Michael Fleischer Traction Control for Railway Vehicles

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Michael Fleischer

Traction Control for Railway Vehicles

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To my parents, Eduard and Renate Fleischer, Christine, and our twins, Niklas and Tim

“Innovationen sind Gewohnheitsbrecher.” “Innovations are habit breakers.”

Dr. Heike Hanagarth Head of Rail Technology and Member of Management Board at Deutsche Bahn AG Keynote speech of the 42nd Conference of Modern Rolling Stock, Graz, Austria, 2014 xii Preface

The work presented henceforth was conducted at the Chair of Electrical Drives of the Friederich-Alexander university in Erlangen and at the Railway Technical Research Institute of Japanese Railways in Tokyo and additional privately financed studies were added. Further, this thesis is jointly approved by both Siemens AG and Japanese Railways. The cornerstone of this thesis was laid at the Chair of Electrical Drives sponsored by a research project with Siemens AG. Thanks to Dr.-Ing. An- dreas Jöckel, Wolfgang Fetter and Dr.-Ing. Werner Breuer, the first simulation model of the traction drive-train was quickly set up for slip-stick vibrations and was verified with measurements from conducted test-runs of an European high-performance locomotive. After various control schemes were tested with the traction drive-train model, my first international publication, which was completely financed by the DFG (German Research Foundation), was presented at the IEEE Advanced Motion Control workshop in Kawasaki, Japan. Shortly after my conference presentation, I was invited to the worldwide biggest Railway Technical Research Institute (RTRI) of Japanese Railways in Tokyo, Japan. As German supervisors, I would like to thank Prof. Bern- hard Piepenbreier on part of the university and both Dr.-Ing. Gerald Amler and Dr.-Ing. Matthias Hofstetter on part of Siemens AG. The main research topic at the RTRI in Tokyo was the development of a new control scheme for damping slip-stick vibrations in Japanese high- performance locomotives. Further, I joint the RTRI’s project team for the first fuel cell and lithium-ion battery train and was supporting the International Affairs Division for foreign guests of Japanese Railways. Besides the control scheme, a novel parameter estimation method was proposed for Japanese traction drive-trains during my extended stay. Thanks to the Japan Society for the Promotion of Science (JSPS) for partially sponsoring my stay at the RTRI. As Japanese supervisors, I would like to thank Prof. Keiichiro Kondo and Hiroshi Hata. Furthermore, several privately financed ideas and research topics such as for example, the virtual filters, were added when composing this thesis. xiv Preface

During my stay at the RTRI, I participated worldwide in three conferences. One of these published papers won the “Innovations for Europe”-award. I would like to thank both Prof. Rik W. De Doncker and Prof. Dirk Abel from RWTH Aachen university for finally supervising this thesis. A version of Chapter 4 and a part of Chapter 6 were filed for patent pending [114,118] and were published in [115–117] and in [119–121]. I would also like to thank Prof. Gerhard Pfaff, Priv.-Doz. Dr.-Ing. habil. Christoph Wurmthaler and Dr.-Ing. Peter Hippe for their constructive comments to the thesis. Finally, I would also like to thank my family and friends for always being helpful and supportive.

Tokyo/Japan and Erlangen/Germany July 19, 2019 Preface xv Abstract

So far the traction drive-train was not considered in the traction control software as all mechanical parameters were unknown. The latter are subject to significant wheelset wear and ageing of rubber elastic joints during the drive-train’s operational life. The known vibration behavior of the drive- train, namely the mode shapes, facilitates an ingenious and simple parameter identification scheme. As a result, an appropriate three-inertia virtual model is derived which is applied for parameter estimation, for system monitoring and for control. Based on this model, virtual sensors are introduced for all relevant signals of the drive-train. The signals are analyzed by the use of histograms to mainly determine both utilization of the wheel-rail contact and load cycle spectrum of the wheelset shaft. The wearless virtual sensors out- match their physical counterpart in means of reliability, robustness, cost and space requirements. Further for anti-vibration control, two state of the art control schemes, namely the standard and the passive readhesion controller, are discussed and improved regarding their anti-windup and prevention performance. Then a novel modal state control scheme is derived in the time domain with its feedforward controllers. Subsequently, an equivalent scheme is developed in the frequency domain. A simple starting procedure of this novel active anti-vibration controller from any standard controller is proposed to raise its acceptance and applicability in the traction application. Due to a low quality of the speed sensor signal, the active anti-vibration controller is designed to be capable of damping slip-stick vibrations up to a certain limit. Beyond this limit, a passive readhesion controller is additionally coupled and intervenes until the active anti-vibration controller can cope with the vibrations. Using the synergy of the active and the passive controllers, the stable operating range of the traction drive is significantly increased without any loss of traction force. To simplify the commission complexity and to further increase the damping performance, a virtual absorber feedback controller with minimized sensor noise amplification is introduced based on the standard speed controller. The absorber is virtually mounted on the indirect-driven wheel of any traction drive-train and for its starting is faded in from the standard control scheme. Its parameters are continuously adapted to wheelset wear and to rubber joint ageing circumventing all drawbacks known from the fixed mechanical absorber installation. On the basis of the virtual absorber, more universally valid virtual filters are applied to the traction drive-train to protect the structure from the vibrational intake as well as to increase the tractive effort utilization. The latter effect is only achieved by active virtual readhesion where on the other hand, the well-known passive approach comes along with a loss of traction force. Virtual protection is obtained by all virtual filters which are mutually exclusively applied to all three inertias of the virtual model. To implement such a multi-feedback controller scheme, a looping state machine is introduced to handle the activation of the several controllers according to the desired state. xvi Preface Zusammenfassung

Bisher wurde der Traktionsantriebsstrang in der Regelungssoftware nicht be- rücksichtigt, da alle mechanischen Parameter unbekannt waren. Letztere un- terliegen dem Radsatzverschleiß und der Gummialterung während der Lauf- zeit des Antriebsstranges. Das bekannte Schwingungsverhalten des Antriebs- stranges ermöglicht ein einfaches und smartes Identifikationsverfahren. Als Resultat wird ein reduziertes Drei-Massen-Model errechnet, welches für Pa- rameterschätzung, Systemüberwachung und Regelung angewendet wird. Ba- sierend auf diesem virtuellen Model werden virtuelle Sensoren für alle re- levanten Signale des Antriebsstranges vorgestellt. Die Signale werden mit Histogrammen analysiert um die Ausnutzung des Rad-Schiene-Kontaktes und das Lastspielspektrum der Radsatzwelle zu bestimmen. Der virtuelle übertrifft den physikalischen Sensor hinsichtlich Verlässlichkeit, Robustheit, Kosten und Einbauraum. Des Weiteren werden für die Schwingungsregelung zwei Standard-Verfahren, nämlich der Standard- und der passive Readhäsion- sregler, erörtert und bezüglich ihrer Anti-Windup und Präventionsleistung verbessert. Dann wird eine neue modale Zustandsregelung im Zeitbereich mit einer Vorsteuerung entworfen. Diese Regelung wird in den Frequenzbereich übertragen. Eine einfache Startprozedur des neuen aktiven Schwingungsre- glers von jeglichem Standardregler wird vorgeschlagen um die Akzeptanz und die Anwendbarkeit für die Traktionsanwendung zu steigern. Aufgrund einer geringen Qualität des Geschwindigkeitsignals ist der aktive Schwingungsre- gler in der Lage, Schwingungen bis zu einem bestimmten Grad zu dämpfen. Jenseits dieser Grenze wird zusätzlich ein passiver Readhäsionsregler gekop- pelt und dieser greift ein, bis der aktive Schwingungsregler wieder die Schwingungen beherrscht. Unter Verwendung der Synergie von aktiven und passiven Reglern wird der stabile Betriebsbereich des Traktionsantriebes ohne Verlust von Zugkraft deutlich gesteigert. Um die Komplexität der Inbetrieb- nahme zu vereinfachen und die Dämpfungsleistung weiter zu erhöhen, wird ein virtueller Tilgerregler mit minimierter Sensorrauschverstärkung auf Ba- sis des Standard-Reglers eingeführt. Der Tilger ist virtuell auf dem indirekt angetriebenen Rad eines beliebigen Antriebsstranges montiert und sein Start wird vom Standard-Regler aus eingleitet. Seine Parameter werden kontinuier- lich an den Radsatzverschleiß und an die Gummialterung angepasst, um alle Nachteile zu vermeiden, die von einem fest installierten Tilger bekannt sind. Auf Basis des virtuellen Tilgers werden universellere virtuelle Filter für den Traktionsantrieb eingesetzt, um den Antriebsstrang vor Schwingungen zu schützen und die Zugkraftausnutzung zu erhöhen. Der letztgenannte Effekt wird nur durch eine aktive virtuelle Readhäsion erreicht, wo andererseits der passive Regler mit einem Verlust der Zugkraft einhergeht. Um ein derartiges Mehr-Filter-Reglerschema zu implementieren, wird eine Zustandsmaschine eingeführt, um die Aktivierung der verschiedenen Regler entsprechend dem gewünschten Zustand einzuleiten. Contents

1 Introduction ...... 1 1.1 Motivation ...... 1 1.2 Stateoftheart ...... 2 1.3 Objective of the thesis ...... 4

2 Traction vehicles...... 5 2.1 Railvehicles ...... 5 2.1.1 System overview ...... 8 2.1.2 Propulsion system modelling...... 18

3 Slip-stick vibrations ...... 27 3.1 Origin of vibrations...... 27 3.1.1 Wheel-rail contact...... 27 3.1.2 Standard control scheme ...... 32 3.2 Locomotive test runs...... 37 3.2.1 Test run conditions ...... 37 3.2.2 Vibrations...... 38 3.2.3 Analysis...... 39

4 Reduced model identiﬁcation ...... 43 4.1 Conventional approach ...... 43 4.2 Mode shape approach...... 49 4.3 Modal approximation approach ...... 52 4.3.1 Approximated mode shape calculation ...... 52 4.3.2 Quality of mode shapes ...... 53 4.3.3 Modal algorithm ...... 55 4.3.4 Eﬀect of ageing and wear...... 56 4.3.5 Speed sensor location ...... 57 4.3.6 Modalscheme ...... 58 4.4 Parameter estimation ...... 60 4.4.1 Basicidea ...... 61 xviii Contents

4.4.2 Estimation criterion ...... 61 4.4.3 Estimation algorithm ...... 63 4.4.4 Estimation error ...... 65 4.4.5 Estimation scheme ...... 66 4.5 Natural identiﬁcation ...... 67

5 Modal estimator ...... 69 5.1 Full-order estimator ...... 69 5.2 Reduced-order estimator ...... 73 5.3 State estimation by transfer functions ...... 77 5.3.1 Indirect approach ...... 78 5.3.2 Direct approach...... 78 5.4 Comparison of estimators...... 81 5.4.1 Simulation studies...... 81 5.4.2 Overview...... 84

6 Anti-vibration control ...... 85 6.1 Passive readhesion controller...... 85 6.1.1 Stateoftheart ...... 85 6.1.2 Advanced concept ...... 86 6.2 Active anti-vibration control scheme ...... 88 6.2.1 Design in the time-domain...... 88 6.2.2 Design in the frequency-domain ...... 93 6.2.3 Simulation studies...... 96 6.3 Advanced anti-vibration control scheme ...... 98 6.3.1 Windup prevention strategies ...... 98 6.3.2 Starting of active anti-vibration controller ...... 99 6.3.3 Coupling of passive and active control schemes ...... 103 6.3.4 Unmodeled dynamics and stability ...... 104 6.4 Virtual absorber...... 104 6.4.1 Stateoftheart ...... 106 6.4.2 Generic controller design ...... 107 6.4.3 Control concept ...... 114 6.5 Virtual protection and virtual readhesion ...... 116 6.5.1 BasicIdea...... 116 6.5.2 Control concept ...... 119 6.6 Comparison of control schemes ...... 121 6.6.1 Simulation studies...... 121 6.6.2 Overview...... 122

7 Research in Japan ...... 127 Contents xix

8 Applications ...... 129 8.1 Natural identiﬁcation ...... 129 8.2 Virtualmodel...... 131 8.2.1 Parameter estimation ...... 131 8.2.2 Reduced generic model...... 132 8.3 Virtualsensors...... 132 8.4 Load cycle histograms...... 135 8.5 Anti-vibration control...... 136 8.5.1 Passive readhesion...... 136 8.5.2 Advanced modal state ...... 138 8.5.3 Virtual absorber ...... 140 8.5.4 Overview...... 142

9 Conclusion ...... 143

A Modelling of traction drive-trains ...... 149

B Curriculum vitae ...... 155 References ...... 157

Acronyms

All abbreviations and global symbols are listed. However, auxiliary variables are neglected due to their minor importance for the overall contribution of the thesis. Abbreviations 4Q Four quadrant AC Alternatingcurrent ARA Amsterdam, Rotterdam and Antwerpen-seaports DB GermanRail(Deutsche Bahn) DC Directcurrent DFG German Research Foundation DSP Digitalsignalprocessing ETG German Association for Energy Technologies FFT Fast Fourier Transformation GTO Gateturn-oﬀthyristor JR JapaneseRailways JRF Japanese Railways Freight company JSPS Japanese Society for Promotion of Science LC Inductiveandcapacitiveinputﬁlter LQR Linear-quadraticregulator MAC Modalassurancecriterion OBB¨ AustrianFederalRailways(Osterreichische¨ Bundesbahn) PI(D) Proportional integral (derivative) PM Protectionmodule,LC-circuit PWM Pulsewidthmodulator R&D Researchanddevelopment RTRI Railway Technical Research Institute SBB SwissFederalRailways(Schweizerische Bundesbahn) SRC Seriesresonantcircuit VDE German Association for Electrical, Electronic and Informa- tion Technologies xxii Acronyms

Unique numericals ❶ Asynchronous machine ❷,❸,❽ Gears ❹,❺ Cardan hollow shaft ❻,❼ Wheelset, direct- and indirect-driven wheel ❾,❿ Brake discs ① Asynchronous machine (Reduced model) ② Direct-driven wheel (Reduced model) ③ Indirect-driven wheel (Reduced model)

General symbols X Actual signal or parameter X˜ Estimated signal or parameter X∗ Reference signal X′ Scaled to nominal value X Matrix x Vector

Greek symbols ∆ˆ(s) Estimator polynomial ∆ˆ0, ∆ˆ1, ... Coefficients of the estimator polynomial ∆0(n, s) Mason cofactor value ∆ϕ Nonlinear backlash ∆(n, s) Mason determinant of the signal graph ΦR Rotor flux δ1 Additional stable pole for windup prevention δ2, δ3 Pole-zero distances δµ/δ(∆v) Adhesion force gradient κ Sliding variable for anti-vibration control µ Adhesion coefficient µ(∆v) Adhesioncharacteristics µmax Maximum available adhesion coefficient ϕ1x, ϕ2x, ϕ3x Mode shapes of the first three vibration modes ϕb Torsional rotation angle σ Damping coefficient indicating rise time of vibration σ2, σ3 Damping ratio σac Damping factor of the adhesion characteristic σcl Damping factor of the closed-loop σpi Damping factor of the standard speed controller σsys Damping factor of the traction drive-train τs Time constant of the lag element for the electric system ω2, ω3 Damped natural frequencies ωi, ωd Speed of the (in-)direct-driven wheel ωnoise Noise signal of the speed sensor/estimator ωr, ω1 Speed of the asynchronous machine Acronyms xxiii

ωs Characteristic frequency of the stopped wheelset ωsensor Speed signal from any sensor in the traction drive-train ωx Speed of the xth inertia

Latin symbols Am, Bz State space matrix B Gradient of the adhesion curve at its origin C Stiffness matrix Dˆ(s) Characteristic closed-loop polynomial Env(T ) Exponential envelope curve of torsional vibrations F2,F3(s),... Feedforward transfer functions Fd, Fi Transmitted traction force at the (in-)direct-driven wheel FH (s) Additional first order lag element for feedforward realization Fm(s) Transfer function of a mechanical three-inertia system Fp(s) Transfer function of the electrical and mechanical system Fres Resistive forces, e.g. air resistance, rolling friction etc. FT (s) Transfer function for the torque feedback Ft Traction force ′ FV , Fq, Fx(s) Auxiliary transfer functions for feedforward determination FΩ(s) Transfer function for the speed feedback Ga(s) Complex transfer function of the actuator dynamics ′ Ga(s) PT1-approximation of the actuator dynamics Gc(s) Transfer function of the speed controller Gna(s) Reducedmodelwithoutabsorber Gtd(s) Transfer function of the traction drive-train Gtd3(s) Transfer function of the reduced model Gwa(s) Reducedmodelwithabsorber Im(s) ImaginaryaxisoftheLaplaceplane J Inertia matrix Ja Inertia of the absorber Js Sum of all inertias within the decoupling plane Jt Total inertia of the traction drive-train Jx Inertia of the xth mass NT (s) Nominator polynomial of FT (s) Nt0, Nt1, ... Coefficients of the NT (s) polynomial Nx(s) Rosenbrockmatrix NΩ(s) Nominator polynomial of FΩ(s) Nω0, Nω1, ... Coefficients of the NΩ(s) polynomial P1, P2 Operating points in the adhesive force characteritics Re(s) RealaxisoftheLaplaceplane T Torque Te Electromagnetic torque within the asynchronous machine ∗ Tp Reference torque as input for passive readhesion controller ∗ Tr Reference torque of the speed controller output Ts Torque signal measured by strain gauges xxiv Acronyms

Txy Shaft torque between inertia x and y Tz Disturbance torques VDC DC link voltage VRST Three phase voltage system a Separation factor for optimal disturbance reaction bi State space input vector cm State space output vector ca Stiffness of the absorber cr Rubber joint stiffness cxy Stiffness between inertia x and y da Damping factor of the absorber dad Variable damping factor in the wheelset shaft dxy Damping value of the stiffness cxy g Gravity constant ¯h Mason absorber loop gains iS Stator current vector iSmod Stator currents vector for sensorless control iSd, iSq Stator current field and torque generating component f1, f2 Braking forces ℓ Mason loop gains l Feedback gain for estimator ml Mass of the high-performance locomotive mt Mass of the complete train n Number of inertias of the reduced model na Number of axles nx Zero of Gtd(s) p0 Mason gain of the forward path px Pole of Gtd(s) t Time in seconds r Control feedback gain vector ra Auxiliary vector for estimator/controller calculation r Inflexion point of the adhesion curve rw Radius of the wheel disc s Laplace variable td Dead time of the actuator dynamics ui State space input u1, u2 Gear transmission ratios uz State space disturbance vector vd, vi Longitudinal difference speed of (in-)direct-driven wheel vl Speed of the high-performance locomotive x State space vector xI Output signal of the speed controller ym State space system output z1, z2 Unknown adhesion forces/torques Chapter 1 Introduction

The development of modern electric locomotives aims at higher traction performance as well as cross national passenger and freight traﬃc. However, the increase in traction performance entails not only higher wheel-rail wear, but also severe vibrations in the traction drive-train.

1.1 Motivation

To give an overview of both modern European and Japanese multi-voltage locomotives according their performance, the continuous ratings under the appropriate voltage systems are given in Tab. 1.1. At lower overhead voltages the current limits come into play reducing the continuous rating of the locomotive. Compared to the first electric locomotive introduced in 1879 with 2.2 kW at DC 150 V, the propulsion power of state of the art high-performance locomotives reaches with over 6 MW the limits of traction force transmission. Therefore in Japan high-performance locomotives with an increased number of powered axles are favored, whereas in Europe the focus is on a high utilization of the tractive effort level. Bad weather conditions like rain, fog or snowfall, however, reduce the available adhesion and at the same time necessitate a reduction in traction drive torque. Consequently, heavy freight trains struggle with both starting and accelerating. Here, the wheelset often operates at a wheel to rail slip higher than the maximum achievable tractive effort value due to fluctuating low adhesion coefficient. In this operating range, the gradual change between stiction and sliding friction at both wheel discs entails self-excited slip-stick vibrations in the torsionally flexible traction drive-train. Besides increased wheelset and rail wear, the alternating forces on the wheels cause high loads on the mechanical drive- train, in particular on the wheelset shaft. Due to considerably high vibration amplitudes, fatigue limit problems can arise in the worst case leading to a rupture of the wheelset shaft. 2 1 Introduction

Tab. 1.1 Continuous ratings of multi-voltage locomotives (in kW).

Siemens Bombardier Alstom Mitsubishi Toshiba ES64F4 TRAXXMS Prima6000 EF510 EH500 EH200

25 kV 6400 4200 6000 20 kV 3390 4000 15 kV 6400 4200 5600 3000 V 6000 4200 6000 1500 V 4200 4000 5000 3390 4000 4520

So the consequences of slip-stick vibrations for freight carriers are increased maintenance costs due to higher wheelset-rail wear and in the long run, an un- acceptable risk of locomotive failure due to rupture of wheelset shaft. To avoid the latter, both sophisticated adhesion and anti-vibration control schemes are necessary.

1.2 State of the art

To achieve a high utilization of tractive effort level, several well-known adhesion control schemes are already in use on the latest European high- performance locomotives. They are applied to operate the wheelset at an optimal tractive effort level without exceeding its achievable maximum. There- fore, a highly dynamic response of the control scheme is essential to adapt to the permanently fluctuating adhesion conditions. Besides optimal tractive effort utilization, the use of an adhesion control scheme also suppresses slip and slide operations of the wheelset to a certain extent. In sequel, three major approaches are discussed and compared according to their applicability. The first developed adhesion control scheme [2,3] is based on a search algorithm. A search strategy varies the reference speed of the asynchronous machine in order to find the maximum motor torque which corresponds to the maximum transmittable traction force. For daily operation, a value close to the maximum adhesion level at a lower wheel to rail slip is chosen to allow some fluctuation of the adhesion conditions. By increasing the control dynamics and improving the search strategy, this safety margin can be further reduced leading to a higher utilization of the tractive effort level [4]. The second control approach [5,6] utilizes the effects of the adhesion phenomena on the propulsion system. Therefore a sinusoidal test signal with a fixed low frequency is injected into the drive system to observe its phase behavior. A standard correlation scheme continuously identifies the phase shift of this test signal by the use of the cross correlation function. Subsequently, the appropriate gradient of the adhesion force characteristic is determined as a function of both the phase shift and the observer frequency. With the know- 1.2 State of the art 3 ledge of the actual gradient, a passing over the maximum achievable tractive effort value is detected. Thus an adhesion control scheme is introduced based on the analysis of the phase shift at a fixed observer frequency. Finally, as a supplement to the above mentioned adhesion control schemes for further improvement of the adhesion utilization, a disturbance observer for the actually transmitted traction force is proposed in [7]. This reduced state-observer is based on a highly simplified linear state-space model. As input signals it requires both known motor reference torque and the actual motor speed. For stand-alone application, the disturbance observer can also be integrated into an appropriate adhesion control scheme [8, 9]. In comparison to the adhesion control concept with searching strategy, the correlation scheme features a direct determination of the adhesion conditions. However, this advantage comes along with test signal injection leading to increased wheelset wear. Both control schemes as well as the additional use of the disturbance observer cannot absolutely assure stable operation at wheel to rail slips smaller than the maximum achievable tractive effort value due to fluctuating adhesion conditions. Consequently, slip-stick vibrations inevitably arise in the traction drive-train where two different vibration frequencies occur. To reduce the high vibration load on the wheelset shaft, the second vibration frequency has to be sufficiently damped. Therefore various investigations were already conducted with minor success. In the following, those state of the art anti-vibration concepts are discussed. As reference scheme, the installed standard (slip) speed controller is assumed, capable of only damping the first vibration mode. However, to cope with arising slip-stick vibrations in the traction drive-train, a passive readhesion controller [10,11] is tuned at the given frequency of the second vibration mode. On its occurrence in the speed signal of the asynchronous machine, the motor torque is reduced at the expense of traction force. Subsequently, the wheel to rail slip decreases and the wheelset operates in the stable range of the adhesive force characteristic, where the standard speed controller achieves the damping of the drive. The installed control solution, namely both standard and readhesion controller, has two major drawbacks. Besides the delayed reaction of the readhesion controller to the already arisen slip-stick vibrations, its intervention entails a loss of traction force. The latter makes it even more difficult to start and to accelerate heavy freight trains under bad weather conditions. To avoid the loss of traction force, a complex mechanical auxiliary system consisting of two vibration absorbers [12] is mounted on the wheelset with the intention of preventing slip-stick vibrations. Besides additional costs, their achievable damping effect for both oscillation modes strongly depends on both temperature and ageing due to rubber joints used. Thus under adverse conditions, the two vibration absorbers fail to perform adequately [12] and inevitably, the harmful slip-stick vibrations still arise. Consequently, further investigations are carried out into active anti-vibration control concepts. First in [13], the emphasis is on a simple control solu- 4 1 Introduction tion. An expansion of the installed slip controller by derivative action to a PID controller is introduced. However, the proposed controller just achieves equivalent damping compared to the already installed PI slip controller. Therefore in [14], the controller order is further increased with the introduction of a state control concept in the time domain. A LQR-designed state controller is proposed using a Kalman filter as a state observer. Both controller and observer are derived on the basis of a reduced-order model of the traction drive-train. Due to poor model quality, only the first vibration frequency is damped to a certain extent, as shown in [15]. More accurate modelling is requested and therefore an evolutionary identification algorithm is proposed for traction drive-trains in [16,17]. This complex algorithm necessitates extensive calculations due to its low speed of convergence and thus involves an enormous computation effort which is not applicable to recent DSP-platforms in traction applications. Nevertheless, it determines an highly accurate reduced two-inertia model. On that basis, a state controller is finally introduced in [16] only for the first vibration frequency. The bottom line of the discussed state of the art schemes is, sufficient damping for the second oscillation mode of the traction drive-train is not successfully achieved and therefore slip-stick vibrations inevitably arise stressing the wheelset to a great extent [59].

1.3 Objective of the thesis

First an appropriate simulation model of a traction drive-train including wheel-rail contact and highly dynamic torque control is provided. Then for validation purposes the origin of slip-stick vibrations is investigated based on test runs with a high-performance locomotive. For control and estimator purposes, a reduced model for the drive-train is identified. Based on that model, several estimator applications are proposed reconstructing various mechanical system variables. Subsequently, three novel anti-vibration control schemes are proposed: passive readhesion-, active modal state- and active virtual vibration-controller. A brief overview about the conducted research at the world’s biggest railway research institute is given in analogy to Japanese high-performance locomotives. Then all newly introduced technologies in this thesis are either tested on a high-performance locomotive or verified by the data acquired in the test runs. And finally, the last Chapter gives a comprehensive conclusion about the integration of the traction drive-train into a novel modular traction control scheme adding a parameter estimation scheme for predictive maintenance, virtual sensors for system load monitoring and several novel anti-vibration control schemes. All applied modules are chosen for a specific traction drive-train based on the optimization of both their performance and synergy effects. Chapter 2 Traction vehicles

As an introduction of traction vehicles, this chapter gives a comprehensive mechatronical system overview of electric rail vehicles. The main focus is put on traction systems of high-performance locomotives which are modelled in great detail.

2.1 Rail vehicles

First of all the utmost discipline of traction is introduced within rail-bound vehicles: modern electric high-performance locomotives. Here in cooperation with Siemens, the well known Eurosprinter family, as shown in Fig. 2.1(a), is considered for this research project. It covers various types of locomotives whereof nowadays nearly one thousands units are successfully operating worldwide in 11 different countries. The track record of this fleet began in 1991 with the development of a prototype locomotive - the Eurosprinter - built at Siemens own expense to extensively test new components. In order to prove its performance capabilities, several test runs were successfully taken under extreme conditions in Norway, Sweden and Switzerland. Finally to round off the prototype’s introduction to the traction world, the Eurosprinter set a new world speed record of 357 kph for three-phase locomotives in 2006. With this impressive demonstration several orders were placed and the Eu- rosprinter got on the track of success. Through the years, the Eurosprinter concept has been continuously enhanced integrating new technology. As a result, new types of locomotives emerged adding to the fleet and the Eu- rosprinter family was born. As last member the class 1216 locomotive for Austrian Federal Railways (OBB)¨ joined the Eurosprinter family. This innovative high-performance locomotive is compatible with the most European voltage systems. Therefore it facilitates cross-border rail travel in 17 countries across Europe without time consuming locomotive changes at the borders. Thanks to the fleet’s large variety of high-performance locomotives all cus- 6 2 Traction vehicles

Tab. 2.1 General data of electric rail vehicles. Suppliers Siemens Toshiba/Mitsubishi Voltage systems AC 25 kV/50 Hz AC 20 kV/50 Hz AC 15 kV/16.7 Hz DC 1.5kV DC 3 kV DC 1.5kV Continuous rating 5000 – 7000 kW 3390 – 6000 kW Starting tractive effort 280 – 420 kN 330 – 460 kN Maximum speed 140 – 230 kph 110 – 120 kph Axle load 20 – 22.5 t 16.8 t Powered axles 4 – 6 6 – 8 Gauge 1435 mm 1067 mm tomer needs could be satisfied for both heavy-duty freight traffic and rapid passenger transport. This wide spectrum is also reflected in the general data of the Eurosprinter family as shown in Tab.2.1. Besides the four standard voltage systems, it covers a wide range of both tractive starting effort for freight service and maximum speed for fast passenger traffic. Consequently, the latest types of the Eurosprinter family are universally applicable for international traffic within the growing European Union. However, Japanese high-performance locomotives are specially designed for the national freight market. Here, large DC electrified networks set the standard for high-performance locomotives for a long time. To improve freight services, the JRF family nowadays also exhibits universal dual-voltage locomotives. Since the Shinkansen fleet exclusively offers high-speed passenger service up to 320 kph, japanese locomotives are only used for heavy-duty freight traffic with a maximum speed of 110 kph. Here as latest member, the series M250, nicknamed Super Rail Cargo, joined the JRF fleet. This new container express train is put into daily service on the main artery line between Tokyo and Osaka. It was developed to reduce the air pollution originating from heavy truck traffic on the freight main line. Besides those environmen- tal considerations, the container express train features a higher service speed of 130 kph than a usual Japanese high-performance locomotive to meet the schedule for overnight delivery. In Japan, track specifications generally limit the maximum axle load for rail vehicles to 17 tonnes. These restrictions make it even more difficult to achieve sufficient traction force at steep track gradients within mountainous areas covering about 75 % of Japan. Therefore an increased number of powered axles is favored for Japanese locomotives, as shown in Fig.2.1(b). For this framework, the use of more powered axles marks a substantial difference to European high-performance locomotives highly utilizing the tractive effort level. 2.1 Rail vehicles 7 (b) JRF family (a) Eurosprinter family Various electric high-performance locomotives. Fig. 2.1 8 2 Traction vehicles 2.1.1 System overview

In the viewpoint of mechatronics, a complete system overview is given for the most powerful European high-performance locomotive, namely the class 1016 and 1116 in service for Austrian Federal Railways. To deliver an insight into the assembly of the main components, Fig. 2.2 shows typical cross sections in top and side views. At the first glance, the symmetry of the component assembly is obvious, e.g. both in driver’s cabs and in pantographs. Depending on the direction of travel, there is only one driver’s cab occupied and the according pantograph, located above this cab, is utilized. Thus, the third unsymmetrical pantograph in-between serves for the locomotive’s use with different structured overhead lines and voltage systems. Referring to the symmetry in the mechanics, the bogies are arranged symmetrical to the main transformer, where each bogie is related to a traction converter, a oil and water cooler, two motor vans and an auxiliary power supply inverter. Due to this symmetry aspects, the traction system is simply described on the basis of one driven bogie. Therefore Fig. 2.3 depicts the main circuit diagram of a bogie which is introduced as follows by the means of the energy flow, starting from the electrical input, the overhead line, towards the mechanical output of the locomotive, the wheel-rail contact. The pantograph, mounted on the roof of the locomotive’s body, collects the electrical energy from the overhead line, feeding it via the main circuit breaker, a grounding switch and a voltage surge protector finally to the main transformer, underfloor- mounted in the center of the railcar body. This single-phase oil transformer, as depicted in Fig. 2.6, consists of a primary high-voltage winding and several secondary windings, where each traction converter is supplied by three of those. The remaining secondary windings supply the auxiliary inverters, the on-board power supply system and the train supply bus, which is utilized e.g. for heating purposes in trailed passenger cars. The traction converter of the class 1016/1116 is shown in Fig. 2.4. Three input terminals are located on its center bottom, directly above the secondary terminals of the transformer. Considering cross-border rail travel, a contactor within the traction converter compensates the line voltage change by appropriate switching of the secondary windings. Therefore the same secondary voltage level maintains for the input converters independent from the primary voltage. The input converter is designed as a four-quadrant chopper in a modular assembly, consisting of two phase modules. Each phase module is based on the fundamental components of power electronics, two high-performance GTO-modules and two anti-parallel free-wheeling diodes. The task of the four-quadrant chop- pers is to provide a rectified DC voltage. Due to its high voltage ripple, the resulting output voltage is filtered with a capacitor to reduce the harmonic content of the inverter-generated DC link voltage. In addition, the capacitor compensates power fluctuations of the DC link because of varying fed- and drawn-power. 2.1 Rail vehicles 9 cab Driver’s motor Traction van Compressed air reservoir LZB continuous automatic train control van motor ment rack Air equip- Traction ) [32]. Oil and side view supply inverter Main transformer Auxiliary power water cooler and top Oil and water cooler power supply inverter Auxiliary Traction converter Door rack equipment Auxiliary van motor Bogie Traction Pantograph van rack motor Traction

Electronic equipment ideview si Cross section of a European high-performance locomotive ( Fig. 2.2 10 2 Traction vehicles

Those both effects are boosted by the coupling of the two parallel DC links for further voltage stabilization and by an additional series resonant circuit (SRC) for suppression of the second harmonic line frequency originated from the four-quadrant chopper. Finally, the four-quadrant chopper in parallel also improves the harmonic content of the DC link voltage by offset pulsing control, increases the redundancy of the electrical system and further, is essential to provide nominal power to the DC link. To protect the capacitor against overvoltage a protection module (PM) adds finally to the DC link. As last component of the traction converter, the pulse-width modulated (PWM) inverter converts the DC link voltage to a three-phase voltage system with variable amplitude and frequency achieved by the drive control to supply a traction motor. Unlike the four-quadrant chopper, the PWM inverter consists of three phase modules conditional on its functionality. Therefore in total twelve phase modules are utilized for a complete traction converter. In case of any faults within a phase module, e.g. a short-circuit in the upper PWM

AC overhead line Pantograph Ft Adhesion Torque Grounding switch control Control

Circuit Drive control breaker DC link 1∼ = PM M = 3 ∼ 4Q chopper PMW inverter 1∼ Traction drive = SRC

4Q chopper PM PMW inverter 1∼ = M

Main transformer = 3 ∼ 4Q chopper DC link Bogie On-board power supply

Auxiliary inverters Drive Train supply bus Control

Fig. 2.3 Main circuit diagram of a European high-performance locomotive [32]. 2.1 Rail vehicles 11

Drive control unit Contactor

Water Phase cooling module circuit

Input terminal

Fig. 2.4 Traction converter [32]. inverter, the DC link coupling is disengaged by high-speed contactors holding up converter-operation at reduced performance. In the framework of this thesis, the mechanical under-carriage of high-performance locomotives, the bogie with its two traction drive-trains, is in the spotlight, as shown in Fig. 2.5. The vehicle body in general is suspended with spiral springs to the bogie to counteract vertical vibrations originated from the wheel-rail contact. Longitudinal forces as the traction force, in contrast, are transferred from the bogie to the vehicle body by a pin - the pivot. This connection and a detailed illustration of the traction drive-train are depicted in an underbody view of a high-performance locomotive in Fig. 2.6. The pivot connection for traction force transmission lies here in the center of the bogie. Point symmetrical to that, the two PWM-inverter supplied traction drives are mounted within the bogie. They are specially developed for high-speed and high-power locomotives with a high proportion of suspended masses due to rail protection. This fully suspended drive, a cardan hollow- shaft drive, becomes very clear in the underbody view. Its branched structure is driven by a three-phase asynchronous machine (❶). In the branching point, namely a two-stage gear (❷,❸,❽), both generated motor speed and torque are transmitted with diﬀerent gear transmission ratios to the output and to the braking shaft. The output shaft leads the torque via the conical hollow shaft (❹,❺), coupled with spheroidal elastic bearings on both ends, to the directly driven wheel (❻) and via the wheel shaft through the hollow shaft to the indirectly driven wheel (❼). Finally, the transmitted torque of the wheelset is converted in traction force at the wheel-rail contact. As braking equipment, the parallel braking shaft consists of two enclosed-ventilated brake discs (❾,❿) whereof each operates with a brake caliper. 12 2 Traction vehicles

Air cooling circuit

Pivot hole

Spiral spring

Traction drive- trains

Fig. 2.5 Bogie with two traction drives [32].

Besides regenerative braking with the induction motor and mechanical braking with the brake discs, the Eurosprinter family naturally features the standard air brake system utilizing the braking forces from trailed railcars. Fig. 2.2 shows the essential parts of this system: the air equipment rack for compression and the compressed air reservoir for storage purposes. Therefore several auxiliary inverters supply fixed- and variable frequency loads such as pumps and as fans e.g. for cooling tasks due to energy conversion losses. Fig. 2.2–2.6 depict the cooling circuits for the three stages of energy conversion within the locomotive. Here three different cooling media are utilized: oil for the main transformer, water for the traction converter and air both for auxiliary inverters and traction drives. To complete the system overview, the drive control concept is discussed first on the basis of Fig. 2.3. Here, the high-performance locomotive operates with individual axle control. Unlike group axle control, each traction drive is independently controlled by a PWM inverter leading to a highly efficient utilization of the wheel-rail contact. The objective of the drive control is to realize the traction force Ft demanded by the locomotive driver. The adhesion control attempts to fulfill this goal by the output of an appropriate reference torque taking the limitations of the wheel-rail contact into account. Finally, a highly dynamic torque control [42] applies this torque with a bandwidth of around 200 Hz to the drive. In comparison to its antecessor, it features substantial improvements of operational performance indispensable for this thesis. In the sequel, those are discussed in descending order of relevance: • The major enhancement of the latest control schemes [42] is the highly increased dynamic torque response. Thus for the first time a drive control 2.1 Rail vehicles 13

Snow plough

Traction drive- train

Pivot Bogie . a due to short distance photographing.

Main Oil transformer Underbody view of a European high-performance locomotive cooling circuit Composition of 18 single camera shots has slight optical distortions Fig. 2.6 a 14 2 Traction vehicles

concept for rail vehicles facilitates active damping of both electrical and mechanical variables. Consequently, on this basis investigations into a anti- vibration control scheme are conducted in this thesis. • The compensation of disturbances originated either at line-side or from the track is achieved to a great extent due to the highly dynamics distinguish- ing the control scheme with a robust performance. • Operation without a speed sensor entails both enhanced reliability due to potential sensor failure and cost reduction of the drive. Further in lower frequency range of the drive, the speed signal resolution is increased above that of an ordinary incremental speed encoder where the mechanical robustness is the limiting factor on the encoder resolution. • The application of optimized pulse patterns and their smooth transitions for the static inverters minimizes torque oscillations below inverter switching frequency and therefore enhances the quality of the reference torque impression. • Automatic commissioning routines identify the inverter and motor parameters with a high accuracy and adapt those in dependency of temperature changes during the drive operation. This self-parameterization considerably reduces the commissioning time and assures a robust control performance over a wide operating range. • Autonomous fault detection and fault analysis rounds out the new features of the drive control scheme. Besides both motor and traction converter, the power electronic components, e.g. the pre- and dis-charging units, the DC link capacitors, the power semiconductors, the contactors, etc., are monitored and protected against overload and overtemperature. In case of any fault, the appropriate component is disengaged from the electric circuit, if possible, to hold up vehicle operation. Finally, Fig.2.7 shows the drive control structure applied for both PWM inverter and traction drive. Based on the DC link voltage VDC , the inverter provides a three-phase voltage system with variable amplitude and frequency, converted by the asynchronous machine into an appropriate torque. The task of the highly dynamic torque control is to assure that the actual torque T complies with the reference torque T ∗ demanded by the adhesion controller. Here, the challenge for the control in particular is to cope with the limitations of static inverters, namely low inverter switching frequencies at high power ratings. In detail, the DC link voltage VDC and the stator currents of the drive are measured. The stator currents are converted to orthogonal components within a coordinate system defined by the rotor flux ψR of the drive. This transformation leads to iSd and iSq decoupling both electromagnetic field and torque. Further, those two components are combined to the stator current vector iS for the well-known rotor-flux orientated control scheme where two independent controllers derive the limited reference stator currents, namely ∗ ∗ iSd and iSq. In addition to this standard control scheme, both inverse inverter and asynchronous machine models are introduced for speed sensorless operation and are adapted by a parameter identification scheme during drive 2.1 Rail vehicles 15 PWM inverter ∼ 3 M drive = Traction RST i DC V Gate signals Smod i model Inverse inverter 3 2 S ∗ RST i V S ψ R Motor ψ model Control of stator mesh T Smod - i + | . | ∗ sd i | adaption parameter R Speed and ψ | Estimated speed s flux | Estimated parameters ω Rotor- control R ψ | ∗ sq ω i + ∗ R ψ R + | ω R ψ | Torque control T Highly dynamic and sensorless traction drive control [42]. ∗ T Fig. 2.7 16 2 Traction vehicles operation. Here, the motor model provides the essential state variables for sensorless control, namely the vector of the stator currents iSmod, the rotor ψR and the stator flux ψS. Consequently, the system behavior of the drive is anticipated by the feed-forward impression of the stator mesh, namely the stator flux. Thus the reference stator currents are appropriately converted ∗ into a reference three-phase voltage system VRST . Subsequently, the inverse inverter model compensates the behavior of the PWM inverter and finally, the modulator realizes the resulting stator voltages by the output of the gate signals. The given insight into a highly dynamic torque control concludes the mechatronic system overview of a European high-performance locomotive for the AC overhead line. For DC voltage systems, the electrical main circuit for locomotives simplifies to a great extent. Both transformer and line-side rectifier become redundant due to the direct provision of the DC voltage. As an example for a pure DC electrified traction vehicle, the main circuit of a typical Japanese high-performance locomotive is investigated in Fig.2.8. In comparison to the prior introduced European locomotive, it traditionally utilizes diamond- shaped pantographs instead of the European single-arm type. Furthermore, the use of both diamond-shaped pantographs on the vehicle ensures a highly continuous current collection but at the same time entails more wear and maintenance costs. Subsequently, the collected current is fed via high-speed circuit breakers, designed to protect the circuits from ground faults by isolat-

DC overhead line Pantograph Circuit breaker = M

3 ∼ M PMW inverter

= M

3 ∼ M PMW inverter

Auxiliary inverters = M PM 3 ∼ M Input ﬁlter PMW inverter Bogie

Fig. 2.8 An example of a main circuit diagram of a Japanese high-performance locomotive. 2.1 Rail vehicles 17 ing large fault currents, to both auxiliary and main inverters. The auxiliary inverters provide various voltage systems for AC and DC loads, such as fans, air conditioners, air compressor, etc. where on the other hand, the three main inverters supply the traction motors. Each main inverter consists of an input filter, namely an LC-circuit, a protection module (PM) against voltage surge and a PWM inverter. The input filter compensates occurring faults in current collection and therefore stabilizes the DC input voltage for the PWM inverter. Unlike the introduced European high-performance locomotive, this particular Japanese locomotive features three bogies due to the given limited axle load according to Tab.2.1. Consequently, six axles have to be inverter- powered and thus, group axle control is introduced for this high-performance locomotive. The drive control scheme applied to one PWM inverter has to operate two asynchronous machines at the same time. For the assignment of two suitable drives, the effect of axle load shift has to be taken into consideration. Therefore the mechanical configuration of both drives and bogies becomes important. Due to their mechanically symmetrical arrangement within the locomotive, the traction drives-trains with equivalent adhesion conditions during occurring load shift are assigned to the same PWM inverter according to Fig. 2.8. However for group axle control, as already mentioned above, it is more difficult to apply an anti-vibration control strategy because of the impression of only one stator voltage system for both asynchronous machines. Therefore the mechanical structure of the traction drive-train has to exhibit sufficient ruggedness so that torsional vibrations in the drive-train are well damped. The axle-hung drive satisfies this requirement. A side view of this drive is depicted in Fig.2.9 as an example for a Japanese high-performance locomotive. Its simple mechanical structure is driven by an asynchronous ma-

Fig. 2.9 Axle-hung drive of a Japanese high-performance locomotive. 18 2 Traction vehicles chine (➊) transmitting the torque via the gears (➋,➌) to the wheelset (➍,➎). As a special axle-hung drive feature, the wheel shaft is hung by two axle bearings to the motor housing, as can be seen from Fig.2.9. Due to its higher portion of unsprung masses compared to fully suspended cardan hollow shaft drive, it only allows freight service up to 110 kph in Japan. The Eurosprinter family also features locomotives with the same type of drive-train, e.g. the class 189 locomotive, where its speed limit is set to 140 kph in Germany. Due to limited service speed, locomotives with axle-hung drive are predominantly used for freight traffic. High-speed passenger traffic up to 230 kph and freight service can only be achieved by locomotives with a fully suspended cardan hollow shaft drive such as the class 1016/1116. Thus, both design and type of traction drive-train define the speed limit of the locomotive, as becomes apparent within the given speed range in Tab.2.1. This completes the mechatronical spectrum of the basic propulsion schemes where the introduced European and Japanese locomotives are completely different in their electrical as well as mechanical system. Furthermore, the multi-voltage system locomotives, e.g. the class 189 of the Eurosprinter family and the EF510 of JRF fleet, utilize both main circuit concepts according to Fig.2.3 and to Fig.2.8 in a jointly optimized approach. In the sequel, multi-purpose locomotives for high-speed passenger traffic as well as for freight service are in the forefront. Unlike axle-hung drives, the applied cardan hollow shaft drive offers only a low damping for torsional vibrations in the drive-train due to its mechanical structure. Therefore individual axle control is inevitable to prevent excessive vibrational loads on the mechanical drive-train. For further investigations into the oscillation behavior, the drive system is modelled in detail.

2.1.2 Propulsion system modelling

In [15, 33, 36, 102–108], the modelling of the locomotives is investigated in great detail starting from electrical side, namely the DC link, both control and switching behavior of the inverter, the asynchronous machine, to the mechanical side, the bogie, the traction drive-train and finally, the wheel- rail contact. However, complex models entail many drawbacks, such as a long simulation duration due to both electrical and mechanical co-simulation. Further, a validation of simulation results becomes more difficult. Therefore a reduced model for each subsystem is derived and validated on the basis of its complex model. In this framework, a suitable simulation model is aimed at reproducing the same effects, namely slip-stick vibrations in the traction drive-train, as on the locomotive. For this purpose, the following boundary conditions are stipulated to reduce the complexity of the simulation model: 2.1 Rail vehicles 19

• Individual axle control decouples the dynamical behavior of the two drives within the bogie with respect to vibrations. • Wheel slip is considered only in longitudinal direction of vehicle motion. Under these two conditions, a simple simulation model is derived as shown in Fig.2.10. Its basic structure consists of the controller, the system and the disturbance model. Here, the system model splits into three parts, namely the highly dynamic torque control, both asynchronous machine and inverter characteristics and finally, the mechanical traction drive-train. The latter is influenced by the unknown adhesions forces, which are determined by the disturbance model. To close the control-loop, the speed sensor signal of the asynchronous machine is fed back. The appropriate reference speed signal is derived by an superimposed adhesion control scheme. Finally, the speed deviation from the reference signal is compensated by the speed controller within the control-loop. In the following, modelling of each subsystem is discussed in more detail where both traction drive-train and adhesion model are in the forefront for investigation into slip-stick vibrations. The speed controller installed on the locomotive is a standard PI-controller with its transfer function 1 Gc(s)= Kp (1+ ). (2.1) TN s As a function of the speed deviation, the controller requests an appropriate ∗ reference torque Tr . However, motor torque is limited given by the specifications of the asynchronous machine. Therefore the torque limitations have to be considered within the control-loop. This is achieved by the implementation of a nonlinearity, namely a saturation element. In case of a saturating nonlinearity, undesired oscillations are triggered in the control-loop. To re- move this effect well-known in control theory as windup, integral-action of the controller has to be prevented during the saturating nonlinearity. There- fore, a simple anti-windup scheme is utilized cutting off the integral-action component. ∗ Subsequently, the actuator realizes the requested reference torque Tr as electromagnetic torque Te within the asynchronous machine. Here, the actuator consists of both electrical inverter and asynchronous machine and the

System Controller Disturbance ∗ ∗ ω T Te r r Torque Inverter/ASM Drive-Train Adhesion - Control Electric Mechanic

ω˜r T˜e ωr

Sensor Fig. 2.10 Basic simulation model. 20 2 Traction vehicles highly dynamic torque control. As additional function, the torque control [42] observes both speed of the asynchronous machineω ˜r for sensorless operation and the electromagnetic torque T˜e. In [15,33], detailed investigations were made into the actuator dynamics and how to derive a suitably reduced model of the actuator. As a result, the closed-loop dynamics of the torque control and the electrical motor characteristics are summarized in an second-order lag element. Due to discrete switching characteristics of the inverter, a dead time td adds to the transfer function of the actuator dynamics

Te(s) 1 −td s Ga(s)= ∗ = 2 2 e . (2.2) Tr (s) s + 2 dTs + T

The actuator applies the electromagnetic torque Te to the traction drive- train, namely to the rotor of the asynchronous machine. Using the example of the most complex traction drive, detailed modelling of cardan hollow shaft drive-trains is introduced. Besides the underbody view of a high-performance locomotive, Fig.2.11(a) also depicts this modern high-performance drive with unmounted wheelset for high-speed passenger and freight traffic. Due to its convoluted construction, Fig.2.11(b) shows a simplified representation of the branched topology. The traction drive-train consists of ten discrete inertias that are connected by torsional stiffnesses to a branched structure. All torsional stiffnesses are defined by the characteristic values of steel except those which connect the hollow shaft (❹,❺) by rubber joints to the gear (❸) and to the direct-driven wheel (❻). Further, a linear color gradient for the inertia size is introduced ranging from light grey to black. Thus, the rotor of the asynchronous machine (❶) is the largest inertia in the drive system. Finally for the simplified model, the arrows referring to inertia (❶) indicate the known signals, the electromagnetic torque Te and the angular rotor speed ωr. The latter is measured by an inertia-less speed sensor, or now even sensorlessly by the latest torque control [42]. Further arrows refer to the unknown adhesion forces (z1, z2) affecting the wheelset (❻,❼) and to the braking forces (f1, f2) which are applied to the brake discs (❾,❿). To derive the simulation model of the mechanical drive-train, differential equations are determined for the angular speed of each inertia and for each shaft torque on the basis of the simplified model

1 ω˙r = ( Te − T12 ) (2.3) J1

T˙12 = c12( ωr − ω2 )+ d12(ω ˙r − ω˙2 ) (2.4) 1 ω˙2 = ( T12 − u1 T23 ) (2.5) J2

T˙23 = c23( u1 ω2 − ω3 )+ d23( u1 ω˙2 − ω˙3 ) (2.6) 1 ω˙3 = ( T23 − T34 + u2 T38 ) (2.7) J3 2.1 Rail vehicles 21

Subsequently, the mechanical parameters of the traction drive-train are calculated on the basis of mechanical drawings: • The inertia size J and the torsional stiffness c are derived by both ge- ometry of the drive structure and characteristical material values for steel and rubber. • The damping value d, however, is difficult to determine due to its frequency- dependent behavior. For investigation of slip-stick vibrations, a constant value related to the occurring vibration frequency is assigned. • The two gear transmission ratios u result from the different number of teeth (❷,❸,❽). In case of the output shaft, the angular speed is reduced and at the same moment the transmitted torque is increased, as becomes obvious from Eq.(2.5) and Eq.(2.6).

(a) Modern high-performance drive with separate brake shaft

f1 f2

ωr

Te z2

(b) Simpliﬁed model of branched traction drive-train

Fig. 2.11 Mechanical multi-inertia cardan hollow shaft traction drive. 22 2 Traction vehicles

• Finally, a nonlinear effect is also considered for the traction drive-train model, namely backlash. It often occurs in mechanical elements of the drive-train, e.g. here in the two-staged gear, due to both manufacturing tolerances and wear. In this case, no drive torque is transmitted within a certain angular range ∆ϕb via the gears to the output and braking shaft. This nonlinear behavior is characterized by a dead band element where the originally occurring angular difference ∆ϕ is mapped to ∆ϕ′. The latter is determined by a linear function of ∆ϕ expect during backslash where ∆ϕ′ is zero. The mathematical description of the angular difference mapping is given in the following equation, where ∆ϕ′ is defined as

∆ϕb ∆ϕb ∆ϕ + 2 : ∆ϕ < − 2

′ ∆ϕb ∆ϕb ∆ϕ =  0 : − 2 <∆ϕ< 2 (2.8)   ∆ϕb ∆ϕb  ∆ϕ − 2 : ∆ϕ > 2  Thus, the dead band element is integrated into the simulation model to the appropriate gear stiﬀness c, where the shaft torque

′ T = ∆ϕ c (2.9)

derives as a nonlinear function of the angular diﬀerence ∆ϕ. This concludes the traction drive-train modelling showing a detailed simulation model of the branched drive in Fig.2.12. For further discussions, a linearized model of the traction drive-train is derived in state-space representation on the basis of Fig.2.12. Therefore two equations, namely the state-space and the output equation, deﬁne the mechanical drive system

˙xm = Am xm + bi ui + Bz uz (2.10)

ym = cm xm (2.11) where the following vectors and matrices are introduced: the state-space vector xm, the input vector bi, the input ui, the disturbance matrix Bz and disturbance vector uz, the system output ym and the output vector cm. The drive system is represented by an equivalent number of states variables as there are discrete energy storage elements, which are in this drive example nineteen. The angular speed of an inertia and the shaft torque contribute each one system state according to Eq.(2.3)–(2.7). For a simple linearized model, steady operation of the traction drive-train is assumed. Consequently, no zero crossing in the transmitted torque within the gears occur and the nonlinear backlash is neglected for this state-space approach. Finally, the input ui is deﬁned by the electromagnetic torque Te and the disturbance vector uz by both braking (f1,f2) and adhesion torques (z1,z2). 2.1 Rail vehicles 23

1/J10 f1 -

d910

c 910 -

1/J1 1/J9 Te ωr f2 - -

d12 d89

T12 c c 12 - 89 -

1/J2 1/J8

- - u1 u1 d38

c 38 - d23

c 23 - u2

1/J3

d34

c 34 -

1/J4

d45

c 45 -

1/J6 z1 - - d67

c 67 -

1/J7 z2 -

Fig. 2.12 Nonlinear model of the traction drive-train with separate brake shaft. 24 2 Traction vehicles

ωr z1

Fig. 2.13 Simpliﬁed model of a axle hung traction drive-train.

In analogy to the complex branched drive-train, the simplified mechanical structure of the Japanese axle hung drive with its five discrete inertias is shown in Fig.2.13. For simplicity, the nose bearings connecting the motor housing with the wheelset shaft are neglected. Compared with the cardan hollow shaft drive of Fig.2.11(b), its simple mechanical structure contains no rubber elements and thus requires less maintenance. Therefore the axle hung drive is highly favored for locomotive application due to its robustness. De- tailed modelling of the axle hung drive, including the appropriate state-space representation, is to be found in [36] on the basis of a class 152 locomotive from German Railways. Besides the traction drive-train, also the wheel-rail contact plays an important role for the occurrence of slip-stick vibrations and therefore a comprehensive adhesion model for traction drive-trains is discussed. In the contact point of the wheel, a transmission of traction force is only possible if a difference speed between wheel and rail occurs. For this kind of slipping motion, the degree of freedom allows a longitudinal and a lateral difference speed. Since the lateral component does not contribute to the available traction force and does not affect the investigations into slip-stick vibrations, it is neglected to obtain a simplified adhesion model. Thus the longitudinal difference speed ∆v derives from the peripheral speed of both direct- and indirect-driven wheel discs (❻,❼), namely vd and vi, and the locomotive speed vl as

∆vd = vd − vl (2.12)

∆vi = vi − vl. (2.13)

On the basis of the diﬀerence speed, the transmitted traction force in the contact point of the direct-driven wheel disc 2.1 Rail vehicles 25

ml g Fd = µ(∆vd) (2.14) 2 na ml g Fi = µ(∆vi) (2.15) 2 na const. is determined by the adhesion characteristics| {z } µ(∆vd) and a constant factor. The latter equals the axial force of one wheel disc and is deﬁned by the mass of the locomotive ml, the gravity constant g and the number of axles na. Further, the speed of the locomotive results from the transmitted traction forces, where it is assumed that all powered axles transmit an equivalent value of traction force. Additionally, the resistive forces Fres, e.g. the air resistance of both locomotive and trailed freight wagons and the speed-independent rolling friction, are considered according to the parameters given in [33]. Finally, the locomotive speed vl derives as a function of the complete train mass mt and the above mentioned variables

vl = mt [na (Fd(∆vd)+ Fi(∆vi)) − Fres] d ∆vd,i). (2.16) Z On the basis of Eq.(2.12)–(2.16), Fig.2.14 depicts the structure diagram of the utilized adhesion model. Here, the indices of both direct- and indirect- driven wheel variables are combined to their appropriate vector representation. Subsequently, the traction drive-train model is combined with the adhesion model feeding back the wheel disc speeds ω as disturbance torques Tz to the drive-train. The basic coupling equations are

Tz = F rw (2.17)

v = ω rw (2.18)

rw rw ωd,i z1,2 Drive-Train Model

vd,i Wheel-rail condition ml g 2 na ∆vd,i Fd,i - Adhesion characteristics vl mt - na P Fres

Fig. 2.14 Wheel-rail contact of traction vehicles. 26 2 Traction vehicles

0.3 →

0.2

0.1 1

adhesion coeﬃcient 0.8 0 → 2.5 0.6 2 0.4 1.5 1 0.2 0.5 ← ∆v 0 0 d,i [m/sec] wheel-rail condition Fig. 2.15 Adhesion characteristics.

where rw is the wheel disc radius. For completion of the adhesion model, the adhesion characteristics are discussed. On the basis of complex contact physics, the adhesion phenomenon between wheel and rail is investigated. As a result, a complex theory for the adhesion coefficient as function of both lateral and longitudinal difference speed is derived in [36] where several advanced analytic models are proposed. However in this framework, a simple and suitable analytic model is aimed at considering only the longitudinal adhesion forces. Such a simplified model is introduced in [31] for the wheel-road contact of electric road vehicles. Here, the adhesion coefficient µ is given as a function of the difference speed ∆v

rs − 0.01 Ke µ(∆v)= µ (1 − e B ∆v) − − 0.01 . (2.19) max (K − 0.01) + 0.01 er ∆v To fit this model to the characteristics of the wheel-rail contact, the parameters are adapted where µmax is the maximum available adhesion coefficient, B defines the gradient at ∆v = 0, K equals µmax −µ(∆v = 1) and r is the inflexion point of the adhesion curve. For investigations of slip-stick vibrations, a variation of the parameter r is assumed as a change of wheel-rail conditions and the remaining parameters are kept constant. Fig.2.15 depicts the resulting curve family. Without a difference speed ∆v, the adhesion coefficient µ is zero. Increasing difference speed leads to the maximum available adhesion coefficient µmax. Beyond the adhesion maximum, the parameter variation results in a change of the negative slope. This operating range is well-known as origin of slip-stick vibrations and will be further investigated in the next chapter. Chapter 3 Slip-stick vibrations

An increased utilization of tractive eﬀort level for high-performance locomotives entails severe oscillations in the drive-train. Therefore the origin of slip-stick vibrations is investigated focussing on the interaction of both ﬂexi- ble traction drive-train and wheel-rail contact. Further, the limitations of the standard control scheme concerning the oscillation suppression are pointed out. Finally, slip-stick vibrations are discussed and analyzed on the basis of both simulation studies and test runs of an European high-performance locomotive.

3.1 Origin of vibrations

In 1967, the phenomenon of slip-stick vibrations in cardan hollow shaft traction drive-trains was studied for the first time in [43]. More detailed investigations followed in [15, 33, 36, 44, 111], extending the knowledge of occurring oscillations in traction applications. Further in [45], the origin of slip- stick vibrations was identified in the characteristics of the wheel-rail contact, namely in the operating range beyond the maximum available tractive effort value. Here, the self-excited slip-stick vibrations arise supported by periodi- cally drawn propulsion power from the asynchronous machine. To suppress slip-stick vibrations, first their triggering effect is investigated taking a closer look at the influences of the wheel-rail contact on the traction drive-train.

3.1.1 Wheel-rail contact

The traction drive-train consists of a flexible mechanical structure due to finite torsional stiffness. Therefore it exhibits several resonant frequencies stressing the mechanical components by different oscillation behavior. To 28 3 Slip-stick vibrations identify the most stressed components during occurring slip-stick vibrations, a modal analysis determines the modal values of the traction drive-train, namely both natural frequencies and mode shapes. In the following, this analysis is carried out on the basis of the most complex drive-train introduced in the last chapter. Due to low damping within the drive-train, the drive system is assumed with no damping. Further, the nonlinear backlash characteristics of the gears are neglected resulting in an undamped linearized mechanical system. Thus the mode shapes for the appropriate natural frequencies are calculated from an eigenvector problem on the basis of a different state space representation of the drive-train than given by the differential equations Eq.(2.3)–(2.7). In the standard approach for modal analysis [46], the torsional rotation angle ϕb of each inertia is considered as state variable. Con- sequently, the derived mode shapes which contain the scaled torsional rota- ∗ tion angle ϕb exhibit the vibration characteristics of the drive-train [109,110] in Fig.3.1, whereas the branch with the ‘o’-symbols represents the braking shaft. Here the axis of abscissa reflects the inertia topology of the branched mechanical system as depicted in Fig.2.11(b). The ten-inertia system contains nine torsional stiffnesses and therefore both nine natural frequencies and modes shapes are derived in addition to one trivial vibration mode due to the mechanically unbounded system. The eigenfrequency of this first trivial mode shape is 0 Hz due to rigid body motion. The second mode shape represents the first vibration mode with the eigenfrequency of 22.78 Hz. Here the wheelset (❻,❼) oscillates against the asynchronous motor (❶) with the hollow shaft (❺,❻) as the nodal point of vibration. The third mode shape reveals the second vibration mode with 65.31 Hz, where the wheelset shaft is twisted by the wheel discs with less influence on the asynchronous motor. For both vibration modes the contribution of the braking shaft (❽–❿) to the oscillatory behavior of the traction drive-train is negligible. However, in the third vibration mode with 117.5 Hz the braking shaft predominantly oscillates against the remaining drive-train almost unaffected by vibrations. The subsequent mode shapes at higher eigenfrequencies also have minor vibrational influences on both asynchronous machine and wheelset, where e.g. the oscillations with eigenfrequencies above 1000 Hz exclusively occur within the gears (❷,❸,❽). Finally on the basis of the derived mode shapes and the parameters of the traction drive-train, the most torsionally stressed stiffness is identified for each eigenfrequency by a simple calculation of the poten-

Tab. 3.1 Eigenfrequencies and maximum torsional load. Frequency/Hz Stiﬀness Frequency/Hz Stiﬀness 0 ❶-❿ 204.5 ❹-❺ 22.78 ❺-❻ 241.3 ❶-❷ 65.31 ❻-❼ 391.6 ❾-❿ 117.5 ❽-❾ 1045 ❷-❸ 138.4 ❸-❹ 1130 ❸-❽ 3.1 Origin of vibrations 29

1 0 Hz 0

-1 1 22.78 Hz 0

-1 1 65.31 Hz 0

-1 1 117.5 Hz 0

-1 1 138.4 Hz ∗ b

ϕ 0

-1 1 204.5 Hz 0

-1 1 241.3 Hz 0

-1 1 391.6 Hz 0

-1 1 1045 Hz 0

-1 1 1130 Hz 0 -1❶ ❷ ❸ ❹❽ ❺❾ ❻❿ ❼

Fig. 3.1 Mode shapes of the branched ten-inertia system at new condition. 30 3 Slip-stick vibrations tial energy distribution of the drive system [46]. As a result, Tab. 3.1 shows those stiﬀnesses for the appropriate eigenfrequencies. In the ﬁrst two vibrations modes, the wheelset shaft (❻-❼) and the rubber connection between the wheelset and the drive-train (❺-❻) are the most stressed components as it also becomes obvious from the mode shapes in Fig.3.1. This concludes the modal analysis of the traction drive-train.

0.3

stable unstable 0.25 range range P2 µ 0.2 P1 (I) (II) 0.15 0.06 (III) δµ = – 0.09 s δ(∆v)  m 0.12 0.1   adhesion coeﬃcient

0.05

0 0 0.5 1 1.5 2 m speed diﬀerence ∆v [ s ] Fig. 3.2 Adhesive force characteristics of wheel-rail contact.

In the sequel, the adhesion characteristics of the wheel-rail contact are discussed [89–96], in particular the operating range, where the slip-stick vibrations arise in the traction drive-train. Therefore Fig. 3.2 depicts various adhesive force characteristics, namely (I)-(III). These three curves reflect the typical adhesion behavior of wet wheel-rail surfaces with different negative adhesion force gradients δµ/δ(∆v) beyond the maximum tractive effort level. Subsequently at higher wheel-to-rail slips, the adhesive force approaches a constant value lower than the maximum tractive effort level. In any operating point on the adhesion characteristics, the gradient of the curve acts as an additional damping factor for the drive-train. To investigate into this variable damping factor, the wheel-rail contact model of Fig.2.14 is considered in its low-level behavior for both input z1,2 and output signals ωd,i. In this case, the speed of the locomotive vl contributes an almost constant offset value

vl = const. (3.1) and thus is neglected for low-level considerations. Consequently, only the proportional elements and the adhesion characteristics, linearized in the op- 3.1 Origin of vibrations 31 erating point, have to be taken into account. As a result, the variable damping factor dad is derived as a function of the adhesion force gradient δµ/δ(∆v).

δz1,2 2 mt g δµ dad = = rw (3.2) δωd,i 2 na δ(∆v)

Therefore operating points at small difference speeds contribute a highly positive damping factor to the wheelset, where dad equals zero at the maximum of traction coefficient. On the other hand, in the operating points beyond the maximum of tractive effort level, located on the negative slope, the damping factor is negative. Those variations of the damping factor affect the dynam-

Im(s) δµ δ(∆v)

~ ω3 ~

δ2 ω2 ~

σ3 σ2 Re(s)

Fig. 3.3 Inﬂuences of the wheel-rail contact on the traction drive-train. ics of the mechanical system and become apparent in the open-loop transfer function of the traction drive-train given according to Fig. 2.10 by

∗ ∗ ωr(s) (s + n2)(s + n2)(s + n3)(s + n3) ... Gtd(s)= = ∗ ∗ (3.3) Te(s) Ks (s + p2)(s + p2)(s + p3)(s + p3) ... where the poles p and zeros n are deﬁned as

p2 = σ2 + jω2 (3.4) ′ n2 = σ2 + j (ω2 − δ2). (3.5)

The open-loop transfer function reveals an integration pole and both ten complex conjugated zeros and poles. To investigate the inﬂuences of the wheel-rail contact on the traction drive-train, the pole-zero diagram in Fig.3.3 shows the relevant part of Gtd(s) as a function of the varying adhesive force gradient. In general, this diagram applies to cardan hollow shaft traction drive-trains, showing only two conjugated complex poles (x) and zeroes (o) to the left of the imaginary axis and the integration pole. The following variables are 32 3 Slip-stick vibrations introduced: damped natural frequencies ω2 <ω3, the damping ratios σ2 <σ3 and the pole-zero distances δ2 ≫ δ3. Their values depend on the mechanical construction of the traction drive-train. In case of a negative adhesion force gradient, both poles and zeroes of the lowest two vibration modes shift towards the right-half plane leading to instability of the drive system. There- fore operating points on the adhesion characteristic beyond the maximum of tractive eﬀort level are unstable as shown in Fig.3.2.

Im(s)

2 1 ~ ~ ω3

1 δ2 2 ω2 ~

σ3 σ2 Re(s)

Fig. 3.4 Root locus diagram: Closed-loop inﬂuences of the standard controller.

3.1.2 Standard control scheme

The standard control scheme is based on the speed of the asynchronous machine where the speed deviations are compensated by a proportional-integral controller as shown in Fig.2.10. This speed controller achieves additional damping to the traction system. Here, the available damping depends on both open-loop characteristics and controller parameters. On the basis of s the open-loop characteristics for δµ/δ(∆v) = 0 m , i.e. without any influence from the wheel-rail contact, the achievable closed-loop dynamics are discussed with the root locus method in Fig.3.4. As in Fig.3.3, only the relevant two mechanical vibration modes are considered in the root locus diagram. The trajectories of the closed-loop poles are derived as a function of the controller gain where the zeroes remain unaffected by the controller parameters. Con- sequently, the controller shifts the closed-loop poles along the trajectories towards the left-half plane and counteracts the influences of the wheel-rail s contact for δµ/δ(∆v) < 0 m . As an example, two sets of closed-loop poles, namely (1) and (2), are shown in Fig.3.4 for different controller gains. A low controller gain (1) leads to maximum damping of the first vibration mode where the achieved damping in the second vibration mode is almost negligi- 3.1 Origin of vibrations 33

1 10 ω

50 45 T

5 6 ω

100 50 67

T 0 -50

5 7 ω

2 i

v 1

10 e

T 5

0 0 1 2 3 4 5 time [s]

Fig. 3.5 Slip-stick vibrations: Simulation studies with adhesion characteristic (III) at rad m various high wheel-rail slips ( ω [ sec ], T [kNm], ∆v [ s ] ). 34 3 Slip-stick vibrations ble. However, a high controller gain (2) significantly reduces the achievable optimal damping for the first vibration mode but on the other hand provides maximum damping in the second vibration mode even though the achieved damping is low in this case. For the application on the locomotive, fixed controller parameters have to be chosen. Therefore a low controller gain is preferred not only due to the higher achievable damping performance but also due to the lower sensor noise amplification. To evaluate the overall performance of the standard control scheme, the influence of the wheel-rail contact, in particular the worst-case of instability according to Fig.3.3, has to be considered. Therefore the damping effects of both pole-zero and root locus diagram are summarized. As a result, the first vibration mode is always stable due to the highly achieved damping by the controller. However, insufficient damping is provided to the second vibration mode independent of the chosen controller parameters. Consequently, the traction system faces instability in the unstable range of the adhesion characteristic and inevitably, slip-stick vibrations arise. On the basis of both simulation model and identically assumed adhesion characteristics for both wheel-rail contacts according to Fig.3.2, the occurrence of slip-stick vibrations is discussed by two simulation studies with the branched traction drive-train. The simulation studies show a start-up of the locomotive from standstill to various operating points at high wheel-rail slips, located on the negative slope beyond the adhesion force maximum. In this operating range, the effects of both operating point and negative adhesion force gradient are pointed out. First in Fig.3.5, three different operating points within the unstable range of the adhesion characteristic (III) are investigated where the adhesion force s gradient is constantly −0.12 m . Starting the locomotive from standstill, the reference value of the asynchronous machine speed is appropriately derived from the superimposed adhesion control scheme to achieve the proposed traction cycle. On the basis of this reference signal, the standard speed controller realizes the actual motor speed ω1 shown in Fig.3.5. Further, the gear transmission ratio (❷,❸) lowers the speeds of both directly and indirectly driven wheel disc, namely ω6 and ω7, and on the other hand increases the torque of the hollow shaft T45 compared to the electromagnetic torque of the asynchronous machine Te to the same extent. The transmitted torque through the wheelset shaft T67 to indirect-driven wheel disc is half the level of the hollow shaft torque due to identically assumed adhesion characteristics for both wheel-rail contacts. Finally, the difference speed ∆vi reveals the operating point of the indirectly driven wheel disc within the adhesion characteristics. s To reach the first operating point at ∆vi = 0.6 m , the maximum of adhesion force characteristic is passed, as indicated by the electromagnetic torque Te. Subsequently, the drive-train operates at this high wheel-to-rail slip in the unstable range of adhesion characteristic and consequently, slip-stick vibrations arise. As calculated with the modal analysis, the predicted vibration behavior of the traction drive-train is verified by this simulation study. 3.1 Origin of vibrations 35

1 10 ω

45 20 T

5 6 ω

50 67 T 0

-50

5 7 ω

2 i

v 1

10 e

T 5

0 0 1 2 3 4 5 time [s]

Fig. 3.6 Slip-stick vibrations: Simulation studies with various negative adhesion force rad m gradients at constant high wheel-rail slips ( ω [ sec ], T [kNm], ∆v [ s ] ). 36 3 Slip-stick vibrations

Both wheel discs oscillate against each other with the same vibrational amplitude in their speed signals. Therefore the wheelset shaft exhibits the highest torsional vibration amplitude in the drive system. However, the vibrational influence to the asynchronous machine is rather low due to the used rubber joints within the torque transmission. In the following, the wheelset shaft torque T67 is in the focus regarding investigations into slip-stick vibrations. Its torsional amplitude increases until the gradual change between stiction and sliding friction of both wheel discs saturates in its dynamics. Due to instability on the negative slope of the adhesion characteristics, this limit cycle results from the highly oscillating difference speeds ∆v of both wheel discs. Here, the low ∆v-values, located on the positive slope, counteract the arising vibrations with a positive damping factor on part of the wheel-rail contact saturating the oscillation amplitudes. After saturating vibration amplitude of the wheelset shaft torque T67, an operating point on the positive slope is chosen to damp the actual vibrations. Subsequently, the second operating point on the negative slope at a s higher difference speed of ∆vi = 0.75 m is reached. Here, also slip-stick vibrations arise, but with a higher saturated torque amplitude in T67. The increase in amplitude results from the operating point at a higher difference speed. Therefore the saturation of vibration necessitates a higher positive damping factor on the positive slope of the adhesion characteristic corresponding to a lower difference speed. Finally, the third operating point with s the highest difference speed of ∆vi = 0.9 m in the unstable range confirms the assumed saturation behavior by the highest vibration amplitude. Thus operating points at higher wheel-to-rail slips at a constant negative adhesion force gradient exhibit higher slip-stick vibration amplitudes. In the second simulation study, the influence of various negative adhesion s force gradients at a constant wheel-rail slip of ∆vi = 0.75 m is discussed. In Fig.3.6, the starting of the locomotive is simulated from standstill to this fixed operating point on the negative slope of the adhesion characteristic (I). s Here, the low adhesion force gradient of δµ/δ(∆v) = −0.06 m contributes a sufficient degree of negative damping to the drive-train and slip-stick vibrations slowly arise. After the achieved vibration damping on the positive slope, the assumed adhesion characteristics for the simulation model are switched from (I) to (III) and subsequently, the operating point is again reached at the same difference speed on the negative slope of (III). The highly negative s adhesion force gradient of δµ/δ(∆v) = −0.12 m , namely twice the negative gradient of the adhesion characteristics (I), increases the vibration dynamics to a greater extent. Consequently, the slip-stick vibrations arise more quickly and the saturated vibration amplitude in T67 is higher, compared to the prior investigated vibration event on (I). This connection is also confirmed by the first stimulation study in Fig.3.5, where the rising speed of each vibration event is the same due to constant negative adhesion force gradient. 3.2 Locomotive test runs 37

To conclude the conducted simulation studies of various wheel-rail contact conditions, the following results are given for the unstable range of the adhesion characteristics: • Higher wheel-to-rail slips entail an increase in the amplitude of torsional vibrations in the traction drive-train. • More negative adhesion force gradients lead to both higher vibration amplitudes and shorter rising times of slip-stick vibrations. In the simulation studies, desired wheel-rail contact conditions were achieved for the vibration investigations assuming both adhesion characteristics and appropriate wheel-to-rail slips. These speciﬁed conditions are impossibly obtained during the test runs on the locomotive due to steadily varying adhesion force characteristics.

3.2 Locomotive test runs

In the following, extensive test runs are conducted with a European high- performance locomotive where all installed anti-vibration countermeasures are deactivated except the essential standard slip controller. Consequently, the natural oscillation behavior of the branched traction drive-train, equipped on this locomotive, is investigated and subsequently analyzed. Due to the stochastic behavior of the wheel-rail contact a large number of consecutive test runs with identical test run conditions have to be conducted to facilitate a comparison of the measured data.

3.2.1 Test run conditions

For identical conditions in the test runs, the decisive factor is definitely the unknown wheel-rail contact depending on several influences, such as • humidity and temperature • separate film between wheel and rail • surface characteristics of wheel and rail. Besides those influencing factors, appropriate wheel-rail contact conditions have to be produced in order to generate slip-stick vibrations in the traction drive-train. Consequently, the adhesion force characteristic of the wheel-rail contact has to exhibit a negative slope for high wheel-to-rail slips. Both negative slope and minimized influencing factors are achieved by watering the wheel-rail contact artificially. On the basis of those conditions, several test runs of a European high-performance locomotive are conducted at a constant locomotive speed. To eliminate track-bounded influences, the measured data are recorded only both in one riding direction and on the same track. 38 3 Slip-stick vibrations 3.2.2 Vibrations

Besides the recorded signals from the standard sensors, namely both measured speed of the asynchronous machine ω1 and electromagnetic torque Te, additional torque sensors are mounted in the mechanical drive-train to investigate into occurring slip-stick vibrations. For the ﬁrst two vibration modes, both hollow and wheelset shaft are the most stressed mechanical components and therefore are chosen as suitable sensor positions already assumed in the simulation studies. In Fig.3.7, these four recorded signals reveal two vibration events at both low asynchronous machine speed and constant locomotive speed. In this case, the vibration occurrence is strongly connected to the electromagnetic torque Te while assuming constant adhesion conditions. Therefore an appropriate increase of Te results in higher wheel-rail slips and subsequently slip-stick vibrations arise as can be seen from the recorded torque signals. Further, the vibration event is also accompanied by a slight rad deviation in the speed signal from its steady state value of ω1 = 10 sec due

1 10 ω

40 45

T 20

67 0 T -50

10 e

T 5

0 0 1 2 3 4 5 time [s]

rad Fig. 3.7 Test run with a European high-performance locomotive (ω [ sec ], T [kNm]). 3.2 Locomotive test runs 39 to slipping wheel discs. Finally, the declining vibrations are the result of a reduction in electromagnetic torque. In analogy, the second slip-stick vibration event occurs. However, the overall vibration duration compared to the ﬁrst vibration event is considerably longer. This extended duration originates from the absence of any anti-vibration countermeasure due to deactivation. Consequently, the second vibration event conﬁrms the importance of a suitable anti-vibration control scheme applied to minimize the vibrational load in the drive-train.

3.2.3 Analysis

All recorded test run data reveal the occurrence of only one slip-stick vibration frequency, namely the frequency of the second vibration mode where the wheelset shaft is twisted by the wheels. Thus the wheelset shaft torque T67 represents the best evaluation criterion for the mechanical load experienced by the drive-train for the following investigations. Furthermore, the frequency of the first vibration mode is evidently sufficiently damped by the standard controller given that it does not occur in the data. In the following, a single vibration event is analyzed in Fig.3.8 to a great extent. Here, the characteristics of the arising slip-stick vibrations are identified by both maximum oscillation amplitude and vibration dynamics. The latter is derived by fitting an exponential envelope curve to the vibration data.

67 40 σt

T e

0 0 0.1 0.2 0.3 0.4 0.5 time [s] Fig. 3.8 Test runs: Investigation into the dynamics of arising slip-stick vibrations (T [kNm]). 40 3 Slip-stick vibrations

Its excellent match results from both constant wheel-rail slip and adhesion conditions during the arising slip-stick vibrations. The exponential damping coeﬃcient σ determines the vibration dynamics where more positive values indicate shorter rise times. To consider the natural damping of the mechanical drive-train and the negligible damping achieved by the standard controller in the second vibration mode, a simple calculation leads to the actual adhesion force gradient for the analyzed vibration event. Due to the huge amount of recorded test run data, a smart tool-box for analysis of the adhesion force gradient is programmed. It also oﬀers an online-calculation of the adhesion force gradient which will be later discussed in Chapter 6.1 in greater detail. As a result, the gradient is depicted in Fig.3.9 for the test run sample of Fig.3.7. For determination of the gradient, the occurrence of an excitation of the traction drive-train, namely slip-stick vibrations, is inevitable. In case of non-excitation, the gradient is here displayed as zero. The spikes from the derived gradient signal result from the nature of the programmed algorithm

67 0 T

-50

80 )

67 60 T ( 40

Env 20 0

0.2 ) ∆v δµ 0 ( δ

-0.2 0 1 2 3 4 5 time [s]

Fig. 3.9 Programmed tool-box: Analysis of adhesion force gradient during slip-stick s vibrations (T [kNm], Env(T ) [kNm], δµ/δ(∆v) [ m ] ). 3.2 Locomotive test runs 41 and have to be neglected. This is achieved in a simple way by ﬁltering the raw gradient signal. A closer look on the total data using the tool-box reveals a variation in the adhesion characteristics in the test runs due to the stochastical behavior of the wheel-rail contact. This also applies to the unstable range of the adhesion characteristic where a wide spectrum of negative adhesion force gradients occurs leading to slip-stick vibrations. This spectrum is investigated by discretized probability distributions for the scaled oscillation amplitude of the wheelset shaft torque T67 and for the negative adhesion force gradient which are both determined for each slip-stick vibration event. In total over three hundred events are studied with the standard control scheme to obtain reasonable probability distributions. As a result, Fig.3.10(a) depicts the probability distribution of the occurring vibration amplitudes where the highest occurrence of about 24% belongs to vibrations with 30% of the maximum wheelset shaft torque amplitude. On the other hand in Fig.3.10(b), the distribution of the adhesion force gradient exhibits as its s most negative value δµ/δ(∆v) = −0.13 m during the conducted test runs. This gradient sets a new worst-case record, compared to to the formerly mea- s sured minimum value of δµ/δ(∆v)= −0.12 m in [47], for occurring negative slopes in the adhesion characteristics. Here, the large number of analyzed slip-stick vibrations facilitates the determination of this highly negative gradient. Thus from the viewpoint of statistical analysis, more negative gradients s than −0.13 m are likely to occur with an extreme low probability. Besides the measured worst-case gradient, most of the slip-stick vibrations arise on s the negative slope of the adhesion characteristics with a gradient of −0.06 m . s More positive gradients than −0.04 m do not occur in the recorded test run data due to the available system damping provided by both mechanical drive- train and standard controller. Finally, the locomotive test runs are also used to verify the derived model of the propulsion system for the simulation studies. However, a simple frequency analysis of both test run and simulation data reveals a considerable diﬀerence in the vibration frequency. This deviation is tracked back to variable parameters in the traction drive-train such as rubber joint ageing and wheelset wear. For the simulation studies, a new drive-train is assumed where for the locomotive test runs the degree of both wear and ageing in the drive- train components is so far unknown. 42 3 Slip-stick vibrations

occurrence in % 5

0 0 0.2 0.4 0.6 0.8 1 wheelset shaft torque [ T67 ] max (T67) (a) Mechanical load on the wheelset shaft

10 occurrence in %

0 -0.04 -0.06 -0.08 -0.1 -0.12 -0.14 s adhesion force gradient [ m ] (b) Wheel-rail contact condition

Fig. 3.10 Probability distribution investigation into slip-stick vibrations on a European high-performance locomotive utilizing the standard control scheme. Chapter 4 Reduced model identiﬁcation

Traction drive-trains of modern locomotives consist of a flexible multi-inertia structure experiencing significant wheelset wear due to wheel-rail-contact and ageing of rubber elastic joints during its operational life. For control design an accurate and suitable reduced model is required. Therefore a universal reduced model identification is proposed, taking branched traction drive-trains and parameter adaption due to wheelset wear and rubber ageing into consideration. Further for system monitoring purposes, a novel estimation scheme is introduced enabling a precise estimation of the variable mechanical parameters. For both schemes, the identification effort is lowered by applying previous knowledge of the flexible structure to comply with the limitations imposed by the traction application.

4.1 Conventional approach

Mechanical components of traction drive-trains are subject to several effects. The wheel-rail slip, inevitable for traction force transmission, causes wheelset wear and thus complies with a steady decrease in wheel disc diameter. Besides wear, ageing affects the flexible rubber joints of the drive-train increasing their torsional stiffness. Both wear and ageing of the mechanical components progress slowly but lead to a major change in the vibration frequencies during the operational life of the traction drive-train. For example, the lowest two natural frequencies of the branched drive-train shift from 22.8 and 65.3 Hz, as given in Chapter 3, to 30.9 and 81.6 Hz while both completely worn-out and aged components are assumed. This considerable change in the vibration frequencies has to be identified and subsequently the anti-vibration controller is adequately adapted. To cope with the complex multi-inertia structure of the drive-train, the control design is based on a suitably reduced model. The latter reflects the lowest two vibration modes of the traction drive-train with appropriate natural frequencies. Such a reduced model is represented by an 44 4 Reduced model identification inertia system with the order of three. Only a few papers have been published on how to derive a suitably reduced three-inertia model. All papers have in common that they only deal with unbranched traction drive-trains. They basically suggest two different approaches: • The first approach [15,33,35] determines the desired three-inertia model with special reduction algorithms known from mechanical engineering. The algorithms are mostly based on a step-by-step inertia reduction of the multi-inertia drive-train accompanied by extensive computations. The results of the proposed algorithms were satisfactory, but the match between the natural frequencies was mostly poor. Further, the application of those algorithms implies a continuous knowledge of all mechanical system parameters over the whole wear and ageing time. Due to wear, the inertias of the wheelset significantly decrease and due to ageing the stiffness of the rubber joints increases considerably, making it impossible to apply the algorithms to real-time operation. Therefore both effects are not considered in those publications and only the traction drive-train in new condition is assumed for control purposes. • The second approach involves an evolutionary identification algorithm [17,18] resulting in a two-inertia system, which is proposed for traction drive-trains in [16]. This complex algorithm also necessitates extensive calculations due to its low speed of convergence. Therefore with the higher inertia system order of three, the speed of convergence declines all the more extremely. Both approaches involve an enormous computation effort which is not applicable to recent DSP platforms in traction applications. As a further drawback, the former publications only consider unbranched drive-train topologies. Therefore in this Chapter, a novel identification scheme is proposed using previous knowledge to comply with the computation effort limitations imposed by the traction application. Further, it determines the optimal reduced model in a universal valid way from both branched and unbranched cardan hollow shaft drive-trains with respect to wheelset wear and to rubber ageing. For this purpose in Fig. 4.1, three structurally different multi-inertia traction drive-trains are provided to verify the universal applicability of the reduced model identification scheme. The mechanical topology for standard cardan hollow shaft drive-trains is shown in Fig.4.1(a). It slightly differs in its mechanical structure from the already discussed branched drive-train of Fig.4.1(b). The braking shaft is removed leading to an unbranched structure where the braking forces are directly applied on the wheel discs. Fi- nally, Fig. 4.1(c) depicts the third drive-train topology. Within its unbranched structure, the commonly utilized rubber joints are replaced by highly-stiff curved-tooth couplings. Consequently, this drive is only affected by wheel wear reducing the maintenance costs due to absence of rubber joint ageing. In all figures of mechanical topologies, such as those in both Fig. 4.1 4.1 Conventional approach 45

f1 f2 ωr

Te z2

(a) Unbranched seven-inertia system with rubber joints

f1 f2

ωr

Te z2

(b) Branched ten-inertia system with rubber joints

f1 f2 ωr

Te z2

Fig. 4.1 Various multi-inertia traction drive-trains. 46 4 Reduced model identification and Fig. 4.3, a linear color gradient for the inertia size is introduced, ranging from light grey to black. Scaling the inertias by gear transmission ratios to wheelset speed, the gears were removed. As a result, Fig. 4.3(a) reveals three characteristic dominant inertias (❶>❻>❼). For control design, especially for observer design, a reduced three-inertia model is required reflecting those characteristics of the multi-inertia system. The three-inertia system of Fig. 4.3(c) meets the inertia characteristics (①>②>③) and thus represents a motor-wheelset model. Such a suitably reduced model describes both the oscillatory behavior of the real system to a sufficient degree and the unknown adhesion forces are taken into consideration for an extended observer design. First of all, the simplest method for deriving a reduced three-inertia model is investigated, fitting the reduced model to the frequency response of the multi-inertia system. The pole-zero diagram in Fig.4.6 applies to all mechanical topologies of cardan hollow shaft traction drive-trains. It shows only the relevant two conjugated complex poles (x) and zeroes (o) as well as an integration pole where the influence of the wheel-rail contact as in Fig.3.3 is not considered. Neglecting the very low damping ratios σ2 and σ3, zero damping with unchanged natural frequencies is assumed for the reduced system. Thus the transfer function Gtd3(s) for the reduced model derives as

2 2 2 2 ωr(s) 1 (s +(ω2 − δ2) )(s +(ω3 − δ3) ) Gtd3(s)= = 2 2 2 2 = (4.1) Te(s) Jt s (s + ω2)(s + ω3) 4 2 b4 s + b2 s + b0 = 4 2 (4.2) s (a4 s + a2 s + a0) where so far both root-locus variables and total inertia Jt of the drive system are assumed to be known. Additionally, a comprehensive representation of

Im(s)

ω3 δ2 ω2 ~

~ ~

σ3 σ2 Re(s)

ωr (s) Fig. 4.2 Transfer function Gtd (s)= . 3 Te(s) 4.1 Conventional approach 47

f1 f2

ωr

Te z2

(a) Scaled inertias by gear transmission ratio to wheelset speed

ωr

(b) Frequency-response-based

ωr

z2 (c) Modal-approximation-based

Fig. 4.3 Mechanical multi-inertia cardan hollow shaft traction drive-train and its reduced three-inertia models without gears. 48 4 Reduced model identiﬁcation

Gtd3(s) is given with its numerator and denominator coeﬃcients. A comparison of coeﬃcients for the transfer functions of Eq.(4.1) and (4.2) leads to

2 2 a0 = ω2 ω3 (4.3) 2 2 a2 = ω2 + ω3 (4.4)

a4 = 1 (4.5) 2 2 ω2 ω3 b0 = (4.6) Jt 1 1 b = b + (4.7) 2 0 (ω − δ )2 (ω − δ )2 2 2 3 3 b0 b4 = 2 2 (4.8) (ω2 − δ2) (ω3 − δ3)

As proposed in [48], a further coefficient comparison of the comprehensive representation in Eq.(4.2) with the appropriate transfer function, based on the mechanical parameters of the three-inertia model, reveals a simple computational solution for each inertia and torsional stiffness 1 J1 = (4.9) b4 α1 = b0 J1 (4.10) a α = J 0 − 1 (4.11) 2 1 K 1 2 c1 = a2 J1 − b2 J1 (4.12) 2 c1 J2 = (4.13) b2 c1 J1 − K1 K2 J3 = K2 − J2 (4.14) K1 J2 J3 c2 = (4.15) c1 where two auxiliary variables, namely α1 and α2, are introduced to shorten the depiction. Finally, the parameter calculation based on the approximated frequency characteristics and the known total inertia Jt leads to the three- inertia system shown in Fig.4.3(b). It can easily be seen from the color gradient that the dominant inertia characteristics of the real system (❶>❻>❼) are not reflected by the reduced frequency-response-based model (①>③>②) compared with the desired motor-wheelset model of Fig.4.3(c) (①>②>③). Further, the inertia difference in both reduced models also becomes apparent in the model structure. Unlike in Fig.4.3(b), the inertia size of both wheel discs nearly coincides in Fig.4.3(c) facilitating a consideration of the wheel- rail contact in the control design. 4.2 Mode shape approach 49 4.2 Mode shape approach

To investigate why the frequency response fitting has those inertia characteristics, additional information about the drive system is required. Thus the appropriate mode shapes for the first three natural frequencies are calculated from an eigenvector problem for both branched and unbranched drive-trains. As a result, the derived mode shapes of Fig.4.4 facilitate a direct comparison of the vibration characteristics where the influence of the highly-stiff couplings becomes apparent. The curved-tooth couplings, unlike the rubber joints, achieve a highly-stiff integration of the hollow shaft within the drive-train. This obviously affects not only both natural frequencies of the unbranched drive ω2u > ω2b and ω3u > ω3b, but also the vibration behavior in both mode shapes in (❸,❹) and in (❺,❻) due to their high stiffness. Therefore the hollow shaft (❹,❺) experiences an increased torsional load. As a major advantage however, the highly-stiff couplings increase the oscillatory influence on the asynchronous motor (❶) in the second vibration mode and also the pole-zero gap δ3 in its frequency response according to Fig.3.4 leading to improved control characteristics of the highly-stiff drive. Regard- ing the branching in the drive-train, the contribution of the braking shaft to the oscillatory behavior of the traction drive-train is negligible for the first three vibration modes. Consequently, only the vibration characteristics of the asynchronous machine (❶) and the output shaft (❷-❼) have to be considered. This fact facilitates a simple and universal identification approach for both branched and unbranched traction drive-trains.

1 0 Hz 0.5 ω2b x ∗ 2 ω2u ϕ

, 0 x ∗ 1 ϕ -0.5

-1 ❶ ❷ ❸ ❹❽ ❺❾ ❻❿ ❼

1 ω3b 0.5 ω3u x

∗ 3 0 ϕ -0.5

-1 ❶ ❷ ❸ ❹❽ ❺❾ ❻❿ ❼

Fig. 4.4 First three mode shapes of a new un-/branched traction drive-train. 50 4 Reduced model identiﬁcation

To reduce the complex multi-inertia to a three-inertia model in a modal approach, the mode shape values of the three dominant inertias (❶,❻,❼) in Fig.4.4 are extracted to two reference vectors

∗ ∗ ∗ T ϕ21r ϕ26r ϕ27r [ϕ2r ϕ3r]= ∗ ∗ ∗ (4.16) " ϕ31r ϕ36r ϕ37r # where the first trivial mode shape is neglected. As a result, the reference mode shapes for both first and second vibration mode are introduced in Fig.4.5 for investigations into the inertia deviations from the desired reduced model. Subsequently, the mode shapes of the frequency-response-based three- inertia model are calculated by solving its eigenvector-problem and are also depicted in Fig.4.5. Thus a comparison could be made between the reference- and the frequency-response-based mode shapes revealing minor deviations in ϕ22 but significant in ϕ33. Those mode shape deviations, especially in ϕ33, are responsible for the deviation of the inertia characteristics (❻>❼) and (②<③) according to Fig.4.3(b). For this reason, not only the variables given by the frequency response characteristics but also the vibration characteristics, namely the mode shapes, have to be jointly considered to obtain a suitably reduced three-inertia model. Consequently, an initial attempt on the basis of the reference mode shapes is made realizing a three-inertia model with the appropriate natural frequencies ω2 and ω3 and the desired inertia-characteristics. Therefore the three mode shape vectors for a three-inertia system are defined as

T 1 1 1

[ϕ1 ϕ2 ϕ3]=  ϕ21 ϕ22 ϕ23  (4.17)  ϕ ϕ ϕ   31 32 33    The eigenvector problem for a three-inertia model is described by

2 (C − ωx J) ϕx = 0 (4.18) where x is the index of the mode shape vector, J is the inertia matrix

J1 0 0 J = 0 J 0 (4.19)  2  0 0 J3   and ﬁnally C the torsional stiﬀness matrix

c1 −c1 0 C = −c c + c −c . (4.20)  1 1 2 2  0 −c2 c2   4.2 Mode shape approach 51

From Eq.(4.18) the following realization conditions for a three-inertia model are derived

ω ϕ (ϕ − ϕ ) 2 =! 31 21 22 =! (4.21) ω3 sϕ21 (ϕ31 − ϕ32) ϕ (ϕ − ϕ ) =! 33 22 23 (4.22) sϕ23 (ϕ32 − ϕ33) where the appropriate mode shapes are conditional on the given ratio of both natural frequencies. Due to system order reduction, those conditions are not fulﬁlled by the given reference mode shapes and by the appropriate natural frequencies. And therefore a three-inertia system could not be realized with the exact reference mode shapes.

0.5 x

2 0 ϕ

-0.5 reference modal approximation -1 frequency response ① ② ③

(a) ω2

1 reference modal approximation 0.5 frequency response x

3 0 ϕ

-0.5

-1 ① ② ③

(b) ω3

Fig. 4.5 Mode shapes of various reduced three-inertia models. 52 4 Reduced model identiﬁcation 4.3 Modal approximation approach

Since an exact realization of the reference mode shapes failed, an approximation of them has to be used to obtain the desired dominant inertia- characteristics in the reduced model. Several variations of the known frequency response characteristic showed that with a more accurate mode shape approximation, unlike with the frequency-response-based mode shapes, a mi- nor deviation in ϕ33 could only be achieved by a smaller pole-zero distance δ3 for the reduced model. On this basis, ﬁrst of all an algorithm is described for calculation of the approximated mode shapes.

4.3.1 Approximated mode shape calculation

For approximated mode shape determination the two vectors of Eq.(4.17) have to be calculated, where both values ϕ23 = 1 and ϕ32 = −1 will be fixed parameters. The remaining four unknown parameters will be derived from the same number of equations. The first two are the conditions, namely Eq.(4.21) and Eq.(4.22), which realizable mode shapes have to fulfill. Since the pole-zero distance δ3 is a variable parameter, only knowledge of ω2, ω3 and δ2 is required to obtain a reliable mode shape calculation without having to determine the small pole-zero distance δ3. The third equation is provided by a match between the frequency response zero ω2 −δ2 of the real system and the reduced model. Thus the frequency response of the three-inertia system is symbolically calculated for the given input Te and output signal ωr on the basis of the mechanical parameters as

s4 + c1 + υ s2 + c1c2 ω (s) J2 J2J3 r = (4.23) c c J Te(s) J s s4 +(µ + υ) s2 + 1 2 t 1 J1J2J3 h i where Jt is the total inertia of the reduced three-inertia system equals the multi-inertia structure and 1 1 1 1 µ = c + υ = c + (4.24) 1 J J 2 J J 1 2 2 3 are auxiliary variables for shorter depiction. Finally the match of the zero is derived from Eq.(4.23) as third equation by

2 c1 c1 c1 c2 J + υ − υ − J + 4 J 2 ! v 2 2 2 ω − δ = u r (4.25) 2 2 u 2 u t 4.3 Modal approximation approach 53

∗ For a very close reference mode shape approximation, identity of ϕ33 = ϕ37 is assumed and thus the approximated mode shapes are calculated. These are as good as equal to the reference and therefore cannot be depicted in Fig.4.5. However, a nearly identical approximation of the reference mode shapes leads to the minimum pole-zero distance of δ3, namely δ3r as shown in Fig.4.6. This significant deviation from δ3 affects the controllability of the resulting modal three-inertia model adversely. So the two goals for the modal identification, picking a close approximation to the reference mode shape and having an exact match of the frequency response, namely δ = δ3, are mutually exclusive. This correlation of the design variables is shown in Fig.4.7, where in connection with both Fig.4.5 and Fig.4.6, all depicted data is referenced in between these three figures in a colour-coded way. Hence, there is an optimal approximation mode shape that covers both goals in a sophisticated manner. This optimal mode shape provides a trade-off between a deviation in the pole- zero distance δ3r < δ3a < δ3 as shown in Fig.4.6 and an excellent reference mode shape approximation according to Fig.4.5. Finally, ϕ33 achieves this trade-off simply as the fourth equation

ϕ33 = Trade-oﬀ parameter (4.26)

This algorithm provides the optimal approximated mode shape whose quality is veriﬁed in the following.

4.3.2 Quality of mode shapes

To evaluate the quality of the various mode shapes given in Fig. 4.5, the modal assurance criterion known from structural dynamics [49] is considered. It is usually applied to verify the calculated mode shapes derived by a theoretical ﬁnite element model with its experimentally measured oscillation behavior. However in this case, a quality comparison of the calculated mode shapes (ϕf , ϕa) for both reduced models of Fig.4.3 with the reference mode shapes ϕr from the multi-inertia model is made. The value of the modal assurance criterion (MAC) determines the degree of orthogonality, respectively the correlation, of two scaled mode shape vectors by their scalar product. Consequently, Eq.(4.27) states the modal assurance criterion for the reference ϕr and the frequency-response-based mode shapes ϕf

T 2 (ϕir ϕjf ) MAC(i,j)= T T i,j = 2, 3 (4.27) (ϕir ϕir)(ϕjf ϕjf ) where the indices i,j indicate the appropriate vibration mode. The MAC value ranges between 0 and 1, where 1 equals identity of both vectors. For comparison, the MAC values for both vibration modes are determined, re- 54 4 Reduced model identiﬁcation sulting in two MAC matrices of the appropriate vector sets

0.982 0.076 MAC{ϕr, ϕf } = (4.28) " 0.041 0.842 # 0.999 0.002 MAC{ϕr, ϕa} = . (4.29) " 0.010 0.996 #

In the ideal case, the main diagonal reveals only ones and the remaining matrix elements are zero. Since the exact reference mode shapes are not realizable, MAC values very close to one and zero are expected. The matrix deviations of MAC{ϕr, ϕf } from its expected values originate from the poor

δ3 = δ3f ≥ δ3a ≥ δ3r Im(s)

δ3r δ3a

δ3

ω3

~ ~

σ3 Re(s)

Fig. 4.6 Pole-zero diagram eﬀects of the various reduced three-inertia models.

δ3x/δ3 1

0.75 ~

~ ~ 0.4 1 ϕ33

Fig. 4.7 Correlation of both mutually exclusive variables δ3x and ϕ33. 4.3 Modal approximation approach 55 match in both mode shape values, namely ϕ22 and especially ϕ33. Compared to that, the MAC matrix of the approximated mode shapes conﬁrms the optimal approximation results, despite the parameter trade-oﬀ in ϕ33 in the third vibration mode. Consequently, the modal algorithm is developed on the basis of the approximated mode shapes.

4.3.3 Modal algorithm

Besides the realization conditions for mode shapes, a set of only four linearly independent equations is derived from the eigenvector problem of Eq.(4.18) for determining the ﬁve unknown mechanical parameters of the modal three- inertia model. The equations only describe proportions of the mechanical parameters deﬁned by the natural frequencies and the optimal approximated mode shapes

c1 ϕ31 2 = ω3 (4.30) J1 ϕ31 − ϕ32

c2 ϕ33 2 = ω3 (4.31) J3 ϕ33 − ϕ32

c1 ϕ22 (ϕ32 − ϕ33) 2 ϕ32 (ϕ22 − ϕ23) 2 = ω2 − ω3 (4.32) J2 ξ ξ

c2 ϕ22 (ϕ31 − ϕ32) 2 ϕ32 (ϕ21 − ϕ22) 2 = ω2 − ω3 (4.33) J2 ξ ξ where the denominator ξ is

ξ =(ϕ22 − ϕ23)(ϕ31 − ϕ32) ...

− (ϕ21 − ϕ22)(ϕ32 − ϕ33). (4.34)

As the ﬁfth equation, the actual total inertia Jt of the branched system is determined by a parameter estimation scheme to

Jt = J1 + J2 + J3. (4.35)

Finally, as the result the optimal modal three-inertia parameters according to Fig.4.3(c) are determined with only the knowledge of ω2, ω3 and Jt as variable parameters to be identified. 56 4 Reduced model identification 4.3.4 Effect of ageing and wear

For application on the vehicle signiﬁcant wheelset wear and rubber ageing have to be taken into consideration. Therefore in Fig.4.8 the mode shapes of the branched ten-inertia system are reconceived with a completely worn-out wheelset and aged rubber joints. In this case, the mechanical parameters, namely the rubber stiﬀness cr and the wheel disc radius rw, are assumed scaled to their nominal value in new condition.

′ cr = 2.25 (4.36) ′ rw = 0.898 (4.37)

′ ′ As a result, both natural frequencies increase ω2 > ω2 and ω3 > ω3 due to both effects. The ageing effect complies with a considerable increase in the rubber joint stiffness cr predominantly affecting the ϕ-values of the hollow shaft (❹,❺) in both mode shapes as can be easily seen in Fig. 4.8. Where on the other hand the wheelset wear, namely the decrease in wheel disc radius rw, predominantly causes a slight drift of the second mode shape in the inertias (❶-❺) and (❽-❿). However, the third mode shape, mainly defined by the twisted wheelset, is unaffected due to simultaneous wear on both wheels. Nevertheless the deviations in both mode shapes vary only negligibly ∗ ∗ in their values ϕ26 and ϕ37, which are essential for the proposed algorithm. The reference and therefore the optimally approximated mode shape vectors

1 0 Hz 0.5 ω2 x ′ ∗ 2 ω

ϕ 2

, 0 x ∗ 1 ϕ -0.5

-1 ❶ ❷ ❸ ❹❽ ❺❾ ❻❿ ❼

1 ω3 ′ 0.5 ω3 x

∗ 3 0 ϕ -0.5

-1 ❶ ❷ ❸ ❹❽ ❺❾ ❻❿ ❼

Fig. 4.8 First three mode shapes of the new and worn-out branched ten-inertia system showing wheelset wear and rubber-ageing eﬀect. 4.3 Modal approximation approach 57

ϕ2 and ϕ3 are neither aﬀected by wear nor ageing. Thus the optimal approximated mode shapes have to be calculated only once at the design stage of the traction drive-trains series. They are characteristic for every type of traction drive-train and determine the optimal three-inertia model independently of wheelset wear and rubber ageing.

4.3.5 Speed sensor location

The proposed scheme is also applicable with a speed sensor at the directly or indirectly driven wheel (❻,❼), unlike the frequency response ﬁtting method, which can not be applied due to the lack of information in the frequency

Im(s)

∗ δ2 ω3

ω2 ~

~ ~ σ3 σ2 Re(s) (a) G′ (s)= ωd(s) td3 Te(s)

Im(s)

ω3

ω2 ~

~ ~ σ3 σ2 Re(s) (b) G′′ (s)= ωi(s) td3 Te(s)

Fig. 4.9 Rootlocus diagrams for various speed sensor locations. 58 4 Reduced model identiﬁcation response at those sensor positions. First, the speed sensor at the directly driven wheel (❻) deriving the speed signal ωd is investigated. Besides the conjugated complex poles at the natural frequencies ω2 and ω3, this frequency ∗ response exhibits only one conjugated complex zero ω2 + δ2 , in contrast to Fig. 4.2. For cardan hollow shaft drive-trains this zero is typically located between both poles as shown in the root locus diagram of Fig. 4.9(a).

c1 s2 + c2 ′ ωd(s) J2 J3 Gtd3(s)= = (4.38) c c J Te(s) J s s4 +(µ + υ) s2 + 1 2 t 1 J1J2J3 h i ∗ c ω + δ = 2 (4.39) 2 2 J r 3 With Eq.(4.39) replacing Eq.(4.25), the approximated mode shape calculation at the design stage is simplified to a great extent since the square of ∗ ω2 + δ2 directly describes the correlation of two mechanical parameters. By analogy to the above, the complete identification scheme is carried out in a simplified way predominantly reducing the calculational effort at the design stage using ϕ31 as the trade-off parameter in this case. However, the frequency response of Eq.(4.40) with a speed sensor at the indirectly driven wheel (❼) measuring ωi does not exhibit any zero as shown in its root locus diagram in Fig.4.9(b). Therefore this frequency response provides only information about both vibration frequencies sufficient for the proposed reduced model identification.

c1 c2 ′′ ωi(s) J2 J3 Gtd3(s)= = (4.40) Te(s) J s s4 +(µ + υ) s2 + c1c2Jt 1 J1J2J3 h i So at the design stage the complexity of the approximated mode shape calculation is greatly reduced by using the zero from the frequency response of the directly driven wheel. Further for operation on the vehicle only the natural frequencies ω2 and ω3 need to be adapted by identiﬁcation from any available speed sensor position in the traction drive-train as long as the sensor is not located at any vibration node of either mode shape according to the vibration characteristics of Fig.4.8.

4.3.6 Modal scheme

In Fig.4.10 a comprehensive overview of the proposed modal scheme for reduced model determination is depicted. To cope with the computational limitations given by the traction application, substantial identiﬁcation eﬀort is anticipated by the optimal approximated mode shapes once calculated at 4.3 Modal approximation approach 59

ωsensor Optimal approx. mode Natural shapes

Design stage identiﬁcation

ϕ2 ϕ3 ω2 ω3 ω2 ω3

Jt Modal algorithm Parameter estimation

J1 c1 J2 c2 J3 cr rw

System monitoring

Real-time operation Optimal three-inertia model

Fig. 4.10 Computation-time minimized modal identiﬁcation scheme.

the design stage of the traction drive-train series. The modal characteristics ϕ2 and ϕ3 are independent of both rubber ageing and wheelset wear and thus represent a unique signature of the drive-train. This ingenious procedure not only simplifies the reduced model identification but also minimizes the computational effort of the proposed scheme for a tailor-made application on the vehicle. Consequently, at real-time operation only the two vibration frequencies ω2 and ω3 have to be determined from the speed signal. Here the identification scheme is generally applicable to any speed sensor location within the drive-train unless both vibration frequencies occur in its derived signal ωsensor. Additionally, a parameter estimation scheme provides the total inertia of the drive-train Jt as last input variable for the modal algorithm. The final results are the parameters of an accurate and suitably reduced three-inertia model, which is the basis for every control concept concerning the traction drive-train. 60 4 Reduced model identification 4.4 Parameter estimation

Besides both signature and vibration frequencies, the reduced model determination necessitates the knowledge of at least one physical mechanical parameter, namely in this case the total inertia of the drive-train. For this purpose, a simple estimation scheme is proposed in analogy to the previously introduced modal scheme. In addition to the total inertia, it also estimates variable mechanical parameters within the traction drive-train such as the rubber joint stiﬀness cr and the wheel disc radius rw. With continuous knowledge of these parameters [68–71], the mechanical traction system can be monitored precisely. Consequently, maintenance of the traction drive-train currently carried out at ﬁxed intervals can be scheduled adaptively according to requirements. This major feature of the novel parameter estimation scheme entails considerable savings in maintenance costs for railway vehicles.

f1 f2

ωr

Te z2

(a) Scaled inertias to wheelset speed with decoupling plane

ωr

z2 (b) Modal three-inertia system for parameter estimation

Fig. 4.11 Parameter estimation by decoupling rubber ageing and wheel wear. 4.4 Parameter estimation 61 4.4.1 Basic idea

In Fig. 4.11, the basic idea of the parameter estimation scheme is described on the basis of the complex branched drive experiencing wheel wear as well as rubber aging. Here in Fig. 4.11(a), and in Fig. 4.11(b), a linear colour gradient for the inertia size is introduced, ranging from light gray to black. Scaling the inertias by gear transmission ratios to wheelset speed, the gears are removed. As a result, Fig. 4.11(a) exhibits three characteristic dominant inertias (❶>❻>❼) of the branched traction drive-train. Hence the parameter estimation is accomplished with a reduced three-inertia model reflecting those characteristics of the multi-inertia system. The three-inertia system of Fig. 4.11(b) meets those inertia characteristics (①>②>③) and thus represents a motor-wheelset model. With this model all parameters of the multi-inertia system are estimated at once. Here, the torsional stiffness connecting the motor inertia (①) and the direct-driven wheel (②) of the reduced model depends mainly on the rubber stiffnesses cr and the sum of all three inertias equals the actual total inertia Jt of the branched drive. To obtain such a model the wear and aging effects have to be decoupled. This is visualized in Fig. 4.11 for each model by introducing decoupling planes that separate the inertia loss at the wheelset from the aging of the rubber joints. If we consider standstill of the drive, a change in torque ∆Te leads to oscillations of all decoupled inertias against the stopped wheelset with a characteristic frequency ωs, which sheds light on the aging status of the rubber joints. This frequency, constant for the unbranched drive because there is no aging effect, is simply determined at the design stage of the traction drive series against its two lowest eigenfrequencies. Further, the constant inertia size Js of all inertias within the decoupling plane is derived at the design stage. Applying this decoupling idea to the reduced model of Fig. 4.11(b), the motor inertia (①) equals Js and the rubber stiffness cr is directly determined by Js and ωs. Finally, in a modal approach, the remaining parameters are calculated from the known vibrational behavior of the cardan hollow shaft traction drive-train, independently of wear, aging and of its structural complexity.

4.4.2 Estimation criterion

To minimize the calculation effort of the estimation approach, previous knowledge of the multi-inertia system is also anticipated at the design stage. In analogy to the foregoing modal approach, the mode shapes of the drive-train are consulted due to their persistent characteristic despite wear and aging. Here the given realization conditions of Eq.(4.21) and Eq.(4.22), however, are insufficient to determine the approximated mode shape values. Therefore additional conditions need to be postulated to derive the appropriate mode shapes for parameter estimation. Both the mode shape of rigid body motion 62 4 Reduced model identification

ϕ11 = ϕ12 = ϕ13 = 1, (4.41) and the following mode shape values

ϕ23 = 1 ϕ32 = −1 (4.42) will be fixed parameters. The remaining four variable mode shape values will be determined from the same number of equations. Besides the realization conditions of Eq.(4.21) and Eq.(4.22), the described estimation idea provides an additional equation to facilitate the calculation of the appropriate mode shapes with the use of a fine-tuning parameter. To distinguish both variable parameters, namely the wheelset wear and the rubber joint ageing, the decoupling plane approach is applied as illustrated in Fig.4.11. At drive standstill any change of torque excites the asynchronous machine oscillating at a characteristic frequency ωs against the stopped wheelset. In this operating point the frequency ωs is just influenced by the rubber stiffness and is independent of wheelset wear. This frequency is determined once at the design stage of each traction drive series. It is constant for the unbranched drive because there is no variable parameter within the decoupling plane and it has to be adapted against its two lowest eigenfrequencies ω2 and ω3 for the branched drive as shown in Fig.4.12. In Fig.4.11(b) this estimation idea is transferred to the reduced model. Therefore a simple estimation criterion is derived on the basis of the known vibration frequency ωs as a function of the mode shapes.

2 c1 ! 2 ! ω3 ϕ31 = ωs = (4.43) J1 ϕ31 − ϕ32

120 →

100

[rad/sec] 80 st ω 60 220 550 180 500 ← → ω 140 3 [rad/sec]450 [rad/sec] 400 100 ω2

Fig. 4.12 Characteristics of the vibration frequency ωst at drive standstill. 4.4 Parameter estimation 63

Furthermore, Eq.(4.43) also defines the proportion of two estimated parameters of the modal three-inertia model, namely the inertia J1 and the torsional stiffness c1, due to the bounded one-inertia system at drive standstill. And as fourth equation the mode shape value ϕ33 is set as a fine-tuning parameter to minimize the total estimation error within the life span of wheels and of rubber joints. Consequently, the four unknown mode shapes values are symbolically derived.

2 ωs ϕ31 = − 2 2 (4.44) ωs − ω3 2 ωs ϕ21 = 2 2 ϕ22 (4.45) ωs − ω2 ω2 1 ϕ = 1 − 2 1+ (4.46) 22 ω2 ϕ 3 33 ϕ33 = Fine-tuning parameter (4.47)

Finally Fig.4.13 shows the results of the approximated mode shape calculation. Here, the gray shaded area depicts the variations of mode shapes in relation to the ﬁne-tuning parameter ϕ33 and within this area the mode shapes with the optimal choice of ϕ33 regarding the parameter estimation results are shown. The ﬁne-tuning procedure will be discussed with the investigation into the estimation error.

4.4.3 Estimation algorithm

To simplify the parameter calculation of the reduced model, here the decoupling plane approach is also applied. Besides the already discussed estimation

0.5 x 3 ϕ , x

2 0 ϕ , x 1 ϕ -0.5 0 Hz ω2 -1 ω3 ① ② ③

Fig. 4.13 Set of approximated mode shapes. 64 4 Reduced model identiﬁcation

0.67

0.66 33 ϕ

0.65

0 20 40 60 80 100 wheelset wear [%] Fig. 4.14 Fine-tuning parameter for the unbranched drive. criterion of Eq.(4.43), no inertia change appears within the decoupling plane and therefore inertia (①) is assumed constant as Js, namely the sum of all inertias located within the decoupling plane of Fig.4.11(a). On the basis of this decoupling approach and of the calculated mode shapes, the physical parameters of the three-inertia model are symbolically derived from the solution of the appropriate eigenvector-problem

J1 = Js = const. (4.48)

2 ϕ31 ! 2 c1 = J1 ω3 = J1 ωs (4.49) ϕ31 + 1 ξ J2 = c1 2 2 (4.50) ω3 (ϕ22 − 1) − ω2 ϕ22 (ϕ33 + 1) ω2 ϕ (1 + ϕ ) − ω2 (ϕ − ϕ ) c = J 2 22 31 3 22 21 (4.51) 2 2 ξ 1 − ϕ22 J3 = c2 2 (4.52) ω2 where ξ is an auxiliary variable for shorter depiction

ξ =(ϕ22 − 1) (ϕ31 + 1) ...

... − (ϕ22 − ϕ21)(ϕ33 + 1). (4.53)

This ingenious approach achieves a simple and stepwise calculation of the physical parameters in ascending order of Eq.(4.48)–(4.52). In the same simple manner the parameters of the traction drive are estimated on the basis of the reduced model 4.4 Parameter estimation 65

Jt = J1 + J2 + J3 (4.54)

cr ∼ c1 (4.55) 1 J + J 4 r ∼ 2 3 (4.56) w 2 where the actual total inertia Jt of the complex drive is represented by the total inertia of the estimated three-inertia model. Furthermore, the rubber joint stiffness cr is derived in a proportional relation to its equivalent stiffness c1 of the reduced model and finally the wheel disc radius rw is determined as a function of the wheelset inertia.

4.4.4 Estimation error

The estimation results for the unbranched highly-stiff drive in Fig. 4.15 reveal only a slight estimation error for the total inertia and for the wheel disc radius over the whole range of wheelset wear. These deviations result from the lower model order applied during estimation as well as the fixed previous knowledge derived only once at the design stage. Furthermore, the identification error ∆ω in both natural frequencies is also taken into consideration, leading to a tight errorband around the estimated parameters. The position

4 2 zero estimation error 0

] -2 h -4 -6 total inertia

2 parameter deviations [ errorband 0

-2

-4 wheel disc radius 0 20 40 60 80 100 wheelset wear [%]

Fig. 4.15 Parameter estimation results of the unbranched drive. 66 4 Reduced model identification of the zero estimation error is chosen at the stage of mode shape calculation by the fine-tuning parameter ϕ33. Therefore Fig.4.14 shows the optimal mode shape value ϕ33 for zero estimation error against wheelset wear. A fix value for ϕ33 is selected at the design stage in such a way that the estimation deviations including the identification errorband are minimized for a new as well as a worn-out wheelset. In case of the unbranched drive, the fine-tuning parameter ϕ33 = 0.66 is chosen for a zero estimation error at 54% of wheel wear minimizing the estimation deviations over the whole range of wheelset wear. As a final result the desired parameters are precisely estimated with negligible total deviations in the per mil range, e.g.the indirect-driven wheel with zero 2 2 wear J7 = 110.7 kgm is estimated in the worst case as J˜7 = 110.2 kgm .

4.4.5 Estimation scheme

In analogy to the modal scheme, the estimation approach is carried out universally valid for branched and unbranched drive-trains. Here, previous knowledge is also anticipated at the design stage of the traction drive series to minimize the calculational eﬀort. Therefore, the parameter estimation scheme shown in Fig.4.16 is divided into two sections: design stage and real-

ωsensor Optimal mode shape Natural

Design stage value identiﬁcation

ϕ33 ω2 ω3 ω2 ω3

Js Jt Modal ω Parameter estimation s algorithm

J1 c1 J2 c2 J3

Model for control design

cr System rw monitoring

Real-time operation

Fig. 4.16 Parameter estimation scheme for traction drive-trains. 4.5 Natural identification 67 time operation. At the design stage, the fine-tuning parameter, namely the optimal mode shape value ϕ33, and the decoupling parameters Js and ωs are determined. Hence, only the identification approach providing the actual vibration frequencies ω2 and ω3 of the traction drive-train and the parameter estimation algorithm are calculated in real-time operation. Finally, the complete estimation scheme supplies the total inertia Jt for the reduced model calculation for control design and both rubber joint stiffness cr and wheel disc radius rw for system monitoring and adaptive maintenance scheduling.

4.5 Natural identiﬁcation

Besides the constant signature of the traction drive-train, both modal and estimation algorithms necessitate an actual information [61–67] from the drive- train to cope with wear and ageing. For this purpose, the knowledge of the two slip-stick vibration frequencies is sufficient. To minimize the identification effort, a simple identification scheme is introduced based on the standard closed-loop control. For excitation of both vibration modes, no test signal injection is required here. Instead, natural effects of the traction drive-train are utilized: slip-stick vibrations. For this purpose, Fig. 4.17 shows the standard control concept. It consists of a PI-speed controller, the highly dynamic torque control [42], the inverter and the asynchronous machine characteristics, the traction drive-train and a superimposed readhesion controller. The novel idea is to reduce the closed- loop dynamics of both vibration modes by decreasing the proportional value of the speed controller as depicted in a simplified way in the root locus diagram of Fig. 4.18. Subsequently, the closed loop-poles of the drive-train shift to their corresponding open-loop poles on parabolic trajectories. With reduced system dynamics, slip-stick vibrations of both vibration frequencies originating from the wheel-rail contact arise. The vibration frequencies are determined by a Fast Fourier Transformation (FFT) of the speed sensor signal ωr in combination with polynomial filtering and additional calculations. The identified values match the natural frequencies of the traction drive-train, namely ω2

∗ ∗

ω T Te

r r Torque ASM - Inverter - PI Control Electric

z1 Slip z2 Readhesion ωr Traction drive-train

Fig. 4.17 Natural identiﬁcation scheme. 68 4 Reduced model identiﬁcation

Im(s)

~ ~ ω3

δ2 ω2 ~

σ3 σ2 Re(s)

Fig. 4.18 H′(s): Closed-loop pole trajectories: Decreasing the proportional value of the speed controller.

and ω3, to a sufficient extent, where the damping factors σ2 and σ3 are not taken into consideration. Finally, the passive readhesion controller minimizes ∗ the stress of the mechanical drive-train reducing the reference torque Tr and simultaneously the parameters of the PI-speed controller are increased to the original standard values. This concludes the natural identification scheme. Due to parameter modifications of the speed controller, additional torque sensors are mounted on both hollow and wheelset shaft to monitor the vibrational load (T45, T67) on the drive-train for the first identification test run. As a result, the slip-stick vibrations stay within their limits, so there is no need for additional sensors for the natural identification scheme. The natural identification procedure is carried out only at certain intervals during slip-stick operation deriving an accurate reduced model as a basis for the anti-vibration control. Furthermore based on the identified natural frequencies, the parameter estimation provides wear and ageing characteristics, namely the rubber joint stiffness cr and the wheel disc radius rw, for system monitoring and adaptive maintenance scheduling. Chapter 5 Modal estimator

To apply state-space control to the traction drive-train, all system states must be available for feedback. Therefore a modal estimator is required to reconstruct the state variables which are not measured by a sensor. On the basis of the identiﬁed modal three-inertia model of the traction drive-train, two diﬀerent estimator designs in the time domain are introduced, namely a full-order estimator with an extended disturbance model and a reduced- order estimator. Subsequently, the estimator design in the frequency domain is presented in a ingenious simple way. Finally, the estimators are compared in view of their performance and suitability.

5.1 Full-order estimator

The classical full-estimator approach based on Luenberger [50, 72–74] was ﬁrst applied by [51] for the traction application. This approach will be discussed in the following, further simpliﬁed as well as enhanced by an improved disturbance model. In the basic simulation model of Fig.2.10, the state of the electrical system is already reconstructed by the highly dynamic torque control [42] using an electromagnetic torque observer. This reduces the order of the necessary estimator by one. Hence only a modal full-order estimator is required which estimates the states of the mechanical three-inertia system from the estimated electromagnetic torque u = T˜e and from the measured rotor speed y = ωr as depicted in Fig.5.1. The state-space representation of the undamped three inertia-system is derived as

˜x˙ = A ˜x + b u y = c ˜x (5.1) T ˜x = ω˜1 T˜12 ω˜2 T˜23 ω˜3 (5.2) h i where T˜12 = T45 of the hollow shaft and T˜23 = T67 of the wheelset shaft. 70 5 Modal estimator

0 − 1 0 0 0 J1

 c1 0 −c1 0 0  A =  0 1 0 − 1 0  (5.3)  J2 J2     0 0 c 0 −c   2 2     0 0 0 1 0   J3   T  b 1 0000 = J1 (5.4) c = 1 0000 (5.5) Due to the functional similarity of the traction drive-train and its reduced modal three-inertia model of Fig.5.1, the adhesion forces are also taken into account for more accurate estimation results. Therefore the classical Luen- berger full-order estimator is extended with an internal disturbance model for step-shaped signals according to Johnson [52]: an integrator. This disturbance model estimates the sum of the adhesive forces (z1 + z2) and feeds it back to the wheelset. Subsequently, the modified vector of the state variables is T ˜x = ω˜1 T˜12 ω˜2 T˜23 ω˜3 z˜1 +z ˜2 (5.6) h i A reconstruction of the real adhesive forces z1 and z2 out of the sumz ˜1 +z ˜2 z˜1+˜z2 is impossible. Thus in [51] it was assumed that the mean value 2 is fed back to each wheel. Several simulations showed that such a feedback choice affects the disturbance reaction of the control loop negatively due to the slight difference between the inertia values J2 and J3. So a separation factor a is introduced for Eq.(5.8) which describes the above statement from [51] for a = 1. A variation of this parameter reveals an optimal disturbance reaction of the closed control loop for

ω˜1 ωr ω˜2 ω˜3 T˜12 T˜23

Fig. 5.1 Model and system variables of the full-order estimator. 5.1 Full-order estimator 71

Traction drive-train

u = T˜e y = ωr

z1 z2

˜x T y˜ ˜x˙ = A ˜x + bu + r c -

r y − y˜ l

Fig. 5.2 Full-order estimator design.

2 J a = 3 (5.7) J2 + J3 The physical interpretation of Eq.(5.7) is that both wheel inertias experience the same acceleration from any disturbance change. So only the ﬁrst vibration mode will be excited in the estimator and no torsion of the wheel shaft will occur. This optimal choice is used for the entire range of wheelset wear.

0 − 1 0 0 0 0 J1

 c1 0 −c1 0 0 0  −  0 1 0 − 1 0 a 2   J2 J2 J2  A =   (5.8)  0 0 c 0 −c 0   2 2     0 0 0 1 0 − a   J3 J3     00 0 0 0 0     T  b 1 00000 = J1 (5.9) c = 1 00000 (5.10) According to Fig.5.2, the full-order estimator with the extended disturbance model is designed using l as feedback gain for the diﬀerence between the measured and estimated output. It continuously corrects the undamped modal three-inertia model with the error signal ˆx = x − ˜x.

˜x˙ = A ˜x + b u + l (y − cT ˜x) (5.11) where the estimator gain l is

T l = l1 l2 l3 l4 l5 l6 (5.12) 72 5 Modal estimator

Undamped modal three-inertia model

ωr 1/J1

T˜e ω˜1 ω˜1 - - l1

c12

- l2 T˜12

1/J2 ω˜2 - l3

a−2 a c23

- l4 T˜23

1/J3 ω˜3 - a l5

l6 z˜1 +z ˜2

Disturbance model Estimator gain l

Fig. 5.3 Block diagram: Full-order estimator design with extended disturbance model.

The error signal ˆx has to decay to zero

ˆx˙ =(A − lcT) ˆx (5.13) revealing the characteristic equation for the error

det sI − A + lcT = 0 (5.14) with reasonably fast and stable eigenvalues by the pole placement technique. Here, the characteristic estimator polynomial ∆ˆ(s) is assigned to have the desired zeros. ∆ˆ(s) =! det sI − A + lcT (5.15) 5.2 Reduced-order estimator 73

The solution of the Eq.(5.15) determines the proportional gain l and concludes the full-order estimator design as depicted in Fig.5.3.

5.2 Reduced-order estimator

As the speed of the asynchronous machine ωr is already directly measured, a full-order estimator with the estimated stateω ˜1 = ωr, as shown in Fig. 5.1, is not necessary. Subsequently, the order of the estimator can be further reduced by the number of the measured state variables, namely in this case by one. Regarding the estimator complexity and thus the computational eﬀort, the lowest estimator-order shall be achieved neglecting also the sophisticated disturbance model from the latter full-order estimator approach. A generic design of a reduced-order estimator is well known in the literature [53] and applied in the following to traction drive-trains for the ﬁrst time. The tailor-made approach of a reduced-order estimator for the traction application bases on a reduced modal model which is obtained by partitioning the state-space description Eqs.(5.1)–(5.5) of the three-inertia system. Here, the state space vector x is split into two parts: xa is the measured state variable, namely ωr, and xb contains the remaining unmeasured state variables.

x˙ Aaa Aab x b a = a + a u (5.16) ˙xb xb bb " Aba Abb # x y = 1 0 a (5.17) xb

ωr ω˜2 ω˜3 T˜12 T˜23

Fig. 5.4 Model and system variables of the reduced-order estimator. 74 5 Modal estimator

Traction drive-train

u = T˜e y = ωr = xa

z1 Aba − l Aaa l z2

˜x˙ b − l y˙ ˜xb − l y ˜xb bb − l ba 1/s ˙xc xc

Abb − lAab

Fig. 5.5 Reduced-order estimator design.

In Fig.5.4 the reduced modal model with the unmeasured state variables xb is depicted which is derived from the partitioning process. The partial system equation for xb is given by

˙xb = Abb xb + Aba xa + Gb u (5.18) known measurement where the term with xa = y = ωr and| u-term{z are} both measurable and are therefore considered as input values. This also applies to the measured xa-state variable equation

x˙ a =y ˙ = Aaa y + Aab xb + ba u (5.19) which is analysed in the same way and all known input terms are collected on the left side. y˙ − Aaa y − ba u = Aab xb (5.20)

known input

On the right side only the| unknown{z state} variables ˜xb remain. Comparing the latter equations with the full-order estimator design of Eq.(5.11), several terms can be substituted in analogy to the full-order estimator. These substitutions reveal the reduced-order estimator design as

˜x˙ b = Abb ˜xb + Aba y + bb u + l (˙y − Aaa y − ba u − Aab ˜xb) (5.21) input measurement T l = l1 l2 l3 l4| {z } | {z } (5.22)

The error signal ˆxb = xb − ˜xb from Eqs.(5.18) and (5.21) has to decay to zero ˆx˙ b =(Abb − l Aab) ˆxb (5.23) 5.2 Reduced-order estimator 75 revealing the characteristic equation for the error

det[sI − Abb + l Aab] = 0 (5.24) with reasonably fast and stable eigenvalues by the pole placement technique. Here, the characteristic estimator polynomial ∆ˆ(s) is assigned to have the desired zeros. ! det[sI − Abb + l Aab] = ∆ˆ(s) (5.25) Finally, Eq.(5.21) is rewritten as

˜x˙ b =(Abb − l Aab) ˜xb +(Aba − l Aaa) y +(bb − l ba ) u + l y˙ (5.26) and implies the derivative of the measured y = ωr. The derivative is avoided in a simple way by introducing a new controller state ˜xc = ˜xb − l y. On this basis, the reduced-order estimator is designed ˙ ˜xc =(Abb − l Aab) ˜xc +(Aba − l Aaa) y +(bb − l ba ) u (5.27) and is depicted in Fig.5.5. For the traction application, the corresponding matrices and vectors are

Aaa = 0 (5.28) A − 1 0 0 0 ab = J1 (5.29) T Aba = c1 0 0 0 (5.30) 0 −c1 0 0 1 0 − 1 0  J2 J2  Abb = (5.31)  0 c2 0 −c2       0 0 1 0   J3  1  ba = (5.32) J1 T bb = 0000 (5.33) resulting in the block diagram of Fig.5.6 for the reduced-order estimator. The characteristic equation for the calculation of the estimator gain l is

s − l1 c 0 0 J1 1  − 1 − l2 s 1 0  J2 J1 J2 ! det[sI − Abb + l Aab] = det = ∆ˆ(s) (5.34)  l3   − J −c2 s c2   1   l4 1   − 0 − s   J1 J3    with its simple symbolic solution due to the many zeros in the matrices. 76 5 Modal estimator

xa = ωr Estimator gain l

l1 l2 l3 l4

1/J l 1 1 ˜ T˜e T12 -

c12

l2 ω˜2

1/J2

l3 T˜23

c23

l4 ω˜3

1/J3

Undamped modal reduced model

Fig. 5.6 Block diagram: Reduced-order estimator design.

l J J c + J J c + J J c + J J c l det[...]= s4 − 1 s3 + 1 2 2 1 3 1 1 3 2 2 3 1 2 s2 ... J1 J1 J2 J3 J c l + J c l + J c l J c c + J c c l + J c c l − 2 2 1 3 2 1 3 1 3 s + 1 1 2 2 1 2 2 3 1 2 4 (5.35) J1 J2 J3 J1 J2 J3 Assuming the desired estimator poles are chosen according to Fig.4.2 to −σ2 ± iω2 and to −σ3 ± iω3, the desired estimator polynomial derives as

ˆ 2 2 2 2 2 2 ∆(s)=(s + 2 σ2 s + σ2 + ω2)(s + 2 σ3 s + σ3 + ω3)= 4 3 2 2 2 2 2 = s + (2 σ2 + 2 σ3) s +(σ2 + 4 σ2 σ3 + σ3 + ω2 + ω3) s + 2 2 2 2 2 2 2 2 (2 σ3 (σ2 + ω2) + 2 σ2 (σ3 + ω3)) s +(σ2 + ω2)(σ3 + ω3) (5.36)

Matching the coefficients of both Eqs.(5.35) and (5.36), the estimator gain l is determined based on a symbolic design. As can be seen from the coefficient 5.3 State estimation by transfer functions 77 comparison, the estimator gain l is not only a function of the parameters of the desired estimator polynomial but also of both inertia and stiffness values. Subsequently, all parameters of the block diagram of Fig.5.6 are known and the reduced-order estimator can be implemented for all four unmeasured system states.

5.3 State estimation by transfer functions

However, if only one system statex ˜, such as e.g. the torque of the wheelset shaft T˜23 = T67, is required, it can be reconstructed by two transfer functions from the estimated electromagnetic torque and the measured asynchronous machine speed. The two transfer functions of the estimator generally exhibit the same denominator ∆ˆ(s), namely the desired estimator polynomial, and only the nominator polynomials N(s) have to be calculated.

NT (s) NΩ(s) FT (s)= FΩ(s)= (5.37) ∆ˆ(s) ∆ˆ(s)

This single state estimator design is depicted in Fig.5.7 with the output of one scaled state variablex ˜. For system monitoring purposes, the state variable has to be de-scaled again by the transmission ratio of the gears

ω˜ T˜ ω˜ T˜ ω˜ z + z = 1 u 1 u 1 u (5.38) 1 12 2 23 3 1 2 1 u1 1 u1 1 h i The estimator design can be obtained in two diﬀerent ways. The indirect approach derives the nominator polynomials on the basis of the known estimator gain l where in the direct approach this knowledge is not required.

Traction drive-train

u = T˜e y = ωr

z1 FΩ (s) z2

x˜ FT (s)

Fig. 5.7 Estimator design for a single state variablex ˜ by two transfer functions. 78 5 Modal estimator 5.3.1 Indirect approach

Using the determinant of the Rosenbrock matrix [54], the nominators Nx(s) are derived in a simple way.

sI − A −b N (s) = det (5.39) x c d where d is the feed-through and with the vector c the desired state variable is chosen. The application of the Rosenbrock matrix to derive NT (s) e.g. for the wheelset shaft torque is based on the Eq.(5.27) of the reduced-order estimator

T˜ sI − A + l A −(b − l b ) N 23 (s) = det bb ab b a (5.40) T ˜ c 0 Te ! where the vector choosing the appropriate state variable, e.g. the wheel shaft torque, is c = 0010 (5.41)

For calculation of the corresponding NΩ(s), the input vector equals

b = Aba − l Aaa (5.42) and the appropriate feed-through

d = l3 (5.43) has to be considered. Subsequently, both nominators and denominators for any state variable of the reduced-order estimator are calculated and shown in Eq.(5.44). On the basis of the known estimator gain l, this indirect approach for two transfer functions can also be applied in analogy to any estimator determining one single state.

5.3.2 Direct approach

Compared with the indirect approach, the direct approach, discussed in [55] for a generic estimator, requires a considerably reduced design eﬀort as the estimator gain l is not being calculated. Therefore the direct approach is the preferred method for deriving a single state estimator by two equivalent transfer functions. As the denominator is already given by the desired estimator polynomial ∆ˆ(s), only the nominator polynomials NΩ(s) and NT (s) have to be calculated. For the introduced full-order estimator with the disturbance 5.3 State estimation by transfer functions 79 2 2 . . . c (5.44) J 1 + c 1 s J 4 l + 2 c 2 1 3 1 c J c J 1 4 c 1 l J 2 l − . . . + 2 . . . c + 3 2 1 1 c J s + J c 1 2 s 2 c J l 1 2 c + 1 J 3 c 1 l l 1 2 1 4 2 c 2 J l J l J + 3 3 l 1 l s + 1 J l − 2 2 c 1 − 2 2 J c 2 2 c J l 1 2 J 3 4 3 l c 1 1 J l 1 l J 1 J 3 1 J l l 1 l + 1 J l − − 2 3 c 3 − 2 2 J 3 J 2 c c l 1 2 J 1 1 2 1 4 1 J J l l c 1 J l ) 1 1 J 2 3 l l J − 12 + c ) − 2 2 3 − 3 1 c 2 l J J c 3 1 2 J 1 1 J 4 3 c c 2 c + l l J 1 J 3 1 l J c + + 23 c 2 2 − 1 J c 3 l + + 2 ( J 2 1 l s c 2 3 2 2 ( 3 23 3 4 4 s s l l J J 2 c J J J 2 2 1 1 23 23 2 2 1 l l 1 c c 12 J J 1 + 1 c J J 1 1 c 1 J J 2 2 1 c c l 1 12 4 l + + s 1 1 J l c J 3 4 l l J J 2 2 + + 2 2 − + J J − 4 4 l l 1 l l 2 s s s 1 1 s 1 c 2 c J J ) J + + 2 J 2 l l 12 1 1 2 + + 2 3 c 1 1 c c 12 23 23 J + J J + c J J c 3 3 3 c 3 4 1 l 1 l l 1 J J 2 3 2 4 2 4 2 1 J l J l J l l l c J 2 c 1 2 + + + J J J 2 + + l + + 2 2 23 2 2 c c + + c 3 2 2 2 1 + 23 J J 12 23 J 3 4 J 1 c J J c 3 2 3 l l c l c 4 1 2 1 ( 1 J J J l 1 4 2 2 3 4 c J l J l l l + J l 2 + + 2 1 1 3 J 2 l J J J J l 3 2 1 1 1 + + 1 c J 2 2 c c J + + + + − 2 J J 3 3 4 23 l l 23 J J 2 3 3 2 1 c c 3 − 1 s 1 1 c 1 l J J J l 2 2 1 2 4 2 + + ) J J 2 l l l J l 2 2 2 1 1 1 l 1 s l 2 2 + l J J J ) 3 3 c c 3 2 + J J − 3 4 l s + + + − − l l 2 3 1 2 2 3 3 3 l − c J s ( s J J J 3 4 1 1 1 2 1 l l + + 2 3 2 2 2 l J ( 12 3 J J J J J 1 3 3 3 c 1 s l 1 J s s 1 1 23 − 2 J J J l c 2 3 4 1 l 1 1 1 2 2 2 l l c c c c J 1 1 1 J J + + + − l l l 1 1 1 3 3 3 3 c c c s s s s + − − − 1 1 1 1 4 4 4 4 + + + 2 4 1 3 l l l l s s s s J J J J 1 3 2 4 − − l l l l − − ...... ====== ! ! ! ! 3 3 2 2 r r e e r r e e ˜ ˜ 23 12 12 23 ˜ ˜ ˜ ˜ T T ω ω ω ω ω ω ˜ ˜ T T ω ω ˜ ˜ ˜ ˜ T T T T

T T Ω Ω T T Ω Ω N N N N N N N N 80 5 Modal estimator model for traction application, the six-order polynomials are

5 4 NT (s)= Nt5 s + Nt4 s + ... + Nt1 s + Nt0 (5.45) 5 4 NΩ(s)= Nω5 s + Nω4 s + ... + Nω1 s + Nω0 (5.46) 6 5 ∆ˆ(s)= s + ∆ˆ5 s + ... + ∆ˆ1 s + ∆ˆ0 (5.47)

As a basis for the following design, a vector ra in analogy to the latter approach is introduced also choosing the desired state variable, e.g. again the wheelset shaft torque

ra = 000100 (5.48) Subsequently, the polynomial coefficients of NΩ(s) are calculated by matching the coefficients. c A5 4  c A  . Nω5 Nω4 Nω3 Nω2 Nω1 Nω0  .  =      c A     c     6 ra A r A5  a  ˆ ˆ ˆ ˆ ˆ ˆ . = 1 ∆5 ∆4 ∆3 ∆2 ∆1 ∆0  .  (5.49)      ra A     ra      After that, the following equation determines the coefficients of NT (s) directly.

Nt5 Nt4 Nt3 Nt2 Nt1 Nt0 = ˆ ˆ ˆ = ra (∆5 ra − Nω5 c)(∆4ra − Nω4 c) ··· (∆1 ra − Nω1 c) ... 2 5 b A b A b ··· A b 0 b Ab ··· A4 b   ......  . . . . .  (5.50)    00 0 bAb       00 0 0 b      Due to the sparsely occupied matrix A and vectors b, c and ra the equations can even be solved symbolically with a relatively simple solution. 5.4 Comparison of estimators 81

To reduce the computation eﬀort, the proposed equations (5.49)–(5.50) for the direct approach have to be adapted to the estimator equation providing a tailor-made approach for the two transfer functions.

5.4 Comparison of estimators

The two modal estimators, namely the full- and reduced-order estimator, are discussed in the following on the basis of their simulation studies in reference to the operational cycle of Fig.3.6 with the real system variables. Finally, recommendations for various designs of estimators are given regarding both their open-/closed-loop suitability and their applicability.

5.4.1 Simulation studies

First, the simulation studies of the full-order estimator with the advanced disturbance model are investigated in Fig.5.8. A direct comparison with Fig. 3.6 reveals on first sight, that all system variables are reconstructed to a great extent. In detail, the measured rotor speed ωr = ω1 is filtered by the estimator where the sensor noise is significantly reduced in the estimated rotor speedω ˜1. The estimated direct- and indirect-wheel speedω ˜2 andω ˜3 are reduced by the gear transmission ratio and show a increased slip-stick vibration amplitude equal to the given mode shapes of the modal three-inertia model. Further, the estimated hollow T˜12 and wheelset shaft torque T˜23 are also a good match with their estimated vibration amplitudes slightly higher due to sensor noise. Finally, the estimated transmittable traction force of the indirect wheel is shown on the basis of integrated advanced disturbance model which is used to reconstruct the wheel-rail contact. Second, the simulation studies of the reduced-order estimator without a disturbance model are shown in Fig.5.9. Here, a direct comparison with the latter simulations reveals, that only the vibrational behavior of the traction drive-train is reconstructed in a sufficient manner. In detail, an estimated signal of the rotor speedω ˜1 is redundant as the measured signal ωr with sensor noise is used. The estimated torques T˜12 and T˜23 do not match the reference action of the real system torques and thus only the slip-stick vibration amplitudes are correctly reconstructed. In all estimated signals of the reduced-order estimator there is a higher noise level compared to the full- order estimator as the estimator order is lower which results in Fig.5.6 in a direct feedthrough of the measured rotor speed and respectively amplifies the sensor noise. Finally due to the lack of a disturbance model, the transmittable traction force and respectively the wheel-rail contact is not reconstructed and thus not considered in the estimated torque signals T˜12 and T˜23 which is the reason for the deviation in their reference action. 82 5 Modal estimator

20 1

˜ 10 ω

40 12 ˜ T 20

8 6 2 ˜

ω 4 2 0

100 23 ˜ T 0

-100

8 6 3 ˜

ω 4 2 0

40 ) 2 z

+ ˜ 20 1 z (˜ a 0 0 1 2 3 4 5 time [s]

Fig. 5.8 Studies of a full-order estimator with disturbance model with various negative rad adhesion force gradients at constant high wheel-rail slips ( ω [ sec ], T [kNm]). 5.4 Comparison of estimators 83

1 10 ω

20 12 ˜ T 0

-20

8 6 2 ˜

ω 4 2 0

100 23 ˜ T 0

-100

8 6 3 ˜

ω 4 2 0 0 1 2 3 4 5 time [s]

Fig. 5.9 Studies of a reduced-order estimator without disturbance model with various rad negative adhesion force gradients at constant high wheel-rail slips ( ω [ sec ], T [kNm]).

For a simple and direct comparison of the real system variables in Fig.3.6 and the estimated ones in both Fig.5.8 and Fig.5.9, the same scaling of the signals is applied in their depiction. 84 5 Modal estimator 5.4.2 Overview

On the basis of several criteria, various estimator designs are discussed in Tab. 5.1: • Estimator order results from both traction drive-train and disturbance model • Noise amplification and computation effort are strongly connected to the realized estimator order • Reference action of the estimated system variables and adhesion estimation are based on the reconstruction of the wheel-rail contact which is achieved by the advanced disturbance model • Reference vibration and mode shape design are obtained with the identified reduced modal model of the traction drive-train which represents the vibration characteristics of the real system.

Tab. 5.1 Comparison of various modal estimators.

Full & Full Reduced & Reduced disturbance disturbance model model

Estimator order six five five four Computation effort high medium medium low Noise amplification low medium medium high Reference action yes no yes no Adhesion estimation yes no yes no Reference vibration yes yes yes yes Mode shape design approx. approx. approx. approx.

Loop suitability closed open open open open Application state-space system vibrational system vibrational control monitoring load monitoring load

Finally, a recommendation for the modal estimator use regarding both suitability and applicability is given:

The full-order estimator with the advanced disturbance model is the best choice for both closed-loop control and open-loop system monitoring. On the other hand, if only the vibration behavior in open-loop has to be monitored, the reduced-order estimator is suﬃcient despite its high noise ratio in the estimated output signal. Chapter 6 Anti-vibration control

First, the state of the art passive readhesion control is enhanced to a great extent by introducing the novel modal estimator for traction applications. Second, applying the estimator, a state space control scheme with a feed- forward control is proposed in the time domain. Subsequently, this control scheme is simpliﬁed by a frequency domain approach. Therefore a new design method is introduced. The limits of the active control scheme are pointed out and circumvented by coupling with the passive readhesion control. Further, the coupling entails an ingenious anti-windup prevention scheme. Third, on the basis of a mechanical absorber study, a novel virtual absorber control scheme is proposed for both road and rail vehicles. Finally, the various control schemes are compared in view of their performance and suitability in simulation studies.

6.1 Passive readhesion controller

To suppress slip-stick vibrations in traction drive-trains, passive readhesion controllers [6,97–101] are usually utilized at the expense of traction force.

6.1.1 State of the art

There are two different approaches of starting the passive controller. The first one is just simply based on bandpass filtering the speed sensor signal of the asynchronous machine ω1. However in the second approach [10], a complex Kalman-Bucy estimator in combination with a bandpass filter is applied to identify slip-stick vibrations. On the occurrence of the slip-stick ∗ vibration frequency, both approaches reduce the reference torque Tr and thus the traction force which is the major drawback of this simple type of control 86 6 Anti-vibration control scheme. The second approach is more sophisticated, but on the other hand more pedestrian as it identifies the exact wheelset-shaft torque T˜67 to finally bandpass filter it again obtaining only the alternating component, namely the slip-stick vibrations.

6.1.2 Advanced concept

To improve the starting of the passive readhesion controller, a novel front end is shown in Fig.6.1. The modal estimator without disturbance model from the previous chapter is applied only reconstructing the alternating component T˜67, namely the slip-stick vibrations in the wheelset shaft. The resulting signal waveforms are displayed at the output of each function block. In the following, the absolute value of the vibration signal and subsequently the envelope curve is derived. The maximum allowable vibration amplitude sets the upper limit for the vibrations that may occur without triggering the passive readhesion controller. Due to severe sensor noise amplification, originated from both rotor speed of the asynchronous machine and use of the estimator, a tolerance band is introduced by the nonlinearity. The latter creates the tolerance band ∗ by suppressing negative output values. The resulting reference torque Tp is the output signal of the front end towards the standard passive readhesion ∗ controller. The task of the controller is the reduction of the Tp -signal to zero. In parallel, the damping factor of the closed-loop σcl is obtained by filtering the envelope curve signal. Subsequently, a mapping scheme derives the adhesion force gradient δµ/δ(∆v) where the damping factors of the traction drive-train σsys and of the standard speed controller σpi are subtracted and the result σac is multiplied by a constant factor. Several examples for the mapping scheme are shown in the adhesion characteristics curve in Fig. 6.1 on the basis of both branched traction drive-train and standard control parameters. The adhesion force gradient is provided as output signal of the front end towards the traction force controller. With this information, the traction force controller can determine the optimal operating point in the adhesion characteristics curve more efficiently.

Tab. 6.1 Output signal evaluation of advanced passive readhesion front end. ∗ δµ Tp δ(∆v) Recommendation 0 > 0 No action 0 < 0 Attention ∗ > 0 > 0 Tp -evalution ∗ > 0 < 0 Tr -reduction

Finally, both output signals of the front end, namely the reference torque ∗ Tp and the adhesion force gradient δµ/δ(∆v) are evaluated with an AND- 6.1 Passive readhesion controller 87 passive readhesion controller controller Standard Traction force ) ∗ p ∆v δµ ( T δ ac pi σ σ - - + limit ) | sys Vibration cl σ 67 σ ˜ T | ( Env Filter Envelope | 67 ˜ T | Mapping ) 67 ˜ cl T ∆v δµ σ ( δ r ] ω ) = s m s [ ( Ω 09 ∆v 5 F . . 0 7 035 − 0 . drive-train Traction 0 − 2 z ) s ( 5 T 0 speed diﬀerence −

µ 1 coeﬃcient n dhesio a z Advanced passive readhesion front end control scheme. e ˜ T Fig. 6.1 88 6 Anti-vibration control relation and recommendations are outlined in Tab.6.1 where the attention ∗ and the Tp -evalution case are discussed in following in greater detail. • Attention has to be paid, in case the adhesion force gradient is negative but ∗ Tp = 0. The vibration amplitude is still in the tolerance band. Depending on the negative value of the adhesion force gradient, the reference torque ∗ Tr can be predictively reduced to prevent the slip-stick vibrations. ∗ • Unlikely Tp > 0, then considerable slick-stick vibrations occur, but the adhesion force gradient is positive. The vibration amplitude is already ∗ declining. Depending on the Tp -evalution, the reduction of the reference ∗ torque Tr can be predictively avoided to prevent a loss of traction force. Compared to state-of-the-art controllers, the novel advanced controller can predict the behavior of slip-stick vibrations and prevent either a loss of traction force or even the slip-stick vibration itself.

6.2 Active anti-vibration control scheme

To prevent the loss of traction force entirely, an active anti-vibration control [75–88] is introduced on the basis of a state space control approach to provide additional damping for the two slip-stick vibration modes of the traction drive-train.

6.2.1 Design in the time-domain

First of all, the modal state control will be developed in the time domain as shown in Fig.6.2. The controllable and observable system consists of an electric and a mechanical part. All nonlinearities of the electric system, like the closed-loop of the torque control, the inverter characteristic and the asynchronous motor, are approximated for control purposes by a linear ﬁrst order lag element with time constant τs. The mechanical part is the branched traction drive-train and is approximated by a modal undamped three-inertia system as discussed in Chapter 4. Thus the linearized system is described by the following state-space equation

∗ ˙xc = Ac xc + bc Tr yc = cc xc (6.1) T xc = ω1 T12 ω2 T23 ω3 Te (6.2) where the state vector xc consists of three angular frequencies, two shaft torques and the electromagnetic torque. 6.2 Active anti-vibration control scheme 89

0 − 1 0 0 0 1 J1 J1

 c1 0 −c1 0 0 0  1 1  0 0 − 0 0   J2 J2  Ac =   (6.3)  0 0 c2 0 −c2 0       0 0 0 1 0 0   J3     00 0 0 0 − 1   τs   T  b 00000 1 c = τs (6.4) cc = 1 00000 (6.5) The unknown adhesion forces are treated as disturbance inputs for the linearized system. Because of them, robust tracking could only be obtained by introducing integral control in Fig.6.2.

˙xc Ac 0 bc ∗ ˙xic = = xic + Tr (6.6) " x˙ i # " −cc 0 # " 0 #

:=Aic :=bic

As a result, a zero steady-state error| {z is achieved} in| the{z } controlled variable ω1. Assuming all states are known, the control feedback gain vector r is calculated using the simple pole placement technique.

! Dˆ(s) = det[sI − Aic + bic r] (6.7)

r = r1 r2 r3 r4 r5 r6 −r7 (6.8)

This technique assigns the chosen root locations to the characteristic closed- loop polynomial Dˆ(s). To apply state control, all system states must be available for feedback. The state of the electrical system is already reconstructed by a highly dynamic torque control [42] using an electromagnetic torque estimator. Hence only an observer is required which estimates the states of the mechanical three-inertia system from the estimated electromagnetic torque T˜e and the rotor speed. Here, the modal full estimator with the advanced disturbance model from the previous chapter is applied. With this extended observer all essential signals of the ten-inertia system are estimated. To complete the time domain concept, a classical feed-forward control is derived for the controlled variable ω1 based on the parameters of the three- inertia system. The feed-forward controller is developed in two steps: ﬁrst for the superimposed integral controller and second for the inner state controller. ω1(s) As a basis for this purpose, the transfer function Fm(s)= of the three- Te(s) inertia system is calculated symbolically, where Jt is the total inertia of the mechanical system. 90 6 Anti-vibration control i.6.2 Fig. ω ˙ oa tt oto ntetm oanwt edfradcontrol. feed-forward with domain time the in control state Modal 1 ∗ 1 s ω 1 ∗ F H ( s ) ω 1 ∗ h F F F - 4 2 6 ( ( ( F s s s V ) ) ) r ( ω s 7 s 1 ∗ ) h x F F I - 5 3 T r ( ( 12 ∗ 1 s s ) ) - T r ∗ - ω r 2 2 ∗ T - r 23 ∗ 3 ω ˜ Control 1 Torque - ω r 4 3 ∗ PT1-approximation T ˜ e - T r e 5 ∗ Inverter - r 6 T T ω ω ω ˜ ˜ T ˜ ˜ ˜ ˜ 12 23 1 2 3 e Electric ASM xeddobserver Extended T T ˜ e e ω z ˜ 1 1 ˜ + T z ˜ 2 e rcindrive-train Traction z 1 z 2 6.2 Active anti-vibration control scheme 91

s4 + c1+c2 + c2 s2 + c1c2 J2 J3 J2J3 Fm(s)= (6.9) c c +c c c c J J s s4 + 1 + 1 2 + 2 s2 + 1 2 t 1 J1 J2 J3 J1J2J3 h i Taking also the electrical system into account the transfer function for the entire system is

ω1(s) 1 Fp(s)= ∗ = Fm(s) (6.10) Tr (s) τss + 1 The best realization of the feed-forward control is achieved if the derivation of ∗ the reference input variable ω1 , which is provided by a superimposed state of the art slip controller, is included in the design. However, that would make it impossible to implement FV (s) without introducing an additional ﬁrst order lag element FH (s) according to [56].

∗ ω1h(s) 1 FH (s)= ∗ = (6.11) ω1 (s) τhs + 1 1 − F (s)= F (s) 1F (s) (6.12) V s p H

To minimize the eﬀect of FH (s) for the control performance, its dynamic is chosen to be faster than the closed-loop systems poles. The ﬁnal stage is to develop the feed-forward controller for the inner state controller. To this end, all transfer functions of the xc state variables to the ∗ reference electromagnetic torque Tr have to be calculated symbolically.

xc(s) 1 Fq(s)= ∗ , whereas Fq (s)= Fp(s) (6.13) Tr (s) ′ x Fx(s)= Fq (s) FV (s) with x = 1, 2,..., 6 (6.14)

Here the index x indicates the chosen state variable of xc. Equation(6.14) describes the design law for the feed-forward transfer functions. In the following, those were optimized and, for brevity, the nominator N(s) and denominator polynomials D(s) of Fm(s) were introduced.

2 c2 c2 c1s s + J + J F (s)= 2 3 (6.15) 2 N(s)

c1 2 c2 J s + J F (s)= 2 3 (6.16) 3 N(s) c1c2 s F (s)= J2 (6.17) 4 N(s) 92 6 Anti-vibration control i.6.3 Fig. iuain ffcso h edowr oto eedn nteopera the on depending control feedforward the of Effects Simulation: Te T67 T45 0.1 0.2 0.2 0.2 0.6 0.4 0.4 ω1 ω7 ω6 10 a edowr oto o l tt variables state all for control Feedforward (a) 0 0 0 0 0 0 2 2 5 4 4 0 ie[s] time 2 1 igpit( point ting b ieec fosre ttsadfefradcontrol feedforward and states observed of Difference (b)

δ7 δ6 δe -0.5 -0.5 δ45 δ67 δ1 10 10 -1 -1 -1 -2 0 0 0 0 0 0 2 5 5 6 4 0 ω [ rad sec ], T [ kNm ie[s] time ]). 2 1 6.2 Active anti-vibration control scheme 93

c1c2 F (s)= J2J3 (6.18) 5 N(s) F (s) D(s) F (s)= H (6.19) 6 N(s)

This is used to carry out the first simulation taking account of the effects of the feed-forward control. Fig. 6.3 shows a start-up of the locomotive to the operating point P1 of Fig. 3.2 with resuming constant speed. As a result, the feed-forward for the angular frequencies is excellent: the observed states and the feed-forward signals are nearly identical according to Fig. 6.3(a). Only Fig. 6.3(b) reveals a slight difference signal δ1,2,3 at the start-up due to the huge change in the adhesion forces, whereafter at constant speed the difference signal is almost zero. The remaining noisy signal is a consequence of the noise added to the adhesion forces for more realistic adhesion and of the sensor noise, which is amplified by the observer to a certain degree. By contrast, the torque feed-forwards exhibit a considerable difference signal corresponding to the unknown adhesion forces. Therefore to take computation time limitations into account, the torque feed- forwards and also FV (s) are chosen to be zero.

FV (s)= F2(s)= F4(s)= F6(s) = 0 (6.20)

Here FH (s) becomes redundant and is set to FH (s) = 1. The feed-forward control could be further reduced, if the dynamics of the three-inertia system are neglected for control purposes.

F3(s)= F5(s) = 1 (6.21)

Thus with Eq.(6.21) the simplest feed-forward structure, which could be gen- eralized for mechanical systems, is derived and will be applied to the controller design in the frequency-domain.

6.2.2 Design in the frequency-domain

Compared with the time domain design of state controllers, the frequency approach requires a considerably reduced design eﬀort. Therefore a direct design method is introduced for deriving a frequency domain controller with equivalent performance. In Fig. 6.4, the feedback controller is based on the modal three-inertia system of Fig. 4.3(c) and consists of two transfer functions:

NT (s) NΩ(s) FT (s)= FΩ(s)= (6.22) ∆ˆ(s) ∆ˆ(s) 94 6 Anti-vibration control i.6.4 Fig. ﬀr-iiie oa tt oto ntefeunydmi ihfeed with domain frequency the in control state modal Eﬀort-minimized ω ˙ 1 ∗ 1 s 1 s ω 1 ∗ - controller Speed r s 7 x I - T r ∗ N N ∆ ∆ ˆ ˆ Ω T ( ( ( ( s s s s ) ) ) ) ω ˜ 1 Control Torque T ˜ e Inverter fradcontrol. -forward Electric ASM T e - ω ω 1 1 ∗ rcindrive-train Traction z 1 z 2 6.2 Active anti-vibration control scheme 95

Here, the denominator is already given by the desired estimator polynomial ∆ˆ(s) according to Eq. (5.15) and only the nominator polynomials NΩ(s) and NT (s) have to be calculated.

6 5 NT (s)= Nt6 s + Nt5 s + ... + Nt1 s + Nt0 (6.23) 5 4 NΩ(s)= Nω5 s + Nω4 s + ... + Nω1 s + Nω0 (6.24) 6 5 ∆ˆ(s)= s + ∆ˆ5 s + ... + ∆ˆ1 s + ∆ˆ0 (6.25)

As a basis for the following design, the feedback gain vector r according to Eq. (6.8) has to be determined. From this, a new vector ra is deﬁned for the controller design:

ra = r1 r2 r3 r4 r5 0 (6.26) Thus the final controller coefficients of NΩ(s) are calculated by a comparison of coefficients in Eq. (6.28). After that, Eq. (6.29) determines NT (s) directly except for

Nt6 = r6 (6.27)

Due to the sparsely occupied matrix Ao and vectors bo and co, the equations can even be solved symbolically with a relatively simple solution. The proposed design equations (6.28)–(6.29) were modiﬁed from [55] to ﬁt the state control with extended observer design, although they were originally used for another purpose.

5 co Ao 4  co Ao  . Nω5 Nω4 Nω3 Nω2 Nω1 Nω0  .  =      co Ao     c   o   6 ra Ao r A 5  a o  ˆ ˆ ˆ ˆ ˆ ˆ . = 1 ∆5 ∆4 ∆3 ∆2 ∆1 ∆0  .  (6.28)      ra Ao     ra      96 6 Anti-vibration control

(Nt5 − ∆ˆ 5 r6 )(Nt4 − ∆ˆ 4 r6 ) ··· (Nt0 − ∆ˆ 0 r6 ) = ˆ ˆ ˆ = ra (∆5 ra − Nω5 c)(∆4 ra − Nω4 c) ··· (∆1 ra − Nω1 c) ... 2 5 bo Ao bo Ao bo ··· Ao bo 0 b A b ··· A 4 b  o o o o o  ......  . . . . .  (6.29)      00 0bo Ao bo     00 00bo      Fig.6.4 shows the proposed active anti-vibration control concept. The observable and controllable system splits into three parts, namely the highly- stiff traction drive-train, the asynchronous machine and inverter characteristics and finally the highly dynamic torque control. For the design of the modal state controller, all nonlinearities of the electrical system including the torque control are approximated by a linear first order element and the mechanical system is represented by the modally reduced drive-train model. The unknown adhesion forces are treated as disturbance inputs for the linearized system. Thus robust tracking can only be obtained by introducing integral control. Consequently, the speed controller emerges as a simple integrator to provide a zero steady state error in the controlled variable ω1. To comply with the limitations imposed by the traction application, the controller order must be kept as small as possible. Using the estimated electromagnetic torque T˜e, the order of the state controller can be reduced by one. The modal feedback controller is directly designed in the frequency domain. Its denominator is already fixed by the desired characteristic observer polynomial ∆˜(s) whereas the nominator polynomials NΩ(s) and NT (s) have to be computed to assure the desired closed-loop dynamics. Finally the feed-forward of the reference ∗ speed ω1 reduces the action of the feedback controller.

6.2.3 Simulation studies

Assuming the same simulation conditions as in Fig.3.6, the first simulation results of the anti-vibration controller are depicted in Fig.6.5. Here, the controller is designed to damp slip-stick vibrations originating from an adhesion s force gradient of up to δµ/δ(∆v) = −0.1 m . Thus the first vibration with a s negative adhesion force gradient of δµ/δ(∆v)= −0.06 m is efficiently damped s where the second vibration based on δµ/δ(∆v)= −0.12 m still arises. 6.2 Active anti-vibration control scheme 97

1 10 ω

45 20 T

5 6 ω

50 67

T 0

-50

5 7 ω

2 i

v 1

10 e T 5

0 0 1 2 3 4 5 time [s]

rad m Fig. 6.5 First simulation of the active anti-vibration control ( ω [ sec ], T [kNm], ∆v [ s ] ). 98 6 Anti-vibration control 6.3 Advanced anti-vibration control scheme

So far the effects of the system input constraint, namely the reference torque saturation, are not considered. Due to the saturating nonlinearity, undesired oscillations are triggered in the anti-vibration control scheme of Fig. 6.4. This effect is well-known in control theory as windup and it has to be removed for the practical application. Here two different phenomena have to be considered, namely the controller and the plant windup. Both are discussed in the sequel with the main focus on the controller windup.

6.3.1 Windup prevention strategies

Increasing the closed-loop dynamics beyond certain limits, plant windup occurs. Remedy is found either in the design of an additional dynamic network or in an appropriate reduction of the closed-loop dynamics. To comply with the application limitations, namely an effort-minimized control structure, an upper limit for the closed-loop dynamics is introduced using a simple design- aid for plant windup prevention [57]. Additionally, a controller windup can be triggered both in the speed and in the feedback controller. It is due to unstable controller dynamics and is easily removed by a structural modification in the controller realization. The observer-based control structure of Fig.6.4 naturally provides controller windup prevention by a feedback of the saturated plant input, namely the reconstructed state T˜e of the electrical system. On the other hand, a high utilization of tractive effort causes reference torque saturation, leading to an integral windup in the speed controller. In the standard control concept

∗ ω1 xI kP s+kI - s

ω1

(a) Classic

∗ ω xI 1 kP s+kI - s+δ1 ~ s+δ1 s ω1 δ1 s+δ1 (b) Modiﬁed

Fig. 6.6 Speed controller with anti-windup prevention schemes. 6.3 Advanced anti-vibration control scheme 99

Tab. 6.2 Speed controller with nonlinearity.

Nonlinearity Windup prevention results Linear kP s+kI s+δ1 s+δ1 s Saturation kP s+kI is stable s+δ1 shown in Fig.6.6(a), windup prevention is achieved by cutting off the integral part of the speed controller in the event of control signal saturation. In order to obtain joint windup prevention for the speed and the feedback controllers, the classic windup prevention is modified. Windup prevention is achieved in the speed controller by introducing a stable controller pole at −δ1. To substitute this pole −δ1 by integral action in the linear case, an appropriate first order element is arranged around the nonlinearity. This modified controller is a PI-controller in the linear case and exhibits no integral windup when saturation becomes active. Fig.6.6(b) shows this modified anti-windup structure and the windup prevention outcome is given in Tab. 6.2. The application of this structure to the anti-vibration control scheme of Fig.6.4 concludes the windup prevention, where kp = 0 and kI = −r7 are applied.

6.3.2 Starting of active anti-vibration controller

If the active anti-vibration control is the primary control scheme on the vehicle, the standard control concept has to be integrated as a fallback system. Consequently, the starting of the active anti-vibration controller can also be carried out from the standard controller during operation. In the event of unexpected system states at the first attempt at starting, the conventional PI-speed controller with its original closed-loop dynamics assures operational functionality of the vehicle. Using the proposed PI-controller modification, both control concepts can easily be coupled. Thus both speed controllers share the same windup prevention structure, as indicated by the dashed box in Fig.6.6(b). The modified control scheme shown in Fig.6.7 provides effort-minimized joint windup prevention for both controllers. Compared with Fig.6.4 it contains an additional transfer function in the feedback path com- pensating for the dynamic behavior of the shared speed controller. The transition structure simply consists of a sliding factor κ. Thus the two control concepts, namely the standard controller (κ = 0) and the active anti-vibration controller (κ = 1), are the two limit values of this sliding factor. To facilitate applicability, the design of the two control concepts is decoupled. Hence the feedback controller design is independent of the parameters of the standard speed controller and therefore remains unchanged. To achieve this, the output of the standard speed controller has to be compensated when κ = 1. Feeding 100 6 Anti-vibration control it additively to the speed controller output of the anti-vibration controller solves this problem in a simple way. Consequently, the anti-vibration scheme is applicable with any standard speed controller. For control design, the first adjusting control loop is the standard controller where the dynamics of the windup prevention scheme are tested. The important constant δ1 is chosen as a stable pole between the imaginary axis and the pole of the approximated torque control. The second adjusting control loop is the anti-vibration controller. In the sequel, the behavior of the complete control scheme is demonstrated by simulation studies with the adhesive force characteristic (II) of Fig.3.2. The active anti-vibration controller is appropriately designed to damp the traction drive-train in this unstable range of the adhesion force characteristic. Besides the damping behavior, soft and hard switched transitions between both control concepts are shown in Fig.6.8. Starting with the standard controller (κ = 0), the maximum of the adhesion force characteristic (II) is passed, as indicated by the electromagnetic torque Te, and the traction drive- train reaches the desired operating point P1 in the unstable range. For all the investigations, it operates at this constant wheel-to-rail slip ∆vi of the indirect-driven wheel. In this operating point the standard controller provides insufficient damping and therefore slip-stick oscillations occur which mainly stress the wheelset shaft T67 with less influence in the speed of the asynchronous machine ω1. At t = 0.8 s a slow transition to the anti-vibration controller is carried out, only providing full damping when κ = 1 is met. Slow transitions therefore apply an unnecessary amount of stress to the wheelset shaft. Consequently, the transition speed is increased and the transition to the active anti-vibration controller is further delayed to test the damping behavior at higher vibration amplitudes. The faster transition to κ = 1 at t = 2.6 s obviously leads to a quicker application of the full damping. The delayed transition at higher vibration amplitude, however, entails a longer duration of active damping. To investigate the effects of transition speed under the same conditions, soft (t = 1.1 s) and hard switching (t = 4.9 s) are compared at the same vibration amplitude. As a result, the stress in T67 and in Te significantly reduces with increasing transition speed. Finally, the worst case of vibration amplitude and delayed transition is considered with hard switching. Due to extreme conditions the actuated variable Te instantly saturates. Because of windup prevention its mean value decreases. As a result, the slip ∆vi declines, temporarily operating the wheelset in the stable range of the adhesive force characteristic until the torque saturation wears off. Subsequently, active damping is effective. The saturating nonlinearity is accompanied by both a loss of traction force and an undershoot of ω1. Both effects are well-known from the use of readhesion controllers. 6.3 Advanced anti-vibration control scheme 101 2 z 1 z Traction drive-train ∗ 1 1 ω ω - e T ASM Electric active anti-vibration control. Inverter e ˜ T Slip Readhesion 1 Torque ) ) Control ) ) s ˜ s ω s s ( ( ( ( T Ω ˆ ˆ ∆ ∆ N N ∗ r T 1 δ s 1 + δ s 1 1 δ + δ s s + s - κ - I k 1 δ + 1 s δ + 7 Anti-vibration controller s r P + k s - ∗ 1 Standard control concept with passive readhesion controller coupled with ω Standard controller with anti-windup Fig. 6.7 102 6 Anti-vibration control i.6.8 Fig.

ω1 κ Te T67 -10 ∆vi -50 30 20 10 10 50 0 0 0 0 0 1 1 2 wthn ﬀcso h tnadadmdlsaecnrle ( controller state modal and standard the of eﬀects Switching 0 1 2 3 ω [ ie[s] time rad sec 4 ], T [ kNm ], ∆v [ 6 5 m s ). ] 7 8 6.3 Advanced anti-vibration control scheme 103 6.3.3 Coupling of passive and active control schemes

For application on the vehicle, the minimum occurring gradient δµ/δ(∆v) of the adhesion force characteristic has to be taken into consideration. To achieve efficient damping at this highly negative gradient, the closed-loop dynamics of the proposed active anti-vibration control have to be increased by a newly designed feedback controller. Higher dynamics, however, also entail an increased sensor noise amplification, producing a permanently highly oscillating variable Te. Here, the sensor noise notably originates in the speed sensor featuring an incremental encoder with a very low resolution imposed by the robustness of traction applications. Furthermore, the increase of the closed-loop dynamics is bounded to the previously derived upper limit due to plant windup prevention. Considering both limitations, the decisive factor on the closed-loop dynamics is definitely the speed sensor noise amplification. To comply with this restriction, the active anti-vibration controller is designed s for a minimum gradient of δµ/δ(∆v) = −0.1 m , capable of damping most of the slip-stick vibrations. However, at operating points with a more negative gradient, the feedback controller provides insufficient damping. Besides ∗ slip-stick vibrations, the reference torque Tr also exhibits vibrations until saturation occurs. A solution of this problem is found by considering state of the art traction drive control systems [42]. Here, the standard speed controller only achieves s sufficient damping up to a gradient of δµ/δ(∆v)= − 0.04 m . At higher negative gradients a passive readhesion controller is utilized to suppress the arising slip-stick vibrations. In detail, the passive controller is tuned at the given ∗ vibration frequency. On its occurrence in ω1, the reference torque Tr is reduced at the expense of traction force. Subsequently, the slip decreases and the wheelset operates in the stable range of the adhesive force characteristic where the damping of the drive is achieved. By analogy, this strategy is applied to the active anti-vibration control scheme for highly negative gradients δµ/δ(∆v) > −0.1 s . Thus the passive readhesion controller is utilized when m ∗ the active anti-vibration controller fails, causing Tr to oscillate with the vibration frequency. Finally for coupling to the anti-vibration control scheme, the behavior of the shared speed controller is also compensated here by an appropriate transfer function in the feedback branch. In the sequel, the complete control scheme is discussed by means of simulation studies shown in Fig.6.9. All three controller types, namely the standard, the active and the passive controllers, are now utilized with various gradients of the adhesive force characteristics, as given in Fig.3.2. First the standard controller carries out the starting of the traction drive based upon the adhesive force characteristic (I) in the lower speed range. Subsequently, the active anti-vibration controller is gradually started in the stable range of (I), easily s providing sufficient damping at the gradient of δµ/δ(∆v)= −0.06 m at operating point P1. Due to the step-by-step decline of the gradient in the adhesive force characteristic (II) and finally in (III), the active anti-vibration controller 104 6 Anti-vibration control

s can not cope with the highly negative gradient of δµ/δ(∆v)= −0.12 m and consequently slip-stick vibrations arise. Instantly the passive readhesion con- ∗ troller reduces the reference torque Tr and therefore Te until the active controller can cope with the gradient. Here, the new operating point P2 is still in the unstable range of (III) with minor wheel-to-rail slip ∆vi. Using the synergy of active and passive controllers, the stable operating range of the s traction drive-train is successfully expanded to δµ/δ(∆v)= −0.1 m without any loss of traction force.

6.3.4 Unmodeled dynamics and stability

Considering only an unbranched three-inertia model of Fig. 4.3(c) for a complex unbranched seven- or even for a branched ten-inertia system of Fig.4.1 several vibration modes were neglected. Operating points beyond the maximum tractive effort level in Fig.3.2 contribute a negative damping factor to only the first two vibration modes of any cardan hollow shaft traction drive- train and subsequently slip-stick vibrations of these vibration modes arise. The first vibration mode with the vibration frequency ω2 mainly twists the cardan hollow shaft (❹,❺) and the second vibration mode with ω3 = 2.8 ω2 the wheelset (❻,❼) for the most complex branched drive-train as depicted in Fig.4.1(b). Higher vibration modes, such as the gear-brake shaft (❽,❾) with ω4 = 5.2 ω2 and as the gear-hollow shaft (❸,❹) with ω5 = 6.1 ω2 are very well damped and will not be triggered by slip-stick vibrations. Thus the suitably reduced model has just to reflect the first two vibration modes of the complex drive-train and therefore a reduced three-inertia model is sufficient for this purpose. Further unmodeled dynamics originating from branching, wear- and ageing effects are discussed in more detail in Fig.4.4. Stability of both standard and modal state controller is theoretically proven by a complex Nyquist-criterion analysis. Here, the sliding parameter κ is used as a variable parameter transitioning both Nyquist plots. Finally, the stability of both controllers is empirically proven for any value of κ in its range between κ = 0 and κ = 1 by the simulations carried out in Fig.6.8 and in Fig.6.9.

6.4 Virtual absorber

The latter control scheme improves not only the damping of the two vibration modes, but also the dynamics of the whole traction system introducing both novel feedback controller and a modiﬁed speed controller. Faced with this fact, several locomotive manufacturers reported, that the design of their standard speed controller evolved from numerous locomotive test-runs, is not likely to be changed and therefore the chances of getting this control scheme 6.4 Virtual absorber 105 8 7 2 P (III) 5 6 ] ). s m [ ∆v ], 4 kNm [ time [s] T ], (II) sec rad [ ω 3 2 (I) 1 1 P 0 Coupled active and passive anti-vibration controllers ( 1 1 0 0 0 0 0

10 10 20 30

67 i e T ∆v T

1 ω Fig. 6.9 106 6 Anti-vibration control from a prototype locomotive to a series-produced one are almost neglectible. Thus in the following, a more locomotive-manufacturer-friendly and easy- to-handle anti-vibration control scheme implying the design of the standard speed controller is introduced. Here, the main idea is based on a mechanical vibration absorber for railway vehicles [12] published in the late ’90s. The newly proposed anti-vibration control scheme simulates this mechanical absorber virtually by a novel feedback controller circumventing all drawbacks known from the mechanical absorber application.

6.4.1 State of the art

To efficiently suppress the second vibration mode of a traction drive-train, a mechanical auxiliary system, namely a vibration absorber, is mounted to the traction drive-train. Here, the position of the absorber is determined by the maximum torsional rotation angle of the second vibration mode shape as given in Fig.3.1. Both wheels exhibit almost the same absolute value of the torsional rotation angle. Therefore other means such as space requirements for the absorbers define the mounting position at the indirect-driven wheel. Fig.6.10 depicts a vibration absorber system for railway vehicles as published in [12] where the absorber mounted at the direct-driven wheel is unnecessary for highly-stiff traction drive-trains as the first vibration mode could be easily damped by the applied standard controller. The indirect-driven absorber itself consists of a mass-spring-system and has three relevant parameters [12], the inertia Ja, the stiffness ca and the damping factor da: • As there is also limited space at the indirect-driven wheel, the absorber inertia Ja cannot be chosen arbitrarily. A greater value results in higher damping, but also increases the unsprung mass. To define a sophisticated reference value for the absorber inertia, the latter has to be at least ten percent of the indirect-driven wheel inertia according to [12] in order to achieve a reasonable damping effect for the second vibration mode. • The value of the stiffness ct is simply determined by the frequency of the second vibration mode which should be suppressed. • Finally, analysing the eigenvalues of a root-locus chart for the variable damping factor dt, only a rubber-spring element provides sufficient damping for the second vibration mode. A steel spring is here not feasible. However, the application of a mechanical vibration absorber for railway vehicles entails several drawbacks [12] where the following are originating from the traction-drive train • Wear of the wheelset reduces each wheel disc inertia shifting the frequency of the second vibration mode towards a higher vibration frequency. The considerable frequency shift is usually within the range from 50 Hz to 90 Hz. 6.4 Virtual absorber 107

• Temperature-dependent nonlinear shift in the stiffness of the rubber elastic joints used for coupling purposes • Age-dependent nonlinear shift in the stiffness of the rubber elastic joints due to rubber embrittlement. and these originate from the mechanical vibration absorber [12] itself • Temperature-dependent nonlinear shift in the stiffness of the rubber spring utilized for the vibration absorber • Age-dependent shift in the stiffness of the rubber spring due to rubber embrittlement • Loosening of the spring-inertia connection of the mechanical vibration absorber. This frequency shift weakens the damping efficiency of the mechanical vibration absorber to a great extent which has to be tuned for a certain fixed vibration frequency. All these drawbacks are counteracted introducing the novel virtual vibration absorber in an ingenious way.

6.4.2 Generic controller design

In Fig. 6.11(a) the most complex branched traction drive-train is depicted with a mechanical absorber mounted to its indirect-driven wheel. The goal for the design of the virtual absorber controller is to provide the same damping behavior and performance. Therefore the controller design is derived on the basis of the identiﬁed reduced three-inertia model, which optimally reﬂects both modal and frequency-response characteristics of the complex branched drive-train. Further using a frequently updated reduced model, the wear,

ωr

Fig. 6.10 Mechanical vibration absorber system at the wheelset [12]. 108 6 Anti-vibration control ageing and temperature-dependent eﬀects of the branched traction drive- train are taken into consideration for the virtual controller and therefore all previously discussed drawbacks of the mechanical absorber application are easily eliminated by this novel control approach. The basic idea for this novel approach is to virtually mount the same absorber by a feedback controller to the reduced model at the same location in the drive-train, namely the indirect-driven wheel. This is achieved by assuming the same control performance in two diﬀerent basic control schemes as given in Fig.6.12. On the one hand in Fig.6.12(a), the mechanical absorber is mounted to the reduced model as shown in Fig.6.11(b) and the resulting structure is applied to the state of the art standard control scheme as given in Fig.2.10. On the other hand in Fig.6.12(b), a feedback controller, namely a virtual absorber controller, based on the estimated and measured signals Te and ωr is applied to the reduced model without a mechanical absorber. A

f1 f2

ωr

Te z2

(a) Branched ten-inertia system with mechanical absorber

ωr

z2 (b) Reduced model with mechanical absorber

Fig. 6.11 Branched traction drive-train and reduced model with mechanical absorber. 6.4 Virtual absorber 109

Gwa(s)

′ Ga(s)

∗ Te ω1 Speed Torque ASM Inverter - controller Control Electric

z1 ω˜1 T˜e z2

ω Traction 1 drive-train

ωnoise

(a) Mechanical absorber mounted at indirect-driven wheel

Gna(s)

′ Ga(s)

∗ Te ω1 Speed Torque ASM Inverter - controller - Control Electric z1 ω˜1 T˜e z2

ω Traction Virtual 1 drive-train absorber

ωnoise FT (s)

FΩ (s)

(b) Virtual absorber as feedback controller

Fig. 6.12 Basic control schemes with identical control performance. performance comparison of both basic control schemes of Fig.6.12 derives the design equation for the virtual absorber controller. As there are two unknown transfer functions FT (s) and FΩ(s), the solution of the design equation has one degree of freedom

′ Ga(s) Gna(s) ′ ′ 1+ Ga(s) FT (s) Ga(s) Gwa(s)= ′ (6.30) Ga(s) 1+ Gna(s) FΩ(s) ′ 1+ Ga(s) FT (s) where a simpliﬁed depiction is 110 6 Anti-vibration control

Gna(s) − Gwa(s) FT (s)+ Gna(s) FΩ(s)= ′ (6.31) Ga(s) Gwa(s) The transfer functions in the latter equation are the PT1-approximation ′ Ga(s) of the actuator dynamics, the reduced model without absorber Gna(s) and the reduced model with absorber Gwa(s) mounted to the end of the traction drive-train. In the following the two last named are symbolically derived based on their signal graph representation in order to obtain a generic controller for both road and rail vehicles with a low complexity. In Fig.6.13, the signal ﬂow graphs of the two transfer functions are depicted connecting both reduced model and absorber to the corresponding nodes in their respective signal graphs. For a generic n-inertia reduced model in it signal ﬂow diagram representation these variables are introduced 1 1 a1 = ... a2 n−1 = (6.32) J1 Jn a2 = c1 ... a2n−2 = cn−1 (6.33) and according to the Mason’s gain formula [58]

ωr(s) ∆0(n, s)na Gna(n, s)= = p0 (6.34) Te(s) ∆(n, s)na where the loops ℓ are

a1 a2 c1 ℓ1 = − 2 = − 2 (6.35) s s J1 . . (6.36) a2n−2 a2n−1 c2n−2 − ℓ2n 2 = − 2 = − 2 (6.37) s s J2n−2 The transfer function for the reduced model with the absorber is based on a inertia system of the order n+1

ωr(s) ∆0(n + 1,s)wa Gwa(n + 1,s)= = p0 (6.38) Te(s) ∆(n + 1,s)wa where the additional variables for the absorber are

a2n = cn = ca (6.39) 1 a2n+1 = (6.40) Jn+1 da b1 = (6.41) Jn Jn + Jn+1 b2 = da (6.42) Jn Jn+1 6.4 Virtual absorber 111

ωr

1 a2 −a2 a4 −a4

1 ℓ 1 ℓ 1 ℓ 1 ℓ 1 s 1 s 2 s 3 s 4 s

−a1 a3 −a3 a5

ωr

1 a2 −a2 a4 −a4 a6 −a6

1 1 1 1 1 1 ¯h 1 ¯h s s s s s s 2 s 1

−b2

−a1 a3 −a3 a5 −a5 a7

a1 b1 Te

Fig. 6.13 Signal ﬂow diagram for a reduced three-inertia model (n = 3) without and with a mechanical absorber mounted at the end of the traction drive-train. 112 6 Anti-vibration control

Tab. 6.3 Case-wise solution of the virtual absorber design equation Eq. 6.31 Case Constraints Solution I FT (s) = 0 FΩ (s) II FΩ (s) = 0 FT (s) Te(s) III = min FΩ (s), FT (s) ωnoise(s) and the absorber loopsh ¯ are

b a a b a − a − ¯h = 1 4 5 = 1 2n 1 2n 2 (6.43) 1 s3 s3 −b ¯h = 2 (6.44) 2 s Due to the one degree of freedom in the design equation Eq.(6.31) for the virtual absorber, three diﬀerent constraints for a simple and sophisticated solution are assumed in Tab.6.3 where there are only two simple and generic symbolical solutions for a n-inertia reduced model available for the ﬁrst two cases:

Case I: Speciﬁc design equation

Gna(n, s) − Gwa(n + 1,s) FΩ(n, s)= ′ (6.45) Ga(s) Gna(n, s) Gwa(n + 1,s) for the corresponding virtual absorber controller

2n−3 1 h1 + l2n−2 l2n−1 FΩ(n, s)= − ′ lm(6.46) p G (s) ∆ (n)na ∆ (n + 1)wa 0 a 0 0 m=1 Y Case II: Speciﬁc design equation

Gna(n, s) − Gwa(n + 1,s) FT (n, s)= ′ (6.47) Ga(s) Gwa(n + 1,s) just diﬀers slightly from the previous case for the corresponding virtual absorber controller and thus the solution is quite simple

2n−3 1 h1 + l2n−2 l2n−1 FT (n, s)= − ′ lm (6.48) G (s) ∆(n)na ∆ (n + 1)wa a 0 m=1 Y Case III: The most sophisticated solution of the complete design equation Eq.(6.31) for the virtual absorber is to be found by considering the natural sensor noise ωnoise(s) generated by a typical speed sensor used for the railway application. Here, the main design goal is to minimize the sensor noise ampliﬁcation in the ω- feedback path by choosing an optimized FΩ(s). This is achieved 6.4 Virtual absorber 113

by providing a second equation, in this case is the sensor noise amplification T (s) e = minimal (6.49) ωnoise(s) to solve the design equation Eq.(6.31) with its two unknown variables. However, the result of FT (s) has to be still realizable. No simple generic symbolical solution is available here as different types of speed sensors are applied in road and rail vehicles generating a complete different sensor noise spectrum.

Due to wear, ageing and temperature-dependent eﬀects, the parameters of the virtual absorber Ja, ca and da are automatically adapted to the reduced three-inertia model each time the latter is updated by the natural identiﬁca- tion process. This frequent absorber update procedure guarantees an optimal

ωr

z1 (a) Reduced model

ωr

z1 (b) Reduced model with virtual absorber

Fig. 6.14 Reduced traction drive-train model of an electric road vehicle (n = 2). 114 6 Anti-vibration control damping performance compared to that of a standard mechanical absorber. The main reason is that the mechanical absorber is designed to a fixed pre- defined damping frequency which can not be altered or adapted to both wear and ageing effects of the traction drive-train. The only drawback of the virtual absorber controller is that its maximum damping performance is dependent on the traction drive-train characteristics. Thus not any mechanical absorber can be realized in the same way as a virtual absorber. However, state of the art drive-trains such as the unbranched one with its highly-stiff transmission to the indirect-driven wheel as shown in Fig.4.1(c) offers excellent characteristics for an almost complete slip-stick vibration extinction based on just the application of a virtual absorber. To sufficiently suppress tip-in vibrations in electric road vehicles, the virtual absorber controller can also be utilized. Here, the complex traction drive- train can be reduced to a two inertia system of Fig.6.14. The reduced model simply consists of the asynchronous machine and both rim and rubber tire. Thus the virtual absorber design for the traction system of electric road vehicles is much simpler, e.g. for case I, it is derived according to Eq.(6.45) by its signal flow graph parameters:

s l1 (¯h1 + l2 l3) FΩ(n = 2,s)= − ′ a1 Ga(s) (1 − l2) (1 − l2 − l3 − l4 − ¯h1 − ¯h2 + l2 l4 +h ¯2l2) (6.50) All other cases for electric road vehicles are solved in analogy to the railway application which is of higher complexity. In the following, an overall control concept with the virtual absorber controller is introduced.

6.4.3 Control concept

The basic control concepts of Fig.6.12 were just two simple auxiliary schemes to introduce the main idea of the virtual controller design as well as to reduce the design complexity. These concepts are not suitable for the real application on the traction vehicle but only for the determination of the two transfer functions of the virtual absorber controller FT (s) and FΩ(s). On the basis of Fig.6.7, the novel control scheme for the use of the virtual absorber is developed. The initial condition for the speed controller is fulﬁlled, as already discussed, by an unmodiﬁed standard controller with the given parameters from the rolling stock manufacturers. Here, the windup pre- ∗ vention scheme of Chapter 6.3.1, the feed-forward of the ω1 -signal of Chapter 6.2.1 and a coupled passive readhesion controller of Chapter 6.1.2 are applied. Again κ is used as a sliding variable for the starting of the virtual absorber controller as given in Tab. 6.4. Finally, a novel control scheme for the virtual absorber application is presented in Fig.6.15 meeting all the expectations of the railway vehicle manufacturers. 6.4 Virtual absorber 115 2 z 1 z Traction drive-train ∗ 1 1 ω ω - e T ASM Electric virtual absorber. Inverter e ˜ T Slip Readhesion 1 Torque Control ) ) ˜ s ω s ( ( T Ω F F ∗ r T 1 δ s 1 + δ s 1 1 δ + δ s s + s Virtual absorber - κ - I k 1 δ + s + s P k - ∗ 1 ω Standard controller with anti-windup Standard control concept with passive readhesion controller coupled with Fig. 6.15 116 6 Anti-vibration control

Tab. 6.4 Eﬀects of the sliding variable κ. κ Eﬀect on virtual absorber 0 Deactivated 0.5 Performance is only 50% 1 Full performance is reached1.

6.5 Virtual protection and virtual readhesion

The introduced virtual absorber approach is interpreted in the following in a more universally way. Instead of a simple mechanical absorber, any ﬁl- ter function can be virtually applied to the traction drive-train. Besides the indirect-driven wheel, the direct-driven wheel as well as the asynchronous machine are also available for the application of a virtual ﬁlter based on the known reduced three-inertia model.

6.5.1 Basic Idea

The reduced modal three-inertia model offers three inertia positions matching with the equivalent positions in the multi-inertia traction drive-train for applying virtual filters: • The indirect-driven wheel features the yellow virtual absorber for active slip-stick vibration suppression in Fig.6.16(b) as discussed in the latter. Besides active vibration suppression, this virtual filter also increases the tractive effort level utilization in an active readhesion way. • The direct-driven wheel corresponds with the first vibration mode stressing mostly the hollow shaft and the rubber joints. To protect both components from extensive vibrational load, a filter function can be applied as shown in Fig.6.16(c) by the green absorber. • The asynchronous machine finally offers the rigid body motion mode for applying a filter function similar to the advanced passive readhesion controller of Chapter 6.1.2 as depicted in Fig.6.16(a) by a red absorber. In this way, each of the three inertias of the reduced model has one virtual filter function mutual exclusively applied for a specific purpose as given in Fig.6.17. Thus three different states are defined for the traction drive-train which are related to a pedestrian traffic light system in the following: • Go! is the standard operation protecting all traction drive-train components between the direct-driven wheel and the asynchronous machine from the vibrational intake originating from the wheel-rail contact.

1 Designed virtual absorber is fully realizable by a feedback controller. 6.5 Virtual protection and virtual readhesion 117

f1 f2 ωr

Te z2

(a) Virtual passive readhesion at the asynchronous machine

f1 f2 ωr

Te z2

(b) Virtual active readhesion at the indirect-driven wheel

f1 f2 ωr

Te z2

Fig. 6.16 Three diﬀerent virtual ﬁlter applications for traction drive-trains. 118 6 Anti-vibration control

Switching History Blending GO!

Virtual Absorber

Virtual Readhesion

STOP! PROCEED

with caution! ita Protection Virtual

Fig. 6.17 Looping state machine with three states and two diﬀerent types of transitions with a basic guideline given by a pedestrian traﬃc light system.

• Proceed with caution! is defined by arising slip-stick vibrations in the traction drive-train. The yellow virtual absorber is applied to contain the arising vibrations. • Stop! is initiated in the case, that the yellow virtual absorber can not cope with the negative adhesion force gradient. After the slip-stick vibrations are declined, usual standard operation is resumed. Pedestrian symbols are used to connect the various virtual filter applications of the traction drive-train in Fig.6.16 with the ones of the reduced model in Fig.6.17 in a simple way. For state transition, two different methods 6.5 Virtual protection and virtual readhesion 119 are available – switching and blending. The differences and effects of both methods are discussed in Chapter 6.3.2 by simulation studies in great detail. As a result, three novel virtual technologies, are presented by the looping state machine in Fig.6.17 based on the application of the virtual filters: • Virtual absorber correlates to a virtualized mechanical absorber mounted at the indirect-driven wheel. • Virtual readhesion splits up into an active as well as a passive method where the filters are mutually exclusively mounted at both ends of the traction drive-train. First, the active method is achieved by the virtual absorber suppressing the slip-stick vibrations and thus increases the tractive effort level utilization. Second, the passive method results from the application of a virtual filter at the asynchronous machine substituting the proposed advanced passive readhesion controller. • Virtual protection is obtained by all three states where the vibrational load of the traction drive-train is considerably lowered. Finally, the history of various anti-vibration control schemes is discussed in regard to Fig.6.17. In the year 1994, the first passive readhesion controller (inferior to red filter) was patented in [10] for a heavily-decoupled motor- wheelset traction drive-train as shown e.g. in Fig.4.1(a). Then four years later, the first vibration mode shape (inferior to green filter) was damped in [4] which is applicable for medium-stiff traction drive-trains like in Fig.4.1(b). Finally now, the yellow virtual absorber is partially realizable for a highly-stiff traction drive-train such as in Fig.4.1(c). A full application will be possible with new traction drive-trains in the near future. So over the years, the development of traction drive-trains with increasing stiffness between the asynchronous machine and the wheelset facilitates a deeper realization of virtual filters into the traction drive-train. In the following, a novel overall control concept is introduced based on both looping state machine and virtual filters.

6.5.2 Control concept

In Fig.6.15, a simple virtual absorber control is presented with one feedback controller for the torque- as well as for the speed-signal. To realize the novel control scheme with the three virtual filter states of the looping state machine, the same number of controllers is necessary for each feedback path as shown in Fig.6.18. The fourth black filter-free model in the center of Fig.6.17 is simply defined similar by the κ = 0 condition of Fig.6.15 within the looping state machine. So all three feedback controller pairs FTx(s) and FΩx(s) have to be designed in analogy to the given equations in Chapter 6.4.2. Finally, the looping state machine is just putting through the corresponding feedback controller signals according to its actual given state. 120 6 Anti-vibration control a 6.18 Fig. “ h ecdsaogtetato oto schemes. control traction the among Mercedes The tnadcnrlcnetwt ita lesadloigsaemac state looping and filters virtual with concept control Standard ihanti-windup with tnadcontroller Standard ω 1 ∗ - k P s + s + δ 1 k I s - + s δ 1 filters Virtual s + δ 1 δ 1 T r ∗ ” F F x T Ωx ( ( ω ˜ s s Control Torque 1 ) ) T ˜ e Inverter hine a . Electric ASM T e - ω ω 1 1 ∗ rcindrive-train Traction z 1 z 2 6.6 Comparison of control schemes 121 6.6 Comparison of control schemes

The three vibration suppression concepts, namely the mechanical absorber and both state and virtual absorber control coupled with a passive readhesion controller, are discussed in the following on the basis of their simulation studies in reference to the operational cycle of Fig.3.6 with both standard controller and real system variables. Finally, recommendations for various designs of control schemes are given for railway manufacturers regarding both their performance and their applicability.

6.6.1 Simulation studies

For comparison of the standard PI-controller with the proposed vibration suppression concepts, the same boundary conditions were assumed like the operating point in the unstable range of the adhesive force characteristic in Fig.3.2 and therefore the wheel-to-rail slip. Consequently, for a start-up from initial standstill to the operating point, the final angular frequencies coincide. However as shown in Fig. 6.19, the maximum of the adhesive force characteristic is earlier reached at t = 0.6 s, instead of t = 1.3 s as in Fig. 3.6, due to the increased system dynamics by the proposed modal state controller. For the first slip-stick vibration with a adhesion force gradient of s δµ/δ(∆v) = −0.06 m , the state controller provides significant additional damping to prevent the oscillations. The slight vibrations in T67 result from the assumed stochastic noise in the adhesion forces. As plots for Te reveal, sensor noise amplification differs significantly for both feedback controls. For the second slip-stick vibration with a very steep adhesion force gradient of s δµ/δ(∆v) = −0.12 m , the state controller does not provide sufficient damp- s ing as it was just designed for up to δµ/δ(∆v) = −0.1 m due to increased sensor noise amplification. Thus the passive readhesion controller triggers and subsequently reduces Te until the slip-stick vibrations decline. In Fig.6.20, a mechanical vibration absorber is mounted at the indirect- driven wheel to suppress slip-stick vibrations. Here, the system dynamics remain unchanged where the maximum of the adhesive force characteristic is reached at t = 1.3 s. Due to the absence of any feedback controller, there is no significant sensor noise amplification. Both slip-stick vibration events are well damped with the design of a sophisticated mechanical absorber. However, the already discussed drawbacks of a mechanical solution with fixed damping parameters and involving maintenance are not the preferred method for vibration suppression. The most sophisticated control scheme is obtained by combining the advantages of both mechanical absorber and state controller resulting in a virtual absorber realized by a feedback controller. Here, the simulation of Fig.6.21 depicts the original system dynamics. In comparison with the state 122 6 Anti-vibration control controller, the sensor noise amplification is considerably lowered where at the s same time higher active damping is provided up to δµ/δ(∆v)= −0.12 m . The latter facilitates the suppression of both slip-stick events where the second one is just slightly damped illustrating the limits of the control scheme with state of the art traction drive-trains.

6.6.2 Overview

In Tab.6.5 all proposed vibration suppression concepts are discussed with their advantages and drawbacks and ﬁnally, recommendations for both their suitability and applicability for various traction drive-trains are given:

The design goal of the modal state control is the optimization of the entire system dynamics, where on the other hand the virtual absorber control just focuses on the slip-stick vibration suppression. The latter scheme is suitable for state of the art drive-trains providing the highest achievable damping considering the sensor noise ampliﬁcation and at the same time oﬀering both a simple design method based on the mechanical absorber parameters and a headache-less commissioning procedure on the vehicle. 6.6 Comparison of control schemes 123

1 10 ω

45 20 T

5 6 ω

30 20

67 10 T 0 -10

5 7 ω

2 i

v 1

10 e

T 5

0 0 1 2 3 4 5 time [s]

Fig. 6.19 Simulation studies with the advanced active-passive anti-vibration control rad m ( ω [ sec ], T [kNm], ∆v [ s ] ). 124 6 Anti-vibration control

1 10 ω

45 20 T

5 6 ω

67 10 T

5 7 ω

2 i

v 1

10 e

T 5

0 0 1 2 3 4 5 time [s]

rad Fig. 6.20 Simulation studies with the mechanical anti-vibration absorber ( ω [ sec ], m T [kNm], ∆v [ s ] ). 6.6 Comparison of control schemes 125

1 10 ω

45 20 T

5 6 ω

67 10 T

5 7 ω

2 i

v 1

10 e

T 5

0 0 1 2 3 4 5 time [s]

rad Fig. 6.21 Simulation studies with the virtual absorber control ( ω [ sec ], T [kNm], m ∆v [ s ] ). 126 6 Anti-vibration control Recommendation a.6.5 Tab. Drawback Advantage Simulation damping Maximum amplification Noise maintenance Mech. time Commissioning force traction of Loss dynamics System controller Speed scheme Control goal Design concepts Control vriw irto upeso oto ocpsfrtato drive-tr traction for concepts control suppression Vibration Overview: ia parameters rical empi- with the is utilized controller where speed train o every for active low damping very scheme control simplest installed, 3.6 Fig. till low no low no original original 2.10 Fig. default Standard − 0 . 035 s/m drive- 1]Fg 6.1 [10]/Fig. sdo not whether or used control concept backup & emergency as every train for force loss traction of in results tion reduc- torque motor traction systems all for ble applica- universally all none no low yes original – reduction torque motor smart Passive i.6.19 Fig. drive- rdb h active scheme the by ered cov- are slip-stick vibrations all not recommendedas not noise amplification sensor high vibrations, all with dealing uncapable active available damping increased till high no high no increased modified 6.4 Fig. optimization system active overall state Modal i.6.5 Fig. − 0 . 1 alone s/m ains. c r increased are ics dynam- system the achievable and is ing damp- active sufficient where trains future for controller speed dified mo- noise amplification, sensor high tions vibra- slip-stick all with deals coupling till high no high no/yes increased modified passive & active of coupling Passive & state Modal i.6.19 Fig. 6.7 Fig. − 0 . 1 subsequently s/m all & drive- 1]Fg 6.10 [12]/Fig. tt fteart is installed the control traction no of where state or decoupled heavily are wheelset and century motor where last drive-trains for parameters absorber fixed unsprung mass, increased maintenance, amplification no noise damping, mechanical unlimited all none yes low no original original wheel indirect-driven the at mounted absorber Mechanical i.6.20 Fig. mlfiainis noise minimized sensor amplification time same the the and at provided is ing damp- adaptive and achievable highest where art the drive-trains of state for amplification noise sensor ef- medium fort, design highest absorber adaptive ping, dam- active highest no original suppression vibration achievable best Passive & absorber Virtual till medium medium no/yes original i.6.15 Fig. i.6.21 Fig. − 0 . 12 s/m all & Chapter 7 Research in Japan

European and Japanese high-performance locomotives differ substantially in their construction as described in Chapter 2. Unlike European locomotives, Japanese locomotives such as both type Blue Thunder EH200 and Kintaro Eco-Power EH500, as shown in Fig. 7.1, mainly operate in large DC 1.5 kV electrified networks. These networks necessitate a different propulsion system technology, such as e.g. an additional LC-input filter for the traction converter, as depicted in Fig.2.8. Besides the differences in the electrical main circuit, also another mechanical traction drive-train topology is applied, namely axle-hung drive-trains. To adapt the novel applications discussed in this thesis to the Japanese railway technology, first, the compatibility of the proposed anti-vibration control schemes was verified with the control scheme installed on the Japanese high-performance locomotive. Here, a detailed analysis derived the interaction of the proposed anti-vibration control scheme with both flux system of the asynchronous machine and the inverter LC-input filter. The research results prove the applicability of the proposed anti-vibration control scheme for Japanese high-performance locomotives and thus facilitate a highly efficient utilization of the tractive effort level. During my extended stay at the Railway Technical Research Institute, a sophisticated novel method for the diagnostics of the traction drive-train was introduced. Several parameters of the Japanese traction drive-trains change due to both wear and ageing effects. These variable parameters, e.g. the stiffness of the rubber elements and the wheel diameter, are estimated in a tailor-made solution based on the analysis of the vibration characteristics. On this basis, a suitable method for the predictive maintenance of the Japanese high-performance locomotives was invented. 128 7 Research in Japan

(a) DC locomotive EH200: Blue Thunder

(b) AC/DC locomotive EH500: Kintaro Eco-Power

Fig. 7.1 High-performance locomotives of Japanese Railways Freight Company [60]. Chapter 8 Applications

All virtual technologies of the last preceding chapters are veriﬁed based on several test runs of a high-performance locomotive as far as possible. The results prove the applicability of the presented control schemes and point out that those are ready for commercial implementation. In the following, the applications are discussed in the order of their implementation sequence on a high-performance locomotive. In the beginning, the two natural frequencies of the traction drive-train are derived by natural identiﬁcation. On this basis, the mechanical parameters of the drive-train, subject to both wear and ageing, are estimated and a complete modally approximated three-inertia model is derived. This model is used to design an estimator including a disturbance model for the unknown traction forces for the most important system variables providing virtual sensors in the multi-inertia drive-train. Based on this estimator, load histograms are introduced not only to measure the torsional impact of the slip-stick vibrations to both cardan hollow- and wheelset-shaft, but also to understand the wheel-rail contact based on the estimated traction forces. Finally, the novel passive and active control schemes are discussed on the basis of several locomotive test runs with the standard speed controller exclusively and then, with the latter plus an additionally installed standard passive readhesion controller. The massive slip-stick vibrations reduction of the novel control schemes becomes apparent in their appropriate probability spectrums.

8.1 Natural identiﬁcation

At first, the traction drive-train has to be identified. Therefore a simple identification method is proposed in Chapter 4.5 by utilizing the natural slip-stick effect. Decreasing the proportional value of the speed controller, slip-stick vibrations arise as shown in Fig.8.1(a) in the measured speed sensor signal ωr. 130 8 Applications

14 13 12 11 [kNm] ∗ r 10 T 9 8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

12 [rad/sec] r

ω 11

10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 time [s] (a) Experimental data by standard sensors

− 10 1

− 10 2

− 10 3 amplitude [dB]

− 10 4

− 10 5 100 150 200 250 300 350 400 450 500 frequency [rad/sec]

(b) FFT analysis of ωr-signal based on test run data

Fig. 8.1 Experimental results of the natural identiﬁcation scheme from an European locomotive and their FFT analysis. 8.2 Virtual model 131

At t = 0.4s, the traction drive-train starts to oscillate with the frequency of the first vibration mode, where the wheelset slips against the asynchronous machine stressing predominantly the hollow-shaft. Then at t = 0.6s, this vibration mode wears off and the frequency of the second mode, which twists the wheelset, becomes apparent. Having obtained sufficient information about the two vibration frequencies of the traction drive-train, the passive readhe- ∗ sion controller reduces the reference torque Tr at t = 0.7s and simultaneously the parameters of the PI-speed controller are restored to their original standard values. As a result, the vibrations decline and the traction drive-train stabilizes again continuing normal operation. Subsequently, the recorded slip-stick vibration data of the ωr-signal is analyzed by a FFT in combination with polynomial filtering. Additional calculations reveal both natural frequencies with a maxima-search, namely rad ω2 = 157.4 and ω3 = 445.4 sec . The two identified slip-stick vibration frequencies are the basis for all novel control technologies introduced in the following.

8.2 Virtual model

Two diﬀerent reduced models of the multi-inertia traction drive are derived which are used either for parameter estimation or for system monitoring and control applications.

8.2.1 Parameter estimation

Wear and ageing of the traction drive-train components lead to an increase in both vibration frequencies. Therefore the above discussed identiﬁcation procedure has to be carried out at certain intervals in order to adapt the frequencies to the continuously changing condition of the traction drive-train. The latter permits the reverse conclusion that the changing frequencies also contain information about wear and ageing. Investigating in this correlation, a novel parameter estimation scheme is introduced in Chapter 4.4. It determines the parameters of a reduced modal three-inertia model based on the approximated mode shapes for only estimation purposes of Fig. 4.13 and the above determined two natural frequencies. The derived estimated parameters are the total inertia of the traction drive-train Jt, the rubber joint stiﬀness cr and the wheel disc radius rw as a function of the wheelset inertia J7,

′ ′ ′ ′ [ Jt cr rw J7 ]=[0.927 1.55 0.944 0.78 ] (8.1) 132 8 Applications where the parameters are each scaled to their nominal value of the reduced model at new condition. The estimated parameters reveal the actual condition of the drive-train utilized in the test runs, where the deviations to the easy measurable parameters

′ ′ ′ [ Jt rw J7 ]=[0.926 0.943 0.77 ] (8.2) are as anticipated in the per mil range. The estimation result of the actual total inertia of the drive-train is mandatory for deriving the reduced three- inertia model for control purposes.

8.2.2 Reduced generic model

For control design, an actual reduced model of the traction drive-train is inevitable considering wear and ageing of its components. In Chapter 4, such a novel scheme for the reduced three-inertia model determination is proposed. It is based on the knowledge of the approximated mode shapes of Fig.4.5, the estimated total inertia and the identiﬁed two natural frequencies of traction drive-train. The resulting parameters of the three-inertia model reﬂect an optimal asynchronous machine-wheelset model for control purposes

′ ′ ′ ′ ′ [ J1 c1 J2 c2 J3 ]=[1.04 1.23 0.87 2.85 0.84 ] (8.3) where also the same scaling is applied as in Eq.(8.1). By introducing this standardization, the effects of wheelset wear, rubber ageing and the modal ′ ′ identification become apparent: the motor inertia J1 is constant, c1 increases ′ ′ mainly due to rubber ageing and both wheel inertias J2 and J3 are affected ′ by wheelset wear. Finally, the increase of c2 is caused by the fixed trade- off between the reference mode shape approximation and frequency response characteristics in the modal identification scheme. Pointing out the differences between both modally reduced models, namely for estimation purposes in Eq.(8.1) and for control in Eq.(8.3), a direct comparison reveals significant deviations not only in the parameters of the ′ ′ ′ indirect-driven wheel J3 and J7, but also in the parameters of the stiffness c1 ′ and cr. Thus both reduced models are optimized for their own purpose: for estimation only and for control application.

8.3 Virtual sensors

The requirements for physical sensors installed in traction drive-trains are manifold and nearly impossible to achieve at same time: 8.3 Virtual sensors 133

• low cost and compact design • high precision • high reliability • robustness against both temperature and shocks Therefore control technology is often applied to substitute physical sensors by virtual sensors which outtake the physical sensors regarding their fulﬁllment of the above mentioned requirements to a great extent. In the standard traction application [42], two variables of the traction drive-train are nowadays already measured sensorlessly. Both variables are depicted as green arrows in Fig.8.2 – the electromagnetic torque Te and the speed of the asynchronous machine ωr.

f1 f2

ω 6 ω7

ωr

T45 z1 T67 z Te ω1 2

(a) Model of the branched traction drive-train

ωr ω˜1 ω˜2 ω˜3 T˜12 T˜23

(b) Reduced model of the branched traction drive-train

Fig. 8.2 Available virtual sensor positions and signals in a branched traction drive-train. 134 8 Applications

Introducing the novel estimator technology of Chapter 5 for the mechanical branched traction drive-train, additional five new variables become available by the virtual sensors. These are depicted in Fig.8.2(a): the speed of both wheels ω6 and ω7 and both hollow- T45 and the wheelset-shaft torque T67 and finally the sum of the adhesion torques z1 +z2. The virtual sensor technology is based on the identified modal three-inertia model for control purposes as given in Eq.(8.3) and is shown in Fig.8.2(b). The observed variables of Fig.8.2(b) coincide with the ones from the branched traction drive-train of Fig. 8.2(a):

ω˜1 = ωr T˜12 = T45 ω˜2 = ω6 T˜23 = T67 ω˜3 = ω7 (8.4) h i To prove the applicability of the virtual sensors, the natural identiﬁcation test run of Fig.8.1 is reviewed with additional sensors installed in the traction drive-train for the hollow- and the wheelset-shaft torque. The estimation results of the virtual sensors compared to the measured variables are depicted in Fig.8.3. The noise ampliﬁcation of the virtual sensors leads to slightly noisy estimated torque amplitudes. Further, the chosen reduced three-inertia

60 50 40 30 20 measured 10 hollow shaft [kNm] estimated 0

100

-50 measured estimated wheelset shaft [kNm] -100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 time [s]

Fig. 8.3 Experimental test run results of the virtual sensors compared with installed sensors from the high-performance locomotive. 8.4 Load cycle histograms 135 model for the virtual sensors does not exhibit the identical modal behavior as the branched traction drive-train, as depicted in Fig.4.5. Thus the slip- stick oscillations of the second vibration mode are slightly underestimated in their torsional amplitudes. Besides that, both estimated and measured shaft torques are almost identical. The estimation performance of the virtual sensors improve to a great extent with the highly-stiﬀ unbranched traction drive-train. Finally with the proven applicability, the novel virtual sensors are available for every kind of traction drive-train as shown in Fig.4.1.

8.4 Load cycle histograms

Applying virtual sensors to the traction drive-train, the torque variables of in- terest are constantly monitored sensorlessly during slip-stick operation. This data is collected to produce torsional load spectrometers for both hollow- and wheelset-shaft. On the basis of these load cycle histograms, the torsional shaft loads of all driven railway vehicle axles are analysed, then accordingly, the predictive maintenance is scheduled and as a result, the maintenance intervals are maximized considerably lowering the downtime of railway vehicles. Further regarding the adhesion control, the traction drive-train with the most signiﬁcant torsional load histograms is considered as the slip-free run- ning axle in order to minimize its future vibration exposure. Fig. 8.4 gives an example of a torsional load histogram of the wheelset-shaft from the test-runs of a European high-performance locomotive at both wet and dry wheel-rail conditions. Here, this standard histogram uses the wheel-rail slip ∆v as a basis and the amplitude of torsional wheelset-shaft torque as output value. Higher torsional amplitudes occur at dry rather than at wet wheel-rail conditions where the maximum amplitude is determined at T67 = 120 kNm for this particular high-performance locomotive [59].

120 100

wet dry

[kNm]

67 50

0 0 1 2 3 4 5 ∆v [m/s] Fig. 8.4 Various wheelset-shaft load spectra from a European high-performance locomotive operating at diﬀerent conditions of the wheel-rail contact [59]. 136 8 Applications 8.5 Anti-vibration control

Working at the limits of adhesion, the wheelset occasionally operates at a wheel-to-rail slip higher than the maximum achievable tractive effort value due to fluctuation of the adhesion coefficient. This is the operating range where the adhesion force characteristics exhibits a negative gradient and thus slip-stick vibrations arise in the traction drive-train. In Chapter 3, an investigation into the probability distributions of slip- stick vibrations is discussed for the standard control scheme in Fig. 3.10. Based on both identical parameters and test run conditions, the various anti- vibration control schemes proposed in Chapter 6 are reviewed in the following by specialized histograms. In this way, the damping performance of the novel anti-vibration controllers compared to the standard control scheme becomes apparent in two separate histograms. The first histogram displays the amplitude spectrum which indicates the torsional load reduction of the wheelset shaft. The second histogram shows the gradient spectrum revealing the achievable damping effort for the corresponding controller within the negative gradient range of the adhesion force characteristic. Additionally, the correlation of both amplitude and gradient spectrums derives interesting novel insights into the nature of slip-stick vibrations in high-performance locomotives.

8.5.1 Passive readhesion

At the expense of traction force, passive readhesion controllers are usually utilized to suppress slip-stick vibrations in the traction drive-trains by reducing the reference motor torque. The impact of the passive readhesion controller on the vibration amplitude is shown in Fig.8.5(a). Compared to the standard control scheme, a minor amplitude reduction in the vibration occurrence is achieved for amplitudes above 30% of the maximum value. This effect leads to a left shift in the amplitude spectrum which is one quality criterion of a readhesion controller: the bigger the left shift, the better is the performance of the passive readhesion controller. However as already discussed in Chapter 6, the passive readhesion controller only deals with arisen vibrations, thus the total occurrence of 100% is here identical to the one of the standard control scheme. Fig.8.5(b) depicts a minor occurrence reduction in the sidebands of the negative gradient spectrum. Here, the application of a passive readhesion controller leads to a center shift. Both high and low negative gradients oc- s cur less often shifting those to the −0.06 to −0.08 m range. This is a great effect for the coupling of both active and passive control schemes as the max- s imum occuring negative gradient of −0.13 m can not be damped efficiently in the highly-stiff traction drive-train even by the best active control scheme proposed in this thesis. 8.5 Anti-vibration control 137

standard standard & passive ts 15

left shift occurrence in % 5

0 0 0.2 0.4 0.6 0.8 1 wheelset shaft torque [ T67 ] max (T67) (a) Mechanical load on the wheelset shaft

standard 25 standard & passive

10 center shift occurrence in %

0 -0.04 -0.06 -0.08 -0.1 -0.12 -0.14 s adhesion force gradient [ m ] (b) Wheel-rail contact condition

Fig. 8.5 Probability distribution investigation into slip-stick vibrations on a European high-performance locomotive utilizing only a passive readhesion control scheme. 138 8 Applications 8.5.2 Advanced modal state

To prevent the loss of traction force, an active anti-vibration control is introduced on the basis of a modal state space control approach. This novel schemes provides additional damping for the two slip-stick vibration modes of the traction drive-train. In the gradient spectrum of Fig.8.6(b), the damping effect of the novel control scheme is depicted. Unlike the damping performance of the standard s controller till −0.04 m , the modal state controller is providing active damping s for the highly-stiff traction drive-train up to a negative gradient of −0.1 m . The remaining high negative gradients are taken care of by the passive readhesion controller. As a result, 90% of all occurring slip-stick vibrations are now actively damped where the use of the passive readhesion controller is reduced by the factor of ten. The corresponding amplitude spectrum for the modal state controller is shown in Fig.8.6(a). With its application, a small amplitude spectrum still remains with a right shifted occurrence maximum of 0.5 instead of 0.3. Also interesting is that the highest torsional amplitude in the wheelset shaft is significantly reduced by 30% compared to the standard control scheme. 8.5 Anti-vibration control 139

standard ts standard & passive 15

10 occurrence in % 5

novel 0 0 0.2 0.4 0.6 0.8 1 wheelset shaft torque [ T67 ] max (T67) (a) Mechanical load on the wheelset shaft

standard 25 standard & passive

passive installed 20 control method novel active passive 15

10 occurrence in %

0 -0.04 -0.06 -0.08 -0.1 -0.12 -0.14 s adhesion force gradient [ m ] (b) Wheel-rail contact condition

Fig. 8.6 Probability distribution investigation into slip-stick vibrations on a European high-performance locomotive utilizing both advanced modal state and passive readhesion control schemes. 140 8 Applications 8.5.3 Virtual absorber

A newly proposed anti-vibration control scheme simulates a mechanical absorber at the indirect driven wheel virtually by a novel feedback controller circumventing all drawbacks known from the mechanical absorber application. Giving up additional damping for the ﬁrst vibration mode as well as reducing the noise ampliﬁcation in the feedback controller lead to an increased damping range in the negative gradient spectrum of Fig.8.7(a). The virtual absorber scheme provides active damping up to a negative gradient of s −0.12 m . As a result, 99% of all occurring slip-stick vibrations are now actively damped where the use of the passive readhesion controller is reduced by the factor of one hundred. This minimizes the amplitude spectrum of the virtual absorber feedback s controller to only one spectral line of −0.13 m vibrations in Fig.8.7(b). The remaining vibrations are at a 66% amplitude of the maximum occurring vibration amplitude. 8.5 Anti-vibration control 141

standard standard & passive 15

10 occurrence in % 5

novel 0 0 0.2 0.4 0.6 0.8 1 wheelset shaft torque [ T67 ] max (T67) (a) Mechanical load on the wheelset shaft

standard 25 standard & passive

passive installed 20 control method novel active passive 15

10 occurrence in %

0 -0.04 -0.06 -0.08 -0.1 -0.12 -0.14 s adhesion force gradient [ m ] (b) Wheel-rail contact condition

Fig. 8.7 Probability distribution investigation into slip-stick vibrations on a European high-performance locomotive utilizing both virtual absorber and passive readhesion control schemes. 142 8 Applications 8.5.4 Overview

The application of active feedback controllers for the highly-stiﬀ traction drive train reduces the maximum occurring torsional amplitude of the wheelset shaft up to 34% according to Tab.8.1, where at the same time the expected value of the amplitude increases as most of the smaller vibration amplitudes can be actively damped. On the other hand, passive readhesion controllers reduce the expected value of the vibration amplitude by 25%, the so-called amplitude left shift, and lead to a gradient center shift where the expected value of the negative gradient remains almost identical. Finally, the synergy eﬀects of both active and passive controllers result in a 90% vibration suppression for the modal state and respectively 99% for the virtual absorber feedback controller coupled with the passive readhesion controller.

Tab. 8.1 Overview: Analysis of both amplitude and gradient spectrums of various anti- vibration controllers for the highly-stiﬀ traction drive-train.

(Standard) Standard Standard Modal state Virtual absorber Standard Passive Passive Passive

Expected value 0.44 0.33 0.47 0.66 Maximum 1 0.95 0.75 0.66 tude

Ampli- Maximum occurrence 0.35 0.25 0.45 0.66

s s s s Expected value -0.071 m -0.070 m -0.109 m -0.124 m s s s s Damping up to -0.04 m -0.04 m -0.1 m -0.12 m Gra- dient Vibration occurrence 100 % 100 % ∼ 10 % ∼ 1% Chapter 9 Conclusion

A highly sophisticated traction control scheme necessitates not only a highly dynamic torque response and a speed sensorless operation as its major core features, but also several additional key control components. The latter are deﬁned as

• Basic integration of the traction drive-train – identiﬁcation, parameter estimation, monitoring and predictive maintenance • Anti-vibration control of both electrical line- and mechanical traction drive-train side • Wear reduction of the mechanical traction drive-train components and reduced rail abrasion • Maximum tractive eﬀort level utilization.

The utmost goal for a state of the art traction control scheme is to achieve the maximum performance of both core features and all key components at the same time. This can only be obtained by a modular control concept. Here, each module represents one unique approach for a key control component ex- ploiting either a novel control method or a special traction effect. As some of these modules are partially interdependent, it is a complex optimization approach to determine the best module match for one traction drive-train series. The matching process itself is predominantly based on the characteristics of the traction drive-train as the most influencing parameters and thus focuses on the maximum synergy of all modules applied. For such a tailor- made traction control, several modules of every key control component are in the pool. The more modules there are, the more sophisticated the synergy match will be for a particular traction drive-train series. This doctoral thesis is the keystone for a novel modular traction control scheme where just a few new modules for the above mentioned key control components are presented in the following for mechanical traction drive-trains. For a sophisticated traction control scheme, it is inevitable to obtain knowledge about the actual parameters of the mechanical drive-train which are subject to significant wheelset wear due to wheel-rail contact and ageing of 144 9 Conclusion rubber elastic joints during its operational life. To comply with the limitations imposed by the traction application, the identification effort is lowered by applying previous knowledge of the flexible multi-inertia structure. Once in the design stage (green) of the drive-train series, its vibrational behavior, namely the mode shapes, are calculated as shown in Fig.9.1. Using the latter as a convergence criterion guarantees adaption to wheelset wear and to rubber joint ageing for branched as well as unbranched traction drive- trains which is the basic idea for this newly proposed identification procedure. For an accurate and suitable reduced model, the characteristic mode shape values are extracted. On this basis, two optimal approximated mode shapes are derived. These represent a mechametrical signiture of the traction drive-train series like a biometrical fingerprint for a human being. So every type of traction drive-train has its own signiture – the optimal approximated mode shapes. This ingenious procedure reduces subsequent additional calculation effort for real-time operation by optimizing the computation time. Due to the application of the optimal approximated mode shapes, only the simple identifiable part of the frequency response, namely the two eigenfrequencies, has to be determined from any given speed sensor location in the traction drive-train. This is achieved at defined intervals (blue) by natural identification utilizing the natural effects of the traction drive-train, namely slip-stick vibrations. Therefore only minor modifications of the implemented traction control scheme were necessary to quickly identify those two vibration frequencies. However in the long run, the huge drawback of the proposed natural identification method, unlike test signal injection, is the unpredictable slip-stick vibration exposure of the drive-train. Therefore test signal injection is highly recommended for commercial use to identify the two required eigenfrequencies at predetermined conditions. To derive a reduced model for generic purpose, one natural given parameter, namely the total inertia, is mandatory which is obtained by a parameter estimation algorithm. The latter additionally necessitates two characteristic parameters of the complex traction drive-train from the design stage to decouple the two effects – ageing and wear. Besides the estimated total inertia, a highly precise estimation of the variable mechanical parameters, namely the rubber joint stiffness and the wheel disc radius, are supplied for the predictive maintenance of the traction drive-train. On the basis of the generic reduced model, virtual sensors are introduced for several speed and torque variables of the complex traction drive-train, e.g. the wheelset shaft torque, and further, the unknown adhesion forces as well. These signals are estimated in real-time (red) and are subsequently analyzed by the use of histograms to mainly determine both utilization of the wheel-rail contact and load cycle spectrum of the wheelset shaft. The latter spectrum is also being consulted for predictive maintenance where each wheelset axle of the high-performance locomotive is assigned its corresponding histogram. The wheelset axle with the highest load cycle impact is predictively determining the next underfloor maintenance. 9 Conclusion 145

Aquisition of preset Identification of variable traction drive-train data vibration frequencies Defined intervals Decoupled Mode shape Natural identification modal extraction parameters

Design stage

Virtual Parameter Generic model estimation model

Real time Predictive Rubber Wheelset Virtual Histo- System main- ageing wear sensors gram moni- tenance toring

Anti- Standard Passive Modal Virtual vibration readhesion state absorber control

Modular approach Active > −0.04 s > −0.04 s > −0.1 s > −0.12 s damping m m m m

Amplitude 100 % 100 % ∼ 10 % ∼ 1 % spectrum

Applications based on test-run data of a European high-performance locomotive.

Fig. 9.1 Overview of all traction drive-train related research covered in this thesis. 146 9 Conclusion

Finally for real-time application, four different anti-vibration control modules are presented in this thesis besides the mechanical vibration absorber approach. In the following, these are discussed in terms of their coupling strategy, their active damping limit and their performance by the vibration amplitude spectrum of the wheelset shaft. First, the standard speed controller installed on every high-performance locomotive provides negligible active damping apart from the traction drive- train’s own damping. Second, adding an advanced passive readhesion controller, which is based on a virtual sensor for the wheelset shaft torque, does not influence the active damping performance in anyway as the controller only reduces the torsional amplitude once the vibration arises. Thus, the vibration count of the wheelset shaft is identical. However, the amplitude spectrum itself exhibits a left shift with a slight reduction of the high vibration amplitudes compared to the standard controller’s spectrum. Further, the gradient spectrum experiences a center shift with reduced side bands. This effect of the passive controller is ideal for coupling with the active anti-vibration controller as the most negative adhesion force gradients are reduced in their occurrence. Third, a modal state controller is derived based on the present standard control scheme. A simple starting procedure of this novel active anti-vibration controller from any standard controller is proposed to raise its acceptance and applicability in the traction application. The resulting control scheme is enhanced by removing several windup effects. Furthermore, the limits of the active anti-vibration controller are determined by the quality of the speed sensor signal and of the applied traction drive-train. To comply with these limitations imposed by the traction application, the active anti-vibration controller is designed to be capable of damping slip-stick vibrations up to a certain minimum gradient of adhesion force characteristic. For more negative gradients, a passive readhesion controller is additionally coupled and intervenes until the active anti-vibration controller can cope with the gradient. Using the synergy of the active and the passive controllers, the stable operating range of the traction drive is significantly increased without any loss of traction force. As a result, the vibration amplitude spectrum shrinks not only by the factor of ten, but also the maximum occurring vibration amplitude is considerably reduced compared to the standard controller’s spectrum. Fourth, the latter state controller results in a sophisticated vibration damping performance and in an increased dynamics of the whole traction system achieved by a modified speed and a feedback controller. Here, the commissioning engineer has to deal with a new modified speed controller as well as with the complex pole placement technique to derive the new feedback controller at the same time. To simplify the commission complexity and to further increase the damping performance, a virtual absorber feedback controller with minimized sensor noise amplification is introduced based on the standard speed controller. The absorber is virtually mounted on the indirect- driven wheel of any traction drive-train and for its starting is faded in from 9 Conclusion 147 the standard control scheme. Its parameters are continuously adapted to wheelset wear and to rubber joint ageing circumventing all drawbacks known from the fixed mechanical absorber installation. As a result, the amplitude spectrum shrinks again by the factor of ten, i. e. by a factor of one hundred compared to the standard controller’s spectrum, as well as the maximum occurring vibration amplitude is again slightly reduced. However, a coupling with the passive readhesion controller is still necessary for very rarely occurring highly negative adhesion force gradients beyond the capabilities of the active controller. On the basis of the virtual absorber, more universally valid virtual filters are applied to the traction drive-train to protect the structure from the vibrational intake as well as to increase the tractive effort utilization. The latter effect is only achieved by active virtual readhesion where on the other hand, the well-known passive approach comes along with a loss of traction force. Virtual protection is obtained by all virtual filters which are mutually exclusively applied to all three inertias of the virtual model. To implement such a multi-feedback controller scheme, a looping state machine is introduced to handle the activation of the controllers according to the desired state. Finally, all presented virtual modules are summarized within the virtual technology programme in Fig. 9.2. From the viewpoint of slip-stick vibrations only, there are two very simple and efficient vibration suppression schemes discussed in this thesis: • Mechanical absorber is mounted on the indirect-driven wheel of last century, asynchronous machine-wheelset decoupled, soft traction drive-trains • Virtual absorber control is applied to state of the art, highly-stiff traction drive-trains.

VIRTUAL TECHNOLOGY PROGRAMME

VIRTUAL BASICS VIRTUAL FILTERS

Virtual Virtual Virtual Virtual Virtual Model Sensors Readhesion Protection Absorber

Parameter Signal Increased Wear Vibration Monitoring Monitoring Traction Reduction Suppression

Predictive Maintenance Best Traction Performance

Fig. 9.2 The Virtual Technology Programme for traction drive-trains improves the predictive maintenance and provides the best traction performance. 148 9 Conclusion

Anti-vibration Synergy Branched traction path drive-train Basic integration Tractive effort level

Wear reduction Synergy path

Fig. 9.3 Best traction control: Synergy path of the novel Virtual ﬁlters-traction control scheme connecting several modules of diﬀerent key control components.

Both schemes are based on the derived absorber parameters according to the traction drive-train characteristics where these are either realized in a mechanical structure or just implemented into a software controller. For an overall high-performance traction control scheme, the traction drive- train has to be integrated by several basic modules which are executed once at the design stage (green), at a given interval (blue) and in real-time (red) as shown in Fig.9.3. Further, the synergy effects of the key control components, namely the anti-vibration control, the wear reduction and the maximum tractive effort level utilization, have to be taken always into consideration based on a specific traction drive-train. To achieve the latter, a modular control concept for the key control components is necessary. Here in Fig.9.3, a synergy path including the modules for the virtual absorber (top), for the virtual protection (lower left) and for the active virtual readhesion (lower right) is given for the branched traction drive-train. This synergy path connects several modules of each key control component and is evaluated on the basis of both their performance and synergy effects. So finally, the contribution of this thesis towards the best overall traction control scheme becomes evident covering both basic integration of the traction drive-train and several modules for all key control components with the main focus on the anti-vibration control. Appendix A Modelling of traction drive-trains

Two modern traction drive-trains are discussed in this chapter. On the basis of their mechanical cross-section model, two discrete inertia models are derived. Subsequently, their mode shapes are calculated for comparison purposes. 150 A Modelling of traction drive-trains

(a) Cross section model [112, 113]

f1 f2

ωr

Te z2

(b) Discrete inertia model

Fig. A.1 Traction drive-train: Branched ten-inertia system with rubber joints. A Modelling of traction drive-trains 151

1 0 Hz 0

-1 1 22.78 Hz 0

-1 1 65.31 Hz 0

-1 1 117.5 Hz 0

-1 1 138.4 Hz ∗ b

ϕ 0

-1 1 204.5 Hz 0

-1 1 241.3 Hz 0

-1 1 391.6 Hz 0

-1 1 1045 Hz 0

-1 1 1130 Hz 0 -1❶ ❷ ❸ ❹❽ ❺❾ ❻❿ ❼

Fig. A.2 Mode shapes of the branched ten-inertia system at new condition. 152 A Modelling of traction drive-trains

(a) Cross section model [13]

f1 f2 ωr

Te z2

(b) Discrete inertia model

Fig. A.3 Traction drive-train: Unbranched seven-inertia system with curved-tooth couplings. A Modelling of traction drive-trains 153

1 0 Hz 0

-1

1 31.39 Hz 0

-1

1 68.22 Hz 0

-1

1 193.61 Hz ∗ b

ϕ 0

-1

1 560.84 Hz 0

-1

1 577.97 Hz 0

-1

1 1059.61 Hz 0

-1❶❷❸❹❺❻❼

Fig. A.4 Mode shapes of the unbranched seven-inertia system at new condition.

Appendix B Curriculum vitae

Personal information Name MichaelFleischer Date of birth 17. August 1975 Place of birth Erlangen, Germany Nationality German

Work experience 03/2019- International e-mobility and railway consulting 01/2018-02/2019 Head of e-mobility, Baum¨uller GmbH, Germany 05/2012-12/2017 International railway consulting 05/2011-04/2012 Project leader for strategy projects, Strategy infra- structure department, Deutsche Bahn AG, Germany 03/2006-04/2011 Technical sub-project leader and system engineer for commuter, underground and high speed trains, System engineering department, Siemens Mobility, Germany 03/2005-11/2005 Guest researcher, Railway Technical Research Institute (RTRI), Japan Railways, Tokyo, Japan 03/2002-03/2005 Research associate, Chair of Electrical Drives, University of Erlangen-Nuremberg, Germany

Fellowship and promotion 11/2008 Invited honorary guest, VDE-Kongress 2008, “Zukunfts- technologien”/“Future technology”, Munich, Germany 10/2006 VDE/ETG-award for the best publication of the year 2006 in the ﬁeld of railways 09/2005 Invited honorary guest as representative of Japan Rail- ways, Schienenfahrzeug Tagung, Graz, Austria 03/2005-09/2005 Pre-/Post-doctoral Fellowship (short-term), Japan Soci- ety for Promotion of Science 156 B Curriculum vitae

03/2004 Conference sponsoring by German Research Foundation (DFG), IEEE Advanced Motion Control Workshop in Kawasaki, Japan 09/2000-11/2000 USA-Internship, Marketing and Sales at the Supermodiﬁ- cation Center, Siemens Energy and Automation Inc., Lit- tle Rock, Arkansas, USA Preparation to USA-internship in the Quotation Center and in the Marketing and Sales America department, Frauenaurach, Germany 07/1995 Emmy-Noether award, Germany

Studies and education 04/2002 Award of Electrical Engineering-Diploma certificate 06/2001-12/2001 Diploma thesis, An Efficient Braking Method for Con- trolled AC Drives with a Diode Rectifier Front End, Re- search and Development department, Siemens Automa- tion and Drives, Motion Control Group 09/2000-11/2000 USA-Internship, Marketing and Sales at the Supermod- ification Center, Siemens Energy and Automation Inc., Little Rock, Arkansas, USA 11/1999-02/2000 Internship, Inverter development, Siemens Automation and Drives, Standard Drives Group, Erlangen, Germany 05/1998-07/2000 Tutor at the Chairs of Applied Mathematics and at the Chair of Electrical Drives, University of Erlangen- Nuremberg, Germany 09/1995-10/1995 Internship, Siemens ANL, Erlangen, Germany 09/1995-12/2001 Studies of Electrical Engineering at the Friedrich- Alexander-University of Erlangen-Nuremberg, Germany 07/1995 High school diploma with distinction 02/1995 High school thesis, “DC and AC drives” 09/1986-06/1995 Emmy-Noether-High school, Erlangen, Germany 09/1982-07/1986 Elementary school Brucker Lache, Erlangen, Germany

Organizational participation IEEE IAS Institute of Electrical and Electronics Engineers, Indus- try Application Society (since 2004) VDE ETG German Society of Electrical Engineers, Power Engineer- ing Society (since 2004) JSPS-Club Alumni Club of the Japanese Society for Promotion of Science References

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So far the traction drive-train was not considered in the traction control software as all mechanical parameters were unknown. Thus an innovative parameter identification and estimation scheme is presented and facilitates predictive maintenance of the traction drive-train. Based on the identified parameters, several active and passive control schemes are developed featuring an ingenious control approach – the application of a looping state machine. The latter achieves an optimal symbiosis of active and passive traction controllers and provides an outstanding traction performance. Finally, the natural identification scheme, parameter identification and estimation method, virtual sensors and the traction control schemes are experimentally validated with data from a high-performance locomotive.

ISSN 1437-675X