the quadratic function is fitted as hyperbolic function of blade Mach number, while a compact quadratic form is proposed in [58]. Similarly, a simplified linear relation of new defined efficiency and flow parameter is presented in [61]. These kind of models have limited extrapolation ability. A minimum constant efficiency is imposed to prevent negative value at low mass flow region. To address this issue, a new approach incorporating turbomachinery physics is developed in [62], where the extrapolation quality is ensured by physical laws.

2.2.2 Turbine Model

The mass flow rate through the turbine can be modeled as isentropic orifice flow, where the effective flow area is a function of the VGT position and pressure ratio [57, 62, 63]. In actual turbine, where the expansion is produced in two steps, critical flow conditions are

14 reached for an expansion rate of approximately 3, whereas a nozzle reaches choke conditions with an expansion rate of approximately 1.89. So the standard orifice flow model is not accurate in choked flow conditions. Although a model based on two nozzles in series [64,

65] can solve this issue, such wave action model is not suited to control-oriented engine models. Therefore, a modified orifice equation is proposed in [66], where a tunable polytropic coefficient is introduced to account for all the unmolded effects. Simplified turbine flow models can be found in [58, 61, 67].

The turbine efficiency characteristic map is the most difficult one to model, due to the dependence of the efficiency on pressure ratio, turbocharger speed, and VGT position, rendering the map four-dimensional [63]. Methods based on blade speed ratio are widely used, for instance see [57,58,68,69]. The dependence on speed prevents this type of model being used when the limited data is not sufficient for a reliable characterization. Therefore, power-based efficiency model is introduced in [24,70], which is independent on turbocharger speed. Other simplified approaches are found in literature, such as constant efficiency [67], efficiency independent on VGT opening [71].

2.3 Diesel Engine Air-Path Control

During the last decades, a considerable research eﬀort has been dedicated to the control of modern Diesel engines, which are all equipped with an exhaust gas recirculation system and a variable geometry turbocharger. Key issue in Diesel engine air-path control is to coordinate the various actuators in meeting desired boost pressure and ﬂow rates of air and residuals.

2.3.1 EGR-VGT Control of Diesel Air-Path System

The nonlinearity of the engine model and the coupling among actuators and controlled outputs make the Diesel engine air-path control challenging. Several methodologies have

15 been explored for the well-known EGR-VGT control problem. Coordinated decentralized and multivariable control strategies were ﬁrst proposed to manipulate VGT and EGR openings for intake manifold pressure and air mass ﬂow rate regulation [72–75]. Nonlinear control approaches were then developed, including constructive Lyapunov function approach [76], adaptive control [77], dynamic feedback linearization [76, 78, 79], and sliding mode control [79–81]. Furthermore, H∞ controller based on the linear parameter varying models is studied in [67, 82]. Other control designs, for instance, optimal control based on dynamic optimization and neural networks [69], motion-planning strategy [83], PID control for pumping work minimization [84], are also investigated in literature.

The choice of feedback variables deﬁnes the overall controller structure, and the most common choice in the literature are compressor air mass ﬂow and intake manifold pressure

[67, 72, 73, 79–82]. Other choices are exhaust manifold pressure and compressor air mass

ﬂow [76], intake manifold pressure and cylinder air mass-ﬂow [77], or compressor air mass

ﬂow and EGR ﬂow [83], oxygen/fuel ratio and EGR-fraction [84]. Even though oxygen/fuel ratio and EGR-fraction cannot be directly measured by production sensors, they are used for performance variables as they are directly related to the emissions.

2.3.2 Model Predictive Control for Diesel Air-Path System

More recently, the advance in model predictive control (MPC), in particular the development of explicit MPC schemes [85], and computational power, cause an growing interest for Diesel engine air-path control application. MPC is considered as a systematic approach to control development and calibration along with providing a multivariable control scheme that automatically accommodates actuator and state constraints [86].

In [87, 88], an explicit MPC is developed to regulate air mass ﬂow and intake manifold pressure. In [89], an online MPC is applied to a real-world Diesel engine, by employing an extension of the online active set strategy. In [90, 91], nonlinear MPC approaches are proposed for the air-path control problem to solve the system nonlinearities and constraints.

16 The simulation results show improved performance, but the required computational power is far more than is available on production ECU. In [92], two MPC controllers using different outputs and a PID based controller are evaluated in simulation, showing that the proposed design has performance improvements. Furthermore, issues of practical importance were also discussed, including integral action of the EGR-fraction to handle model errors, prediction of engine load and speed, and prevention of input oscillations.

2.3.3 Integrated Control of Engine and Aftertreatment Systems

Integrated control of engine and aftertreatment systems is a novel concept, which ex- ploits the synergy between engine and aftertreatment systems to optimize the overall system fuel economy and emissions. The previous research effort is mainly on two problems, namely active thermal management [93–95] and active NOx control [96–98]. In [93], a control-oriented temperature dynamic model for a modern Diesel engine equipped with a complete set of aftertreatment systems including Diesel oxidation catalyst (DOC), Diesel particulate filter (DPF), and selective catalytic reduction (SCR) is developed. The influences of in-cylinder post injection, both fuel injection rate and injection timing, on the temperature dynamics are investigated. In [94], different methods of temperature increase upstream catalyst are compared, including intake air throttling, re- tard of main injection, early and late post injection, and exhaust cam-phasing. In [95], 4 in-cylinder strategies and 1 in-pipe strategy were employed for the purpose of thermal management of a DPF. Experiments results show that at the low temperature of (150 ◦C), early

injection strategies that create temperature rise from both combustion and light reductant

exotherms are preferred.

In [96], a backstepping-based active NOx control method is proposed by treating the

engine-out NOx concentration as a control input for the ammonia coverage ratio control of a two-cell SCR system. The engine brake speciﬁc fuel consumption (BSFC) can be reduced

signiﬁcantly by actively relaxing the constraint on the engine-out NOx concentrations. [97]

17 proposed an integrated emission management strategy based on Pontryagin’s minimum principle, which optimizes engine-out NOx emissions by online adjustment of EGR/VGT

control settings based on the actual SCR NOx reduction capacity. In [98], a steady-state optimization and MPC is developed for a combined EGR/SCR control problem. The con-

troller minimizes BSFC including urea cost by simultaneously optimizing engine out NOx and urea dosing, while maintaining emission levels.

2.4 Combined Design and Control Optimization

When optimizing a plant (artifact) and its controller, one could adopt a sequential

strategy whereby the plant is optimized ﬁrst, followed by the controller. However, the

plant and controller optimization problems are coupled in the sense that their sequential

solution is not guaranteed to be a combined optimum [99]. Design and control integration

reported in literature are in diverse areas including ﬂexible structures [100–102], vehicle

suspensions [103, 104], electric motors [105, 106], robotics [107–109], and various types of

mechanisms and machine tools [110–112].

Solving combined design and control optimization problems, or co-design problem, has

received substantial attention from researchers. Fathy et al. [99] classiﬁed the various opti-

mization strategies into sequential, iterative, bi-level (nested), and simultaneous strategies,

and showed mathematically that system-level optimality is guaranteed with the nested and

simultaneous strategies. Tanaka et al. [113] proposed a general formulation to the co-design

problem and reduced the formulation to a bilinear matrix inequality (BMI) using a de-

scriptor form. Shi et al. [114, 115] proposed an iterative redesign strategies, which have

the advantage of guaranteeing a locally convergent solution to the optimization problem.

Peters et al. [116] presented a modiﬁed sequential approach utilizing a control proxy func-

tion, which can provide near-optimal solutions to the co-design problem, while allowing the

problem to be decomposed into an artifact design problem and a control design problem.

18 Chapter 3

Parametric Model of Turbocharger

ARAMETRIC model for the compressor and turbine maps is developed in this P chapter. First, a brief overview of basic turbomachinery theory is given, where the dimensionless parameters and relevant geometric parameters for the characterization of flow and efficiency maps are defined. Then, a formal derivation of the parametric maps for families of compressors and turbines is discussed together with the model results. The parametric modeling approach is applied to an engine model with two-stage turbocharger

(2ST), which is validated against steady-state and transient experimental data. Preliminary analysis is conducted to show the inﬂuence of 2ST conﬁgurations on system performance.

3.1 Basic Turbomachinery Theory

3.1.1 Dimensionless Representation

For the compressor and turbine characteristic maps, the performance parameters are given in the form of pseudo-dimensionless variables, which is a standard representation method for turbomachinery. The variables deﬁned in Table 3.1 are normalized by the inlet condition. Typically, turbocharger manufacturers provide data and performance maps in terms of corrected variables. For this reason, and the fact that the corrected variables are closer to the physical meaning of each parameter, the parametric turbocharger model will be developed using the corrected variables.

In addition to the above variables, the dimensionless parameters are often used for the performance characterization of both compressor and turbine, which are deﬁned in Table 3.2.

19 Table 3.1: Pseudo-dimensionless Representation

Corrected Variables Reduced Variables p √ Tin/Tref Tin Mass Flowm ˙ corr =m ˙ m˙ red =m ˙ pin/pref pin pout β = for compressor Pressure Ratio pin pin = for turbine pout N N Shaft Speed Ncorr = p Nred = √ Tin/Tref Tin P P Power Pcorr = p Pred = √ pin/pref Tin/Tref pin Tin

Table 3.2: Dimensionless Representation

Compressor Turbine

m˙ corr m˙ corr Flow Parameter Φ = π 2 Φ = π 2 ρin 4 Dc Uc ρin 4 Dt Ut γ−1 1−γ cpTin β γ − 1 cpTin 1 − γ Head Parameter Ψ = 1 2 Ψ = 1 2 Uc U π 2 π 2 t Blade Tip Speed Uc = DcNcorr Ut = DtNcorr 60 60 Uc Ut Blade Mach Number M = √ M = √ γRTin γRTin Uc Ut Blade Speed Ratio BSR = γ−1 BSR = 1−γ 2cpTin β γ − 1 2cpTin 1 − γ

3.1.2 Similitude Theory

The theory of similitude (or similarity) allows one to compare the performance of different turbomachines that belong to the same family [117]. This theory, well-known in the general ﬁeld of ﬂuid mechanics, was originally used to predict performance of full-scale systems based on the analysis of scaled prototypes. The similarity theory recognizes three categories:

20 Geometric similarity: it occurs when two turbomachines (model-m and prototype-p)

diﬀer only in scale, so corresponding dimensions have the same ratio.

L L = (3.1) D m D p

Kinematic similarity: it occurs when the streamline patterns in two machines are

the same. This implies that the machines have the same velocity coeﬃcients, hence

similar velocity triangles.

Φm = Φp (3.2)

Dynamic similarity: it occurs if the ratios of force components at corresponding points

in the ﬂow through two machines are equal. This implies equality of the load coeﬃ-

cients.

Ψm = Ψp (3.3)

This theory will be used here for interpolating turbocharger maps data and for the analysis of the relevant parameters.

3.1.3 Turbocharger Geometric Parameters

Traditionally, the performance of turbochargers is determined by three geometric parameters [118], namely diameter, Trim, and A/R (Area/Radius) as shown in Figure 3.1.

The diameter is generally deﬁned as the outer diameter of the wheel. In this sense, the compressor diameter refers to the exducer wheel diameter, while the turbine diameter refers to inducer wheel diameter. Since the ﬂow capacity is approximately proportional to the size, the diameter is a major design parameter.

The Trim is deﬁned as the ratio between the inducer and exducer areas of both turbine and compressor wheels: 2 Din Trim = 100 (3.4) Dex

21 (a) (b) (c)

Figure 3.1: Deﬁnition of the Main Geometric Parameters

The Trim affects performance by shifting the airflow capacity. Under the same conditions, a higher Trim wheel will allow more flow than a smaller Trim wheel. However, it is very unlikely that a larger Trim can be obtained without changing any of the other geometric parameter and hence affecting the overall turbocharger performance. Hence, there is no guarantee that a larger Trim will allow more flow.

Another important parameter is the A/R, which is defined as the ratio between the inlet cross-sectional area and the distance between the axis of rotation and the centroid of that area. The A/R affects the swallowing capacity of a turbine. Larger A/R provides more flow through the turbine, hence improved boost at higher engine speed conditions by increasing the flow capacity and reducing the exhaust gas back pressure. Compared to the effects of A/R on the turbine performance, the sensitivity of the compressor performance to this parameter is rather limited.

3.2 Compressor Model

This section will describe the equations used to model a family of geometrically similar compressors relevant for the two-stage turbocharger system. In particular, the focus is on the scalability of the compressor mass ﬂow rate and eﬃciency outputs, with respect to its design parameters, such as diameter, A/R and Trim.

22 Table 3.3: GT Compressors Speciﬁcation

No. Model Diameter [mm] Trim [-] A/R [-] 1 GT2859R 59.4 56 0.42 2 GT2860R(1) 60 55 0.42 3 GT2871R(1) 71 52 0.60 4 GT2871R(4) 71 56 0.60 5 GT3071R(1) 71 56 0.50 6 GT3076R 76.2 56 0.60 7 GT3582R(2) 82 56 0.70 8 GT4088 88 54 0.72

Based on the analysis of the most relevant publications in this ﬁeld, several modeling approaches have been identiﬁed as potential candidates for characterizing and scaling the compressor maps. All these approaches are based on dimensionless representation of the

flow, speed, pressure and efficiency variables. Among the various methods found in literature, it is necessary to select a model structure that maintains sufficient accuracy in representing a group of similar compressors when one of the geometric parameter is modi-

ﬁed.

Based on the above considerations, eight sets of compressor flow and efficiency maps were selected from the Garrett GT Series, which is the same family as the two compressors of the two-stage turbocharger. Two approaches for the mass flow model and three approaches for the efficiency model are considered. A comparative study is proposed here to evaluate each modeling method.

3.2.1 Assumptions and Compressor Selection

To illustrate the advantages of adopting a dimensionless representation of the compressor performance maps, the corrected maps for the three sample compressors listed in Table 3.3 are shown in Figure 3.2, together with their dimensionless representation. Although the three compressor maps are quite diﬀerent, their dimensionless representations condense to

23 3.5 3.5 3.5

3 3 3

2.5 2.5 2.5 β β β 2 2 2

1.5 1.5 1.5

1 1 1 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 m [kg/s] m [kg/s] m [kg/s] corr corr corr

1.1 1.1 1.1

1 1 1

Ψ 0.9 Ψ 0.9 Ψ 0.9

0.8 0.8 0.8

0.7 0.7 0.7

0.6 0.6 0.6 0 0.05 0.1 0.15 0 0.05 0.1 0.15 0 0.05 0.1 0.15 Φ Φ Φ

Figure 3.2: Example of Compressor Maps: Corrected Dimensional Maps (Top) and Dimensionless Maps (Bottom)

4 HPC LPC GT2859R 3.5 GT2860R(1) GT2871R(1) GT2871R(4) 3 GT3071R(1) GT3076R GT3582R(2) GT4088

β 2.5

1.5

1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 m [kg/s] corr

Figure 3.3: Compressor Maps of the Original Two-Stage Turbocharger and Similar Models of the GT Series the same region. Therefore, it is reasonable to assume that the kinematic similarity and dynamic similarity hold for a group of geometric similar compressors. Because the A/R has little eﬀect on compressor performance, it is possible to reduce the number of design

24 parameter considered for the model development.

The speciﬁcations of the eight compressors in the Garrett GT family considered in this study are shown in Table 3.3, while their maps are shown in Figure 3.3 together with the maps of the original high and low pressure compressors of the two-stage turbocharger. The performance maps of the GT compressors are similar to the original two-stage compressors.

Hence, they represent a relevant family for the application of the parametric modeling methodology.

3.2.2 Compressor Mass Flow

The first proposed compressor flow model is the Jensen-Kristensen (JK) model, based on the work presented in [57]. In this work, the head parameter is expressed as a hyperbolic function of the flow parameter. The function parameters are fitted as quadratic functions of the blade Mach number:

K1 + K2Φ 2 Ψ = ,Ki = ki1M + ki2M + ki3, i ∈ {1, 2, 3} K3 − Φ (3.5) P = k11 k12 k13 k21 k22 k23 k31 k32 k33

The above modeling approach results into nine fitting parameters that are identified on the data extracted from the available compressor maps. The results of the curve fitting process using the JK model is shown in Figure 3.4 for the GT3071R(1) compressor.

An alternate approach that was used to fit the compressor flow maps is the Ellipse model, originally proposed in [61]. Here, a new flow parameter is defined to explicitly take into account the compressor speed. In this model, a third order polynomial is used for curve

ﬁtting. However, it has been shown that using this approach leads to numerical issues in the extrapolation. For this reason, an extension of the model was proposed in [58], where

25 −1 0.21 0.24 0.2 −1.1 0.22 0.19

1 0.2 2 −1.2 3 0.18 K K K

0.17 0.18 −1.3 0.16 0.16 −1.4 0.15 0.8 1 1.2 1.4 1.6 0.8 1 1.2 1.4 1.6 0.8 1 1.2 1.4 1.6 Mach Mach Mach

4 1.3 4 data 1.2 3.5 fit 3.5 1.1 3 3 1 β β 2.5 Ψ 2.5 0.9 2 2 0.8

1.5 0.7 1.5

1 1 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0 0.1 0.2 0.3 0.4 0.5 m [kg/s] Φ m [kg/s] corr corr

Figure 3.4: Identiﬁcation of the JK Flow Model

4 4 data 3.5 fit 1.2 3.5

1.1 3 3 1 c c

2.5 Ψ 2.5 η η 0.9 2 2 0.8

1.5 0.7 1.5

1 0.6 1 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0 0.1 0.2 0.3 0.4 0.5 m [kg/s] Φ" m [kg/s] corr corr

Figure 3.5: Identiﬁcation of the SE Flow Model the Ellipse model is used to correlate the ﬂow parameter with the head parameter:

0 2 2 Φ Ψ 0 Φ + = 1, Φ = γ−1 k1 k2 M γ (3.6) P = k1 k2

The above model results in only two regression parameters and the results of the curve ﬁtting

26 process using the Shaver-Ellipse (SE) model is shown in Figure 3.5 for the GT3071R(1) compressor.

According to similarity theory, the dimensionless parameters for a turbomachine are expressed functionally as:

f(Φ, Ψ,M, Re, γ, ··· ) = 0 (3.7)

Each of the models presented has a unique set of parameters P , which is identiﬁed through a nonlinear least square method. The parameter set is introduced to account for modeling error associated to neglecting the inﬂuence of non-dimension terms, including Re, γ, and other geometry parameters on the dimensionless characteristic curves. Therefore, above relationship can be reduced to:

f(Φ, Ψ,M) = 0 (3.8)

For a family of compressor listed in Table 3.3, a reasonable assumption is that the set of parameters P is the same for all compressors. To evaluate the above assumption, as well as the accuracy of the two modeling approaches, a comparative study is presented. First, the ﬂow data from all compressors listed in Table 3.3 except for No. 6 are used to identify the parameter set for two models. The result of this procedure is one set of regression parameters P for each of the seven compressors. Finally, a single set of regression parameters representative of the compressor family is obtained by averaging among the seven sets.

The same procedure is applied to both the JK model and the SE model, and are then used to represent the ﬂow characteristics of all compressors. Performance maps for all the eight compressors were generated, and results are shown in Figure 3.6 for two of the compressors in the calibration set (No. 2 and 4) and for the validation set (No. 6). The results show that the JK model is able to well predict the performance of the GT3076 compressor (No. 6), which is reserved for validation.

In general, there are two ways for implementing the compressor mass ﬂow model in

27 GT2860R(1), JK model GT2871R(4), JK model GT3076R, JK model

3 4 3 3.5 2.5 2.5 3 β β β 2 2 2.5 2 1.5 1.5 1.5

1 1 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0.5 m [kg/s] m [kg/s] m [kg/s] corr corr corr

GT2860R(1), SE model GT2871R(4), SE model GT3076R, SE model

3 4 3 3.5 2.5 2.5 3 β β β 2 2 2.5 2 1.5 1.5 1.5

1 1 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0.5 m [kg/s] m [kg/s] m [kg/s] corr corr corr

Figure 3.6: Extrapolation of the JK and SE Models an engine air-path system model, depending upon the causality relationships between the models of the system components. A ﬁrst approach consists of modeling the mass ﬂow rate through the compressor as function of the pressure ratio and turbo shaft speed:

m˙ c = f1(β, Ntc) (3.9)

A diﬀerent approach consists of considering the mass ﬂow rate as the input to the compressor model, which is then used to compute the pressure ratio:

β = f1(m ˙ c,Ntc) (3.10)

When viewing compressor model in isolation, Type 2 model appears to be a better choice, because:

In the region where the speed lines are almost ﬂat, coinciding with a large part of

28 Table 3.4: Comparison of the Two Compressor Flow Models

JK model SE model Low speed High High Extrapolation Low ﬂow High High High ﬂow High Low Accuracy on data points High Low at high speed Diameter sensitivity Low High

engine operating region, Type 2 model is insensitive to input errors. Conversely, the

characteristic curves become very steep when pressure ratio is used as input.

Type 2 model can be easily incorporated into a dynamic model that exhibits surge

behavior [119].

When the compressor model is connected to the engine model, Type 2 model is equivalent to Type 1 model at steady-state, but more stable in transients, due to the fact that the pressure ratio changes much faster than the compressor ﬂow rate.

The two modeling approaches are evaluated by analyzing the error on data points and extrapolation quality, particularly in the low speed region, low mass flow region (surge), and high mass flow region (choking). The results are summarized in Figure 3.7 to Figure 3.10, and in Table 3.4. It is evident that the JK model presents an overall higher accuracy of both interpolation and extrapolation and it appear to be less sensitive to the compressor diameter compared to the SE model. For this reason, the JK model is selected as the parametric compressor flow model. Moreover, Type 2 model is adopted due to its better prediction ability. The fully parameterized compressor flow model is validated on the eight compressor maps, as shown in Figure 3.11.

The implementation of this model in the engine air-path system is achieved through the introduction of a new state. Considering the inertia eﬀects of the air massm ˙ c in the pipe connecting the compressor outlet and the volume after the compressor, as shown in

29 GT2860R(1) GT2871R(4) GT3076R

3 4 3 3.5 2.5 2.5 3 β β β 2 2 2.5 2 1.5 1.5 data 1.5 fit 1 1 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0.5 m [kg/s] m [kg/s] m [kg/s] corr corr corr

0.5 0.3 0.4 0.4 0.25 0.3 0.2 0.3

0.15 0.2 Model Model Model 0.2 0.1 Mean = 38.1 [%] 0.1 Mean = 5.37 [%] 0.1 Mean = −17.4 [%] 0.05 Std = 30.5 [%] Std = 23.6 [%] Std = 52.5 [%] 0 0 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0.5 Data Data Data

Figure 3.7: Prediction Error for JK Model (Type 1: Pressure Ratio as Input)

GT2860R(1) GT2871R(4) GT3076R

3 4 3 3.5 2.5 2.5 3 β β β 2 2 2.5 2 1.5 1.5 data 1.5 fit 1 1 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0.5 m [kg/s] m [kg/s] m [kg/s] corr corr corr

0.5 0.3 0.4 0.4 0.25 0.3 0.2 0.3

0.15 0.2 Model Model Model 0.2 0.1 Mean = 33.8 [%] 0.1 Mean = 5.26 [%] 0.1 Mean = 9.26 [%] 0.05 Std = 26.8 [%] Std = 20.1 [%] Std = 21.8 [%] 0 0 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0.5 Data Data Data

Figure 3.8: Prediction Error for SE Model (Type 1: Pressure Ratio as Input)

30 GT2860R(1) GT2871R(4) GT3076R

3 4 3 3.5 2.5 2.5 3 β β β 2 2 2.5 2 1.5 1.5 data 1.5 fit 1 1 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0.5 m [kg/s] m [kg/s] m [kg/s] corr corr corr

3 4 3 3.5 2.5 2.5 3

Model 2 Model Model 2 2.5 2 1.5 Mean = 5.88 [%] 1.5 Mean = −2.04 [%] Mean = −1.55 [%] 1.5 Std = 3.7 [%] Std = 2.22 [%] Std = 4.77 [%] 1 1 1 1 1.5 2 2.5 3 1 1.5 2 2.5 3 1 1.5 2 2.5 3 3.5 4 Data Data Data

Figure 3.9: Prediction Error for JK Model (Type 2: Mass Flow as Input)

GT2860R(1) GT2871R(4) GT3076R

3 4 3 3.5 2.5 2.5 3 β β β 2 2 2.5 2 1.5 1.5 data 1.5 fit 1 1 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0.5 m [kg/s] m [kg/s] m [kg/s] corr corr corr

3 4 3 3.5 2.5 2.5 3

Model 2 Model Model 2 2.5 2 1.5 Mean = 6.7 [%] 1.5 Mean = −0.82 [%] Mean = 4.21 [%] 1.5 Std = 4.73 [%] Std = 4.04 [%] Std = 9.45 [%] 1 1 1 1 1.5 2 2.5 3 1 1.5 2 2.5 3 1 1.5 2 2.5 3 3.5 4 Data Data Data

Figure 3.10: Prediction Error for SE Model (Type 2: Mass Flow as Input)

31 GT2859R GT2860R(1) GT2871R(1) GT2871R(4) 3.5 3 3 3 3 2.5 2.5 2.5 2.5 β β β β 2 2 2 2

1.5 1.5 1.5 1.5

1 1 1 1 0 0.1 0.2 0.3 0 0.1 0.2 0.3 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 m [kg/s] m [kg/s] m [kg/s] m [kg/s] corr corr corr corr

GT3071R(1) GT3076R GT3582R(2) GT4088 4 4 4 3.5 3.5 3.5 3.5 3 3 3 3

β 2.5 β β 2.5 β 2.5 2.5 2 2 2 2

1.5 1.5 1.5 1.5

1 1 1 1 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 m [kg/s] m [kg/s] m [kg/s] m [kg/s] corr corr corr corr

Figure 3.11: Compressor Flow Parametric Model Validation (JK Model, Type 2)

Compressor Volume

pc Pipe p ,T p1,T1 2 2

m c NTC

Figure 3.12: Schematic Diagram for Momentum Equation

Figure 3.12, the momentum equation is given by:

dm˙ c A(p2 − pc) = mv˙ = ρlAv˙ = l (3.11) dt hence dm˙ c A = (p2 − pc) (3.12) dt l where A and l are the cross sectional area and length of the connecting pipe.

32 3.2.3 Compressor Eﬃciency

Together with the modeling methodology for the compressor flow maps, [57] also introduces a modeling approach for the compressor efficiency maps. Application of this method, however, showed low accuracy and poor ability to extrapolate the efficiency maps to regions where experimental data are not available.

For this reason, an alternative approach is considered here, based on the study presented by Eriksson [71]. In this work, the eﬃciency is represented as a quadratic function of ﬂow parameter and corrected speed:

 T     Φ − Φ c c Φ − Φ T  max   1 2  max  ηc = ηmax − χ Qηχ = ηmax −       N − Nmax c2 c3 N − Nmax (3.13) Q = c1 c2 c3 ηmax Φmax Nmax

This modeling approach requires the determination of six parameters and sample results for one of the compressors considered in this study are shown in Figure 3.13.

0.8 0.8 1

0.75 0.75 0.8

0.7 0.7 0.6 c c c η η η 0.65 0.65 0.4

0.6 0.6 0.2 data fit 0.55 0.55 0 0 0.1 0.2 0.3 0.4 0 0.05 0.1 0.15 0.2 0 0.1 0.2 0.3 0.4 m [kg/s] φ m [kg/s] corr corr

Figure 3.13: Identiﬁcation of the Eriksson Eﬃciency Model

33 0.8 2 1

0.75 1.8 0.8

0.7 1.6 0.6 " c c c η η η 0.65 1.4 0.4

0.6 1.2 0.2 data fit 0.55 1 0 0 0.1 0.2 0.3 0.4 0 0.05 0.1 0.15 0.2 0 0.1 0.2 0.3 0.4 m [kg/s] Φ" m [kg/s] corr corr

Figure 3.14: Identiﬁcation of the Shaver Eﬃciency Model

5 5 x 10 x 10 2 −1 1

1 0.8

a −2 1.5

] 0.6 −2 −3 s

2 0.5 1 1.5 1 5 c 5 η x 10 x 10 h [m 3

∆ 0.4

2 0.5 2

a 0.2 data 1 fit 0 0 0 0 0.1 0.2 0.3 0.4 0.5 1 1.5 0 0.1 0.2 0.3 0.4 m [kg/s] N [rpm] 5 m [kg/s] corr x 10 corr

Figure 3.15: Identiﬁcation of the Martin Eﬃciency Model

A second approach is proposed by Shaver [61], where the new efficiency is defined as a linear function of the new flow parameter:

0 0 0 Φ 0 Ψ ηc = c1Φ + c2, Φ = γ−1 , ηc = M γ ηc (3.14) Q = c1 c2

Using this approach, the parameters to be identified are now reduced to two. The result of this fitting method for generating the efficiency maps is shown in Figure 3.14.

The last modeling approach considered in this study is based on a direct application of the conservation laws to the compressor, as proposed by Martin [62]. Here, the enthalpy rise is modeled as a linear function of the mass ﬂow rate, and the parameters are assumed to

34 Table 3.5: Comparison of the Three Compressor Eﬃciency Models

Eriksson model Shaver model Martin model Low speed High Low High Extrapolation Low ﬂow Middle Low High High ﬂow High Middle Low Accuracy on data points Low at high speed Insensitive to input High Diameter sensitivity Low Low High

GT2860R(1), Eriksson model GT2871R(4), Eriksson model GT3076R, Eriksson model 1 1 1

0.8 0.8 0.8

0.6 0.6 0.6 c c c η η η 0.4 0.4 0.4

0.2 0.2 0.2

0 0 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0.5 m [kg/s] m [kg/s] m [kg/s] corr corr corr GT2860R(1), Shaver model GT2871R(4), Shaver model GT3076R, Shaver model 1 1 1

0.8 0.8 0.8

0.6 0.6 0.6 c c c η η η 0.4 0.4 0.4

0.2 0.2 0.2

0 0 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0.5 m [kg/s] m [kg/s] m [kg/s] corr corr corr GT2860R(1), Martin model GT2871R(4), Martin model GT3076R, Martin model 1 1 1

0.8 0.8 0.8

0.6 0.6 0.6 c c c η η η 0.4 0.4 0.4

0.2 0.2 0.2

0 0 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0.5 m [kg/s] m [kg/s] m [kg/s] corr corr corr

Figure 3.16: Extrapolation of 3 Eﬃciency Models be a quadratic function of the corrected speed. The compressor eﬃciency is then calculated

35 GT2860R(1), Eriksson Method GT2871R(4), Eriksson Method GT3076R, Eriksson Method 1 1 1 Mean = −0.461 [%] Mean = 0.795 [%] Mean = 1.02 [%] 0.8 Std = 10.6 [%] 0.8 Std = 2.64 [%] 0.8 Std = 5 [%]

0.6 0.6 0.6 Model Model Model

0.4 0.4 0.4

0.2 0.2 0.2 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Data Data Data

GT2860R(1), Shaver Method GT2871R(4), Shaver Method GT3076R, Shaver Method 1 1 1 Mean = 6.15 [%] Mean = 1.9 [%] Mean = 3.19 [%] 0.8 Std = 5.79 [%] 0.8 Std = 6.26 [%] 0.8 Std = 8.19 [%]

0.6 0.6 0.6 Model Model Model

0.4 0.4 0.4

0.2 0.2 0.2 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Data Data Data

GT2860R(1), Martin Method GT2871R(4), Martin Method GT3076R, Martin Method 1 1 1 Mean = −39 [%] Mean = −7.46 [%] Mean = 18 [%] 0.8 Std = 2.79 [%] 0.8 Std = 3.76 [%] 0.8 Std = 9.44 [%]

0.6 0.6 0.6 Model Model Model

0.4 0.4 0.4

0.2 0.2 0.2 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Data Data Data

Figure 3.17: Prediction Error for 3 Eﬃciency Models as the ratio between the isentropic enthalpy rise and the actual enthalpy rise as follows:

γ−1 ∆his 1 2 ηc , ∆his = cpTin β γ − 1 = ΨU , ∆h 2 c 2 ∆h = a1m˙ c + a2, ai = ci1N + ci2N + ci3, i ∈ {1, 2, 3} (3.15) Q = c11 c12 c13 c21 c22 c23

Since the shape of the efficiency curves is the same among different compressors (in light of the similitude theory), the set of parameters Q, determined through the fitting of the compressor efficiency maps, is assumed constant for all compressors listed in Table 3.3.

To conﬁrm this assumption and compare the accuracy of the three models, a comparative study similar to the one presented above, is conducted for the compressor eﬃciency. The

36 GT2859R GT2860R(1) GT2871R(1) GT2871R(4) 1 1 1 1

0.8 0.8 0.8 0.8

0.6 0.6 0.6 0.6 c c c c η η η η 0.4 0.4 0.4 0.4

0.2 0.2 0.2 0.2

0 0 0 0 0 0.1 0.2 0.3 0 0.1 0.2 0.3 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 m [kg/s] m [kg/s] m [kg/s] m [kg/s] corr corr corr corr

GT3071R(1) GT3076R GT3582R(2) GT4088 1 1 1 1

0.8 0.8 0.8 0.8

0.6 0.6 0.6 0.6 c c c c η η η η 0.4 0.4 0.4 0.4

0.2 0.2 0.2 0.2

0 0 0 0 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 m [kg/s] m [kg/s] m [kg/s] m [kg/s] corr corr corr corr

Figure 3.18: Compressor Eﬃciency Parametric Model Validation (Eriksson Model) results are summarized in Figure 3.16, Figure 3.17, and Table 3.5.

Since the Eriksson model presents higher accuracy in both the interpolation and extrapolation of eﬃciency data with the least sensitivity to the compressor diameter, it will be selected as the parametric compressor eﬃciency model. The Eriksson model is validated on the eight compressor maps, as shown in Figure 3.18.

3.2.4 Surge and Choke Model

Surge is a dangerous instability and can occur during a gear shift under acceleration, when a throttle closing causes a fast reduction in mass ﬂow. The reduction of mass ﬂow produces an increase in pressure ratio over the compressor, which may lead to reversed

ﬂow. Such oscillating mass ﬂow can cause undesired noise, and damage, or even destroy, the turbo.

Choke is a situation where the pressure ratio of a speed line drops rapidly (vertically) with little or no change in ﬂow. In most cases the reason for choke is that close to Mach

1 velocities have been reached somewhere within the impeller and/or diﬀuser generating

37 a rapid increase in losses. Therefore, a maximum flow line is defined to avoid compressor operate at low efficient choke region.

Since surge line and choke line represent the limit of compressor operation, their model is important when investigating control design for turbocharged engines. In [59], a ﬁrst order polynomial is used to model maximum ﬂow line:

m˙ max = k11N + k12 (3.16)

where k12 > 0 gives a positive ﬂow also for N = 0. Power functions are used to model surge mass ﬂow and pressure ratio lines:

k22 m˙ surge = k21N + 0 (3.17) k32 βsurge = k31N + 1 where the 0 and 1 emphasize zero ﬂow and unity pressure ratio for N = 0.

To get a surge and choke model for a family of compressors, Mach number M is used to replace corrected speed N. The modiﬁed model becomes:

k13 m˙ surge = (k11Dc + k12)M (3.18) k22 βchoke = k21M + 1

We only need to model surge mass flow or surge pressure ratio, the other variables is obtained via compressor flow model. The reason of choosing mass flow rather than pressure ratio for surge line model is that constant speed line is flat in surge region, so this model is less sensitive to input or modeling errors. Similarly, pressure ratio is used to model choke line.

The results of the curve ﬁtting process is shown in Figure 3.19. The integrated compressor

ﬂow model and surge-choke model is validated on the eight GT Series compressor maps, as shown in Figure 3.20.

38 0.5 0.5

0.4 0.4

0.3 0.3 surge surge

m 0.2 m 0.2

0.1 0.1

0 0 0.4 0.6 0.8 1 1.2 1.4 1.6 0.6 0.8 1 1.2 1.4 1.6 1.8 N [rpm] 5 Mach x 10

GT2859R 3.5 3.5 GT2860R(1) GT2871R(1) 3 3 GT2871R(4) GT3071R(1) 2.5 GT3076R 2.5 GT3582R(2) choke choke β 2 GT4088 β 2

1.5 1.5

1 1 0.4 0.6 0.8 1 1.2 1.4 1.6 0.6 0.8 1 1.2 1.4 1.6 1.8 N [rpm] 5 Mach x 10 Figure 3.19: Identiﬁcation of the Surge and Choke Model

GT2859R GT2860R(1) GT2871R(1) GT2871R(4) 3.5 surge line 3 choke line 3 3 3 2.5 2.5 2.5 2.5 β β β β 2 2 2 2

1.5 1.5 1.5 1.5

1 1 1 1 0 0.1 0.2 0.3 0 0.1 0.2 0.3 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 m [kg/s] m [kg/s] m [kg/s] m [kg/s] corr corr corr corr

GT3071R(1) GT3076R GT3582R(2) GT4088 4 4 4 3.5 3.5 3.5 3.5 3 3 3 3

β 2.5 β β 2.5 β 2.5 2.5 2 2 2 2

1.5 1.5 1.5 1.5

1 1 1 1 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 m [kg/s] m [kg/s] m [kg/s] m [kg/s] corr corr corr corr

Figure 3.20: Integrated Compressor Flow Model and Surge-choke Model

3.3 Turbine Model

This section presents the approach and equations used to model a family of geometrically similar turbines, for the two-stage turbocharger system. The objective is to develop a

39 scalable ﬂow model for ﬁxed geometry turbine (FGT), which can predict the outputs based on a set of design parameters such as diameter, A/R and Trim. Methods to extend the model to variable geometry turbine (VGT) are also proposed.

In literature, the standard orifice equation [120] and its modified versions are widely used for turbine flow model, for instance see [57], [71], [62], [67], [66]. Among others, the modified orifice equation presented in [66] exhibits high accuracy in predicting the turbine mass flow rate in a wide range of operating conditions. This modeling methodology relies on the identification of two tuning parameters, which are representative of certain physical characteristics of the turbine flow. This approach is selected here to parameterize a family of turbine maps.

Similar to the previous section, a set of eight fixed geometry turbines was extracted from the Garrett GT Series, with performance map similar to the turbines of the original two-stage turbocharger. The parametric flow model for these FGTs is developed based on the assumption of geometric similarity. Then, three variable geometry turbines are analyzed with the objective of extracting a simplified model that represents the general relationship between the flow rate and the VGT opening position. The outcome of this analysis is a simple VGT scaling method that allows to estimate how the flow characteristic maps will change is a VGT actuation is considered.

For turbine efficiency, methods based on blade speed ratio are widely used, for instance see [57], [71], [68], [69]. Turbine efficiency is greatly affected by VGT opening, and there is limited data available, which make it challenging to get a fully parameterized turbine efficiency model. Some simplified approaches are seen in literature, such as constant efficiency [67], efficiency independent on VGT opening [71] or speed [70]. Since the manufac- turer does not provide efficiency data for Garrett GT Series turbines, the original efficiency data is used, which include both FGT and VGT. Two modeling methods are developed and evaluated. The relation between efficiency and the VGT opening is obtained by interpolation.

40 3.3.1 Turbine Selection

Due to its critical eﬀect on turbine performance, the A/R must be considered in the parametric model of the turbine. This introduces an additional degree of complexity to the problem, compared to the case of the compressor. A set of turbine data from the Garrett

GT Series that match with the previous eight compressors are considered in this section, and their speciﬁcations are listed in Table 3.6.

Figure 3.21 shows that the performance maps of the family of GT turbines considered provides a good coverage of the operating range of the original two-stage turbine maps.

Considering the typical range of operations of a turbine for engine charging applications, the set of turbines used for the parametric model development can be further reduced from eleven to eight (No. 2 to No. 9) without loss of accuracy.

Table 3.6: GT Turbines Speciﬁcation

No. Turbine model Compressor model Diameter [mm] Trim [-] A/R [-] GT2859R 1 GT28(1) 53.9 62 0.64 GT2860R(1) 2 GT2871R(1) 0.86 GT28(2) 53.9 76 3 GT2871R(4) 0.64 4 1.06 GT3071R(1) 5 GT30 60 84 0.82 GT3076R 6 0.63 7 1.06 8GT35 GT3582R(2) 68 84 0.82 9 0.63 10 1.34 GT40 GT4088 77.6 83 11 1.19

41 0.35 HPT LPT 0.3 GT28(1) GT28(2) GT30 0.25 GT35 GT40

0.2 [kg/s] corr

m 0.15

0.1

0.05

0 1 1.5 2 2.5 3 3.5 ε

Figure 3.21: Turbine Maps of the Original Two-Stage Turbocharger and Similar Models of the GT Series

3.3.2 Turbine Mass Flow

Turbine Flow Model

The equation describing the flow of compressible fluids through a restriction (also known as the isentropic orifice equation) is defined as:

pin √ m˙ t = CdA√ γf1() RTin  r γ  2 − 2 − γ+1 1 2 γ−1  γ − γ if ≥ (3.19)  γ − 1 γ + 1 f1() = s γ+1 γ  2 γ−1 1 2 γ−1  if <  γ + 1 γ + 1

where = pin is the pressure ratio across the orifice. pout In general, the flow through a turbine is considerably more complex that an orifice flow from a physical standpoint. The pressure ratio across the turbine at which chocked flow occurs is actually lower than the expected value of 1.8 (depending on γ only), because the turbine effectively behaves as a series of two nozzles (inlet nozzle vanes and rotor passages),

42 7

crit 4 ε

1 1 2 3 4 5 6 7 8 9 10 m (a) (b)

Figure 3.22: Interpretation of the Turbine Flow Model Parameters which each individually experience lower pressure ratios than the total across the turbine.

Therefore, a modified orifice equation is considered, where a polytropic coefficient m is introduced as a tunable parameter that accounts for all the unmolded effects. This leads to the following equation:

pin √ m˙ t = CdA√ γf1() RTin m  r m−1 2 − 2 − m+1 1 2  m − m if ≥ (3.20)  γ − 1 m + 1 f () = 1 1 r m  2 m−1 2 m − 1 1 2 m−1  if <  m + 1 γ − 1 m + 1 m + 1

− m 2 m−1 The critical pressure ratio, crit = m+1 represents the limit conditions at which the flow through the turbine reaches sonic speed, therefore leading to the well-known choked behavior. Note that the critical pressure ratio can be approximated as a linear function of the parameter m without loss of accuracy, as shown in Figure 3.22a. In this sense, the parameter m can be interpreted as a scaling factor that linearly influences the choking limit of the turbine. This information can be useful to establish a set of constraints that allow one to simplify and automate the turbine flow model calibration process.

In (3.20), the product CdA is named the equivalent ﬂow area, which generally relates to

43 the throat area of the turbine. As shown in Figure 3.22b, the throat area is defined as the cross-sectional area (normal to the direction of the flow) corresponding to the first point where the flow is in contact with the runner. Assuming that geometric similarity holds for the GT turbine family, the throat area results proportional to the turbine diameter and

A/R parameters as follows: A A A = R ∝ D (3.21) R R

The physical consistency of the two calibration parameters CdA and m allows one to obtain a direct relation between the turbine geometric parameters and the model parameters, which

facilitates the calibration and extrapolation of the results.

Fixed Geometry Turbine Flow Parametric Model

The procedure for ﬁtting the turbine ﬂow maps is similar to the one used for the compres-

sor. As a ﬁrst attempt, the eight GT turbines considered were ﬁtted individually, without

considering any constraint on the parameter m. However, it was found that performing an unconstrained optimization led to values of m that ultimately made the critical pressure ratio unreasonably large. For this reason, the identiﬁcation procedure was modiﬁed by considering the physical limits of the pressure ratio at which choking occurs. The results of

0.3 5

4 crit ε 0.25 3

2 30 35 40 45 50 55 60 65 70 75 0.2 (A/R)D 6

[kg/s] 0.15 m 4 corr m

GT35 A/R = 1.06 2 0.1 30 35 40 45 50 55 60 65 70 75 GT35 A/R = 0.82 −4 x 10 (A/R)D GT35 A/R = 0.63 8 GT30 A/R = 1.06 0.05 GT30 A/R = 0.82 6 GT30 A/R = 0.63 GT28 A/R = 0.86 CdA 4 GT28 A/R = 0.64 0 2 1 1.5 2 2.5 3 3.5 30 35 40 45 50 55 60 65 70 75 ε (A/R)D

Figure 3.23: Identiﬁcation of the Turbine Flow Model with Constraints on the Parameter m

44 0.3

0.25 6

4 m 0.2 2

0 70 [kg/s] 0.15 65 1.4

corr 1.2

m 60 1 55 0.8 GT35 A/R = 1.06 0.6 0.1 D [mm] 50 0.4 A/R GT35 A/R = 0.82 −4 x 10 GT35 A/R = 0.63 8 GT30 A/R = 1.06 0.05 GT30 A/R = 0.82 6 GT30 A/R = 0.63 GT28 A/R = 0.86 CdA 4 GT28 A/R = 0.64 0 2 1 1.5 2 2.5 3 3.5 30 35 40 45 50 55 60 65 70 75 ε (A/R)D

Figure 3.24: Turbine Flow Parametric Model Validation

this regression procedure are shown in Figure 3.23. Once the parameters m and CdA have been determined for each of the turbines considered, a linear ﬁtting is used to schedule the two parameters as function of the turbine diameter and the A/R as follows:

A m = k1 + k2 + k3D R (3.22) A C A = k4 D + k5 d R

Using the results of the ﬁtting procedure, the turbine geometry parameters are used to generate the ﬂow maps for the turbines listed in Table 3.6. The results for the eight GT

ﬁxed geometry turbines are obtained and shown in Figure 3.24.

The drawback of this modeling process is that the constraints of each turbine need to be tuned individually, which is time consuming. An alternative way proposed here is so-called global fitting, meaning that the data set [m ˙ t D A/R] of 8 turbines can be fitted at one time, by combining (3.20) and (3.22). The extrapolation quality and error of two approaches are compared in Figure 3.25, noting that the coefficients of determination, denoted R2, are almost the same.

45 Fitting with constraint Global Fitting

0.25 0.25

0.2 0.2

[kg/s] 0.15 [kg/s] 0.15 corr corr

m GT35 A/R = 1.06 m GT35 A/R = 0.82 0.1 0.1 GT35 A/R = 0.63 GT30 A/R = 1.06 GT30 A/R = 0.82 0.05 0.05 GT30 A/R = 0.63 GT28 A/R = 0.86 GT28 A/R = 0.64 0 0 1 1.5 2 2.5 3 3.5 1 1.5 2 2.5 3 3.5 ε ε

0.25 0.25

0.2 0.2

Model 0.15 Model 0.15

0.1 R2 = 0.9912 0.1 R2 = 0.9899

0.1 0.15 0.2 0.25 0.1 0.15 0.2 0.25 Data Data

Figure 3.25: Comparison of Constrained Fitting and Global Fitting

Variable Geometry Turbine Flow Model

The parametric turbine flow model can be extended to the case of a turbine with variable geometry. The objective is to determine a parametric model that accounts for the influence of a VGT actuator on the output mass flow rate of a given fixed geometry turbine. As a starting point to develop the model, the data for the high pressure turbine (HPT) of the original two-stage turbocharger system were selected. The flow curve corresponding to each

VGT position was interpolated using (3.20). This procedure leads to seven values for the parameters m and CdA (one per VGT position), shown in the right plots of Figure 3.26. The two parameters were then obtained and ﬁtted as quadratic functions of the VGT opening:

2 m = k11xvgt + k12xvgt + k13 (3.23) 2 CdA = k21xvgt + k22xvgt + k23

As a veriﬁcation step, the characteristic maps obtained through the model given by

(3.20) and (3.23) are compared to the original data set, and the results are shown in the left

46 0.14 15

0.12 10 m

0.1 5

0.08 0 0 0.2 0.4 0.6 0.8 1

[kg/s] x vgt

corr −4 x 10 m 0.06 6

5 rack = 0% 0.04 4 rack = 20% rack = 40% 3 rack = 60% CdA 0.02 2 rack = 80% rack = 90% 1 rack = 100% 0 0 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.2 0.4 0.6 0.8 1 ε x vgt

Figure 3.26: Variable Geometry Turbine Flow Model Validation (HPT)

GT32 Turbine with VGT GT17 Turbine with VGT 0.2 0.12

0.18

0.1 0.16

0.14 0.08

0.12

[kg/s] 0.1 [kg/s] 0.06 corr corr m m 0.08

0.04 0.06

rack = 20% 0.04 rack = 10% rack = 25% 0.02 rack = 40% rack = 50% rack = 60% 0.02 rack = 70% rack = 80% rack = 100% rack = 100% 0 0 1 1.5 2 2.5 3 3.5 4 4.5 5 1 1.5 2 2.5 3 3.5 4 4.5 5 ε ε

Figure 3.27: Variable Geometry Turbine Flow Parametric Validation (GT32 and GT17) plot of Figure 3.26. It can be seen that the parametric VGT ﬂow model preserves accuracy in interpolating the given data.

Figure 3.27 illustrates the results obtained when the same method is applied to other two variable geometry turbines, indicating that this approach can be in principle generalized to diﬀerent VGT systems.

47 Scaling Method for Variable Geometry Turbine Flow Parametric Model

Since the availability of experimental data and geometric information for variable geometry turbines is rather limited, it is extremely challenging to generate a fully parametric

VGT flow model that accounts for the real geometry of a VGT actuator (particularly, the relation between the throat area and VGT position). Instead, a different approach is proposed, namely consisting of finding a representation of the parameters m and CdA as a function of the VGT position by averaging across several sets of data obtained from dif-

ferent turbines with VGT actuators. The ultimate goal of this modeling approach is to

qualitatively predict how the ﬂow map of a given ﬁxed geometry turbine would be scaled

if the turbine were equipped with a VGT actuator. This would enable one to conduct vir-

tual design studies, evaluating and comparing performance of variable and ﬁxed geometry

turbines using simulation tools.

The proposed approach consists as follows. First, the parameters m and CdA for the three VGTs considered above are summarized and shown in the left plots of Figure 3.28.

The three curves are then ensemble-averaged to deﬁne a qualitative behavior representative

15 10 HPT GT32 8 10 GT17 6 m m 4 5 2

0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 x x vgt vgt −4 x 10 6 1

5 0.8 4 0.6 3 CdA C dA 0.4 2

1 0.2

0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 x x vgt vgt

Figure 3.28: Turbine Flow Model Parameter as Function of VGT Position: 3 VGTs (left) and Normalization (Right)

48 of all the turbines considered. Deﬁning the normalized parameter as:

m(xvgt) m = m(xvgt)|x =1 vgt (3.24) CdA(xvgt) CdA = CdA(xvgt)|xvgt=1

the data from the three turbines will condensed to the same point at 100% VGT opening.

A modiﬁed version of the quadratic function in (3.23) is then used to ﬁt the normalized

parameters: 2 m = k11(xvgt − 1) + k12(xvgt − 1) + 1 (3.25) 2 CdA = k21(xvgt − 1) + k22(xvgt − 1) + 1

The complete parametric ﬂow model for a family of variable geometry turbines is ﬁnally summarized as:

A A m xvgt, ,D = m(xvgt) · m ,D scale R fix R (3.26) A A C A xvgt, ,D = C A(xvgt) · C A ,D d scale R d d fix R

A veriﬁcation of the parametric model was conducted by applying the model to two

ﬁxed geometry turbines taken from the GT family, and extrapolating the ﬂow maps as if

GT30 A/R = 1.06 with VGT GT28 A/R = 0.64 with VGT 0.25 0.25 rack = 0% rack = 20% rack = 40% 0.2 0.2 rack = 60% rack = 80% rack = 90% 0.15 0.15 rack = 100% [kg/s] [kg/s] corr corr

m 0.1 m 0.1

0.05 0.05

0 0 1 1.5 2 2.5 3 3.5 4 4.5 5 1 1.5 2 2.5 3 3.5 4 4.5 5 ε ε

Figure 3.29: VGT Flow Parametric Model Validation (Extrapolated VGT Maps from Fixed Geometry Turbine Data)

49 the turbines were equipped with VGT actuator. The results are illustrated in Figure 3.29, proving that the method leads to extrapolated maps with physically consistent behavior.

3.3.3 Turbine Eﬃciency

The first proposed turbine efficiency is based on blade speed ratio, which has the same form as Eriksson compressor efficiency model. The efficiency is represented as a quadratic function of blade speed ratio and Mach number:

 T     BSR − BSR c c BSR − BSR T  max  1 2  max ηt = ηmax − χ Qηχ = ηmax −       M − Mmax c2 c3 M − Mmax (3.27) P = c1 c2 c3 ηmax BSRmax Mmax

This modeling approach requires the determination of six parameters for one ﬁxed geometry

turbine. Figure 3.30 illustrates the result of the BSR-based turbine eﬃciency model. Since

the data of original high pressure turbine (HPT) contains 7 VGT positions, the result

of this procedure is one set of regression parameters P of each of the 7 VGT positions.

0.8 0.8 0.8

0.6 0.6 0.6 t t t

η 0.4 η 0.4 η 0.4

0.2 0.2 0.2

0 0 0 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 ε ε ε

rack = 0% 0.8 0.8 0.8 rack = 20% rack = 40% rack = 60% 0.6 0.6 0.6 rack = 80% rack = 90% rack = 100% t t t

η 0.4 η 0.4 η 0.4

0.2 0.2 0.2

0 0 0 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 ε ε ε

Figure 3.30: BSR-based Turbine Eﬃciency Model Validation (HPT)

50 7000 1 rack = 0% rack = 20% 0.9 6000 rack = 40% rack = 60% 0.8 rack = 80% rack = 90% 5000 0.7 rack = 100% 0.6 4000 [W] t

η 0.5 corr P 3000 0.4

2000 0.3

0.2 1000 0.1

0 0 1 1.5 2 2.5 3 3.5 4 4.5 5 1 1.5 2 2.5 3 3.5 4 4.5 5 ε ε

Figure 3.31: Power-based Turbine Eﬃciency Model Validation (HPT)

Finally, eﬃciency at arbitrary VGT opening is obtained by interpolation among 7 eﬃciency submodels.

Although this model has high accuracy on turbine steady-state data points, it is not a good choice when connected to the engine model. Since pressure drop is faster than speed drop in simulation, and the speed line is very steep at low pressure region, in some cases efficiency will decrease to saturation points. Therefore, a robust turbine efficiency model is introduced, which is power-based and independent on speed. The corrected turbine power is defined as: 1−γ Pcorr = ηtm˙ corrcpTin 1 − γ (3.28) which can be approximated as a power function of pressure ratio:

c2 Pcorr = c1 + c3 (3.29)

Then the eﬃciency is calculated as:

c c1 2 + c3 ηt = 1−γ (3.30) m˙ corrcpTin 1 − γ

The result of the power-based turbine eﬃciency model is shown in Figure 3.31.

51 1 rack = 0% rack = 20% 0.9 0.25 rack = 40% rack = 60% 0.8 rack = 80% rack = 90% 0.7 0.2 rack = 100%

) 0.6 − 1 γ 1/ t ε

0.15 η 0.5 ( 1− t

η 0.4 0.1 0.3

0.2 0.05 0.1

0 0 1 1.5 2 2.5 3 3.5 4 4.5 5 1 1.5 2 2.5 3 3.5 4 4.5 5 ε ε

Figure 3.32: Modiﬁed Power-based Turbine Eﬃciency Model Validation (HPT)

0.9

0.8

1 0.7

0.8 0.6 0.6 t

η 0.5 t η 0.4 0.4 0.2 0.3 0 5 0 0.2 4 0.2 0.1 0.4 3 0.6 2 0 0.8 1 1.5 2 2.5 3 3.5 4 4.5 5 1 1 ε ε x vgt

Figure 3.33: Variable Geometry Turbine Eﬃciency Model

Notice that in (3.30),m ˙ corr is obtained from turbine flow model. Since the power-based efficiency model is coupled with flow model, it can not be used for scalable turbocharger model. In [121], a power function is used to model the turbine mass flow. Therefore,m ˙ corr as well as constant term cpTin can be removed from (3.30), which yields:

c c1 2 + c3 ηt = 1−γ (3.31) 1 − γ

The result of the modified power-based turbine efficiency model, shown in Figure 3.32, is almost the same with that in Figure 3.31. The advantage is this efficiency model is decoupled with flow model, thus it can be used in the case the flow model is scaled with

52 geometry.

Similar as BSR-based model, one set of regression parameters P = [c1 c2 c3] of each of the 7 VGT positions is obtained, and eﬃciency at arbitrary VGT opening is computed by

interpolation. The eﬃciency map for variable geometry turbine is shown in Figure 3.33.

3.4 Mean-Value Model of Diesel Engine

In this section, the Mean-Value Model (MVM) of Diesel engine is introduced, focusing

on the characterization of the engine air-path system, in order to study and control the

two-stage turbocharger dynamics and its interaction with engine breathing dynamics. The

engine model, based on the early work documented in [70], is updated with parametric

model of turbocharger and thermal model of exhaust system.

The system considered is a GM Duramax medium-duty Diesel engine, equipped with a

two-stage turbocharger, intercooler and high pressure cooled EGR. The air-path conﬁgura-

tion is shown in Figure 3.34, while the engine characteristics are in Table 3.7.

The turbocharger system includes a smaller high pressure (HP) turbocharger arranged

in series with a larger low pressure (LP) turbocharger. The HP stage includes a controlled

bypass valve which allows for switching between single stage and two-stage operations. The

fresh air is ﬁrst compressed in the LP compressor and then fed to the HP compressor. In

order to increase the air density, the air passes through an intercooler, and before entering

the engine cylinders, it mixes with exhaust gases coming from the high pressure EGR

system. The two turbines convert the exhaust gas energy into shaft work, through a common

shaft drives the connected compressor. On the HP stage, the ﬂow through the turbine is

controlled through the VGT and the bypass actuator. The bypass, VGT, and EGR positions

determine the operating points of the two-stage turbocharger and control the back pressure

in the exhaust to drive the EGR ﬂow.

53 Table 3.7: GM Duramax Engine Characteristics

Displacement 6.599L Valvetrain OHV 4-V Injection Common-rail Rated Torque 820 Nm @ 1800 rpm Rated Power 231 kW @ 3200 rpm

Intake LPC HPC Manifold charge cooler

EGR cooler

EGR valve

VGT Exhaust LPT HPT Manifold

LP waste-gate HP bypass valve

Figure 3.34: Air-Path Conﬁguration of Diesel Engine with Two-Stage Turbocharger

The engine is divided into subsystems, as shown in Figure 3.35. The engine system model contains the intake and exhaust manifold dynamics coupled by EGR system, and model describing engine breathing, combustion, and torque production. The engine system model connected with the turbocharger model at the compressor side (mass flow rate and outlet temperature feeding the intake manifold model), and at the turbine side (the exhaust manifold states being the inputs of the high pressure turbine). The flow and energy outputs of the compressors and turbines are typically modeled as quasi-steady elements. In order to respect the cause and effect relationships, the two compressor stages must be separated by receivers in order to estimate the intermediate pressure (the same concept holds for the turbine stages).

54 Turbocharger Model Engine System Model

Two-stage Turbo: Intake Manifold Compressors System

Engine TC Shaft EGR Combustion and Dynamics Torque Output

Two-stage Turbo: Exhaust Manifold Turbines System

Figure 3.35: Functional Scheme of the System Model

3.4.1 Valves and Flow Restrictions

The flow restrictions are modeled as steady-state, adiabatic reversible flow through an orifice. The model can be applied to the EGR valve, HP bypass valve, and LP waste-gate.

pin √ pout m˙ = CdA√ γf1 RTin pin v  u " 2 γ+1 # γ  u 2 p γ p γ p 2 γ−1  t out − out out ≥  if (3.32) pout  γ − 1 pin pin pin γ + 1 f1 = pin  s γ+1 γ  γ−1 γ−1  2 pout 2  if < γ + 1 pin γ + 1

where the eﬀective area CdA needs to be identiﬁed on experimental data for each component.

3.4.2 Receivers

The receivers are treated as mass and energy accumulators, and their dynamics is char-

acterized by the conservation laws. In a mean-value engine model, the typical components

55 represented by receivers are intake and exhaust manifolds. Besides, intermediate volumes of the two-stages turbocharger should be modeled as receivers to maintain consistency with the dynamics of the engine system and the cause and eﬀect relationships.

The isothermal approach is used to model the intermediate volumes of the two-stage turbocharger: d RT p = (m ˙ in − m˙ out) (3.33) dt V

The adiabatic approach is applied to the intake manifold:

d γR pIM = [m ˙ HP cTcc +m ˙ egrTegr − m˙ engTIM ] dt VIM d γRTIM TIM TIM = m˙ HP cTcc +m ˙ egrTegr − m˙ engTIM − (m ˙ HP c +m ˙ egr − m˙ eng) dt pIM VIM γ (3.34)

In exhaust manifold model, a heat loss term is introduced:

d γR hA(TEM − Twall) pEM = (m ˙ eng +m ˙ fuel)Texh − (m ˙ egr +m ˙ HP t)TEM − dt VEM cp d γRTEM TEM = (m ˙ eng +m ˙ fuel)Texh − (m ˙ egr +m ˙ HP t)TEM (3.35) dt pEM VEM TEM hA(TEM − Twall) − (m ˙ eng +m ˙ fuel − m˙ egr − m˙ HP t) − γ cp

3.4.3 Mass Flow through Engine

Speed-density equation is used to model the mass flow from the intake manifold into cylinders: λvpIM VdNeng m˙ eng = (3.36) 120RTIM where Vd is the engine displacement, λv is the engine volumetric efficiency. The volumetric efficiency is modeled as:

λvpIM = k1pIM + k2 (3.37)

56 where k1 and k2 are constant parameters, identiﬁed on experimental data.

3.4.4 Engine-out Temperature and Torque Production

The application of MVM techniques to the engine combustion and torque production assumes that the processes occurring in the engine cylinders are cycle-averaged. Therefore, the in-cylinder processes is typically reduced to quasi-steady, algebraic relationships.

The exhaust gas temperature is estimated through the deﬁnition of a temperature in- crement ∆T , which is a regression function of the charge composition:

Texh = TIM + ∆T (AF R, EGR, TIM ) (3.38)

The torque model is based on estimation of the indicated mean effective pressure (IMEP) from the experimental brake mean effective pressure (BMEP), and estimation of the friction mean effective pressure (FMEP) by the Willans line approach [122]. Then the indicated efficiency can be calculated as:

IMEP · Vd · Neng ηind = (3.39) 120 · m˙ fuel · LHVf

where LHVf is the fuel lower heating value. Based on the principles of Diesel combustion and on the analysis of engine data, a phenomenological model for engine indicated eﬃciency is proposed, which is a function of the engine speed, air fuel ratio, and the diﬀerential pressure:

2 k5 ηind = η0(k1 + k2Neng + k3Neng)(1 + k4AF R )[1 + k6(pEM − pIM )] (3.40)

The FMEP is modeled as a quadratic function of engine speed:

2 FMEP = c1 + kcNeng + c3Neng (3.41)

57 Then the torque is calculated as:

120ηindm˙ fuelLHVf 2 Vd Teng = − (c1 + kcNeng + c3Neng) (3.42) VdNeng 4π

3.4.5 Turbocharger Speed

The turbocharger speed is modeled using Newton’s second law for rotating systems:

2 d 30 Pt − Pc Ntc = (3.43) dt π JtcNtc

where Jtc is the inertia, Pt is the power delivered by the turbine, and Pc is the power required to drive the compress.

3.5 Integration and Validation of Two-Stage Turbocharged

Engine Model

The parametric model of turbocharger is integrated with the Diesel engine air-path system model, and used to simulate the system behavior at steady-state and transient conditions and validate against a set of experimental data.

In order to identify the parameters of the engine system model, a sparse set of steady- state engine data is used, covering the engine operating range in terms of speed and torque, shown in the left plot of Figure 3.36. The validation of the model is done on the complete set of available data, considering the engine speed and torque as the inputs for the model.

To this extent, the model was completed with the introduction of a simple control strategy, which operates the fuel, EGR, VGT and HP turbine bypass based on the desired engine speed and torque.

The structure of the control strategy is shown in Figure 3.37. The fuel mass ﬂow rate is determined by a feedforward maps of desired torque and engine speed. The EGR, VGT

58 700 100

80 600 54 65 87 76 60 107 64 [%]

97 egr

75 x 40 42 53 86 500 52 106 74 115 32 41 51 63 96 20 62 73 85 114 312 40 50 61 724 84 95 105 400 23 0 10 20 30 40 50 60 70 80 90 100 110 94 104 113 30 39 49 60 71 Data Points [Nm] 22 83

eng 70 103 112 T 29 38 48 59 82 300 15 21 100 81 93 102 111 x 373 47 58 69 vgt 20 28 9 14 80 101 80 x 110 bp 200 1 36 46 57 68 8 13 19 27 79 60 9291 45 56 67 90 100 109 7 12 18 26 35 78 89 108 40 100 6 17 25 34 44 55 66 77 11 88 20 98 5 10 16 24 33 43 HP Turb. Actuators [%] 0 99 0 1000 1500 2000 2500 3000 10 20 30 40 50 60 70 80 90 100 110 N [rpm] Data Points eng

Figure 3.36: Engine Operating Conditions and Control Inputs for Model Calibration

T des m Open-loop Fuel fuel Command pIM Neng

x p Open-loop VGT vgt EM Control Engine Model m HPc x Open-loop EGR egr Control

zegr

x Open-loop bp Bypass Control

Figure 3.37: Block Diagram of Engine Model with Open-loop Control Strategy and bypass positions are controlled using a simple feedforward strategy, based on static maps depending on the desired engine speed and fuel mass ﬂow rate. The positions of three actuators are shown in the right plots of Figure 3.36.

3.5.1 Steady-State Validation

The parametric model of turbocharger is used to replace the previously developed, look- up table based model of the two-stage turbocharger contained in the Diesel engine air-path

59 system. Note that, due to the modiﬁed causality of the compressor model (Type 2 model), the overall model results with a higher number of states.

Starting from the original maps, the parameters for the compressor and turbine models were identiﬁed by following the procedure highlighted in previous sections. The models were calibrated on the original two-stage turbocharger data, and then integrated with the air-path system model of the 6.6L Duramax Diesel engine.

Simulation results were conducted with respect to 115 steady-state points covering the entire engine operating range. The results obtained with the parametric turbocharger model are compared with the results from original look-up table based turbocharger model, as summarized in Figure 3.38 and Figure 3.39. From the analysis of the results it can be observed that the parametric model achieves the same accuracy as the original model.

In particular, an analysis of the operating points on the compressors maps indicates that the two models are almost equivalent, hence showing the validity of the new parametric modeling approach in predicting the system behavior with full engine system simulation.

5 x 10 0.4 4 data look−up table 3.5 0.3 parametric model 3 [kg/s] 0.2 [Pa] 2.5 IM HPc p

m 2 0.1 1.5

0 1 10 20 30 40 50 60 70 80 90 100 110 10 20 30 40 50 60 70 80 90 100 110 Data Points Data Points

5 x 10 60 8

50 7 6 40 5 [%] 30 [Pa] egr EM

z 4 p 20 3

10 2

0 1 10 20 30 40 50 60 70 80 90 100 110 10 20 30 40 50 60 70 80 90 100 110 Data Points Data Points

Figure 3.38: Steady-State Validation of the Engine Model

60 4 4 HPC

LPC 0.75 0.75 3.5 3.5

0.65 0.65 0.7 0.7 0.65 0.65

0.65 0.65

3 0.75 3 0.75

0.7 0.7

0.6 0.6

β 2.5 β 2.5 0.7 0.7

0.65 0.65

2 0.6 2 0.6

0.7 0.7

1.5 0.65 1.5 0.65 0.650.75 0.650.75 0.7 0.6 0.7 0.6

1 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 m [kg/s] m [kg/s] corr corr

Figure 3.39: Comparison of Compressor Operating Points: Original Look-up Table Model (Left) and Parametric Model (Right)

3.5.2 Simulation of FTP Cycle

After steady-state calibration and validation, the engine model is tested for transient simulations on a FTP cycle. The target proﬁles shown in Figure 3.40 for engine torque and speed are assigned to the model, while using the controller structure illustrated in

Figure 3.37, regulate fuel, EGR, VGT and bypass commands. A feedback torque control is introduced, which adjust the fuel command to track the desired torque proﬁle.

The left of Figure 3.41 shows the output of the torque controller, representing the feedforward command and the correction made by the feedback portion. The fuel ﬂow to the engine is determined mostly by the open loop mapping, while the PI controller operates small changes around the set points to accommodate for plant errors. The right of Figure 3.41 shows the EGR, VGT and bypass actuators positions during the cycle. In particular, because of the low power demand during FTP operations, the HP turbine bypass valve is always closed.

Finally, Figure 3.42 shows the evolution of the intake and exhaust manifolds pressure, air mass ﬂow, and EGR ratio during the FTP cycle. All the variables are in reasonable range, showing the validity of the integrated model in predicting the system behavior at transient operations.

61 700 2500

2000 600

[rpm] 1500 eng 500 N 1000

400 500 0 200 400 600 800 1000 1200 Time [s] [Nm] eng T 300 600

500

200 400

[Nm] 300 eng 100 T 200

100

0 0 1000 1500 2000 2500 3000 0 200 400 600 800 1000 1200 N [rpm] Time [s] eng

Figure 3.40: Engine Speed and Torque Proﬁle during FTP Cycle

−3 x 10 7 100 x FF vgt 6 PI x 80 egr 5 x bp

4 60

[kg/s] 3 fuel

m 40 2 Actuators [%]

1 20 0

−1 0 0 200 400 600 800 1000 1200 0 200 400 600 800 1000 1200 Time [s] Time [s]

Figure 3.41: Commands during FTP Cycle: Fueling Rate (Left) and Actuators Positions (Right)

5 x 10 0.2 3

0.15 2.5

0.1 [Pa] 2 IM p MAF [kg/s] 0.05 1.5

0 1 0 200 400 600 800 1000 1200 0 200 400 600 800 1000 1200 Time [s] Time [s]

5 x 10 70 5

60 4

50 [%] [Pa] 3 egr EM z 40 p

2 30

20 1 0 200 400 600 800 1000 1200 0 200 400 600 800 1000 1200 Time [s] Time [s]

Figure 3.42: Simulation Results of the Engine Model during FTP Cycle

62 Chapter 4

Air-Path Control of Diesel Engine for Rapid Warm-up

EGISLATED emission limits for medium and heavy duty trucks are constantly re- L duced while there is a significant drive for good fuel economy. In order to meet the stringent regulations, aftertreatment systems have become necessary for Diesel-powered vehicles. In specific, Diesel oxidation catalyst (DOC) is a key component for removing the engine-out CO, total hydrocarbon emissions, and volatile organic fraction. The chemical reactions inside DOC are suppressed under cold operating conditions until the light-off temperature (200 − 250 ◦C) is reached. Therefore, active temperature control for rapid catalyst warm-up during cold start is necessary for emission reduction. Aftertreatment system thermal management based on fuel-path control can be found in [93–95]. However, the opportunity of controlling the air-path to achieve rapid catalyst warm-up has not been explored.

Therefore, an air-path control strategy, based on a two-stage turbocharged Diesel engine for rapid catalyst warm-up, is investigated in this chapter. As shown in Figure 4.1, the engine is equipped with a VGT and a high pressure EGR loop. A bypass valve is placed across the HP turbine, while a waste-gate is placed across the LP turbine, to regulate the available power to the corresponding compressor. The control actuators are the EGR valve, the VGT, the bypass, and the waste-gate. Typically, due to the relative low power demand during cold start, the bypass and waste-gate are fully closed. So all the exhaust gas energy is consumed by turbines for boosting. However, part of the exhaust energy could go through the waste-gate for catalyst warm-up, result in reduced boost performance. Meanwhile, the total amount of exhaust gas energy is aﬀected by the boost performance, and the EGR

63 p m p c,int HPc Intake IM LPC HPC Manifold charge cooler EGR cooler

N LT N HT EGR x valve egr T x cat p vgt p t,int Exhaust EM DOC LPT HPT Manifold xwg xbp

LP waste-gate HP bypass valve

Figure 4.1: Air-Path Conﬁguration of Two-stage Turbocharged Engine with DOC ratio should be kept at desired level. The nonlinearity of the engine system, the presence of multiple actuators, and the coupling eﬀects, make the air-path control challenging. Since model predictive control (MPC) is considered as a systematic multivariable control scheme to accommodate actuators and state constraints, it is chosen for this Diesel engine air-path control study.

This chapter is organized as follows. First, thermal models for waste-gate and DOC are introduced. Preliminary study on the FTP city driving cycle is conducted to analysis the inﬂuence of bypass and waste-gate on catalyst warm-up. Then, model order reduction and linearization are applied, resulting in a discrete-time piecewise aﬃne (PWA) system. The rapid warm-up problem is formulated as MPC tracking control, then the controller and state observer are designed on the PWA system. Finally, MPC and observer are implemented on the full order nonlinear engine model, and a heuristic control for bypass opening is extracted from the solution of MPC.

64 4.1 Thermal Model of Exhaust System

4.1.1 Bypass Valve Thermal Model

The bypass valve flow is modeled as steady-state, adiabatic reversible flow through an orifice. However, due to the sudden expansion after the throat, a large portion of kinetic energy is converted to internal energy, which makes the outlet temperature higher than the one determined from adiabatic approach. Therefore, a bypass valve thermal model is developed, which take temperature recovery and heat transfer into account.

As illustrated in the left side of Figure 4.2, bypass valve is divided into two parts: inlet to throat (1 to 2) is modeled as converging nozzle, where the flow is characterized by adiabatic orifice flow; throat to outlet (2 to 3) is modeled as diverging nozzle, where the kinetic energy is converted to internal energy. One widely adopted assumption is that the kinetic energy of 1 and 3 can be ignored (c1 = c3 = 0), since the cross sectional area of 1 and 3 are much larger than 2. To decouple the bypass valve thermal model and flow model, another assumption is made, such that the pressure in the diverging nozzle are roughly the same

(p2 = p3).

T2, p2 T1

T1, p1 T3, p3

Turb.

Tt T3 T1 T3

T2 Tout (a) (b)

Figure 4.2: Sketch of Bypass Valve Thermal Model: Converging-Diverging Nozzle (Left) and Turbine with Bypass (Right)

65 The energy equation of converging nozzle 1 to 2 is:

2 c2 h1 = h2 + (4.1) 2 where h stands for enthalpy and c stands for velocity. Since 1 to 2 is adiabatic process, the velocity of 2 is obtained as:

v u " γ−1 # u p3 γ c2 = t2cpT1 1 − (4.2) p1

The oriﬁce equation is:

p1 √ p3 m˙ wg = CdA√ γf1 RT1 p1 v  u " 2 γ+1 # γ  u 2 p γ p γ p 2 γ−1  t 3 − 3 3 ≥  if (4.3) p3  γ − 1 p1 p1 p1 γ + 1 f1 = p1  s γ+1 γ  γ−1 γ−1  2 p3 2  if < γ + 1 p1 γ + 1 where CdA is a function of bypass valve opening. The energy equation of diverging nozzle 2 to 3 is: 2 ˙ c2 Q23 h2 + = h3 + (4.4) 2 m˙ wg where Q˙ 23 is the heat transfer loss,m ˙ wg is the bypass valve mass ﬂow determined above. The heat transfer term can be expressed as:

Q˙ 23 = hA(T3 − Tw) (4.5)

where hA is the production of heat transfer coeﬃcient and eﬀective area, Tw is the wall temperature.

66 Assuming turbulent pipe ﬂow, based on the Dittus-Boelter equation, the heat transfer coeﬃcient can be expressed explicitly as:

hD = 0.023Re0.8Pr0.3 (4.6) k where D is the average inside diameter of the diverging nozzle, k is the thermal conductivity of exhaust gas, Re is the Reynolds number and Pr is the Prandtl number. The Reynolds number can be expressed as: c2D Re = (4.7) ν where ν is the kinematic viscosity. Combining (4.6) and (4.7) yields:

kPr0.3 D0.8 hA = Kc0.8,K = 0.023 πDL (4.8) 2 ν0.8 D where K is a calibration parameter which depends on exhaust gas property and temperature

(ﬁrst brackets), and on bypass valve geometry (second brackets).

Finally, plugging (4.5) and (4.8) into (4.4), the outlet temperature is expressed as:

0.8 m˙ wgcp Kc2 T3 = 0.8 T1 + 0.8 Tw (4.9) m˙ wgcp + Kc2 m˙ wgcp + Kc2

It can be seen that bypass valve outlet temperature is a weighted average of inlet temperature and wall temperature.

Since the bypass valve and turbine are mounted in parallel, as shown in the right side of Figure 4.2, there is a mixing of the gases at the outlet. This mixing is modeled as an adiabatic process, which gives the following temperature for the mixture:

m˙ tTt +m ˙ wgT3 Tout = (4.10) m˙ t +m ˙ wg

67 700

680

660

640

620

600 T [K] x = 1% bp 580 x = 2% bp 560 x = 5% bp x = 10% 540 bp T is 520 T t 500 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 p / p 1 3

Figure 4.3: The Bypass Valve Outlet Temperature

wherem ˙ t is the turbine mass ﬂow, and Tt is the turbine outlet temperature expressed as:

" γ−1 # p3 γ Tt = T1 − ηtT1 1 − (4.11) p1

Bothm ˙ t and ηt are obtained from turbine parametric model discussed in Section 3.3.

Given the inlet temperature T1 and pressure p1, the outlet temperatures are shown in Figure 4.3. It can be seen that the temperature Tis obtained from adiabatic approach

is lower than turbine outlet temperature T3. In the proposed bypass thermal model, T3 asymptotically decreases as pressure ratio approaches to 1, since trivial exhaust energy

goes through bypass valve. At high pressure ratio region, due to the heat transfer, T3 will slowly decrease. At a certain pressure ratio, the mass ﬂow rates are the same but

the velocity c2 changes with bypass opening. Since c2 aﬀects heat transfer, larger bypass

opening leads to higher T3.

68 4.1.2 Catalyst Thermal Model

In literature, research conducted on control-oriented catalyst thermal model can be found in [122–124]. In the application of cold start warm-up, a reasonable assumption is that no chemical reaction, so a lumped catalyst thermal model is described by heat transfer model: d (mc)cat Tcat =m ˙ excp(Tex − Tcat) − hA(Tcat − T ) (4.12) dt amb wherem ˙ ex and Tex stand for the mass ﬂow and temperature of exhaust gas upstream of

DOC, respectively. (mc)cat is the thermal inertia, and hA is the production of convective heat transfer coeﬃcient and outer surface of a DOC, which can be identiﬁed experimentally.

4.2 Preliminary Analysis on FTP Cycle

4.2.1 Baseline Scenario

Since the study focuses on cold start warm-up, ﬁrst 200 seconds of the FTP city driving

◦ cycle, during which the catalyst can reach the light-oﬀ temperature (Tthr = 200 C), is used to evaluate the system performance. Because of the low power demand during FTP cycle,

the HP bypass valve and LP waste-gate valve are always closed. A preliminary analysis is

conducted based on the original feedforward control strategy. The result of this baseline

condition is shown in Figure 4.4. It can be seen that the catalyst is light oﬀ at 192s, and the

peak boost pressure is about 2.5 bar. From the plots of turbocharger speed and pressure

ratio, it can be concluded that the boost is mainly contributed by the HP turbocharger.

4.2.2 Inﬂuence of HP Bypass Valve

To analyze the eﬀect of HP bypass valve on warm-up performance, a study is conducted

considering several constant bypass openings, while keeping the original control strategy of

EGR, VGT, and waste-gate. It can be seen in Figure 4.5, the boost pressure is very sensitive

69 5 x 10 3 500

2.5 450

2 [K] [Pa] 400 cat IM T p

1.5 350

1 300 0 20 40 60 80 100 120 140 160 180 200 0 20 40 60 80 100 120 140 160 180 200 Time [s] Time [s]

5 x 10 0.2 2 HP 0.15 1.5 LP

0.1 1 [rpm] tc N MAF [kg/s] 0.05 0.5

0 0 0 20 40 60 80 100 120 140 160 180 200 0 20 40 60 80 100 120 140 160 180 200 Time [s] Time [s]

2.5 3.5 HP HP LP 3 LP 2 2.5 ε β 2 1.5 1.5

1 1

0 20 40 60 80 100 120 140 160 180 200 0 20 40 60 80 100 120 140 160 180 200 Time [s] Time [s]

Figure 4.4: FTP Cycle Analysis: Baseline

5 x 10 3 550 baseline x = 5% bp 500 2.5 x = 10% bp x = 15% bp 450

2 [K] [Pa] cat IM T p 400

1.5 350

1 300 0 20 40 60 80 100 120 140 160 180 200 0 20 40 60 80 100 120 140 160 180 200 Time [s] Time [s]

Figure 4.5: FTP Cycle Analysis: Inﬂuence of Bypass to bypass opening, and the warm-up is improved by penalizing the boost pressure. To maintain certain level of boost, the maximum bypass valve opening should be constrained.

4.2.3 Inﬂuence of LP Waste-Gate Valve

Similarly, constant waste-gate openings are tested with original control strategy of other actuators, and the result is in Figure 4.6. Since LP turbocharger does not eﬀectively

70 5 x 10 3 500 baseline x = 25% wg 2.5 x = 50% 450 wg x = 100% wg

2 [K] [Pa] 400 cat IM T p

1.5 350

1 300 0 20 40 60 80 100 120 140 160 180 200 0 20 40 60 80 100 120 140 160 180 200 Time [s] Time [s]

Figure 4.6: FTP Cycle Analysis: Inﬂuence of Wastegate work in low power operations, waste-gate valve has trivial eﬀect on warm-up temperature improvement. Therefore, the original waste-gate control strategy is reserved, which can reduce one input for control design.

4.3 Piecewise Aﬃne System Development

The engine air-path model is a high order nonlinear system, which can be particularly troublesome for control design. Thus, approximations become necessary, very often in the form of local linear models. In this study, we propose a piecewise aﬃne discrete- time approximation of the system dynamics, which is used for control and observer design.

First, model order reduction techniques is applied to the high-order complex engine model.

Then, the reduced order nonlinear model is linearized at 20 representative operating points.

Finally, a discretized piecewise aﬃne system with a polyhedral partition of the exogenous input space (engine speed and torque) is obtained.

4.3.1 Model Order Reduction

The engine air-path system is characterized by a 11-state dynamic model, where the states include catalyst temperature, intake/exhaust manifold pressures and temperatures, two-stage turbocharger shaft speeds, compressor mass ﬂow rates, and intermediate volume

71 5 x 10 0.35 3.5 11 states 0.3 7 states 3 0.25

0.2 2.5 [kg/s] [Pa] IM HPc 0.15 p 2 m 0.1 1.5 0.05

0 1 0 20 40 60 80 100 120 0 20 40 60 80 100 120 Data points Data points

360 900

350 800

340 700 [K] [K] 330 600 IM EM T T 320 500

310 400

300 300 0 20 40 60 80 100 120 0 20 40 60 80 100 120 Data points Data points

Figure 4.7: Validation of Reduced Order Model at Steady-State Conditions pressures:

T x = Tcat pEM TEM pIM TIM m˙ HP c NHP pt,int m˙ LP c NLP pc,int (4.13)

The compressor mass ﬂow ratesm ˙ HP c andm ˙ LP c can be reduced by using Type 1 model as discussed in previous report. The intake/exhaust manifold temperatures TIM and TEM are modeled as static correlations:

TIM = k1 + k2pIM + k3zegr (4.14) TEM = k4Texh + k5

Therefore, the original engine model can be approximated by a reduced order model with 7 states, which can simplify the control design:

T x = Tcat pEM pIM NHP pt,int NLP pc,int (4.15)

The comparison of the reduced order model and the full order model are shown in

72 5 x 10 3 500 11 states 7 states 2.5 450

2 [K] [Pa] 400 cat IM T p

1.5 350

1 300 0 20 40 60 80 100 120 140 160 180 200 0 20 40 60 80 100 120 140 160 180 200 Time [s] Time [s]

5 x 10 0.2 4

3.5 0.15 3

0.1 [Pa] 2.5 EM p

MAF [kg/s] 2 0.05 1.5

1 0 0 20 40 60 80 100 120 140 160 180 200 0 20 40 60 80 100 120 140 160 180 200 Time [s] Time [s]

Figure 4.8: Validation of Reduced Order Model during FTP Cycle

Figure 4.7 and Figure 4.8. From the ﬁgures, it can be concluded that the reduced order model is capable to characterize the engine system behavior both in steady-state and in transient operations.

4.3.2 Linearization and Discretization

After model order reduction, the dynamics of the remaining 7 states are summarized as: d (mc)cat Tcat =m ˙ excp(Tex − Tcat) − hA(Tcat − T ) (4.16a) dt amb d γR pIM = (m ˙ HP cTcc +m ˙ egrTegr − m˙ engTIM ) (4.16b) dt VIM d γR hA(TEM − Twall) pEM = (m ˙ eng +m ˙ fuel)Texh − (m ˙ egr +m ˙ HP t)TEM − (4.16c) dt VEM cp d RTLP c pc,int = (m ˙ LP c − m˙ HP c) (4.16d) dt Vc,int d RTHP t,mix pt,int = (m ˙ HP t +m ˙ bp − m˙ LP t − m˙ wg) (4.16e) dt Vt,int 2 d 30 PHP t − PHP c NHP = (4.16f) dt π JHP NHP 2 d 30 PLP t − PLP c NLP = (4.16g) dt π JLP NLP

73 480 480

19 20

400 400

16 17 18

320 320

12 13 14 15

[Nm] 240 [Nm] 240 eng eng T T 8 9 10 11

160 160

4 5 6 7

80 80

FTP 0−200s 1 2 3 Key point 0 0 600 900 1200 1500 1800 2100 2400 600 900 1200 1500 1800 2100 2400 N [rpm] N [rpm] eng eng

Figure 4.9: Partition of the Engine Map Based on FTP Cycle (0-200s)

5 x 10 70 1.12 0.04 NL 60 0.039 1.1 Lin 50 0.038 1.08 40 0.037 [kg/s] [Pa] 1.06

30 IM HPc

p 0.036 m

Actuator [%] x 1.04 20 vgt 0.035 x egr 10 1.02 0.034 x bp 0 1 0.033 0 10 20 30 40 0 10 20 30 40 0 10 20 30 40 Time [s] Time [s] Time [s]

5 x 10 1.24 54 380

1.22 52 360 1.2 50 [K] [%] [Pa] 1.18 48 340 cat egr EM T z p 1.16 46 320 1.14 44

1.12 42 300 0 10 20 30 40 0 10 20 30 40 0 10 20 30 40 Time [s] Time [s] Time [s]

Figure 4.10: Validation of Linearized Model at Region 6: Change of Actuator Position

The reduced order engine model is linearized at 20 representative operating points, resulting in a PWA system with a polyhedral partition of the exogenous input space, as illustrated in Figure 4.9. At each operating point,m ˙ fuel, xvgt, xegr are determined by

74 5 x 10 1500 1.1 0.045 NL 1450 1.08 Lin 0.04 1400

1350 1.06 [kg/s] [rpm] [Pa] 0.035 IM eng HPc 1300 p 1.04 N m 1250 0.03 1.02 1200

1150 1 0.025 0 20 40 60 80 0 20 40 60 80 0 20 40 60 80 Time [s] Time [s] Time [s]

140 55 440

130 420 50 400 120 380 [K] [%] [Nm] 110 45 cat egr eng T z 360 T 100 340 40 90 320

80 35 300 0 20 40 60 80 0 20 40 60 80 0 20 40 60 80 Time [s] Time [s] Time [s]

Figure 4.11: Validation of Linearized Model at Region 6: Change of Operating Point

feedforward maps, and xbp = 3%. The continuous PWA system has form:

x˙ = Ai(x − x0) + Bi(u − u0) + Ei(v − v0) = Aix + Biu + Eiv + fi

y = Ci(x − x0) + Di(u − u0) + y0 = Cix + Diu + Fiv + gi (4.17) T if Neng Teng ∈ Pi, i = 1,..., 20 where states x, control inputs u, exogenous inputs v, outputs y are deﬁned as:

T T x = Tcat pEM pIM NHP pt,int NLP pc,int u = xvgt xegr xbp (4.18) T T y = pIM m˙ HP c zegr Tcat v = Neng m˙ fuel

th and Pi denotes the i polytope on the engine map, which is given by:

2 Pi = {x ∈ R : Hx ≤ ki}, i = 1,..., 20 (4.19)

The quality of linearization is evaluated at each region, and the representative examples

75 5 x 10 70 2.4 0.14 NL 60 2.3 Lin 0.13 50 2.2

40 2.1 [kg/s] [Pa] 0.12

30 IM HPc

p 2 m

Actuator [%] x 20 vgt 1.9 x 0.11 egr 10 1.8 x bp 0 1.7 0.1 0 10 20 30 40 0 10 20 30 40 0 10 20 30 40 Time [s] Time [s] Time [s]

5 x 10 3.6 55 500

3.4 50 450 3.2 [K] [%] [Pa] 3 45 400 cat egr EM T z p 2.8 40 350 2.6

2.4 35 300 0 10 20 30 40 0 10 20 30 40 0 10 20 30 40 Time [s] Time [s] Time [s]

Figure 4.12: Validation of Linearized Model at Region 18: Change of Actuator Position

5 x 10 2400 2.4 0.16 NL 2350 2.3 Lin 0.15 2300 2.2 0.14 2250 [kg/s] [rpm] [Pa] 2.1 0.13 IM eng HPc

2200 p N 2 m 0.12 2150

2100 1.9 0.11

2050 1.8 0.1 0 20 40 60 80 0 20 40 60 80 0 20 40 60 80 Time [s] Time [s] Time [s]

390 55 550

380 500 50 370 450 [K] [%] [Nm] 360 45 cat egr eng T z

T 400 350 40 340 350

330 35 300 0 20 40 60 80 0 20 40 60 80 0 20 40 60 80 Time [s] Time [s] Time [s]

Figure 4.13: Validation of Linearized Model at Region 18: Change of Operating Point at low load and high load regions are shown in Figure 4.10 to Figure 4.13. Overall, the linearized model captures the system dynamics well. However, in Figure 4.10, a dc-gain sign reversal from the VGT to the compressor mass ﬂow channel around 5s is found in region

6. This sign reversal is a essential property of this nonlinear system, and is well discussed in [125].

76 5 x 10 3 500 11 states NL 7 states PWA 2.5 450

2 [K] [Pa] 400 cat IM T p

1.5 350

1 300 0 20 40 60 80 100 120 140 160 180 200 0 20 40 60 80 100 120 140 160 180 200 Time [s] Time [s]

5 x 10 0.2 4

3.5 0.15 3

0.1 [Pa] 2.5 EM p

MAF [kg/s] 2 0.05 1.5

1 0 0 20 40 60 80 100 120 140 160 180 200 0 20 40 60 80 100 120 140 160 180 200 Time [s] Time [s]

Figure 4.14: Validation of Piecewise Aﬃne System during FTP Cycle

The zero-order-hold method is used for discretization, and the sampling time is selected to be Ts = 0.05s. Then the discrete-time PWA system is obtained, which has the form:

x(k + 1) = Aix(k) + Biu(k) + Eiv(k) + fi

y(k) = Cix(k) + Diu(k) + Fiv(k) + gi (4.20) T if Neng(k) Teng(k) ∈ Pi, i = 1,..., 20

4.3.3 Piecewise Aﬃne System Validation

To verify the accuracy of the discrete PWA model, both full order nonlinear model and

PWA model are tested on FTP cycle, where feedforward control strategy is used. The results in Figure 4.14 show that the PWA model is a reasonable approximation of the nonlinear plant model, thus will be used for control and observer design.

77 4.4 Model Predictive Controller Design

4.4.1 Introduction of Model Predictive Control

MPC is based on iterative, ﬁnite horizon optimization of a plant model. At time t,

starting at the current state, an optimal control problem is solved over a ﬁnite horizon, as

shown in the top of Figure 4.15. Only the ﬁrst computed optimal control input is applied

to the system during [t, t + 1]. At time t + 1, a new optimal control problem based on new

measurements of the state is solved over a shifted horizon, as depicted in the bottom of

Figure 4.15.

Consider the problem of regulating to the origin for the discrete time linear time invari- Introductionant (LTI) system with polyhedral states and inputs constraints. The following N-horizon

Figure 4.15: A Discrete MPC Scheme 1 At each sampling time, solve a CFTOC. 2 Apply the optimal input only during [t,t + 1] 3 At t +1 solve a CFTOC over a shifted horizon based on new state measurements 78 4 The resultant controller is referred to as Receding Horizon Controller (RHC) or Model Predictive Controller (MPC).

Francesco Borrelli (UC Berkeley) Model Predictive Control April 5, 2011 3 / 61 optimal control problem is solved at time t:

N−1 X min J(x(t),U0) , Jf (xN ) + l(xk, uk) U0 k=0

subj. to xk+1 = Axk + Buk, k = 0,...,N − 1 (4.21)

Axxk ≤ bx,Auuk ≤ bu, k = 0,...,N − 1

Af xN ≤ bf , x0 = x(t)

where U0 = {u0, . . . , uN−1}. Considering 2-norm cost function:

N−1 T X T T J(x(t),U0) = xN Qf xN + xk Qxk + uk Ruk (4.22) k=0 then the problem can be recast into the form of a quadratic programming (QP) problem:

 T     U HF T U  0     0  min JN (x(t),U0) =       U0 x(t) FY x(t) (4.23)

subj. to G0U0 ≤ w0 + E0x(t)

where H,F,G0, w0,E0 can be obtained from the plant model and Q, R (for details, see [85]). Since the optimization problem (4.23) depends on the current state x(t), the implementation

of MPC requires the online solution of a QP at each time step.

∗ In [85], an approach to pre-compute the control law U0 (z) for all possible state location x(t) = z is proposed. With polyhedral state and control constraints, explicit solution

can be obtained through multiparametric quadratic programming (mp-QP), resulting in a

79 piecewise aﬃne control law:

   F1x(t) + f1 if H1x(t) ≤ h1  ∗  . . U0 (x(t)) = . . (4.24)    FM x(t) + fM if HM x(t) ≤ hM

As all Fi, fi,Hi, hi can be stored in look-up tables, the solution to the problem of (4.23) boils down to the search for the active partition and the use of corresponding control law given

by (4.24). This makes the MPC useable for high-speed applications like engine control.

4.4.2 Model Predictive Control for the Engine Air-Path

The objective of air-path control for rapid catalyst warm-up is to improve warm-up

performance with trade-oﬀ on boost pressure, while maintaining desired EGR ratio. This

problem can be cast as MPC tracking problem, and the reference signals are deﬁned as:

T r = Tcat,d pIM,d zegr,d (4.25)

The proposed engine air-path control system is depicted in Figure 4.16. According to

previous discussion, MPC is essentially state feedback control, while scheduling of linearized

points according to operating region (see Figure 4.9) is essentially feedforward control. Since

MPC needs the information of all 7 states in (4.15), observer is designed to estimate the

unmeasured states.

For controller design, Matlab with the Multi-Parametric Toolbox (MPT) [126] is used.

Since the current version of MPT does not support quadratic programming for PWA system,

80 T des m Open-loop Fuel fuel Command

Neng

Feedforward Control Engine Model

x ,x ,x Feedback vgt egr bp Control

xˆ

pIM ,m HPc ,Tcat Observer

Figure 4.16: Diagram of Engine Air-Path Control System the PWA system (4.20) is treated as 20 separated LTI systems:

x(k + 1) = Aix(k) + Biu(k) + Eiv(k)

y(k) = Cix(k) + Diu(k) + Fiv(k) (4.26) T if Neng(k) Teng(k) ∈ Pi, i = 1,..., 20

Since the partition of PWA system only depends on exogenous inputs, the switch between

PWA regions can be easily converted to switch between LTI systems. A supervisory controller is added to select the active LTI system and corresponding controller.

It should be noted that (with a mild abuse of notation) all variables in LTI models represent deviations from the nominal operating condition. In this section, we will omit to distinguish between the variables representing deviations from steady-state and those representing actual values, as this will be clear from the context.

Because engine speed and EGR valve opening has direct eﬀect on EGR ratio, there is

81 direct feedthrough from inputs to EGR ratio in (4.26). To address the feedthrough term, delta input (δu) formulation is introduced, the MPC scheme uses the following LTI model:

x(k + 1) = Aix(k) + Biu(k) + Eiv(k)

u(k) = u(k − 1) + δu(k) (4.27) y(k) = Cix(k) + Diu(k) + Fiv(k) T if Neng(k) Teng(k) ∈ Pi, i = 1,..., 20

In order to use MPC, the engine air-path LTI model (4.27) has to be rewritten in a suitable form:

        x(k + 1) Ai Bi Ei 0 x(k) Bi                  u(k)   0 I 0 0 u(k − 1)  I            =     +   δu(k)         v(k + 1)  0 0 I 0  v(k)   0          r(k + 1) 0 0 0 I r(k) 0 (4.28) | {z } | {z } | {z } | {z } x¯(k+1) A¯i x¯(k) B¯i   Neng(k)   ∈ P y(k) = Ci Di Fi 0 x¯(k), if   i, i = 1,..., 20 T (k) | {z } eng C¯i

where x, u, v are deﬁned in (4.18), and y is the EGR ratio zegr. Then the MPC tracking problem is formulated as:

N−1 T X T T min x¯N Qf x¯N + x¯k Qx¯k + δuk Rδuk (4.29a) ∆U0 k=0

subj. tox ¯k+1 = A¯ix¯k + B¯iδuk (4.29b) T if Neng(0) Teng(0) ∈ Pi, i = 1,..., 20 (4.29c)

x¯min ≤ x¯k ≤ x¯max, k = 0,...,N (4.29d)

82 δumin ≤ δuk ≤ δumax, k = 0,...,N − 1 (4.29e)

umin ≤ uk−1 + δuk ≤ umax, k = 0 (4.29f)

δuk = 0, k = Nc,...,N (4.29g) T x¯0 = x(t) u(t − 1) v(t) r(t) (4.29h)

where ∆U0 = {δu0, . . . , δuNc−1} are the optimization variables, N and Nc denote the prediction horizon and the control horizon. The control input applied to the system is:

∗ u(t) = u(t − 1) + δu0 (4.30)

The optimization (4.29) is repeated at time k + 1 based on the new state x(t + 1), exogenous inputs v(t+1), reference signals r(t+1) and current control inputs u(t), yielding a receding horizon control strategy.

In the MPC tracking problem (4.28)-(4.29), the following assumptions are used:

A1 The control signal is assumed constant for all Nc ≤ k ≤ N (4.29g). This reduces the computational complexity of the MPC.

A2 The exogenous inputs v are assumed constant over the horizon (4.28). If PWA

prediction models for v(k) are available they could be included in the MPC for-

mulation.

A3 Constant region i over the horizon (4.29c), which basically implies that over the

prediction horizon only one linear MPC for one member of the set of LTI systems

is implemented. Ideally, prediction of switches between aﬃne dynamics over the

horizon will improve the perforance. But the problem beocmes a mixed integer

quadratic program, which is more complex than QP.

83 4.4.3 MPC Implementation Details

Reference Signals

The reference boost pressure and EGR ratio are maps of operating conditions based on steady-state simulation results, as shown in the left of Figure 4.17.

pIM,d(t) = f(Neng(t), m˙ fuel(t)), zegr,d(t) = f(Neng(t), m˙ fuel(t)) (4.31)

The map-based reference signals are compared with actual signals from full order engine

model tested on FTP cycle. It can be seen that the reference one has a good match with

baseline results, except EGR ratio transient during deceleration.

The reference temperature of catalyst not only relates to operating condition, but also

relates to time, which makes it diﬃcult to generate a map. Instead, the reference tempera-

ture will update at each sampling time, based on previous actual catalyst temperature plus

an oﬀset:

◦ Tcat,d(t) = Tcat(t − 1) + 5 C (4.32)

5 x 10 5 x 10 3 actual ref 4 2.5

[Pa] 2 [Pa] IM IM p

2 p

1 1.5 3000 0.01 2000 0.005 1 1000 0 0 20 40 60 80 100 120 140 160 180 200 N [rpm] m [kg/s] eng fuel Time [s]

60 55

40 50 [%] [%] 45 egr

z 20 egr z 40 0 3000 0.01 35 2000 0.005 1000 30 0 0 20 40 60 80 100 120 140 160 180 200 N [rpm] m [kg/s] eng fuel Time [s]

Figure 4.17: References Signal Maps (Left) and Comparison with FTP Baseline Results (Right)

84 Cost Function

Since a linear MPC is deﬁned on a speciﬁc operating region shown in Figure 4.9, the cost function should be region dependent. Therefore, a normalized cost function, which normalize the weight on boost, is proposed as follows:

T 2 wp 2 2 x¯ Qx¯ = wT (Tcat − Tcat,d) + 2 (pIM − pIM,d) + we(zegr − zegr,d) (4.33) p¯IM,i

th wherep ¯IM,i denotes the nominal boost pressure at i region. According to (4.28), Q ∈

15×15 R has the form:

    wT 0 0 · · · −wT 0 0 Tcat          0 0 0 ··· 0 0 0  pEM           ··· −     0 0w ¯p 0 w¯p 0  pIM       ..   .  Q =  .  + weQegr, x¯ =  .  (4.34)         −w 0 0 ··· w 0 0 T   T T   cat,d          0 0 −w¯p ··· 0w ¯p 0 pIM,d     0 0 0 ··· 0 0 0 zegr,d

15×15 wherew ¯p is the normalized weight on boost, and Qegr ∈ R is given by:

  C¯T C¯ −C¯T  1,i 1,i 1,i Qegr =   (4.35) −C¯1,i 1

1×14 1×15 where C¯1,i ∈ R is the ﬁrst 14 columns of C¯i ∈ R .

85 MPC Design Parameters

In the literature of Diesel engine air-path control application, the MPC parameters are typically in the following range:

0.02s ≤ Ts ≤ 0.05s, 25 ≤ N ≤ 120 (4.36) 2 ≤ Nc ≤ 5 (online), 1 ≤ Nc ≤ 2 (explicit)

Since the catalyst temperature dynamics is slow, a 0.05s sampling time is chosen. So far,

v and r are assumed to be constant in prediction, then a larger prediction horizon does

not necessarily improve the performance. The simulation results show that once N ≥

10, prediction horizon has little eﬀect on the performance, thus N = 10 is used. Using

constraints on the actuators and states as well as the control horizon 1 yields a polyhedral

partition of the state space in about 300 regions for explicit MPC. A control horizon 2

would yield a partition into more than 2000 regions. Considering the memory occupancy,

Nc = 1 is retained.

Input Oscillation

In PWA system, there are discontinuities in the nominal control inputs (feedforward

portion in Figure 4.16) when switching between operating regions. Although partition into

smaller regions will decrease the magnitude of such discontinuities, it will result in more

frequent switch and high memory occupancy. Slew rate constraints are not convenient to

use for limiting these small oscillations since these input rate constraints have to be very

tight, leading to a slow closed-loop system.

In [92], two diﬀerent QP in diﬀerent operating points are solved at each sampling instant

k. The resulting control inputs are a linear interpolation between the two solutions. In [127],

similarly, a convex combination of two cost functions are proposed. Another way to prevent

these oscillations is to use nonlinear MPC based on linear parameter varying models [128].

86 For the sake of simplicity, in this study, a moving average ﬁlter with 5th order is used to smooth the control inputs.

4.5 Observer Design

Since MPC is state feedback control which required either measurement or estimation of the 7 states deﬁned in (4.15), observer is designed to estimate the unmeasured states.

Starting from the discrete-time PWA system (4.20), check the observability when only MAF

7×7 2×7 and MAP sensors are used. The system matrices Ai ∈ R and Ci ∈ R are given by:

  A A  11 12 6×6 Ai =   ,A22 ∈ R 0 A22 (4.37) 2×6 Ci = 0 C2 ,C22 ∈ R

Then the observability matrix is expressed as:

    Ci 0 C2         CiAi  0 C2A22     O =  .  = . .  (4.38)  .  . .   .  . .      6 6 CiAi 0 C2A22

Since rank(O) = 6, (C2,A22) is an observable pair, but Tcat is unobservable. In fact, the catalyst temperature dynamics is connected to the rest of the system in a cascade

structure, so it will not observable from the dynamics of the other states. Therefore, catalyst

temperature sensor is needed, and observer is designed based on the reduced order PWA

system:

x(k + 1) = Aix(k) + Biu(k) + Eiv(k) + fi (4.39) y(k) = Cix(k) + gi

87 where x, u, v, and y are deﬁned as:

T x = pEM pIM NHP pt,int NLP pc,int T u = xvgt xegr xbp (4.40) T v = Neng m˙ fuel T y = pIM m˙ HP c

It should be noted that (with a mild abuse of notation) all matrices in (4.39) are submar- tices of that in (4.20) with proper dimension. In this section, we will omit this distinction and this will be clear from the context.

4.5.1 Reduced Order Observer

The reduced order observer is designed based on the PWA system (4.39):

xˆ(k + 1) = Aixˆ(k) + Biu(k) + Eiv(k) + fi + Li(y(k) − Cixˆ(k) − gi) (4.41)

The error dynamics is:

e(k + 1) = x(k + 1) − xˆ(k + 1)

= Aix(k) − Aixˆ(k) − Li(Cix(k) + gi − Cixˆ(k) − gi) (4.42)

= (Ai − LiCi)e(k)

6×2 Output injection matrix Li ∈ R are designed such that the closed-loop system has faster convergence rate, and all the eigenvalues are in the unit circle:

eig(Ai − LiCi) = 0.9 · eig(Ai) (4.43)

88 400 150

350 140

300 130 [kPa]

[kPa] 250 120 t,int EM p p 200 110 150 Model Observer 100 100

160 165 170 175 180 185 190 195 200 160 165 170 175 180 185 190 195 200 Time [s] Time [s]

130 200 HP 125 LP 150 120

115

[kPa] 100 [krpm] tc

c,int 110 p N 105 50 100

95 0 160 165 170 175 180 185 190 195 200 160 165 170 175 180 185 190 195 200 Time [s] Time [s]

Figure 4.18: Validation of Observer on PWA System

400 150

350 140

300 130 [kPa]

[kPa] 250 120 t,int EM p p 200 110 150 Model Observer 100 100

160 165 170 175 180 185 190 195 200 160 165 170 175 180 185 190 195 200 Time [s] Time [s]

130 200 HP 125 LP 150 120

115

[kPa] 100 [krpm] tc

c,int 110 p N 105 50 100

95 0 160 165 170 175 180 185 190 195 200 160 165 170 175 180 185 190 195 200 Time [s] Time [s]

Figure 4.19: Validation of Observer on Full Order Engine Model

The observer is ﬁrst implemented on the PWA system and the states can be estimated accurately, as shown in Figure 4.18. Then the observer is applied to the nonlinear engine model. It can be seen in Figure 4.19 that estimation of pc,int has large error in high load region.

89 4.5.2 Observer Design Considering Model Uncertainties

It should be noted that the observer is designed based on PWA model (4.20). Due to model order reduction and linear approximation, the PWA model (4.20) has mismatch with the full order nonlinear engine model, even at linearized points. Consider the modeling error as additive disturbance, (4.39) should be rewritten as:

x(k + 1) = Aix(k) + Biu(k) + Eiv(k) + fi + ∆xi (4.44) y(k) = Cix(k) + gi + ∆yi

Then the estimation error dynamics becomes:

e(k + 1) = x(k + 1) − xˆ(k + 1)

= Aie(k) + ∆xi − Li(Cix(k) + gi + ∆yi − Cixˆ(k) − gi) (4.45)

= (Ai − LiCi)e(k) + ∆xi − Li∆yi

At steady-state, the estimation error is:

−1 e = (I − Ai + LiCi) (∆xi − Li∆yi) (4.46)

It can be seen that the output injection matrix Li will aﬀect steady-state estimation error due to the presence of ∆yi. In the observer design, both the error convergence rate and the norm of Li need to be considered.

Table 4.1: Comparison of 3 Output Injection Matrices

No. Li kLik kI − (Ai − LiCi)k

1 eig(Ai − LiCi) = 0.9 · eig(Ai) 3.0006 2.9810 2 eig(Ai − LiCi) = eig(Ai) 0.0582 1.1061 3 pole placement 0.3251 0.9157

90 2400 300 180

2350 160 2300 250 140 [g/s] [rpm]

2250 [kPa] IM HPc eng 120 p N 2200 200 m

2150 100

2100 150 80 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 Time [s] Time [s] Time [s]

400 180 60

170 55 380 160 50

[Nm] 360 150 45 [krpm] [krpm] eng HP HP T N 140 N 40 340 130 35

320 120 30 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 Time [s] Time [s] Time [s]

500 150 150 Model 450 140 Observer 140 400 130 [kPa] [kPa] [kPa] 350 130 t,int EM c,int 120 p p 300 p 120 250 110

200 100 110 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 Time [s] Time [s] Time [s]

Figure 4.20: Steady-State Estimation Error at Region 18: eig(Ai − LiCi) = 0.9 · eig(Ai)

2400 300 180

2350 160 2300 250 140 [g/s] [rpm]

2250 [kPa] IM HPc eng 120 p N 2200 200 m

2150 100

2100 150 80 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 Time [s] Time [s] Time [s]

400 180 60

170 55 380 160 50

[Nm] 360 150 45 [krpm] [krpm] eng HP HP T N 140 N 40 340 130 35

320 120 30 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 Time [s] Time [s] Time [s]

500 150 150 Model 450 140 Observer 140 400 130 [kPa] [kPa] [kPa] 350 130 t,int EM c,int 120 p p 300 p 120 250 110

200 100 110 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 Time [s] Time [s] Time [s]

Figure 4.21: Steady-State Estimation Error at Region 18: eig(Ai − LiCi) = eig(Ai)

91 2400 300 180

2350 160 2300 250 140 [g/s] [rpm]

2250 [kPa] IM HPc eng 120 p N 2200 200 m

2150 100

2100 150 80 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 Time [s] Time [s] Time [s]

400 180 60

170 55 380 160 50

[Nm] 360 150 45 [krpm] [krpm] eng HP HP T N 140 N 40 340 130 35

320 120 30 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 Time [s] Time [s] Time [s]

500 150 150 Model 450 140 Observer 140 400 130 [kPa] [kPa] [kPa] 350 130 t,int EM c,int 120 p p 300 p 120 250 110

200 100 110 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 Time [s] Time [s] Time [s]

Figure 4.22: Steady-State Estimation Error at Region 18: Li Obtained by Pole Placement

400 150

350 140

300 130 [kPa]

[kPa] 250 120 t,int EM p p 200 110 150 Model Observer 100 100

160 165 170 175 180 185 190 195 200 160 165 170 175 180 185 190 195 200 Time [s] Time [s]

130 200 HP 125 LP 150 120

115

[kPa] 100 [krpm] tc

c,int 110 p N 105 50 100

95 0 160 165 170 175 180 185 190 195 200 160 165 170 175 180 185 190 195 200 Time [s] Time [s]

Figure 4.23: Validation of Redesigned Observer on Full Order Engine Model

Three cases of Li are listed in Table 4.1 and their steady-state estimation errors are illustrated in Figure 4.20 to Figure 4.22. Comparing case 1 with case 2, the pole location is shrunk by the factor of 0.9, but kLik increase more than 50 times, yielding larger steady-

92 state estimation error in Figure 4.20. By tuning the closed-loop pole location, smaller kLik is achieved with reasonably fast error convergence rate. The redesigned observer is implemented on the engine model and the FTP cycle results are shown in Figure 4.23. In this case, the estimation of pc,int is much better than that in Figure 4.19.

4.6 Simulation Results and Discussion

4.6.1 Feedforward Control

In production engines, the desired actuator positions at diﬀerent operating points are calibrated through experiments. The results are stored in lookup tables as feedforward control. For the engine considered, because of low power demand in FTP cycle, the bypass valve is always closed. However, opening bypass will increase the energy of exhaust gas which is transported to the catalyst. In a feedforward control strategy, VGT and EGR positions are obtained by lookup tables, while bypass opening is set to be constant regardless of operating condition. The results of feedforward control with diﬀerent constant bypass openings are compared in Figure 4.24 and Table 4.2.

The catalyst warm-up performance index is deﬁned as ratio between baseline ﬁnal catalyst temperature and actual one

Tcat,b(t = 200s) IT = (4.47) Tcat(t = 200s)

Table 4.2: Performance Indices of Feedforward Control

xbp[%] IT Ip Ie 0 1 0.0258 0.0296 2 0.9817 0.0604 0.0375 4 0.9673 0.0993 0.0564 6 0.9577 0.1287 0.0753

93 520

500

480

[K] 460 cat T 440

420

400 0 20 40 60 80 100 120 140 160 180 200 Time [s]

300 baseline x = 2% 250 bp x = 4% bp 200

[kPa] x = 6% bp IM p ref 150

100

0 20 40 60 80 100 120 140 160 180 200 Time [s]

[%] 45 egr z 40

30 0 20 40 60 80 100 120 140 160 180 200 Time [s]

Figure 4.24: Comparison of Feedforward Control with Diﬀerent Bypass Opening

The boost performance index is deﬁned as the root mean square (RMS) error of desired boost pressure and actual one

v u n 2 u 1 X pIM (k) Ip = t 1 − (4.48) n pIM,d(k) k=1

The EGR performance index in deﬁned as the RMS error of desired EGR ratio and actual one v u n 2 u 1 X zegr(k) Ie = t 1 − (4.49) n zegr,d(k) k=1

It can be seen from the ﬁgure and table, in a small range, large bypass opening leads

to higher catalyst temperature but less boost pressure and worse EGR ratio. Feedforward

control is incapable to control the three actuators coordinately, which makes MPC an

appealing solution.

94 4.6.2 MPC and Pareto Analysis

The controller developed in Section 4.4 and state observer developed in Section 4.5 are implemented on the engine model. First, set the weight on catalyst temperature wT to be 0 to study the case when rapid catalyst warm-up is not considered, as depicted in

Figure 4.25. It can be seen that boost pressure and EGR ratio are close to desired value, but no improvement on warm-up performance.

Then, the weights in the cost function (4.33) are tuned such that the EGR ratio is close to reference one, with a trade-oﬀ between warm-up and boost, as shown in Figure 4.26.

300 60 actual ref 55 250 50

200 [%]

[kPa] 45 egr IM z p 40 150

100 30 0 20 40 60 80 100 120 140 160 180 200 0 20 40 60 80 100 120 140 160 180 200 Time [s] Time [s]

Figure 4.25: MPC for Rapid Warm-up on FTP Cycle (wT = 0, wp = 50, we = 50)

300 actual 500 actual ref baseline 250 450

200 [K] [kPa]

cat 400 IM T p

150 350

100 300 0 20 40 60 80 100 120 140 160 180 200 0 20 40 60 80 100 120 140 160 180 200 Time [s] Time [s]

60 80

55 60 50

[%] 45 40 egr z x vgt 40 Actuator [%] x 20 egr 35 actual x bp ref 30 0 0 20 40 60 80 100 120 140 160 180 200 0 20 40 60 80 100 120 140 160 180 200 Time [s] Time [s]

Figure 4.26: MPC for Rapid Warm-up on FTP Cycle (wT = 2000, wp = 10, we = 50)

95 80 80

70 70

60 60

50 50

40 FF 40 [%] [%] vgt

MPC egr x

30 x 30

20 20

10 10

0 0

−10 −10 0 20 40 60 80 100 120 140 160 180 200 0 20 40 60 80 100 120 140 160 180 200 Time [s] Time [s]

Figure 4.27: Comparison of Feedforward and Feedback Portions of Actuators Positions

400 150

350 140

300 130 [kPa]

[kPa] 250 120 t,int EM p p 200 110 150 Model Observer 100 100

160 165 170 175 180 185 190 195 200 160 165 170 175 180 185 190 195 200 Time [s] Time [s]

130 200 HP 125 LP 150 120

115

[kPa] 100 [krpm] tc

c,int 110 p N 105 50 100

95 0 160 165 170 175 180 185 190 195 200 160 165 170 175 180 185 190 195 200 Time [s] Time [s]

Figure 4.28: State Estimation of Closed-loop System

Since the control inputs in MPC are deﬁned as the deviations from the nominal operating condition, this controller consists of an integrated feedforward and feedback control.

The scheduling of nominal control inputs according to operating region is the feedforward portion, and the deviation obtained from MPC is the feedback portion, which are shown in Figure 4.27. It is found that, in Figure 4.9, region 1, 2, 3 stands for deceleration, while region 4 represents idling. Since boost pressure is close to 1 bar and EGR ratio should be kept, the 3 actuators have very limited control authority in these regions. So MPC is not used in those regions, and the feedback portion is set to be 0.

The reduced order PWA observer is applied to estimate the unmeasured states, and

96 Table 4.3: Performance Indices of MPC

wT wp we IT Ip Ie 2000 50 50 0.9932 0.0355 0.0288 2000 20 50 0.9824 0.0405 0.0310 2000 10 50 0.9696 0.0670 0.0344 2000 5 50 0.9568 0.1132 0.0399

520 baseline 500

480

[K] 460 cat T 440

420

400 0 20 40 60 80 100 120 140 160 180 200 Time [s]

300 w = 5 p 250 w = 20 p ref 200 [kPa] IM p 150

100

0 20 40 60 80 100 120 140 160 180 200 Time [s]

[%] 45 egr z 40

30 0 20 40 60 80 100 120 140 160 180 200 Time [s]

Figure 4.29: Comparison of MPC with Diﬀerent Weights wT = 2000, we = 50 the results are illustrated in Figure 4.28. It can be seen that the state estimator has high accuracy in the closed-loop system.

To analyze the trade-oﬀ between warm-up and boost, the MPC with diﬀerent cost functions are compared in Figure 4.29 and Table 4.3. As the weight for boost wp decreases, catalyst warm-up is improved at the expense of boost pressure, while the EGR ratio, as a soft constraint, is roughly maintained at the desired level.

A comparison of feedforward control and MPC is depicted in Figure 4.30. Clearly,

97 1 MPC 0.995 I = 0.029 feedforward e 0.031 0.99

0.985 0.031 0.037 0.98 T

I 0.975 0.047 0.97 0.034 0.056 0.965 0.066 0.96 0.04 0.075 0.955

0.95 0 0.05 0.1 0.15 I p

Figure 4.30: Comparison of Feedforward Control and MPC

MPC can achieve the same improvement of catalyst warm-up performance as feedforward control, with lower boost pressure and EGR ratio tracking errors. Therefore, a Pareto front is obtained.

4.6.3 Heuristic Control

Since this controller is dedicated for rapid catalyst warm-up during cold start and require large amount memory and computational effort, it is not cost-efficient to implement in production engines. In order to not violate the production engine control for the air-path system, a rule-based heuristic control is developed to only control the bypass opening, which is an “add-on” controller appended to the existing air-path control. From the Pareto front, the case with wT = 2000, wp = 5, we = 50 is chosen to be the candidate. From the MPC results, it is found that the bypass opening can be approximated by a linear function of fuel flow rate in each region, as illustrated in Figure 4.31. In region 5, where desired boost pressure is low, the bypass are almost fully open, because the other two actuators can largely compensate the boost pressure. In other regions, bypass is partially open. When fuel injection increases, bypass opening should decrease to meet higher desired boost pressure.

98 Region 5 Region 9 Region 11

10 10 10 8 8 8 6 6 6 [%] [%] [%] bp bp bp

x 4 x 4 x 4 2 2 2 0 0 0

0.8 1 1.2 1.4 1.6 1.5 2 2.5 2.4 2.6 2.8 3 3.2 m [kg/s] −3 m [kg/s] −3 m [kg/s] −3 fuel x 10 fuel x 10 fuel x 10 Region 15 Region 16 Region 19

10 10 10 MPC fit 8 8 8 6 6 6 [%] [%] [%] bp bp bp

x 4 x 4 x 4 2 2 2 0 0 0

3.8 4 4.2 4.4 4.6 4.8 5 2.8 3 3.2 3.4 3.6 3.8 4 4.2 4.5 5 5.5 6 6.5 m [kg/s] −3 m [kg/s] −3 m [kg/s] −3 fuel x 10 fuel x 10 fuel x 10

Figure 4.31: Identiﬁcation of Heuristic Control of Bypass Valve Opening

520 baseline 500

480

[K] 460 cat T 440

420

400 0 20 40 60 80 100 120 140 160 180 200 Time [s]

300 MPC 250 heuristic ref 200 [kPa] IM p 150

100

0 20 40 60 80 100 120 140 160 180 200 Time [s]

[%] 45 egr z 40

30 0 20 40 60 80 100 120 140 160 180 200 Time [s]

Figure 4.32: Heuristic Control for Rapid Warm-up on FTP Cycle

The comparison of heuristic control and MPC is shown in Figure 4.32, Figure 4.33, and

Table 4.4. Although it has some deviation from the MPC, the heuristic control can largely improve warm-up performance by penalizing boost, while keeping desired EGR ratio.

99 Table 4.4: Comparison of Heuristic Control and MPC

Warm-up Time [s] Final Temp. [ ◦C] Boost RMS [%] EGR RMS [%] Baseline 192.2 212.0 2.6 3.0 MPC 169.5 233.9 11.3 4.0 Heuristic 170.6 229.2 9.6 3.4

1 MPC 0.995 I = 0.029 feedforward e 0.031 heuristic 0.99

0.985 0.031 0.037 0.98 T

I 0.975 0.047 0.97 0.034 0.056 0.965 0.033 0.066 0.96 0.04 0.075 0.955

0.95 0 0.05 0.1 0.15 I p

Figure 4.33: Comparison of Heuristic Control and MPC

100 Chapter 5

Conclusions and Future Work

5.1 Conclusions

HIS thesis presents two main contributions to the research and development of T systematic optimization and control design for downsized boosted engines with advanced charging systems. First, a parametric modeling approach for a family of geometrically similar turbochargers is proposed. Second, a model predictive control approach for a two-stage turbocharged Diesel engine air-path system is designed. The main conclusions of this study are summarized below.

5.1.1 Parametric Model of Turbocharger

Several modeling approaches has been evaluated to characterize the flow and efficiency of families of compressors and turbines in similar, with the objective of defining a fully scalable and parametric model that can represent the performance of a family of turbochargers based on key geometric parameters. From this study, the main conclusions are:

Upon conducting a comparative analysis of several modeling approaches available in

literature, the Jensen-Kristensen model for compressor ﬂow and the Eriksson model for

compressor eﬃciency are selected to develop a parametric compressor model. Speciﬁ-

cally, the compressor performance maps (ﬂow and eﬃciency) are shown to be scalable

according to only one geometric parameter, namely the compressor diameter.

101 Implementation issues related to causality of the compressor performance map is also addressed. Since the compressor is ultimately modeled as a map, the causality can be invertible, assuming either the pressure ratio or the mass ﬂow rate can be used as the input of the sub-model. Using the mass ﬂow rate as the input to the compressor model (hence, calculating the pressure ratio as output) leads to a more stable model for transient simulation. However, coupling such model with the engine air-path system simulator requires the introduction of an additional state variable. Due to the better accuracy in predicting the compressor performance, this approach is ultimately selected.

Simpliﬁed surge and choke model is developed for a family of compressors, which represents the limit of compressor operation. The developed model will be used to integrate with compressor ﬂow model to provide important information for system design and control optimization.

A fully parametric model for the turbine flow is also developed, starting from modified orifice flow equation. Based on the geometric similarity principle, a parametric flow model is obtained to characterize a family of fixed geometry turbines. Then, a scalable variable geometry turbine model is developed by representing the set of turbine flow parameters with respect to the VGT opening. Three different variable geometry turbines are considered, and their averaged behavior is assumed for defining a scalable model representing the effects of VGT actuation on turbine flow maps.

A modiﬁed power-based turbine eﬃciency model is obtained, which is decoupled with

flow model and independent on speed, and thus can be used in the case the flow model is scaled with geometry. Since the set of turbine efficiency parameters is difficult to

ﬁt analytically as function of VGT opening, eﬃciency at arbitrary VGT opening is computed by interpolation.

102 To provide a preliminary veriﬁcation, the parametric modeling approach is applied to a

two-stage turbocharger and integrated with a Diesel engine air-path system simulation

model, which is validated against steady-state and transient experimental data. The

original two-stage turbocharger model, based on look-up tables, is used as benchmark

for comparing the new parametric model. Good agreement on the performance of

engine and compressors, and improved stability of the simulator demonstrate the

advantages of adopting the new modeling approach.

5.1.2 Model Predictive Control of Diesel Engine Air-Path

Starting from the air-path model of a Diesel engine with two-stage turbocharger, an air-path control strategy based on model predictive control is designed, which can achieve rapid catalyst warm-up to improve cold start emission performance. From this study, the main conclusions that have been drawn are:

Waste-gate thermal model is proposed, which take temperature recovery and heat

transfer into account. The ﬂuid temperature at waste-gate outlet is a weighted average

of inlet temperature and wall temperature. Assuming no chemical reaction during cold

start, a lumped catalyst thermal model is developed based on heat transfer model.

According to the preliminary analysis on the FTP cycle, opening the HP bypass will

improve the warm-up performance but deteriorate boost performance, while opening

the LP waste-gate has trivial eﬀect. To prevent large decrease of boost pressure, the

maximum bypass opening should be constrained.

Piecewise aﬃne system is proposed, which is a reasonable approximation of nonlinear

plant model and can be used for control and observer design. Starting from 11-

state model for engine air-path system, the dynamics of compressor mass ﬂow rates

and manifold temperatures can be reduced. Then, the reduced 7th order model is

103 linearized at 20 operating points. Finally, a discrete-time piecewise aﬃne system with a partition on the engine map is obtained.

By properly deﬁning the reference signals, the air-path control for rapid warm-up problem is formulated as MPC tracking control problem. Delta input formulation is introduced to address the feedthrough term from inputs to EGR ratio in the PWA system. The MPC design parameter is selected with the consideration of system performance and computational load. To address the discontinuities in control inputs, a moving average ﬁlter is proposed.

The reduced order observer based on PWA system is designed to estimate the unmeasured states by production MAF and MAP sensors. Because catalyst is connected to the rest of the system in a cascade structure, catalyst temperature sensor is required.

To minimize the estimation error caused by modeling error, both the error convergence rate and the norm of output injection matrix need to be considered in observer design.

The developed model predictive controller and observer are implemented on the engine model. Compared to feedforward control strategy with constant bypass opening, MPC can achieve the same improvement of catalyst warm-up performance while keeping higher boost pressure and less deviation from desired EGR ratio. The eﬀects of diﬀerent weights in cost function are evaluated, and a Pareto front is obtained.

After selecting the candidate MPC on the Pareto front, a rule-based heuristic control is developed. The bypass opening is controlled as a linear function of fuel rate in each region. The heuristic control is a good approximation of MPC and can be implemented as an “add-on” controller appended to the existing air-path control in production engines.

104 5.2 Future Work

Leveraging on the knowledge and results obtained in this thesis, the future work of system optimization and control design for Diesel engine with two-stage turbocharger (2ST) will focus on:

Co-optimization of system design and control for 2ST conﬁgurations will be investi-

gated to improve the system performance in steady-state and transient conditions. To

enable a coupled and systematic model-based optimization, parametric turbocharger

model will be used, which allows to systematically evaluate the inﬂuence of design

parameters on system performance.

Following the previous work on air-path control, a control strategy will be developed

for Diesel engine with 2ST system with focus on improving robustness and smoothness

of response during mode switching. This will be achieved by developing a reference

governor to coordinate the control of the bypass valve and VGT to minimize the

pressure excursion in the intake manifold.

In the next phase, the model-based synthesis work will be extended and generalized to investigate several advanced boosting technologies, such as electrically assisted turbochargers (EAT), electric superchargers (eBooster), or turbo/supercharging systems. The next phase of this research will focus on:

Set up the engine in the test cell, and extend the engine air-path model in presence

of EAT and eBooster systems.

System optimization and control design approach will be extended to EAT and eBooster

systems conﬁgurations, with the consideration of beneﬁts, complexity vs. performance

trade-oﬀ study.

105 Energy-optimal control strategy will be developed for selected electric boosting sys-

tem while minimizing turbo-lag, preventing compressor surge, and meeting emissions

constraints.

This brings us to the conclusion of this thesis, it is hoped that this work will serve as the ground work for future research in this fascinating area.

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119 Appendix A

Nomenclature

A.1 Abbreviations

2ST two-stage turbocharger BMI bilinear matrix inequality BSFC brake specific fuel consumption BSR blade speed ratio DOC Diesel oxidation catalyst DPF Diesel particulate filter EAT electrically assisted turbocharger EGR exhaust gas recirculation FGT fixed geometry turbine FTP federal test procedure HCCI homogeneous charge compression ignition HP high pressure HPC high pressure compressor HPT high pressure turbine LNT lean NOx traps LP low pressure LPC low pressure compressor LPT low pressure turbine LTC low temperature combustion LTI linear time invariant MAF mass air flow MAP manifold absolute pressure MPC model predictive control MVM mean-value model PWA piecewise affine QP quadratic programming SCR selective catalytic reduction SI spark ignition

120 VGC variable geometry compressor VGT variable geometry turbine VVA variable valve actuation VVT variable valve timing

A.2 Subscripts

amb ambient b baseline bp bypass c compressor cat catalyst cc charge cooler corr corrected d displacement/desired e EGR ratio EM exhaust manifold egr exhaust gas recirculation eng engine ex upstream of catalyst exh upstream of exhaust manifold HP high pressure HT high pressure turbocharger IM intake manifold in inlet int intermediate volume is isentropic LP low pressure LT low pressure turbocharger out outlet p boost pressure red reduced ref reference t turbine/throat T catalyst temperature tc turbocharger thr threshold vgt variable geometry turbine w wall wg waste-gate

121 A.3 Variables and Constants

Model

A m2 area AF R - air fuel ratio BSR - blade speed ratio c m/s speed of gas cp J/kg/K specific heat at constant pressure D m diameter h J/kg specific enthalpy h W/m2/K heat transfer coefficient J kg·m2 rotational inertia k W/m/K thermal conductivity L m length m˙ kg/s mass flow rate M - blade Mach number N rpm speed p Pa pressure P W power P r - Prandtl number Q˙ W heat transfer rate R J/kg/K gas constant Re - Reynolds number t s time T K temperature T Nm torque U m/s blade tip speed V m3 volume x % actuator opening zegr % EGR ratio β - pressure ratio for compressor - pressure ratio for turbine γ - specific heat ratio η - efficiency ρ kg/m3 density λv - volumetric efficiency ν m2/s kinematic viscosity Φ - flow parameter Ψ - head parameter

122 Control

e estimation error N predictive horizon Nc control horizon P polytope r reference signals Ts sampling time u control inputs v external inputs x system states y system outputs

123