stabilization of an induction machine drive7532/fulltext01.pdfstabilization of an induction machine...

Stabilization of an Induction Machine Drive

Henrik Mosskull

TRITA–S3-REG-0301ISSN 1404-2150

ISBN 91-7283-555-9

Automatic ControlDepartment of Signals, Sensors and Systems

Royal Institute of TechnologyStockholm Sweden

Submitted to the School of Electrical Engineering, Royal Institute ofTechnology, in partial fulfillment of the requirements for the degree of

Technical Licentiate.

AbstractModern electric trains almost exclusively use induction machines fed byvoltage source inverters for propulsion. To reduce external as well as internaldisturbances, an LC-filter is inserted between the input terminals of the driveand the inverter. In connection with efficient torque control of the motors,these filters however potentially make the drive unstable as excitations of thefilter resonance are amplified by the torque control. The solution to suchstability problems usually is to modify the torque reference according tooscillations in the inverter input voltage. Although this strategy may bemotivated from simple models of the closed loop drive, structured tuning ofthis feed-forward compensation is more difficult. Experience shows that thecompensation must not only be adapted to the varying dynamics of the drive,but also properties of the torque control and applied inverter pulse patternsmust be considered.

This thesis derives an expression for the feedforward compensation stabilizingthe drive in terms of the operating conditions. These conditions include themotor speed, operating point torque and flux as well the actual DC-linkvoltage and time delays of the torque control. From an equivalent feedbackrepresentation of the drive, the stabilization problem is first interpreted asappropriately shaping the inverter input admittance. The exact shape of thecompensation meeting the design requirements is then derived using linearmodels of the drive, including control. The derived linear models are alsoused to obtain tuning rules for the torque and flux controller parameters, givenrequirements on bandwidth and stability margins. It is assumed that torque iscontrolled using the method Indirect Self Control.

Stability of the closed-loop drive with the proposed stabilization is validatedfrom measurements. Using models obtained from frequency domain systemidentification, stability of the non-linear closed-loop drive is verified bycombining stability results for linear systems with the small gain theorem forthe non-linear model errors.

AcknowledgementsThis work is the result of a collaboration between Bombardier Transportationin Västerås and KTH, Stockholm. I would first of all like to express mygratitude to the people who made this work possible in the first place. Theseinclude my supervisor at KTH, Prof. Bo Wahlberg and my superiors atBombardier Transportation, Peter Oom and Dr. Kent Öhrn. Especially duringthe writing of this thesis, Kent helped me hide from much of the daily work atthe company.

My technical supervisor at Bombardier Transportation was Johann Galic.Without his experience and deep knowledge of drives in general, this workwould not have been printed.

I would also like to thank Prof. Stefan Östlund, KTH, for support regardinggeneral issues related to induction motor control and the people introducingme to the subject, Dr. Steffen Richter, Dr. Peter Krafka and Michael Rampe atBombardier Transportation in Mannheim, Germany.

Eva Sandberg was kind to help me with the layout in Microsoft Word.

This work was partly supported by The Swedish Science Foundation, which isgratefully acknowledged.

Finally I would like to thank you Åsa, Albin and Lukas for your love andsupport.

Contents

1 Introduction......................................................................... 1

1.1 Background......................................................................................1

1.2 Notation and Nomenclature .............................................................3

1.3 Outline .............................................................................................7

1.4 Contributions ...................................................................................8

1.5 Drive Data........................................................................................9

2 DC-Link Stabilization ...................................................... 11

2.1 Simplified Stability Analysis .........................................................11

2.2 Stabilization Methods ....................................................................14

2.3 Summary........................................................................................16

3 Stabilization Formulated as Loop Shaping .................... 19

3.1 Feedback Representation ...............................................................20

3.2 Design Objective............................................................................21

4 Linear Process Model ....................................................... 23

4.1 Inverter...........................................................................................24

4.2 Induction Machine .........................................................................304.2.1 Physical Description ...........................................................304.2.2 Mathematical Description...................................................32

4.3 Fieldweakening..............................................................................37

4.4 DC-Link Current............................................................................40

4.5 Linearization ..................................................................................41

5 Indirect Self Control......................................................... 45

5.1 Slip Frequency Control ..................................................................46

viii Contents

5.2 Discrete Time ISC..........................................................................475.2.1 Control Law........................................................................475.2.2 Prediction............................................................................50

5.3 Continuous Time ISC ....................................................................525.3.1 Control Law........................................................................525.3.2 Prediction............................................................................54

5.4 Modulator ......................................................................................54

5.5 Summary........................................................................................55

6 Linearization of Control Law .......................................... 57

6.1 Field-orientation.............................................................................57

6.2 Field-Oriented Control Law...........................................................59

6.3 Linear Model of Prediction............................................................60

6.4 Modulator ......................................................................................62

6.5 Summary........................................................................................63

7 Closed Loop Dynamics ..................................................... 65

7.1 Field-Orientation without Time Delay...........................................65

7.2 Field-Orientation with Time Delay................................................70

7.3 Disturbance Rejection....................................................................727.3.1 DC-Link Voltage Compensation ........................................727.3.2 Motor Speed Compensation................................................73

7.4 Closed-Loop Control .....................................................................74

7.5 Inverter Input Admittance..............................................................76

7.6 Fieldweakening..............................................................................77

8 Controller Tuning............................................................. 79

8.1 Torque Controller ..........................................................................798.1.1 Gain Scheduling .................................................................85

8.2 Flux Controller...............................................................................89

9 Design of Additive Stabilization ...................................... 91

9.1 Loop Gains without Stabilization ..................................................91

Contents ix

9.1.1 Coasting..............................................................................919.1.2 Driving................................................................................939.1.3 Braking ...............................................................................95

9.2 Modified Inverter Input Admittance ..............................................96

9.3 Approximations .............................................................................98

9.4 Stabilization at Zero Torque ..........................................................999.4.1 Low Frequencies...............................................................1009.4.2 High Frequencies ..............................................................1039.4.3 Choice of Band-Pass Filter ...............................................1089.4.4 Torque Controller Gain Scheduling..................................108

9.5 Stabilization at Non-Zero Torque ................................................1099.5.1 Choice of Stabilization Gain.............................................1099.5.2 Choice of Band-Pass Filter ...............................................1129.5.3 Torque Controller Gain Scheduling..................................1139.5.4 Braking .............................................................................114

9.6 Summary......................................................................................116

10 Linear Stability Analysis ................................................ 119

10.1 Identification of Inverter Input Admittance .................................120

10.2 Software Implementation.............................................................12110.2.1 Time Delays......................................................................12110.2.2 Torque Controller .............................................................12210.2.3 DC-Link Stabilization.......................................................122

10.3 Closed Loop without Stabilization...............................................12310.3.1 Coasting............................................................................12310.3.2 Driving..............................................................................12610.3.3 Braking .............................................................................127

10.4 Closed Loop with Stabilization....................................................12810.4.1 Coasting............................................................................12810.4.2 Driving..............................................................................12810.4.3 Braking .............................................................................129

11 Input-Output Stability Analysis .................................... 131

11.1 Robust Stability Analysis using Measurements ...........................131

11.2 Experimental Result.....................................................................136

x Contents

12 Conclusions and Future Work....................................... 139

12.1 Conclusions..................................................................................139

12.2 Suggestions for Future Work .......................................................140

A Approximation of Admittance Phase............................ 143

B Linear Model of Field-Orientation................................ 147

Alternative Expression for Field-Orientation .......................................148

Non-Linear State Space Representation ...............................................150

Stationary Operating Points..................................................................153

Linearization.........................................................................................155

Field-Orientation ..................................................................................158

DC-Link Voltage Disturbance..............................................................159

C Drive Data........................................................................ 163

Drive Parameters ..................................................................................163

Tractive Effort ......................................................................................164

Switching Frequencies and Pulse Periods ............................................164

Operating Points used for Evaluation...................................................167

D Space Vectors .................................................................. 169

Bibliography ........................................................................... 173

Chapter 1

Introduction

1.1 BackgroundThe three-phase induction machine was invented already in the 19th century.The induction machine is an AC motor but to stress that it may just as well beused as a generator, we use the term induction machine instead of inductionmotor. Compared to DC-motors, induction machines have higher powerdensities and are mechanically more robust, which make them the ideal motorin many applications. On the other hand, they require AC power supplies andwere for a long time considered difficult to control. As a matter of fact, firstwith the invention of field-oriented control (or vector control) in the early1970’s, [7][4], control of the induction machine could be compared to that ofa separately excited DC-motor. The need of advanced power electronicsfurther delayed the widespread use of induction machines in industrial andtraction applications till the 1980’s. Today, however, the induction machine isthe standard traction motor and is almost exclusively fed by a voltage sourceinverter. A typical propulsion system in a DC traction application maytherefore be depicted as in Figure 1-1, where the induction machine drive isconnected to the DC supply voltage E(t) either via an overhead line or a thirdrail.

E

+

-

Ud

id

VoltageSourceInverter

+

-

Motor

ωm, T

DC

Figure 1-1: DC propulsion system in a traction application.

2 Introduction

Each inverter feeds one or several induction machines in parallel and powercan flow in both directions, i.e. either from the electric power supply to themotors or vice versa. The torque applied at the motor shaft is denoted T(t) andthe motor speed is represented by ωm(t).

The voltage source inverter generates three AC voltages from a DC-voltage,which is denoted Ud in Figure 1-1 and will be referred to as the DC-linkvoltage. To ensure that this voltage is smooth, the inverter is connected to theDC-power supply via a capacitor bank. Disturbances on the DC-link voltagecome through the input voltage E(t), but are also generated by the inverteritself. Such disturbances may interfere with the signaling system in a tractionapplication and must consequently be suppressed. For this reason the drive isalso equipped with an inductor. By modeling the inductor by an inductance Land a resistance R, and the capacitor bank by a capacitance C, the inductionmachine drive in Figure 1-1 can be represented as in Figure 1-2. Actually thevalues of R and L in Figure 1-2 should also include resistance and inductanceof the overhead line. This will however be neglected in this work as theinductance of the overhead line normally is small compared to the filterinductance. Figure 1-2 also shows the coupling vector k(t) through which theoperation of the inverter is controlled.

E

+

-

Ud

idR

L

CVoltageSourceInverter

+

-

Motor

ωm, T

k

Figure 1-2: Model of an induction machine drive.

The components R, L and C form the input filter of the drive, which can becharacterized by its resonance frequency ω0 and damping factor ζ given by

(1.1)

The resistance R has to be as small as possible in order to reduce powerlosses. From the expression for ζ in (1.1), it however follows that a smallresistance makes the input filter poorly damped. Whereas the inductornormally is determined by requirements on disturbance rejection, the

Introduction 3

capacitor may be chosen more freely. In traction applications, where spaceand weight are critical, the ambition is therefore to reduce the size of thisbulky and heavy component. The effects of reducing the capacitance are thatthe DC-link voltage becomes more sensitive to power fluctuations and theresonance frequency of the input filter increases, see (1.1).

From a control point of view, induction machines are very challenging as theyare non-linear multi-dimensional systems. Controllers, like the field-orientedcontroller mentioned above, were therefore historically designed from purephysical insight. Stability and performance of closed-loop induction machinecontrol have rigorously been treated in a number of recent papers, see forexample [20] and [3]. These contributions however only treat the closed-loopdynamics of the induction machine and not the complete drive. In for exampletraction applications, the main difficulty during commissioning is usually notstability of the internal torque control itself but rather interactions between theinverter and the input filter. Such problems have been analyzed for open-loopcontrol of induction machines in [13] and [1]. The practically more interestingcase with closed-loop torque control is discussed in [2]. Here the analysis ishowever constrained by assuming perfect torque control. This is neverachieved in practice, but the assumption makes it possible to simplyunderstand the stability problems from an electric-circuit point of view. Thecircuit model also naturally leads to the ideas in [2] and [26] for stabilizing thedrive. Although these proposed methods have proven to work in practice, theinvolved tuning often tends to be done through trial and error. As thedynamics of the system strongly vary with the operating points, trial and errortuning becomes a tedious and expensive task, which still may result in non-optimal performance. The intention with this contribution is therefore toreduce commissioning time by deriving tuning rules for the stabilization interms of process data and conditions of operation. The derivation will bebased on improved linear models of the closed loop induction machine drive.It is further assumed that torque control is based on the method Indirect SelfControl, [9].

1.2 Notation and NomenclatureThis thesis treats stability problems due to interaction between the motors andthe input filter of an induction machine drive. We will introduce the term DC-link stabilization for methods aiming at solving such problems with DC-linkstability. As this contribution originates from problems in tractionapplications, the used vocabulary often associates with traction drives. Forexample driving and braking are used to indicate the direction of the power

4 Introduction

flow of the drive. Driving means power flowing from the DC-link to themotors and consequently the word braking is used for power flow in the otherdirection. For operation with zero torque we introduce the term coasting.

Much of the analysis and synthesis is based on linear models of thecomponents of the drive. With zero initial conditions, a linear system withinput u(t) and output y(t) can be described with Laplace transforms as

Here U(s) and Y(s) are the Laplace transforms of the input and output signals,respectively and G(s) is the transfer function of the system. To represent therelation between the time domain signals we will use the differential operatorp = d/dt. The output y(t) of the system with transfer function G(s) and inputu(t) can then be written as

Now, p is conventionally used to represent the number of pole pairs of aninduction machine. In order to distinguish between the differential operatorand the number of pole pairs, the latter will be denoted by pIM.

Subscripts

A, B, C Three-phase components

ref Reference value

Superscripts

s Space vector

* Steady state value

Operators

p Differentiation operator

Transfer functions

A(s) Average over a pulse period

B(s) Band-pass filter

Introduction 5

D(s) Time delay

F(s), Ffw(s) Linear torque and flux controllers

Gc(s) Closed loop system

GUd(s) DC-link voltage disturbance

Gω(s) Motor speed disturbance

P(s) Closed-loop transfer function after field-orientation

Q(s) Disturbance transfer function after field-orientation

S(s) Sensitivity function

Y(s) Inverter input admittance

ZDC(s) Input filter

Signals

δ(t) Load angle

E(t) Input DC-voltage

id(t) DC-link current

is(t) Stator current

k(t) Coupling vector

χµ(t) Angle of stator flux space vector

χr(t) Angle of rotor flux space vector

mµ(t) Magnitude of stator flux space vector

mr(t) Magnitude of rotor flux space vector

Ψµ(t) Stator flux

Ψr(t) Rotor flux [Γ-ECD]

T(t) Torque

uc(t) Output of linear torque and flux controllers

Ud(t) DC-link voltage

us(t) Stator voltage

6 Introduction

ωm(t) Motor speed

ωµ(t) Stator frequency

ωslip(t) Slip frequency

Constants

k1 Upper cut-off frequency of B(s)

k2 Lower cut-off frequency of B(s)

K= Kω + KT Stabilization gain

Kpi, Ki Torque controller parameters

Lµ Stator inductance

Lσ Leakage inductance [Γ-ECD]

n Number of motors per voltage source inverter

pIM Number of pole pairs of the induction machine

ζ Damping factor of input filter

Rinv Equivalent resistance of the inverter

Rr Rotor resistance

Rs Stator resistance

Td Time delay

Tda Total effective time delay due to pure delay and average

Tp Pulse period

Tpull-out Pull-out torque

Ts Sampling time

Tσ Rotor leakage time constant

Ttorque Inverse torque control bandwidth

φm Phase margin

ωc Crossover frequency

ω0 Resonance frequency of input filter

Introduction 7

Abbreviations

DSR Direct Self Control

ECD Equivalent Circuit Diagram

FFT Fast Fourier Transform

IGBT Insulated Gate Bipolar Transistor

ISC Indirect Self Control

PWM Pulse Width Modulation

1.3 OutlineThis thesis is organized as follows.

Chapter 2 - Chapter 3: Introduction to the stability problem

The first two chapters give an introduction to the subject of stabilization of aninduction machine drive. It is shown in Chapter 2 how the assumption ofperfect torque control makes it possible to treat the drive as an ordinaryelectric circuit. The inverter is then simply replaced by an equivalentresistance, which depends on the operating point. Besides providing anintuitive understanding of the problem, this approach also gives the ideasbehind the existing stabilization methods, i.e. to make the equivalentresistance of the inverter positive.

The advantage of assuming perfect torque control is that the ideas forstabilization can be derived without detailed knowledge of the underlyingtorque control. The simple resistance model of the inverter is however notgood enough to derive appropriate expressions for the stabilization working inpractice. The model of the inverter is therefore extended from a constantresistance to a linear transfer function in Chapter 3. The requirement on theinverter is relaxed from acting like a positive equivalent resistance to beingpassive, at least in the vicinity of the resonance peak of the input filter. Beforethe actual stabilization problem can be addressed, we need a linear model ofthe drive. Such a linear model is derived in Chapter 4 to Chapter 7 andconsequently the actual tuning has to be postponed till Chapter 8 and Chapter9.

8 Introduction

Chapter 4 - Chapter 7: Derivation of a linear model of the drive

The process of deriving a linear model of the controlled induction machinedrive is initiated in Chapter 4, where models of the voltage source inverter andthe induction machine are presented and linearized. The closed-loop systemalso depends on the torque controller and the control scheme applied in thiscontribution is the Indirect Self Control, described in Chapter 5. The controllaw is linearized in Chapter 6 and the linear dynamics for the closed-loopsystem are derived in Chapter 7.

Chapter 8 - Chapter 9: Tuning of stabilization

Now the models are available to design the stabilization according to the ideaspresented in Chapter 3. First however the internal torque control is tuned to bestable in Chapter 8 before the DC-link stabilization problem is solved inChapter 9.

Chapter 10 - Chapter 11: Stability analysis using measurements

The linear model of the inverter input admittance and hence also thestabilization are derived subject to a number of approximations. In order toverify the validity of these approximations, the analytical models arecompared to models achieved through identification from measurements inChapter 10. At the same time, the proposed stabilization is verified bychecking stability of the identified models. Still, these tests only consider thelinear dynamics of the underlying non-linear system. Chapter 11 thereforeestimates the effects of the neglected non-linearities on the stability resultsobtained for the linear dynamics.

Finally, conclusions and ideas for future work are given in Chapter 12.

1.4 ContributionsThe basis for most results in this thesis is the simple linear model of theclosed-loop drive, presented in Chapter 7. Using this model, the tuning rulesfor the Indirect Self Controller in Chapter 8 as well as the expression for thestabilization gain in Chapter 9 are derived. The results of these chapters arepresented in the following conference papers

H. Mosskull. Tuning of Field-Oriented Controller for Induction Machines.In Conference on Power Electronics, Machines and Drives, Bath, UK,2002

Introduction 9

H. Mosskull. Stabilization of an Induction Machine Drive. In 10th

European Conference on Power Electronics and Applications, Toulouse,France, 2003

A further contribution is the treatment of non-linear effects when establishingstability of the drive from measurements. This work resulted in the followingconference paper

H. Mosskull, B. Wahlberg and J. Galic. Validation of Stability for anInduction Machine Drive using Measurements. In 13th IFAC Symposiumon System Identification, Rotterdam, The Netherlands, 2003

1.5 Drive DataThroughout the thesis, the derived results are applied to a test case. The testcase is the metro of Guangzhou in China. Data and other relevant informationabout the test set-up are given in Appendix C.

Figure 1-3: Guangzhou metro.

Chapter 2

DC-Link StabilizationIn this chapter stability problems due to interactions between the inverter andthe input filter of a controlled induction machine drive are analyzed. Section2.1 shows that by assuming constant power, the voltage source inverter inFigure 1-2 acts like a negative resistance for small variations around anoperating point in driving. Clearly, this is bad from a stability point of viewand how the behavior of the inverter can be improved is discussed in Section2.2.

2.1 Simplified Stability AnalysisConsider the induction machine drive shown in Figure 1-2. As described inSection 1.1, the damping factor of the input filter is very small and any kind ofexcitation of the filter resonance may hence result in large DC-link voltageoscillations. Large oscillations degrade performance of the drive and mayeven cause the drive to shut down. Proper operation of the drive thus dependson the ability of the inverter to damp such oscillations. It however turns outthat efficient torque control of the induction machines makes the situationeven worse, as oscillations are actually amplified, not damped. To understandthis we may assume perfect torque control. Keeping the torque perfectlyconstant at constant speed implies that the product of DC-link voltage andDC-link current is constant. A decrease of DC-link voltage due to adisturbance therefore must result in an increase of DC-link current, whichtends to lower the DC-link voltage even more. The disturbance in the DC-linkvoltage is thus increased by the control. One may also proceed a little morerigorously. Under the constant power assumption, the product of DC-linkvoltage and DC-link current may be differentiated to give

D

D D INV D

D

(2.1)

12 DC-Link Stabilization

Here we used asterisks to denote steady-state values and introduced theequivalent resistance Rinv as

D D

INV

D

(2.2)

where the steady state power P* is given by P*=Ud*id

*. Equation (2.1) showsthat if we assume perfect torque control, at least for small variations around anoperating point, the drive in Figure 1-2 may be replaced by the circuit shownin Figure 2-1. The inverter is thus exchanged by a resistance, which dependson the operating point. Note that the equivalent resistance Rinv becomesnegative for positive power.

E

+

-

Ud

idR

L

C

+

-

Rinv

Figure 2-1: Equivalent model of the induction machine drive under theassumption of perfect torque control.

We may now treat the drive as an ordinary electric-circuit. For example thecharacteristic polynomial of the system in Figure 2-1 is given by

INV INV

(2.3)

For the drive to be stable, the roots of the polynomial (2.3) must have negativereal parts. In a real application

D

INV

(2.4)

and stability of the drive consequently means that the following relation mustbe satisfied

INV

(2.5)

DC-Link Stabilization 13

As the equivalent resistance is negative in driving, condition (2.5) may not besatisfied for large positive power. If we neglect losses in the inverter and themotor, the inverter input power may be approximated as

D D M (2.6)

where n is the number of motors fed in parallel by one inverter. Now, with(2.2) and (2.6) it follows for positive speed that (2.5) is equivalent to

D

M

(2.7)

Clearly problems satisfying (2.7) may only occur in driving and for theexample specified in Appendix C, the maximum positive torque and the limit(2.7) are shown in Figure 2-2 as functions of motor speed. The torque limit forstability is the solid curve and is exceeded by the dashed curve, representingthe maximum torque, for all but very low motor speeds.

0 5 10 15 20 25 30 35 400

200

400

600

800

1000

1200

1400

1600

1800

2000Torque limit for instability

Tor

que

[Nm

]

Motor speed [Hz]

Figure 2-2: The solid line shows the torque limit for stability under theassumption of perfect torque control, whereas the dashed curve represents themaximum torque defined for the example in Appendix C. The drive may hencebecome unstable for all but very low motor speeds.

From Figure 2-2 it hence follows that the real parts of the roots of thecharacteristic polynomial (2.3) may become positive, which indicate an


unstable system. We may also study the properties of the drive by examiningthe damping factor ζ’ of the circuit in Figure 2-1, which is given by

INV INV INV

(2.8)

The approximation above follows from (2.4) and ζ is the damping factor ofthe input filter, see (1.1). From (2.8) it follows that positive power flow leadsto decreased damping, whereas negative power increases the damping of thedrive. At zero power, the resistance Rinv is infinite and the inverter does noteffect the DC-link voltage at all. In this case the damping factor of the driveequals that of the input filter.

2.2 Stabilization MethodsEquation (2.8) shows that an inverter behaving like a positive resistanceimproves the poorly damped behavior of the input filter. This simply meansthat a decreasing DC-link voltage should result in a likewise decreasing DC-link current. As the power flow of the drive may be effected through thetorque reference, a solution to the stability problem would be to feedoscillations in the DC-link voltage to the torque reference. A decreasing DC-link voltage would then lead to a decrease of the torque reference, which inturn would imply a decrease in DC-link current. This procedure is the ideabehind the solutions proposed in [26] and [2]. In [26], stabilization is done bymultiplying the torque reference by the first or second power of the DC-linkvoltage, i.e.

DREF REF M

DREF

DREF REF M

D

S

S

(2.9)

where the steady-state value Ud* is formed through a moving average of the

DC-link voltage. The design parameter ρ is set to either 1 or 2, referred to aslinear and quadratic stabilization, respectively. It is claimed in [26] that linearstabilization gives a constant current behavior whereas quadratic stabilizationgives a resistive behavior of the inverter. This follows from the assumption ofconstant power, which for positive power and constant motor speed impliesthat


D D M

D DREF M

D D

S S

(2.10)

Here the power losses due to resistances are neglected and it is assumed thatthe torque is identical to the torque reference, i.e. perfect torque control. Now,with (2.10) the equivalent resistance in Figure 2-1 becomes

D

INV

(2.11)

For ρ=1 the equivalent resistance is infinite and consequently the DC-linkcurrent is constant. With ρ=2 the inverter behaves like a positive resistance,which improves the damping of the drive.

Similarly, from (2.9) it follows that the equivalent resistance with negativepower is given by

D

INV

(2.12)

which is positive for both choices of ρ. Under the assumption of perfecttorque control, the analysis hence has showed that the drive remains stable inall modes of operation with the torque reference adapted as in (2.9).

A drawback with stabilization according to (2.9) is however that nostabilization is achieved at zero torque. Under the assumption of perfecttorque control, the equivalent resistance is infinite at zero torque. This meansthat the damping factor of the circuit in Figure 2-1 equals the damping factorof the input filter, which may be unsatisfactory. Actually, it will be shownlater that the drive may even become unstable also at zero torque if we relaxthe assumption of perfect control. To overcome problems at zero torque, thestabilization may be modified to add a correction to the reference torqueinstead of modifying the torque reference through multiplication. Such anadditive stabilization is proposed in [2], where the torque reference isbasically changed according to (2.13), where K is a proportional gain.

REF D REF (2.13)

With stabilization according to (2.13) one faces the problem of choosing thegain K. One approach would be to select the gain to give a desired positive


equivalent resistance independently of the power flow. Using (2.13) andassuming constant power, the following equivalent resistance results

D

INV

M D

(2.14)

Hence, a desired equivalent resistance Rdinv is obtained by choosing K as

D D

D DM INV M D M INV D

(2.15)

Note that expression (2.15) gives large gains for low motor speeds. Largegains lead to strong torque reactions, which result in torque jerks and evenshut-downs of the drive due to large current transients. The gain shouldtherefore be as small as possible, still stabilizing the drive. Practicalexperience shows that setting the stabilization gain is a tricky job, where thegain must be adapted to the operating conditions.

Remark: In the definition of additive stabilization (2.13), variations on theDC-link voltage are directly affecting the torque reference. In for example atraction application with DC-supply, the input voltage varies along the linedue to trains accelerating and braking. These unavoidable variations are ofrather low frequency and should not be compensated for by the stabilization.They are not indications of an unstable drive and reacting on them only causesunnecessary torque jerks. Further, due to limited bandwidth of the torquecontrol, it is probably no use in trying to damp high frequency oscillations onthe DC-link. A practical implementation of additive stabilization thereforeshould remove low and high frequency components of the voltage beforeadding them to the torque reference. Hence (2.13) would be implemented as

REF D REF (2.16)

where B is a band-pass filter. A band-pass filter is proposed also in [2], whereonly the resonance frequency of the input filter is let through.

2.3 SummaryIn Section 2.2 two methods for stabilizing an induction machine drive werepresented, namely (2.9) and (2.13). In the first approach the torque referencewas modified through multiplication whereas the modification in the latterwas added to the torque reference. We refer to these kinds of stabilization asmultiplicative and additive stabilization, respectively. As observed in Section


2.2, it is not possible to improve the properties of the drive at zero torque withmultiplicative stabilization. It therefore seems like additive stabilization offersmore flexibility and could lead to better results with a proper selection of thestabilization gain. As mentioned in Section 2.2, tuning of the gain K in (2.13)is difficult and tends to be done through trial and error in practice. This coststime and money and may still result in non-optimal performance. A structuredway of selecting the gain in (2.13) would therefore be of great value and willbe derived in the following chapters.

Chapter 3

Stabilization Formulated asLoop ShapingThe analysis of DC-link stability as well as the synthesis of stabilization inChapter 2 rely on the assumption of perfect torque control. Perfect torquecontrol is never valid in practice and consequently stability analysis using theequivalent resistance Rinv as in equation (2.3) may not be accurate enough toguarantee stability. As a matter of fact, one can show that the drive maybecome unstable also at zero torque, which not follows from the analysis inChapter 2. Furthermore, selection of the stabilization gain K in terms ofequivalent resistances is not obvious. As shown by expression (2.15), aconstant desired equivalent resistance gives large gains for low motor speeds,which are not needed in practice but only causes problems with torque jerksand large current transients. Replacing the inverter by an equivalent resistancewas enough to motivate the strategy for stabilization. It did however not giveaccurate information about how to efficiently set the gain to make the drivestable.

In order to improve stability analysis and stabilization gain synthesis, we needa better model of the inverter than the equivalent resistance. In Section 3.1 wewill show that the drive in Figure 1-2 can be represented as a non-linearfeedback system. By linearizing the inverter input admittance, i.e. the relationbetween DC-link voltage and DC-link current, stability of the drive may beexamined through the Nyquist stability criterion for linear systems [23]. Ananalytical model of the input admittance of the inverter would also makesynthesis of DC-link stabilization more accurate. This is described in Section3.2. The actual design of the stabilization is however postponed to Chapter 9,where a linear model of the input admittance is derived using results fromChapter 4 to Chapter 8.

20 Stabilization Formulated as Loop Shaping

3.1 Feedback RepresentationThis section shows how the induction machine drive in Figure 1-2 can berepresented as a non-linear feedback system. The linear part of the drive is theinput filter, which is described by the following equation

D % $# D (3.1)

where the transfer functions ZE(s) and ZDC(s) are given by

%

(3.2)

$#

(3.3)

(Note that ZE is not impedance even though the notation suggests so). Hence,(3.1) is an expression for the DC-link voltage, Ud(t), in terms of the supplyvoltage E(t) and the DC-link current id(t). Further, through the inverter, theDC-link voltage Ud(t) affects the DC-link current id(t). Under the assumptionof perfect torque control, the inverter was approximated by an equivalentresistance in Chapter 2. In the general case with realistic control, the inverterwill be represented by the non-linear function f(.), i.e.

< > D D (3.4)

The relation between DC-link current and DC-link voltage in (3.4) is calledthe input admittance of the inverter. By combining equations (3.1) and (3.4)we arrive at the non-linear feedback model shown in Figure 3-1.

+f(.)+

id

-

E

ZDC

UdZE

Figure 3-1: Non-linear feedback model of an induction machine drive

To facilitate stability analysis as well as design of stabilization, we use alinear approximation of the inverter input admittance (3.4), i.e.

D D (3.5)

Stabilization Formulated as Loop Shaping 21

We thus consider the following linear feedback model of the inductionmachine drive.

+Y+

id

-

E

ZDC

UdZE

Figure 3-2: Linear feedback model of an induction machine drive.

3.2 Design ObjectiveIf we assume that the linear model Y of the inverter input admittance is stable,stability of the feedback system in Figure 3-2 only depends on the loop gainYZDC. Stabilizing the DC-link can then be interpreted as shaping the inverterinput admittance to prevent the Nyquist trajectories of the loop gain fromencircling the point –1. The intuitive idea of making the inverter act like apositive resistance is then replaced by a quantitative requirement on theinverter input admittance. Although (3.5) still is an approximation of the inputadmittance, just like the inverse of the equivalent resistance in Chapter 2, it isnot constrained to a constant.

As the filter ZDC is passive, i.e. its phase stays between –90° and 90° for allfrequencies, a sufficient condition for stability is that also the inverter inputadmittance is passive. Even if this goal cannot be completely reached weshould at least try to make the admittance passive where the gain of ZDC islarge, which is around the resonance. This is illustrated in Figure 3-3 showingthe Bode plot of the input filter for the example given in Appendix C. A smallgain of ZDC should result also in a small loop gain YZDC. Hence, stabilityshould not be a problem for low and high frequencies.

We will hence interpret the task of making the drive stable as finding anadditive stabilization gain K that makes the inverter admittance passive, atleast in a region around the resonance frequency of ZDC. Actually we not onlywant a stable system, but also a certain stability margin to efficiently damposcillations in the DC-link.

Remark: By also considering resistance and inductance of the overhead line inFigure 1-1, the resonance frequency of the input filter is not constant butvaries with surrounding trains and distances to feeder stations.

22 Stabilization Formulated as Loop Shaping

100

101

102

10-2

100

102

DC-Link Impedance

Gai

nFrequency [Hz]

100

101

102

-100

-50

0

50

100

Frequency [Hz]

Pha

se

Figure 3-3: Bode diagram of the input filter ZDC.

Chapter 4 to Chapter 7 lead to the linear model Y and the derivation of thegain to make the admittance passive is done in Chapter 9.

Chapter 4

Linear Process ModelThe induction machine drive in Figure 1-2 converts between electrical andmechanical energy. The power flow is a consequence of the torque generatedby the induction machine, which is controlled through the voltages betweenthe three terminals shown in the figure. The voltages in turn are generated bythe voltage source inverter, whose operation is affected by a controllerthrough the coupling vector k(t). The controller either directly sets thecoupling vector or indirectly through reference voltages. In the latter case thereference voltages are mapped to a coupling vector by a process referred to asmodulation. This situation is illustrated in Figure 4-1, representing the drivefrom the controller point of view.

Modulatorref. voltage

Inverter Motork(t)

actual voltage

Ud(t) ωm(t)

torquemotor currents

id(t)

Figure 4-1: Model of an induction machine fed by an inverter seen from thecontroller point of view.

Figure 4-1 also shows that the DC-link voltage and the motor speed influencethe system and that the generated power also affects the DC-link current id(t).The intention with this chapter is to model this process, from referencevoltages to torque and DC-link current, under the influence of the DC-linkvoltage and the motor speed.

Modulation, i.e. the conversion from a voltage reference to the couplingvector k(t) (which is equivalent to switching times for the inverter powersemiconductors) is a vast topic in itself and will only be touched upon in thischapter. The inverter is treated in some detail in Section 4.1 and basicproperties of the induction machine are given in Section 4.2. The inverter putrestrictions on the magnitude of the voltages possible to generate and a

24 Linear Process Model

consequence of this is so-called fieldweakening. Fieldweakening is treated inSection 4.3, followed by a derivation of an expression for the DC-link currentid(t) in Section 4.4. Both the inverter and the motor are non-linear processesand to reach the goal of a linear model of the inverter input admittance, thesystem in Figure 4-1 is linearized in Section 4.5.

4.1 InverterThis section gives a short introduction to voltage source inverters used togenerate AC voltages with variable amplitude and frequency for the inductionmachines. To facilitate analysis of the drive, also a simplified non-linearmodel of the inverter is derived.

A three-phase voltage source inverter is depicted in Figure 4-2, consisting ofthree legs, one for each phase. The legs contain IGBTs (Insulated GateBipolar Transistor) in parallel to diodes.

id

Ud

+

-

ABC

iC

iBiA

Figure 4-2: Voltage source inverter.

The IGBT is a transistor, which can either conduct or block current in thedirection of the arrow in the figure (in the other direction it always blocks). Ineach inverter leg only one of the IGBTs are turned on at a time to prevent ashort-circuit of the DC-link.

Consider the case where the upper IGBT is turned on and the lower is turnedoff. A positive current iA(t) must pass through the upper IGBT whereas anegative current must go through the upper diode in phase A. If we neglectvoltage drops of the IGBTs and diodes, the potential at point A hence equalsthe higher potential of the capacitor in Figure 4-2. For the other case, with theupper IGBT turned off and the lower turned on, the potential at point A equalsthe lower potential of the capacitor. Hence, simplified we may consider one

Linear Process Model 25

pair of arms as a switch as is shown in Figure 4-3. Shown in this figure arealso the three stator windings of the induction machine and the three phasevoltages denoted uA(t), uB(t) and uC(t). These voltages will be referred to as thestator voltages.

iC

A C

B

+

-

Ud

id

+ +

+

---uA

uB

uC

iA iB

0

Figure 4-3: Simplified model of a voltage source inverter.

Through the switches in Figure 4-3, the potentials at points A, B and C caneach be switched between two possible values. If we assign potential zero tothe point 0 in the figure, the potential of each phase switches between±Ud(t)/2. We will denote these potentials vA(t), vB(t) and vC(t) and use thefollowing representation

! ! D (4.1)

where kA(t) only can take the two values ±0.5. By also introducing thenotation vstar(t) for the potential at the star point of the motor, the statorvoltages may be expressed as

Å Å

!! ! STAR

" " STAR " D STAR

# # STAR #

(4.2)

The vector with elements kA(t), kB(t) and kC(t) is the coupling vector k(t),which we have used previously without a formal definition.

Through (4.2) the stator voltages have been expressed using three equations,one per phase. If the motor is Y-coupled and the star point not connected as in


Figure 4-3, it can however be shown that the stator voltages are notindependent as sum of the voltages vanishes [11], i.e.

! " # (4.3)

That this for example also holds for the stator currents follows directly fromFigure 4-3 and Kirchoff’s current law. The constraint (4.3) implies that allinformation about the stator voltages may be captured by a two-dimensionalquantity. For this purpose the so-called space vector representation has beenintroduced, see for example [11]. Space vectors are complex numbers and thetransformation from three-dimensional quantities, like the ones in (4.2), to thecomplex-valued space vectors is given in Appendix D. For example equaloffsets to all three phase quantities, so called zero sequences, are mapped tozero by the space vector transformation. This means that the last term in (4.2)disappears in the corresponding space vector relation (4.4), where spacevectors are denoted with superscripts s.

S S

S D (4.4)

Note that even though the three-phase coupling vector k(t) always contains azero sequence (the sum of the components is non-zero), only the informationcontained by the space vector of the coupling vector is relevant for theinduction machine due to (4.3).

The stator voltages are hence generated by switching the switches in Figure4-3 back and forth. These switchings result in discontinuous stator voltagesjumping between discrete values. This behavior is illustrated in Figure 4-4where a reference stator voltage (phase voltage) and the resulting statorvoltage are shown. The behavior of the inverter depends on the used (average)switching frequency, which is defined per switching component (IGBT). Thestator frequency in the example is 20 Hz and the used switching frequency is550 Hz.

The process of modulation shown in Figure 4-1 has the task to make thefundamental of the stator voltage follow its reference subject to minimizingthe effects of the high frequency harmonics caused by the switchings. Afrequency analysis of the stator voltage in Figure 4-4 shows that the signalbasically consists of a component at the stator frequency (20 Hz) and highfrequency components being multiples of the switching frequency. The maincomponent in the example occurs at twice the switching frequency as seen inFigure 4-5.


0.46 0.47 0.48 0.49 0.5 0.51 0.52 0.53 0.54

-1000

-500

0

500

1000

Actual stator voltage and reference stator voltage

Time [s]

Vol

tage

[V

]

Figure 4-4: Phase voltage generated by an inverter (solid) and referencevoltage (dashed).

2 5 10 20 50 100 200 550 11000

50

100

150

200

250

300

350

400FFT of stator voltage

Frequency [Hz]

Figure 4-5: FFT of a stator voltage.


The modulation scheme used to generate the voltage in Figure 4-4 is calledspace vector modulation [28], which strives to make the average statorvoltage over time intervals called pulse periods equal to the reference. Thisway the fundamental stator voltage is assured to follow the reference. Therewill however be a phase shift between the reference and the fundamental ofthe stator voltage, which will be modeled as half a pulse period. To motivatethis assumption we need some more details of space vector modulation. FromFigure 4-3 it follows that together, the inverter switches can be put in eightdifferent combinations. A closer analysis of the stator voltages caused bythese combinations shows that all switches in the same position result in zerostator voltage. Space vector modulation starts each pulse period with zerostator voltage by putting all switches in either the upper or lower position.Within the pulse period each switch is then switched once giving rise to non-zero stator voltages until all switches have been switched. In this case allswitches are again in the same position and zero voltage is applied to themotor. The non-zero voltage pulses are generated such that the times withzero voltage at the beginning and the end of the pulse periods are equallylong. The pulsing may hence be schematically illustrated as in Figure 4-6,where the switching times Tsw are calculated every pulse period instant tk togenerate voltage pulses centered in the following pulse period. The pulseperiod is denoted Tp.

Tp

tk tk+1

Tsw

t

Figure 4-6: Timing of switching time calculations.

Centered pulses have the positive effect that the sampled values of currents,fluxes and torque at the pulse period instants correspond to the fundamentalvalues [10]. Hence, no additional filtering, accompanied by time delays isneeded to extract the low frequency components of the signals. Butsymmetrical pulsing also means a delay from the reference to the middle ofthe voltage pulses of half a pulse period, which motivates the assumed timedelay above. We have thus reached the following model for the stator voltagegenerated by the inverter

S S S

S SREF P 07- (4.5)


where the signal usPWM(t) is assumed to contain all the high frequency

components of the stator voltage caused by the modulation, c.f. Figure 4-5.The abbreviation PWM stands for Pulse Width Modulation. As the inductionmachine, as any physical system, has small gain for high frequencies, thesignal us

PWM(t) will be neglected when analyzing the dynamics of the drive.This is a standard procedure, see for example [13]. Without the highfrequency components, the stator voltage is hence approximated through

S S

S SREF P (4.6)

For expression (4.6) to hold, it follows from (4.4) that the low frequency partof the coupling factor must be given by

S

S REF PS

,&

D

(4.7)

where subscript LF denotes the low frequency part of the signal. Without ameasurement of the DC-link voltage the best we can do is to replace the DC-link voltage in (4.7) by its nominal value. This means that the stator voltagewill no longer be independent of the DC-link voltage, but is given by

DS S

S S REF D

D

(4.8)

where

D P (4.9)

How a measurement of the DC-link voltage can be used to suppressdisturbances due to a varying DC-link voltage is discussed in Section 5.4.

Remark: Even though the model of the stator voltage generated by the inverter(4.8) was derived under the assumption of space vector modulation, it will beused also with other types of pulse patterns. The switching times are assumedto be calculated to give a desired average stator voltage during a certain timeinterval, called the pulse period. The pulse periods may however not start andend with all switches in the same position, as they do with space vectormodulation.


4.2 Induction MachineThe induction machine is the device performing the actual conversionbetween electrical and mechanical energy in Figure 4-1. We consider thestator voltages as inputs to the induction machine. Under the influence of themotor speed a certain torque is produced at the motor shaft, which isconsidered the output of the motor. Before presenting the mathematicsdescribing the induction machine in Subsection 4.2.2, Subsection 4.2.1 verybriefly explains the physics behind the induction machine. For a detaileddescription, see a book on the topic, e.g. [11].

4.2.1 Physical DescriptionThe induction machine consists of a stationary part, the stator, and a rotatingpart, the rotor. A stator with a distributed stator winding is shown in Figure4-7. Although the winding is distributed, the effect of the winding may beapproximated by the three windings A, B and C in the figure, separated 120°in space.

Figure 4-7: Stator.

The rotor is usually of the squirrel cage type as shown in Figure 4-8, simplyconsisting of a number of rotor bars connected through two end rings. Theelectrical effects of the rotor may however also be modeled as threeequivalent phase windings, see for example [19].


Figure 4-8: Squirrel cage rotor.

The stator and rotor are hence both modeled as three phase windings. This isillustrated in Figure 4-9 where the stator windings are denoted A, B and C, andthe rotor windings a, b and c. Shown in the figure is also the angle α(t)measuring the orientation of the rotor relative to the stator.

c

A

B

a

b C

α(t)

Figure 4-9: Orientation of the stator and rotor of an induction machine seenfrom above. The stator windings are denoted A, B and C and the rotorwindings a, b and c. The angle α(t) measures the orientation of the rotorrelative to the stator.

We will further define the motor speed, denoted ωm(t), as the derivative of theangle α(t), i.e.

M

(4.10)


Remark: We use the term speed although frequency would be moreappropriate as the quantity in (4.10) is represented by an ω and is measured inradians per second. The reason is to avoid confusion with other frequencies.

Applying sinusoidal three-phase voltages to the stator terminals results in arotating flux in the stator. The most part of the flux propagates to the rotorwhere hence the rotor bars see a magnetic flux passing by and consequentlyvoltages across the rotor bars are induced. The voltages in turn give rise torotor currents. Together with the original flux, the additional flux produced bythese currents generates torque, which strives to align the two fluxes. Thetorque hence forces the rotor to follow the applied stator flux. Note howeverthat non-zero torque requires the stator flux to rotate asynchronously to therotor. This also motivates the name asynchronous machine, which also is usedfor the induction machine.

The frequency of the induced electrical rotor quantities equal the frequency ofthe flux seen by the rotor bars, i.e. the difference between the rotating flux andthe mechanical rotation of the rotor, called the slip frequency. It shouldhowever be noted that the phase windings of the stator may be arranged togive multiple poles, see [11]. This means that one physical rotation of therotor corresponds to pIM rotations for the electrical quantities of the rotor,where pIM is the number of pole pairs of the induction machine.

Although torque is the main control variable, i.e. the main quantity we want tocontrol, usually also the flux in the machine is controlled. The inductionmachine is designed to operate at a certain nominal flux and running with forexample larger fluxes drives the machine into saturation, which produceslarge currents in the stator windings.

4.2.2 Mathematical DescriptionMathematically induction machines may efficiently be represented in anumber of ways using space vectors, see e.g. [11]. From a physical point ofview the so-called T-model is most relevant as it models the stator and rotorwindings corresponding to a real machine. The name T-model refers to theway the inductances appear in the equivalent circuit diagram (ECD). Seenfrom the machine terminals, there are however equivalent models that may bebetter suited in certain situations. One such model is the Γ-ECD shown inFigure 4-10, [25]. The name is again derived from the appearance of theinductances in the ECD, which form a Γ. Here the total leakage inductance Lσis put in the rotor mesh as opposed to the T-model, where the physical statorleakage inductance is modeled in the stator mesh (note that this would


generate a T in Figure 4-10). The reason for using the Γ-model is that thenumber of states (inductances) is reduced compared to the T-model and theequations for the stator mesh are simplified. The latter is well suited whenusing stator flux oriented control laws, like the Indirect Self Controlintroduced in Chapter 5.

Rs Lσ

L µ

Rr

jpIMωmΨrsus

s

iss

Ψµs Ψr

s

irs

iµs

. .

Figure 4-10: Γ-equivalent circuit diagram of an induction machine.

In Figure 4-10, ψµs(t) and ψr

s(t) represent the stator and rotor fluxes,respectively, ωm(t) is the motor speed and pIM the number of pole pairs of themotor. Rs and Rr stand for the resistance in the stator and rotor windings andthe stator inductance is denoted Lµ. Finally, is

s(t) is the stator current spacevector and from the figure the following equations describing the fluxes of theinduction machine can be derived

S S S

S S S N (4.11)

R RS S S

R )- M R

N

T T

(4.12)

The output torque can be calculated from the fluxes as [25]

)-

R R

NC B NB C

T

(4.13)

Hence, equations (4.11)-(4.13) completely specify the induction machine forour purposes. The equations however do not directly give the insight neededto motivate the ideas behind the control law presented in Chapter 5. In orderto reveal the properties of the induction machine we introduce a polar notation


of the space vectors, where for example the magnitudes and angles of thefluxes and stator voltage are denoted by

J TS ND

N N (4.14)

R

S J T

R R D (4.15)

U

S J T

S U D (4.16)

Using the polar notation, the electrical torque (4.13) can be expressed as

)-

R

N

T

(4.17)

where δ(t) is the angle between the stator flux and rotor flux space vectors, i.e.

R

N (4.18)

The angle (4.18) is often called the load angle, which makes sense from(4.17) as the torque to a large extent depends on δ(t). The load angle isillustrated in Figure 4-11.

α

Ψµs(t)

β

Ψrs(t)

δ(t)

Figure 4-11: Definition of load angle δ(t).

From equation (4.12) we see that the rotor flux changes with the time constantLσ/Rr, which we will denote Tσ, i.e.

R

T

T (4.19)

The time constant (4.19) is called the rotor leakage time constant and isrelatively large compared to the time constant of stator flux changes. For theexample in Appendix C, Tσ=33ms compared to the time constant of torquecontrol, which is often in the range of 1–10ms. Hence, the load angle should


be possible to control through the frequency of the stator flux space vector.This frequency will be called the stator frequency and is denoted by ωµ(t), i.e.(see (4.14))

N N (4.20)

By increasing the stator frequency the torque can be increased and bydecreasing the stator frequency the torque is decreased. It should be noted thatwhereas (4.20) corresponds to the frequency of the physical stator flux, thederivative of χr(t) in (4.15) is not the frequency of the rotor flux that would bemeasured by an instrument attached to the rotor. The rotor flux in the ECDdoes not correspond to a physical flux but is a result of mathematics.Physically the rotor windings rotate whereas the stator windings are fixed andby using the notation α(t) introduced in Figure 4-9 for the angle between thestator and the rotor, equation (4.15) can be written as

)-S S JP T

R RR B (4.21)

where

R )-

S J T P T

RR R D B (4.22)

(Note the effect of the number of pole pairs pIM in (4.21)). The flux given by(4.22) is the physical rotor flux and is denoted Ψ srr(t). The second subscript rstands for rotor coordinates indicating that the space vector is expressed incoordinate system rotating with the rotor, which is illustrated in Figure 4-12.

Ψrrs(t)

α(t)

α

β

dq

Figure 4-12: Illustration of stator- and rotor coordinates.

The impact of the rotor flux on the stator quantities in (4.21) is however givenby Ψ sr(t ). This space vector is referred to as the rotor flux in statorcoordinates as it is expressed in the fixed coordinate system with coordinates


α and β, see Figure 4-12. The frequency of the rotor flux in rotor coordinateswill be called the slip frequency and denoted ωslip(t), i.e.

SLIP R )- M (4.23)

where ωr(t) is the derivative of χr(t) and ωm(t) is the derivative of α(t), see(4.10). The slip frequency (4.23) is the frequency of the rotor flux relative tothe rotor bars and hence the frequency, which induces the currents in the rotorwindings. It seems natural that the generated torque should depend on thisfrequency. Indeed, from equation (B.20) in Appendix B it follows that the slipfrequency is given by

R

SLIP

R

N

T

(4.24)

Combining this result with equation (4.17) gives

)-R SLIP

R

(4.25)

Equation (4.25) shows that if the rotor flux is constant, the torque isproportional to the slip frequency. If instead of regulating the rotor flux, thestator flux is kept constant, the rotor flux magnitude varies with the load andexpression (4.25) gets a little more complicated. A relation between slipfrequency and torque with constant stator flux may however be obtained insteady state. By using the following trigonometric identity

(4.26)

and inserting equation (B.35) into (4.24) and (4.25), one can show that

SLIP)-

SLIP

T

N

TT

(4.27)

Here asterisks again denote steady state values. Equation (4.27) is plotted inFigure 4-13 as a function of slip frequency. Around the origin the torquedepends almost linearly on slip frequency but eventually reaches a maximumfor positive slip frequencies and a minimum for negative slip frequencies. Themaximum (or minimum) torque, which is called the pull-out torque, occurs at

SLIP T

(4.28)


and equals

)-PULL OUT

N

T

(4.29)

Increasing the slip frequency above the value (4.28) results in a decreasingtorque. The increased slip frequency however increases losses in the rotor,which may damage the motor. The steady state slip frequency shouldtherefore always be limited by (4.28).

-20 -15 -10 -5 0 5 10 15 20-5000

-4000

-3000

-2000

-1000

0

1000

2000

3000

4000

5000Torque as a function of slip frequency

Slip frequency [Hz]

Tor

que

[Nm

]

Figure 4-13: Steady state torque as a function of slip frequency.

Finally, we note that as the load angle (4.18) is zero in steady state, the statorand rotor flux space vectors rotate with the same frequency. This for examplemeans that the slip frequency may also be expressed in terms of the statorfrequency as

SLIP )- MN (4.30)

4.3 FieldweakeningThe stator voltage was given as the product of the coupling vector and theDC-link voltage by (4.4). As the elements of the coupling vector only


switches between finite values, the fundamental of the coupling space vectoris limited. Maximum fundamental voltage is generated when using so calledblock mode (alternatively six-step operation or hex mode), which is illustratedin Figure 4-14 with a stator frequency of 65 Hz. With block mode the voltageamplitudes are not adjustable but directly depend on the DC-link voltage andthe inverter switching frequency equals the stator frequency. By comparingFigure 4-14 with Figure 4-4 we see that the extra switchings in Figure 4-4make it possible to reduce the fundamental but also to reduce low frequencyharmonics.

Remark: The dashed curves in Figure 4-4 and Figure 4-14 represent referencevoltages. According to equation (4.5) the fundamental of the stator voltageequals the reference except for a time delay of half a pulse period. FromFigure C-3 in Appendix C we see that the time delay is larger at a statorfrequency of 65 Hz compared to a stator frequency of 20 Hz. This is why thetime delay is more obvious in Figure 4-14 compared to Figure 4-4.

0.46 0.47 0.48 0.49 0.5 0.51 0.52 0.53 0.54

-1000

-500

0

500

1000

Actual stator voltage and reference stator voltage

Frequency [Hz]

Vol

tage

[V

]

Figure 4-14: Maximum stator voltage generated by an inverter. The solidcurve is the actual stator voltage and the dashed curve is the referencevoltage.

The stator voltage in Figure 4-14 can be expanded in a Fourier series and thefirst term (the fundamental) is given by 2/π Ud

*. The maximum magnitude ofthe fundamental stator voltage is therefore limited by [18]


U D

(4.31)

and relation (4.6) is consequently only valid if (4.31) is satisfied. Note that thelimit is given for peak values of the phase voltages. A consequence of (4.31)is that the flux, which normally is kept constant, has to be reduced for highmotor speeds. To see this we observe that if the voltage drop across the statorresistance in (4.11) is neglected, the stator voltage magnitude may beapproximated by

U

N N (4.32)

With a constant stator flux it follows from (4.32) that the magnitude of thestator voltage increases linearly with stator frequency and eventually reachesthe limit (4.31). The speed where the limit is reached for nominal flux andDC-link voltage (and nominal torque) is called base speed and is denoted ωb.To increase the stator frequency beyond the base speed, the flux has to bedecreased inversely proportional to the stator frequency, see (4.32). This iscalled fieldweakening and is illustrated in Figure 4-15 showing the flux as afunction of motor speed for the three driving scenarios defined in Appendix C.

0 5 10 15 20 25 30 35 40 450

0.5

1

1.5

2

2.5

3

Motor speed [Hz]

Stator flux

Flu

x [V

s]

Figure 4-15: Flux as a function of motor speed in coasting (solid), driving(dashed) and braking (dotted). Above base speed, the flux has to be reduceddue to saturation of the stator voltage amplitudes. This region is called thefieldweakening region.


The relative errors introduced when using the approximation (4.32) for thethree scenarios defined in Appendix C are shown in Figure 4-16. From thefigure it follows that the approximation is good except at low motor speeds.

0 5 10 15 20 25 30 35 40 45-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

Motor speed [Hz]

Relative error in stator amplitude due to approximation

Vol

tage

err

or [

V]

Figure 4-16: Relative stator voltage magnitude error when usingapproximation (4.32) in coasting (solid), driving (dashed) and braking(dotted)

4.4 DC-Link CurrentA quantity of major importance when deriving a linear model of the inverterinput admittance is the DC-link current id(t). If we neglect power losses in theinverter and the motor, expression (2.6) holds, which may be written

D D M (4.33)

Here n denotes the number of motors fed by one inverter, c.f. Section 2.1.Hence, the DC-link current can be written as

M

D

D

(4.34)


Note that treating several machines only by the number of motors implies thatwe assume the applied torque to be equal for all motors. In a tractionapplication this for example requires that all wheels run with the same speed,i.e. we require the same the wheel diameters and coefficients of adhesion.

4.5 LinearizationBoth the model of the voltage source inverter (4.8) and the equations for theinduction machine are non-linear. To simplify analysis of the drive, theseequations are linearized around stationary operating points in this section. Asspace-vectors are not constant at steady state but rotate with constantmagnitude and frequency, it seems appropriate to linearize about magnitudesand frequencies of the involved quantities. For that purpose we introduce areal-valued vector representation, where the first component is the magnitudeand the second component the frequency of a signal. For example the statorvoltage and stator flux will therefore be represented as

U

S

U

N

N

N

(4.35)

where the real-valued signal representation is indicated by removing thesuperscripts s of the space vectors. To simplify notation we will also introducethe vector y(t), containing the control variables torque and flux, i.e.

N

(4.36)

Apart from the vector y(t), the control law to be described in Chapter 5 alsoneeds the complete stator flux vector as well as the stator current and themagnitude of the rotor flux. To handle all these signals we will use thefollowing notation for the linearized induction machine

Y Y

R MR MR

S M

S I I

X

X

N Z XZ

X

(4.37)

where the G’s are transfer functions, see Section 1.2. Only the notation for thelinearized induction machine is introduced by (4.37) but no explicitexpressions for the transfer functions are given. The reason for this is that to


make the linearization as simple as possible, the induction machine will belinearized only after parts of the control law has been applied, see AppendixB. Note also that the signal names in the linear representation are not changedcompared to the original representation, although the signals in (4.37)represent deviations from operating points. Hence, y(t) in (4.37) shouldactually be written ∆y(t) = y(t)-y*, where the asterisk denotes the operatingpoint value. The ∆’s will however be skipped to simplify notation and it isimplicitly understood that signals in linear models are deviations fromoperating points.

In Section 4.1 the inverter was modeled by equation (4.8). The magnitude andfrequency of the space vector us

s(t) can be linearized to give the real valuedvector us(t) as

U

UREF D D

DS SREF D 5D D

UREF D

(4.38)

where we used that m*u=m*

uref, which follows from (4.8). We also introducedthe notation GUd for the transfer function from DC-link voltage to the statorvoltage, i.e.

U

5D

5D D D

N N

(4.39)

where the approximation follows from (4.32).

The linearized equations for the inverter and induction machine, i.e. equations(4.37) and (4.38), can graphically be illustrated as in Figure 4-17. Here Gdenotes the entire equation (4.37) and the block D formally represents thetime delay (4.9). In the Laplace domain the time delay may be represented by

DS4 (4.40)

Finally, the equation for the DC-link current (4.34) can be linearized to yield

M M

D M D

D D D

(4.41)


+

GUd

Ud

+ + us

ωm

Dusref G

ymr

Ψµis

Figure 4-17: Linear model of the inverter and induction machine.

Chapter 5

Indirect Self ControlThe classical field-oriented controller introduced by [7] and [4] meant a majorbreak-through in control of induction machines. With this technique the non-linear multidimensional induction machine could be controlled just like aseparately excited DC-motor. The trick is to express the stator voltage in acoordinate system aligned with the rotor flux. This way the rotor fluxmagnitude may be controlled through the voltage component parallel to therotor flux whereas the voltage component orthogonal to the rotor flux controlsthe torque. The classical field-oriented controller can be interpreted ascontroller achieving asymptotic exact linearization, see for example [17].

Since the advent of the first field-oriented controller, other similar techniqueshave been derived, [5],[27]. For example a control scheme commonly used tocontrol induction machines in traction applications is the Indirect Self Control(ISC), see [9], [8] and [16]. The ISC orients to the stator flux which offerssome advantages compared to the classical rotor flux-oriented control. Statorflux orientation is less parameter dependent than rotor flux orientation [29]and stator-flux orientation is well suited for operation in field-weakening. Thename Indirect Self Control is derived from another stator-flux oriented controlmethod, the Direct Self Control (DSC) [6]. Whereas the DSC directlygenerates the switching commands for the power semiconductors, the ISCgenerates the switching commands indirectly via a stator voltage reference.The stator voltage reference is separately converted to switching commandsby a modulator, see Figure 4-1.

The ISC is a stator flux slip control method following the nomenclatureintroduced in [21]. By this is meant that the torque of the machine iscontrolled by a slip frequency controller while keeping the magnitude of thestator flux constant. This chapter starts by introducing the ideas behind slipfrequency control in Section 5.1 before presenting the ISC control law inSection 5.2. The ISC is a discrete-time controller and to incorporate thecontroller into the continuous time analysis, a continuous time approximationof the ISC is derived in Section 5.3. The chapter ends by describing howvariations in the DC-link voltage can be compensated for in Section 5.4.

46 Indirect Self Control

5.1 Slip Frequency ControlThe idea behind ISC follows from the relation between steady-state torqueand slip frequency given by equation (4.27) and visualized in Figure 4-13.The figure shows that, with constant stator flux, the torque depends almostlinearly on slip frequency as long as the pull-out torque (4.29) is not exceeded.Hence, by keeping the stator flux constant, the torque may be controlledthrough the slip frequency only. Controlling the slip frequency in turn meanscontrolling the stator frequency, which follows from equation (4.30) if weassume the motor speed is available. Hence, torque and flux control basicallyput requirements on the frequency and the magnitude of the stator flux,respectively. How to achieve the desired stator flux follows from equation(4.11) giving the relation between the derivative of the stator flux and theinput to the machine, the stator voltage. Using the polar notation (4.14), thederivative of the stator flux space vector may be expressed as

J T J TS

N ND D

N N N N N (5.1)

By introducing

N N N? (5.2)

it follows from (5.1) that equation (4.11) can be rewritten as

J TS SS S S ND

? (5.3)

Apart from the term canceling the voltage drop across the stator resistance, theexpression for the stator voltage space vector has been decomposed intovoltage components parallel and orthogonal to the stator flux space vector.This part of the stator voltage space vector, which we will denote by us(t), issaid to be expressed in so-called stator flux coordinates, i.e.

S ? (5.4)

which is illustrated in Figure 5-1.

Indirect Self Control 47

us(t)

u//(t)

u⊥ (t) χµ(t)

α

β

dq

Figure 5-1: Illustration of stator flux coordinates.

The stator flux coordinates are denoted d and q, the direct and quadraturecomponent, respectively. The benefit of using flux coordinates follows from(5.2) showing that u//(t) influences the magnitude of the stator flux whereasthe frequency of the stator flux is proportional to u⊥ (t). Combined with thediscussion above it seems natural to let a flux controller affect u//(t) and a slipfrequency controller to set u⊥ (t), i.e. the stator frequency. This is also the ideaof ISC, presented in Section 5.2.

5.2 Discrete Time ISCSubsection 5.2.1 presents the discrete-time ISC control law, generating areference stator voltage space vector as output. The reference voltages arecalculated at times denoted tk in Figure 4-6 and correspond to the requiredaverage voltages during the following pulse periods. However, such asampling strategy may not be possible in a real time implementation as isexplained in Subsection 5.2.2. The calculation of the stator voltage must thenbe done in advance, using predicted information about the induction machineat time tk.

5.2.1 Control LawThe discrete-time ISC control law generates a stator voltage space vector to beapplied in average over the next pulse period. The effect of the averagevoltage is seen by integrating equation (4.11) over a pulse period to give

S S S

K P K S S

S S S

P P

N N N (5.5)


where the bars over the signals indicate average over the pulse period. Hence,for example

K P

K

T 4

SS

PT

(5.6)

From equation (5.5) it follows that in order to produce a desired stator fluxincrement ∆Ψ sµd, the reference stator voltage should be generated as

S

DS S

SREF K S S

P

N (5.7)

The average current is however not known but can be approximated by thecurrent measured at the beginning of the pulse period. An alternative, whichmay be more accurate, is to predict the value of the (fundamental) current tothe middle of the pulse period. This is the approach we will take here and theexpression for the reference stator voltage then becomes

S

DS S

SREF K S S K P K

P

N (5.8)

where the hat above the stator current denotes an estimated value and theargument means that the estimation corresponds to time tk+½Tp giveninformation available up to time tk.

Section 5.1 showed how the magnitude and frequency of the stator flux spacevector could be independently affected in continuous time by expressing thestator voltage in stator flux coordinates. In order to do the same in discretetime, we note that the stator flux increment in (5.5) may be written as

JS S S SK P K K

K

N

N D

N N N N

N

%

(5.9)

where

K P K N N N (5.10)

K P K N N N (5.11)

This is illustrated in Figure 5-2.


ψ µs (t k

+Tp) ∆ψ

µ

∆χµ

Re

Im

ψµs (t k

)

∆mµ

Figure 5-2: Stator flux increment over a pulse period.

Equation (5.9) corresponds to expression (5.1), where (5.10) describes theupdate in flux magnitude and (5.11) the increment of the stator flux angle overthe pulse period. Due to the non-zero pulse period, the components affectingthe magnitude and frequency of the stator flux space vector are however notexactly orthogonal. From equation (5.9) we see that the desired update of thestator flux over the pulse period is related to the stator flux at the beginning ofthe pulse period. Hence the scheme orients to the stator flux.

Equation (5.9) together with (5.8) now motivate the ISC control law, which isgiven by

REFJ SKS S

SREF K S S K P KP

ND

Z N% (5.12)

The relative change of magnitude Kψ and the angle increment ∆χµref in (5.12)are given by

REF K K Z Z N N (5.13)

and


REF )- M K SLIPREF K P

K

P

NR P NR SLIPERROR K

P

N

(5.14)

where q is the shift operator, i.e. q-1ωsliperror(tk) = ωsliperror(tk-1). Equation (5.13)is a P-controller for the flux and (5.14) is the desired stator flux angleincrement over a pulse period. The latter is composed of two terms where thefirst term gives the steady-state stator frequency using equation (4.30). Thesecond term is a PI-controller with gains Knr and Tnr to speed up the dynamicsand force zero steady state error (as a matter of fact [16] only considers a P-controller). The signal ωsliperror(tk) is the slip frequency error at time tk, i.e.

SLIPERROR K SLIPREF K SLIP K (5.15)

The quantity we want to control is however not the slip frequency but thetorque. By using the relation (4.25) we may express (5.14) in terms of torqueas

R

)- M K REF K P

)- R K

K

R P

NR P NR ERROR K

)- R K P

N

(5.16)

where

ERROR K REF K K (5.17)

The slip frequency controller hence becomes a torque controller with gainsthat depend on the rotor flux magnitude. To summarize, the discrete time ISCcontrol law is given by equations (5.12), (5.13), (5.16) and (5.17).

5.2.2 PredictionThe control law presented in the previous subsection generates a referencevoltage us

sref(tk) given information about the motor state at time tk. Thereference voltage in turn is mapped to switching times for the inverter phasesas illustrated in Figure 4-6. If long voltage pulses are needed, the time fromthe beginning of the pulse period to the first switch may be very short. As amatter of fact shorter than the actual time it takes to calculate the switchingtimes in real time. To avoid such problems, the switching times have to be


calculated in advance, using predicted information about the state of the motorat time tk. The standard way of implementing ISC is to pre-calculate theswitching times one entire pulse period. The extra time delay introduced thisway may be too long and cause stability problems and to reduce time delaywe have to run the calculations within the pulse period. For that matter weintroduce the sampling time Ts denoting the pre-calculation of the switchingtimes, see Figure 5-3.

Tp

Ts

tk tk+1

Tsw

t

Figure 5-3: Pulse periods and sampling times.

For the example defined in Appendix C we use Ts = 200µs, but in general

S P (5.18)

Including the extra time delay, equation (5.12) therefore is replaced byequation (5.19)

REFJ SK K SS

SREF K SP

SS S K P K S

ND

Z N%

(5.19)

where the stator flux and stator current are predicted. We will howeverassume that the quantities KΨ and ∆χµref are calculated from non-predictedsignals corresponding to time tk-Ts (although including prediction is not aproblem).

It should however be noted that equations (5.13) and (5.16) involve only thelow frequency components of the signals. In a real implementation, the highfrequency oscillations due to the inverter described in Section 4.1 must beremoved. There are several ways to extract the low frequency componentsfrom real measurements. Most easy is to only work with signals sampled atthe pulse period instants, tk, as they correspond to the fundamentals withsymmetrical pulsing. As the inverter switching times are calculated at time tk-Ts in Figure 5-3, which does not coincide with a pulse period instant we couldtry to predict signals to time tk. An alternative way to estimate the


fundamental of a signal is through a moving average over a pulse period. Thismethod causes phase shift but gives reliable results in practice and we assumethe latter method is used for estimation of the torque. The control error Terror

in (5.16) is hence given by

ERROR K S REF K S K S (5.20)

where (assume Tp/Ts is an integer)

P S4 4

SK S K S

P K

(5.21)

Remark: For flux control, which will be of less importance in thiscontribution, we simply neglect these issues and use the (fundamental) valueat time tk-Ts.

5.3 Continuous Time ISCThis subsection derives a continuous time approximation of the discrete timeISC control law presented in Subsection 5.2. We first consider the casewithout predictions in Subsection 5.3.1 and the effects of predictions aremodeled in Subsection 5.3.2.

5.3.1 Control LawConsider the following controller gains

NR NRP PI I

P P P

Z (5.22)

where PΨ is the P-gain of the discrete time flux controller and Knr and Tnr theparameters for the discrete time torque controller. By using (5.22), acontinuous time approximation of the discrete time control law can beachieved as the limit of (5.12) as the pulse period goes to zero. The resultingcontinuous time control law becomes

S S S

SREF 4 S S Z N (5.23)

where the quantities uT(t) and uΨ(t) are given by


R

4 PI I REF

)- R

REF

P

R)-

)- R REF M

Z N N

(5.24)

Note that we choose to use the average of the torque as introduced by (5.20).In continuous time however the average torque over a pulse period is formedas

P

T

PT 4

(5.25)

In (5.25) we also introduced the formal operator A for the average value overa pulse period. In the Laplace domain, A is therefore given by

PS4

P

(5.26)

By comparing (5.23) with (5.3) and (5.2) it follows that

4

ZN N

? (5.27)

Although uΨ(t) and uT(t) do not equal the components u//(t) and u⊥ (t) (scaledby the actual stator flux), the former two will still be referred to ascomponents in stator flux coordinates in the sequel. As a matter of fact, uΨ(t)and uT(t) are not even voltages but have dimension s-1. The notation uΨ(t) anduT(t) might therefore be a bit misleading but keeps the links with generation ofthe stator voltage space vector.

For future use we also introduce the following notation for the vector in (5.24)

4

C.,

Z

(5.28)


5.3.2 PredictionIn continuous time the effect of applying the voltage pulses in the middle ofthe pulse periods was modeled as a time delay of half a pulse period inSection 4.1, see Figure 4-17. From the discrete time control law (5.12) we alsosee that this time delay is explicitly compensated for by predicting the statorcurrent. However, also the first term in (5.12), the desired stator flux updateover the pulse period, may be interpreted as a predicted value of the stator fluxderivative, since

S S

K P K S

K P

P

N N

N

(5.29)

From equations (5.29) and (5.19) it therefore seems reasonable to include theeffects of time delays into the continuous time ISC by replacing the stator fluxand stator current in equation (5.23) by predictions, i.e.

S S S

SREF 4 D S S D Z N (5.30)

where the time delay Td is given by

D P S (5.31)

(Note that the time delay is modified compared to (4.9)). The quantities instator flux coordinates, uΨ(t) and uT(t) are assumed not predicted and arehence still given by (5.24).

5.4 ModulatorIn Section 4.1 the low frequency component of the stator voltage generated bythe inverter was modeled by equation (4.8). With a varying DC-link voltage,the applied stator voltage differs from its reference. In order to compensate forvariations in the DC-link voltage, the reference stator voltage may bemodified as

DS S

SREF SREF

D

(5.32)

where D is the average DC-link voltage over the last pulse period. Inserting(5.32) into (4.8) gives the following stator voltage


DS SS SREF D

D D

(5.33)

Due to the filtering of the DC-link voltage (the average) and the time delay,perfect disturbance rejection is not possible.

5.5 SummaryIn this chapter a continuous time approximation of the ISC control law wasderived. If neglecting time delays, the control law is given by equations (5.23)and (5.24). In order to suppress variations in the DC-link voltage, the controllaw is modified according to (5.32). Further, if the time delays present in areal application are considered, equation (5.23) is replaced by (5.30),involving predicted values of the stator flux and stator current.

Chapter 6

Linearization of Control LawIn this chapter the continuous time ISC control law presented in Section 5.3 islinearized. The idea with field-oriented control is to generate a referencevoltage in a coordinate system aligned with the flux, where the voltagecomponents are set by torque and flux controllers. The equations for the actualfield-orientation are linearized in Section 6.1 followed by a linearization ofthe control law in stator flux coordinates in Section 6.2. Section 6.3 modelsprediction of signals needed for field-orientation when time delays areconsidered and finally, the equation for rejection of DC-link voltage variationsis linearized in Section 6.4.

6.1 Field-orientationThe coordinate transformation from stator flux coordinates to stator fixedcoordinates as defined in Subsection 5.2.2, i.e. from uT(t) and uΨ(t) to mu(t)and χu(t), is given by the space vector equation (5.23). By introducing thefollowing notation for the actual field-orientation

S S

SREF 4 Z N (6.1)

equation (5.23) may also be written as

S S S

SREF SREF S S (6.2)

This equation can be linearized and put on the real-valued vectorrepresentation introduced in Section 4.5 to give

SREF V SREF S S (6.3)

where Kv and Ks are transfer function matrices. We will not derive explicitexpressions for these transfer function matrices as the closed loop dynamicsmay be achieved in a more efficient way, see Appendix B.

58 Linearization of Control Law

Now, the voltage vssref(t) in (6.1) is a nonlinear function of the controller

outputs and the stator flux space vector. Using the real-valued vectorrepresentation, the linearization of equation (6.1) will be written as

SREF C C., Z N (6.4)

where ucNL(t) was defined by (5.28). To derive the expressions for the transferfunctions Tc and TΨ we introduce the following notation for the magnitude andphase of the space vector vs

sref(t)

VREFJ TS

SREF VREF D (6.5)

Together with (4.14) and (6.1) it now follows that

VREF 4 N Z (6.6)

4

VREF

N

Z

(6.7)

and a linearization of (6.6) and (6.7) gives

4

VREF

VREF

N N N

Z

N

NN

(6.8)

As we want relations between magnitudes and frequencies we rewrite (6.8) as

4

VREF

VREF

N N NZ

N

NN

(6.9)

The matrix describing the transformation (6.9) will be denoted TFO and itfollows that

C

N N

N

(6.10)

Linearization of Control Law 59

N

Z

(6.11)

By combining equations (6.3) and (6.4), the total linearized equation for thefield-orientation consequently becomes

SREF V C C., V S S Z N (6.12)

6.2 Field-Oriented Control LawThe reference voltage in stator flux coordinates was given by (5.24) anddenoted ucNL(t), see (5.28). Before linearizing ucNL(t) we introduce thefollowing transfer function

4

(6.13)

where A(s) is defined by (5.26). The expression (6.13) enables us to stay withthe vector notation, as averaging of the stator flux for the flux controller is notforeseen.

It is now straightforward to linearize (5.24), which gives

C., REF 4 FW REF

M MR R

X

(6.14)

where

R

PI I

)- R

P

(6.15)

R

)- RFW

(6.16)

)- X

(6.17)


R

MR R

T

(6.18)

Here it was used that

REF (6.19)

which can be motivated from the fact that the torque controller containsintegral action. First of all, (6.14) contains the linear flux and torquecontrollers and these terms will be denoted uc(t), i.e.

C REF 4 FW REF (6.20)

Apart from (6.20), equation (6.14) also holds terms depending on the motorspeed and the rotor flux magnitude. The former is a feedforwardcompensation of the disturbance ωm(t) and is considered first in Section 7.3.2.The term depending on the rotor flux magnitude arises from the conversion ofslip frequency to torque, see (5.24).

6.3 Linear Model of PredictionIn Subsection 5.3.2 the continuous time ISC control law was extended to alsohandle time delays due to modulation and real-time control. The extensionrelied on predictions of the quantities needed for field-orientation. In thissection we will derive linear models of these predictions and we start byrewriting the linear model of the induction machine related to stator flux in(4.37) as

SREF D 5D D M N Z Z XZ (6.21)

The prediction of the stator flux at time t+Td given information up to time tk,is assumed to be performed as


D

D

D

D

T 4

D SREF D

T

T 4

5D D D

T

T 4

D M

T

N N

Z

Z

XZ

(6.22)

where Z , 5D Z and XZ are estimated impulse responses of the

systems in (6.21). Further D and

M are predictions of the DC-

link voltage and motor speed respectively at time τ, given information up totime t, where τ ≥ t. Note that for t≤τ ≤ t+Td

SREF D SREF D (6.23)

as the information is available. We will assume that no predictions of the DC-link voltage and motor speed are available, which means that (6.22) and (6.23)lead to

D REF D REF D N N N N (6.24)

where

T

REF D SREF N Z (6.25)

DT 4

REF D SREF N Z

(6.26)

The prediction of the stator flux can hence be represented as in Figure 6-1. Ifwe assume that the estimated model in Figure 6-1 equals the true model, thenthe prediction of the stator flux may be written

D SREF N Z N (6.27)

Similarly, the prediction of the stator current used in (5.30) is given by

S D I SREF S (6.28)


usref(t) GΨ

+

D

+

-

D

GΨ

+

GΨGUd

Ud(t)

+ +

Ψµ(t+Td t)^

Ψµ(t)

Ψµref(t+Td t)

Ψµref(t)

^^ +

ωm(t)

GωΨ

+

Figure 6-1: Linear model of stator flux prediction.

Although we assumed the quantities uΨ(t) and uT(t) not to be predicted, wewill actually use a predicted value of the rotor flux to simplify futurecalculations. Hence, we model the predicted rotor flux as

R D MR SREF R (6.29)

As the rotor flux magnitude is a relatively slowly changing quantity, thepredicted and true values should be more or less equal.

6.4 ModulatorHow to suppress variations in the DC-link voltage was explained in Section5.4 through equation (5.32). This equation contains the average of the DC-linkvoltage formed over a pulse period, which may be represented by the notationA introduced in (5.25). By further using GUd defined by (4.39), thelinearization of (5.32) can be written as

SREF SREF 5D D (6.30)

Here we choose to change the order of the transfer function GUd and theaveraging operator A to simplify notation later. This is possible as the twooperators commute. However to make the dimensions fit, the average operatorin (6.30) must be two-dimensional. Hence A in (6.30) should actually bewritten as diag(A,A). We will however not explicitly make this distinction. Itshould be clear from the context, which form of A is used.


6.5 SummaryThe complete linearized ISC control law is given by equations (6.3), (6.12),(6.14) and (6.30). By using definition (6.20) and neglecting disturbancerejection of the motor speed and the DC-link voltage, these equations combineto

SREF V C V S S V MR R N (6.31)

Equation (6.31) is given for the ideal case with no time delays and in case oftime delays it is replaced by

SREF V C C V D

S S D V C MR R D

Z N

(6.32)

where the predicted values are given by (6.27)-(6.29).

Furthermore, disturbance rejection of motor speed DC-link voltage variationsis done according to (6.14) and (6.30), which we may summarize as

SREF SREF V C M 5D D

X (6.33)

Chapter 7

Closed Loop DynamicsIn this chapter the linear closed loop dynamics of an ISC controlled inverterfed induction machine are examined. The complete linearized ISC control lawis given by (6.31), which will be applied to the linear model of the processshown in Figure 4-17. The derivation of the closed-loop transfer functionswill be done in several steps. First the effect of all terms besides the output ofthe controllers is examined in Section 7.1. That is, the transfer function fromuc(t) to y(t) is derived. This almost corresponds to examining the effects offield-orientation as defined in Subsection 5.2.2. The only difference is theterm depending on the rotor flux magnitude in (6.31), which strictly speakingbelongs to the control law in field-oriented coordinates. For convenience wewill however include this term into the field-orientation and again redefine theconcept of field-orientation. In the linear case we will thus define field-orientation as the transformation from uc(t) to usref(t).

With time delays, equation (6.31) is replaced by (6.32) with predicted valuesaccording to (6.27)-(6.29). The effects of the time delays on the field-orientation as defined above are examined in Section 7.2. Before closing thetorque and flux controller loops in Section 7.4, the effects of disturbancerejection is considered in Section 7.3. The chapter ends by studying the effectsof stator voltage saturation in the so-called field-weakening region in Section7.5.

7.1 Field-Orientation without Time DelayFor the linearized system field-orientation was defined as the transformationfrom the torque and flux controller contributions to the stator voltage, i.e.from uc(t) to usref(t). In the ideal case with no time delays, this transformationis given by (6.31). If we neglect disturbance rejection, the effect of field-orientation on the system is hence described by Figure 7-1. Here it is assumedthat all motor quantities are available for measurement. In a practical

66 Closed Loop Dynamics

application this may not hold and typically all signals but the stator currenthave to be estimated by an observer.

TFO +

GUd

Ud

+ +usref us

Ψµ

+

Fmr

uc +

+G

ωm

ymr

Ψµ

mr

+is

+

+

Ks

Kv

ucNL vsref

Figure 7-1: Linear model of field-orientation without time delay.

From Figure 7-1 it follows that the reference stator voltage satisfies

SREF S I V V C MR MR SREF

S I V V C MR MR 5D D

S I V V C MR MR M V C C

Z Z

Z Z

X Z XZ X

(7.1)

and that the output y(t) can be calculated through

Y SREF Y 5D D Y M

X (7.2)

Inserting the expression for usref(t) into (7.2) gives

C V 5D D M

X (7.3)

where

Y S I V V C MR MR V Z Z

(7.4)

V S I C MR MR Y X X Z XZ X X

(7.5)

It is further shown in Appendix B, that apart from very low motor speeds, thefollowing approximation holds

Closed Loop Dynamics 67

V 5D 5D (7.6)

By using (7.6) in (7.3) and introducing the notation

C (7.7)

the system in Figure 7-1 may be depicted as shown in Figure 7-2.

yP

QGUd

++ +uc

Gω

+

ωm

+ +

Ud

Figure 7-2: Process model after field-orientation.

In Appendix B it is shown that expressions (7.4), (7.5) and (7.7) can beevaluated as

)- R

R

T

T

T

T T

N

(7.8)

)- R

R

T

N N T T

T TN

N N

(7.9)

)- R

R

TX

(7.10)

where


T

T

(7.11)

The transfer function M depends on the torque through the load angle δ∗ ,defined by (4.18). In order to estimate the size of tan2(δ*) we observe thatfrom equations (4.17) and (B.35) in Appendix B, together with the expressionfor the pull-out torque (4.29), the following equality holds

PULL OUT

(7.12)

The expression tan2(δ∗ ) can hence be evaluated as a function of the steady-state torque normalized by the pull-out torque. The result is shown in Figure7-3 where it is seen that the factor tan2(δ∗ ) is very small for most torquelevels.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

tan2(δ) as a function of T/Tpullout

T/Tpullout

Figure 7-3: tans(δ) as a function of T/Tpullout.

In Appendix C the maximum used torque in driving is set to 1780 Nmwhereas the pull-out torque at nominal flux can be calculated to 4600 Nm, see(4.29). Hence, at nominal flux the ratio between torque and pull-out torquedoes not exceed 0.4 and a reasonable approximation is therefore to set


tan2(δ∗ )=0. This in turn means that the transfer function M can beapproximated by one, which leads to the following approximations

)- R

R

T T

T T

N

(7.13)

)- R

R

N N T T

TN T

N N

(7.14)

)- R

R

TX

(7.15)

The field-orientation hence converts the system Gy into the upper triangulartransfer function matrix P. Flux control is consequently affected only by theoutput of the flux controller and, as the flux level is usually kept constant, thetorque may purely be controlled by the output of the torque controller. Wealso see that for zero torque, the linearized dynamics are completelydecoupled.

Above we defined field-orientation as the transformation from uc(t) to usref(t).From equations (7.2) and (7.3) we see that the reference stator voltage can bewritten as

SREF Y C Y 5D D

Y Y M

X X

(7.16)

where we also used the approximation (7.6). Figure 7-1 may hencealternatively by described by Figure 7-4, where the field-orientation isinterpreted as an inverse based controller. Note that except for the statorcurrent, the feedback signals in Figure 7-1 are not measured in a practicalapplication but have to be estimated by an observer. This justifies therepresentation in Figure 7-4.


yGy

GyGUd

++ +

Gωy

+

ωm

+ +

Ud

Gy-1P

uc +

(I-Gy-1Q)GUd

++ - + -

Gy-1(Gωy-Gω)

Figure 7-4: Alternative representation of field-orientation as a method ofinverting the process dynamics.

7.2 Field-Orientation with Time DelayIf time delays are included, the control law changes from (6.31) to (6.32),where the predictions are given by (6.27)-(6.29). Adding time delays andprediction hence results in the diagram shown in Figure 7-5.

From Figure 7-5 it can be shown that the expression for usref(t) is identical toequation (7.1) and usref(t) may therefore be written as (7.16) also in case oftime delays. As the output y(t) in Figure 7-5 is given by

Y SREF Y 5D D Y M

X (7.17)

the following equations describe the field-orientation with time delays

C Y 5D D

Y M

X X X

(7.18)

For small time delays we may approximate (7.18) by

C 5D D M

X (7.19)

where P, Q and Gω are given by (7.13)-(7.15). Hence, the field-orientationwith time delays may be modeled as the system in Figure 7-6.


+

GUd

Ud

+ +usref us G

ωm

ymr

Ψµ+is

+

+KvTFO

uc

+

+

D GΨ

+

+

+

-

++ u’c

Gmr+

+

D

+

+-

+

+

Fmr

Gi+

+

D

+

+

+

-

D

Ks

Figure 7-5: Linear model of field-orientation with time delay.

yP

QGUd

++ +uc

Gω

+

ωm

+ +

Ud

D

Figure 7-6: Field-oriented control with time delay.


7.3 Disturbance RejectionThis section adds disturbance rejection to the system given in Figure 7-6. DC-link voltage disturbances are treated in Subsection 7.3.1 and motor speeddisturbances in Subsection 7.3.2.

7.3.1 DC-Link Voltage CompensationA varying DC-link voltage is compensated through equation (6.30). Thecompensation is added directly to the stator voltage reference, which isillustrated in Figure 7-7.

+

GUd

Ud

+ + usu’sref Dusref +

GUd

+ -

A

G

ωm

ymr

Ψµis

Figure 7-7: Linear model of compensation of DC-link voltage disturbances.

The effect of the disturbance rejection is consequently that Figure 4-17 ischanged to Figure 7-8.

+

(I-DA)GUd

Ud

+ + usDusref G

ωm

ymr

Ψµis

Figure 7-8: Linear process model with compensation of DC-link voltagedisturbances.


From Section 7.2 we further know that when applying field-orientation, thesystem in Figure 7-8 changes to

yP ++ +uc

Ud

D

Q(I-DA)GUd

Figure 7-9: Field-oriented control with time delays and DC-link voltagedisturbance rejection.

7.3.2 Motor Speed CompensationMotor speed disturbances are considered in the ISC control law, see equation(6.14). The input in field-oriented coordinates is hence modified as

C C

X X (7.20)


yP

Gω

+

ωm

+ +uc ++ +

D

Fω

Figure 7-10: Compensation of motor speed variations.

Considering this compensation, the output y(t) can be written as

C M

X X

(7.21)

From the definitions of P, Fω and Gω, given by (7.13), (6.17) and (7.15), itfollows that


X X (7.22)

which means that the effect of compensation of motor speed variations can berepresented as

yPuc

(I-D)Gω

+

ωm

+ +D

Figure 7-11: Effects of motor speed compensation.

7.4 Closed-Loop ControlIn the previous sections it was shown how field-orientation and disturbancerejection affect the original model of the process in Figure 4-17. By alsoclosing the controller loops with equation (6.20), the closed-loop system maybe represented as in Figure 7-12.

uc+

-+ F

yref y+

Q(I-DA)GUd

Ud

++D P+

Ffw

AT

(I-D)Gω

ωm

+

Figure 7-12: Closed loop system.

The output of the closed-loop system can be derived as


C REF 5D D

M

X

(7.23)

Here the sensitivity function S is given by

4

(7.24)

and the closed-loop transfer function Gc equals

C FW

FW

(7.25)

From (7.25) it follows that flux control is decoupled from torque control. Incase of zero operating point torque, P12 = 0 and consequently the torque andflux control loops are completely decoupled, see (7.13).

Further, the transfer function from DC-link voltage to output in (7.23) may beevaluated to

5D

5D

(7.26)

and the transfer function from motor speed to output is given by

X

X

(7.27)

From equations (7.23), (7.25), (7.26) and (7.27) it follows that Figure 7-12can be redrawn as Figure 7-13.


F11 P11++Tref T

-

F22 P22++

-

mµ

++

+

mµref D

D

A

P12

Ffw11

++ +

Q11(1-DA)GUd1

ωmUd

+

+

(1-D)Gω1

Figure 7-13: Closed-loop system.

7.5 Inverter Input AdmittanceIn Chapter 3 it became clear that the inverter input admittance is crucial whenit comes to DC-link stability. Hence, in order to properly analyze DC-linkstability and synthesize stabilization, we need a model of the input admittance,i.e. a transfer function from DC-link voltage to DC-link current, see (3.5).

To derive a linear model of the inverter admittance we start with the linearizedexpression for the DC-link current (4.41). By assuming the motor speed to beconstant and setting the scaling factor n to 1 for notational convenience, thisequation becomes

M M

D D

D D

(7.28)

By adding equation (7.28) to Figure 7-13, while neglecting flux control andassuming that the motor speed is constant, we end up with Figure 7-14.


T++ +

Ud

DTref

+

id-

+

ωm*T*

Ud2*

ωm

Ud

*

*S11Q11(1-DA)GUd1

Gc11

Figure 7-14: Linear model of inverter input admittance.

From Figure 7-14 it follows that the input admittance Y is given by

M M

5D

D D

(7.29)

The second term of (7.29) corresponds to the inverter input admittance withperfect control c.f. (2.2). The first term however also shows the effects of non-perfect compensation of DC-link voltage variations due to inevitable timedelays and limited control bandwidth. In absence of time delays, thedisturbance rejection would give (1-DA)GUd1 = 0 and perfect control wouldgive S11 = 0 and hence the first term in (7.29) would vanish.

For later use we introduce the notation Y0 for the first term of (7.29), i.e.

M

5D

D

(7.30)

7.6 FieldweakeningIn fieldweakening, the stator voltage magnitude saturates. Therefore the statorvoltage magnitude can no longer be influenced by the control andconsequently the control law (7.16) changes to

SREF

Y C Y 5D D Y M

X X

(7.31)

where


(7.32)

This means that the field-orientation is described by

C Y Y 5D D

Y Y M

X X X

(7.33)

which differs from (7.18) due to the transfer function X, given by

Y Y (7.34)

Further, from (6.30) it follows that rejection of DC-link voltage disturbancesare no longer possible as the compensation is done using the magnitude of thestator voltage. The disturbance transfer function (I-DA)GUd in Figure 7-9reduces to the expression (4.39), where

U D

(7.35)

Note that removing the disturbance rejection reduces the phase shift of theadmittance by 90° for small frequencies, which may be advantageous from aDC-link stability point of view.

Chapter 8

Controller TuningIn Chapter 7 the linearized dynamics of the closed loop system were derivedand illustrated in Figure 7-13. From this figure we see that torque and fluxcontrol are partly decoupled as the flux is not influenced by the torquereference. The torque is however affected by the flux, but the flux reference isnormally kept constant and we will not examine the cross coupling anyfurther. Using the linear models of the closed-loop system, this chapterderives expressions for the controller parameters to obtain desired controlbandwidths and stability margins. As the system dynamics vary, the controllerparameters have to be updated according to the operating points. We will callsuch modifications of the controller parameters gain scheduling and twopossible ways to update the parameters, to give either constant bandwidth orconstant phase margin, are considered.

The closed-loop torque transfer function is considered in Section 8.1 and thecorresponding flux transfer function in Section 8.2.

8.1 Torque ControllerFrom Figure 7-13 it follows that the closed-loop transfer function for thetorque control is given by

FW

REF

(8.1)

From the expressions (7.13), (6.15), (6.16) for P, F and Ffw respectively, wesee that several factors are eliminated in the product of P11 and the sum of F11

and Ffw11. That is, the torque controller compensates for a varying gain of P11.To simplify the expressions we therefore introduce the following transferfunctions

4 PI I 4

T

(8.2)

80 Controller Tuning

which means that (8.1) can be written as

4 4

REF4 4

(8.3)

From Figure 7-13, we see that the torque controller consists of a feedback partwith an additional fixed feedforward controller. Whereas stability of thesystem is only determined by the feedback part, the torque control bandwidthis also affected by the feedforward controller. To simplify controller designwe will therefore rewrite the torque controller as a pure feedback controllerwhere tuning only involves shaping the loop gain. This can be achieved bymaking the following definition

4

44

(8.4)

The transfer function (8.3) may then be written as

4 4

C

4 4

(8.5)


++Tref

-

TP’T1+FT D

A

Figure 8-1: Closed-loop torque control.

Bandwidth of the system is proportional to the crossover frequency of theloop gain, i.e. the frequency where the magnitude of the loop gain crosses one.For the system in Figure 8-1 the crossover frequency, denoted ωc, hencesatisfies

C

4 4 S J

X

(8.6)

Stability of the closed-loop system in Figure 8-1 will be expressed in terms ofthe phase margin, φm, defined as

C

M 4 4 S J

X

(8.7)

Controller Tuning 81

Phase margin is hence defined for the open-loop system P’T in Figure 8-1.This means that a certain phase margin according to (8.7) does not necessarilyguarantee the same phase margin for the more physically correct open-loopsystem PT in (8.3).

When numerically investigating the loop gain it will be convenient to makethe following Laplace domain approximation, valid for low frequencies

PP

4 SS4

P

(8.8)

Hence, the average is approximated by a time delay and consequently theeffective time delay in the torque control loop is the sum of the pure timedelay Td and Tp /2 from (8.8). We introduce the notation Tda for this time, i.e.

D P

DA4 4 S

4 S

(8.9)

where

DA D P (8.10)

We may further simplify analysis of the system in Figure 8-1 by making thefollowing approximation

4 4 APPROX

4 44 4 4

T

(8.11)

To evaluate the error made by the approximation (8.11) we form the followingquotient

DAAPPROX S4DA4

4

T T T

(8.12)

As long as the effective time delay Tda is much smaller than Tσ, we can expectsmall errors. From (8.10) and (5.31) it follows that for small sampling timesTs we may approximate

DA P S P P (8.13)

and the effective time delays are hence approximately given by Figure C-3.From this figure we see that Tda ≈ Tp ≤ 2.7 ms whereas Tσ = 33 ms. The effectof the approximations should hence be small. This is also verified by Figure


8-2 showing the exact Bode plot of the quotient (8.12) for the exampledefined in Appendix C.

100

101

102

0.9

0.95

1

1.05

1.1

Frequency [Hz]

Gai

n

P’ approxT /P’T

100

101

102

-4

-2

0

2

4

Frequency [Hz]

Pha

se [

deg]

Figure 8-2: Approximation error made in (8.11).

Using the approximations (8.9) and (8.11), the loop gain in Figure 8-1 is givenby

DA

I

PI PI 4 S4 4

T

(8.14)

From Bodes relation, see e.g. [23], we know that the phase and magnitude of atransfer function depend on each other. In order to guarantee a large phasemargin, the slope of the loop gain must therefore be –1 around the crossoverfrequency ωc. For the loop gain (8.14) this implies that

I

CPI

(8.15)

which also is motivated in Figure 8-3 where the (asymptotic) Bode plot ofP’T(1+FT) is shown. The gain decreases with slope –2 for low frequencies,which implies a phase of –180°. At the break point Ki /(1+Kpi) the slope


changes to –1 and the phase starts to increase towards -90°. The lower thefrequency at the breakpoint, the higher the phase margin.

100

101

102

100

102

Bode plot of P’T(1+FT)

Gai

n

Ki/(1+Kpi) → ← ωc

100

101

102

-180

-90

Frequency [Hz]

Pha

se [

deg]

← φm

Figure 8-3: Influence of the controller parameters on the phase margin.

If requirement (8.15) is satisfied, the following relation approximately holds

C

PI

4 4 S JC

X

T

(8.16)

Hence, a desired crossover frequency, ωcd, is obtained by setting

PI CD T (8.17)

which means that requirement (8.15) is equivalent to

I CD T

(8.18)

Selecting Ki is a trade-off between robustness and small stationary errors. Alarge Ki means large loop gains for small frequencies but decreased stabilitymargins and vice versa. A proposal is to set Ki four times smaller than thelimit in (8.18), i.e.

CD

I

T

(8.19)


With these parameter settings the actual crossover frequency becomes

C CD CD

(8.20)

and the phase margin can be derived as

CM 4 4 S J

CD DA

X

(8.21)

The first term corresponds to the phase margin in absence of time delays andis approximately 76.3°, independently of ωcd. The time delays however reducethe phase margin and put restrictions on the bandwidth. In order to assure adesired phase margin φm

d, we get the following bandwidth constraint

DM

CD

DA

(8.22)

The bandwidth (crossover frequency) is thus limited by an expressioninversely proportional to the time delay. It may therefore be convenient towork with the inverse bandwidth, which will be denoted Ttorque, i.e.

TORQUE

CD

(8.23)

The expressions for the gains (8.17) and (8.19) then become

PI I

TORQUE TORQUE

T T (8.24)

and the bandwidth constraint (8.22) can be stated as

TORQUE AD

D

M

(8.25)

If for example a phase margin of 45° is required it follows that

TORQUE AD AD

(8.26)


If the restriction on Ttorque gets too large, Ki must be decreased to increase thephase margin.

Using parameters as in (8.24) result in the torque step response shown inFigure 8-4, where the time is scaled by Ttorque.

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

1.2

Time/Ttorque

Torque Step Response

Figure 8-4: Torque step response with controller parameters according to(8.24).

The over-shoot may be motivated by the fact that typical torque references areramps rather than steps. To reach small errors at ramps, the low frequencyloop gain must be large, which causes an overshoot at a step response.

8.1.1 Gain SchedulingOften the switching frequency of the inverter is varied as a function of thestator frequency to avoid low frequency torque oscillations and to generatepredictable harmonics. This in turn means that also the time delays vary withthe operating points, see the example in Appendix C. Varying time delaysmean that the controller bandwidth and/or the stability margins change withthe operating points. By setting Ttorque to a constant value, the bandwidth isfixed but the phase margin varies with the time delay according to (8.27),which follows from (8.21) and (8.23).


AD

M

TORQUE

(8.27)

On the other hand, in order to keep the phase margin constant at φmd, the

parameter Ttorque should be updated according to

AD

TORQUED

M

(8.28)

Assume we want a constant bandwidth with a minimum phase margin of 45°.The effective time delay may be approximated by (8.13) and in our examplethe longest pulse period is 2.7 ms as seen in Figure C-3. From (8.28) it thenfollows that

TORQUE (8.29)

In Figure 8-5 the magnitude of the torque transfer function is shown for theexample in Appendix C (zero torque) with Ttorque = 5ms.

Figure 8-5: Magnitude of torque transfer function with constant bandwidth.


In Figure 8-6 the crossover frequency and phase margin have been evaluatedaccording to (8.6) and (8.7) respectively. From the figure we see how thephase margin varies when the crossover frequency is kept constant. Thebandwidth was set to reach a phase margin larger than 45° for all operatingpoints and from Figure 8-6 we see that the goal was met.

0 5 10 15 20 25 30 35 4031.5

32

32.5

33

33.5

34Crossover frequency as a function of motor speed

Fre

quen

cy [

Hz]

0 5 10 15 20 25 30 35 4040

50

60

70

80Phase margin as a function of motor speed

Motor speed [Hz]

Ang

le [

deg]

Figure 8-6: Crossover frequency and phase margin with constant bandwidth.

With a constant bandwidth, the parameter Ttorque hence is limited by thelongest effective time delay Tda. For operating points with shorter time delays,the bandwidth may be increased and an alternative way is to set the controllerparameters to reach constant phase margin. The time constant Ttorque is thendetermined by expression (8.28). In Figure 8-7 the magnitude of the torquetransfer function is shown the operating points in Appendix C (zero torque)with a constant phase margin of 45°.

Figure 8-8 shows the achieved crossover frequency and phase margin with aconstant phase margin of 45°. The phase margin is close to the desired phasemargin and we see that the crossover frequency in increased compared toFigure 8-6.

Remark: The controller gains were set such that the only independentparameter was Ttorque. There are of course other possible ways of setting thecontroller gains but this is not considered here.


Figure 8-7: Magnitude of torque transfer function with constant phasemargin.

0 5 10 15 20 25 30 35 400

50

100

150Crossover frequency as a function of motor speed

Fre

quen

cy [

Hz]

0 5 10 15 20 25 30 35 4046

48

50

52

54Phase margin as a function of motor speed

Motor speed [Hz]

Ang

le [

deg]

Figure 8-8: Crossover frequency and phase margin with constant phasemargin.


8.2 Flux ControllerUnder the assumption that the flux for the flux controller is not predicted, itfollows from Figure 7-13 that the closed-loop transfer function for fluxcontrol is given by

REF

N N

(8.30)


++mµref

-

mµP22F22 D

Figure 8-9: Closed-loop flux control.

The loop gain in Figure 8-9 is given by

D4 S

P

N

(8.31)

and the crossover frequency equals

C P N (8.32)

If we introduce the time constant Tflux as the inverse of the desired crossoverfrequency, i.e.

FLUX

CD

(8.33)

then the bandwidth requirements is reached by setting the controller gain Kp

as

P

FLUX

N

(8.34)

Further, the phase margin for the flux control is given by

C

D

M S JFLUX

X

(8.35)


In order to assure a desired phase margin φmd it hence follows that

D

FLUXDM

(8.36)

If for example a phase margin of 45° is required the following relation mustbe satisfied

FLUX D D

(8.37)

With the flux controller tuned to give constant bandwidth, the phase marginvaries according to (8.35). On the other hand, to obtain a constant phasemargin φm

d, the time constant Tflux should be set as

DFLUX

DM

(8.38)

From (5.31) it follows that if the sampling time Ts is much smaller than thepulse periods then

D P S P (8.39)

and (8.34), (8.38) and (8.39) combine to

DM

P

P

N

(8.40)

Chapter 9

Design of AdditiveStabilizationThis chapter uses the linear model of the inverter input admittance derived inSection 7.5 to analyze stability as well as synthesizing the stabilization gain ofthe feedback system shown in Figure 3-2. Stability is checked using theNyquist criterion, i.e. the Nyquist trajectories of the loop gain YZDC in Figure3-2 are plotted. For the closed-loop system to be stable the trajectories are notallowed to encircle the point –1.

The chapter starts by examining the properties of the drive withoutstabilization in Section 9.1. Section 9.2 derives the expression for the inverterinput admittance with stabilization. In order to derive an expression for theadditive stabilization gain in Sections 9.4 and Section 9.5, someapproximations used during synthesis are proposed in Section 9.3. The resultsof the chapter are summarized in Section 9.6.

9.1 Loop Gains without StabilizationA linear model for the inverter input admittance was given by (7.29). Thissection evaluates this expression for the three different cases; zero torque(coasting), positive torque (driving) and negative torque (braking), defined inAppendix C. During the evaluation, the torque controller is assumed to betuned according to (8.24) in order to give a constant phase margin of 45°.

9.1.1 CoastingThe simplified analysis performed in Section 2.1 modeled the inverter as anegative resistance. For zero power this resistance was infinite and only theinput filter influenced the stability of the drive. As the filter is stable, althoughmaybe poorly damped, also the drive would hence be stable. If instead using

92 Design of Additive Stabilization

the representation of the drive shown in Figure 3-2, the simplified analysisgives an admittance that is identically zero and clearly the feedback loop mustbe stable. The linear model of the inverter input admittance given by (7.29) ishowever non-zero due to the non-perfect properties of the control. At zerotorque (zero power) the admittance reduces to Y0 given by (7.30) for whichthe Nyquist curves as well as the Bode diagram are shown in Figure 9-1.

-1 0 1 2 3 4 5-3

-2

-1

0

1

2

3Nyquist plot of inverter input admittance

Re

Im

100

101

102

10-5

100

Frequency [Hz]

Gai

n

Bode plot of inverter input admittance

100

101

102

-90

0

90

180

Frequency [Hz]

Pha

se [d

eg]

Figure 9-1: Nyquist (a) and Bode (b) plots of the inverter input admittance Y.

From these plots we see that the phase of the inverter admittance is 180° forlow frequencies, which actually means that the inverter acts like a negativeresistance even at zero torque. Note that as the copper losses of the machinewere neglected in equation (4.34), these effects do not arise due to powerconsumed for magnetization. The phase shift of the admittance howeverdecreases with increasing frequency and the Nyquist trajectories eventuallyleave the left half plane in Figure 9-1.a. This means that the admittance getspassive and stabilization is consequently no longer needed. We also see thatthe phase of the admittance approaches -90° for high frequencies, which maycause problems at these frequencies as well.

The fact that the input admittance is not passive for low frequencies meansthat the loop in Figure 3-2 may become unstable, which actually is the case inthis example as seen from the Nyquist trajectories of the loop gain in Figure9-2.

Design of Additive Stabilization 93

-5 -4 -3 -2 -1 0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

2

3

4

5Nyquist plot of loop gain

Re

Im

Figure 9-2: Nyquist plots of original loop gain YZDC.

The fact that the induction machine drive may become unstable at zero torquecan not be shown using the simplified analysis done in Section 2.1 byassuming constant power flow.

Remark: With the standard ISC structure, i.e. Ts = Tp, the system wouldbecome unstable at high frequencies as well.

9.1.2 DrivingFrom the expression for the input admittance (7.29) we see that a positivepower flow shifts the Nyquist trajectories in Figure 9-1.a to the left, whichmakes the admittance less passive. This is also seen in Figure 9-3 showing theNyquist and Bode plots of the inverter input admittance in driving. Notehowever that the power is not constant for all operating points (see Figure C-1.b), which means that Figure 9-3.a is not simply achieved by shifting the axisin Figure 9-1.a. Nyquist trajectories corresponding to different powers areshifted different amounts. Further, Y0 is not completely independent of thetorque. Anyway, the influence of the positive power makes the Nyquisttrajectories of the loop gain even worse, see Figure 9-4.


-1 0 1 2 3 4 5-3

-2

-1

0

1

2


Re

Im

100

101

102

10-5

100

Frequency [Hz]

Gai

n


100

101

102

-90

0

90

180

Frequency [Hz]

Pha

se [d

eg]

Figure 9-3: Nyquist (a) and Bode (b) plots of the inverter input admittance Ywith positive torque.

-5 -4 -3 -2 -1 0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

2

3

4


Re

Im

Figure 9-4: Nyquist plots of original loop gain YZDC with positive torque.

From Figure 9-3.a we also see that even with maximum torque, the inverterinput admittance is non-passive only for low and high frequencies. Inbetween, the Nyquist trajectories are in the right half plane and herestabilization is actually not needed.


9.1.3 BrakingIf a positive power made the inverter input admittance less passive comparedto zero power, the opposite holds for negative power, which is seen in Figure9-5.

-1 0 1 2 3 4 5-3

-2

-1

0

1

2


Re

Im

100

101

102

10-5

100

Frequency [Hz]

Gai

n


100

101

102

-90

0

90

180

Frequency [Hz]

Pha

se [d

eg]

Figure 9-5: Nyquist (a) and Bode (b) plots of the inverter input admittance Ywith negative torque.

-5 -4 -3 -2 -1 0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

2

3

4


Re

Im

Figure 9-6: Nyquist plots of original loop gain YZDC with negative torque.


The Nyquist trajectories are shifted to the right in Figure 9-5.a making theadmittance almost passive. The phase is however close to 90° and even if theNyquist curves of the loop gain do not encircle the point –1, see Figure 9-6,the stability margins are small.

9.2 Modified Inverter Input AdmittanceThe results of Section 9.1 showed that the induction machine drive may beunstable or at least poorly damped in all modes of operation. In order tostabilize the drive we apply additive stabilization according to equation (2.16).In this section we derive the influence of the stabilization on the inverter inputadmittance.

We will assume that the band-pass filter B in (2.16) is implemented as

(9.1)

where k1 and k2 are constants satisfying k1 >> k2. The band-pass filter Bremoves low and high frequency components of the DC-link voltage but alsointroduces a phase shift as shown by the Bode diagram in Figure 9-7. Here k1

and k2 set to 2000 and 20, respectively.

100

101

102

10-1

100

Frequency [Hz]

Gai

n

Bode Plot of B(s)

100

101

102

-100

-50

0

50

100

Frequency [Hz]

Pha

se [

deg]

Figure 9-7: Bode plot of band-pass filter B.


Stabilization according to equation (2.16) hence modifies Figure 7-14 intoFigure 9-8.

Gc11T’ref D

Tref ++ +

BK

T++ +

Ud

+

id-

+

ωm

Ud

*

*

ωm*T*

Ud2*

S11Q11(1-DA)GUd1

Figure 9-8: Structure of DC-link stabilization.

From Figure 9-8 it follows that the inverter input admittance with stabilizationis given by

M M

C

D D

(9.2)

where Y0 is defined through (7.30).

As the original admittance consists of two terms, we will also split thestabilization gain K into two parts, i.e.

4 X

(9.3)

where Kω will be used to stabilize the first term of the original admittance, i.e.Y0 and KT will be used to handle the second term. Using the definition (9.3)we observe that equation (9.2) can be written as

M 4 M

STAB C

D D

(9.4)

where

STAB C FW

5D

X

(9.5)

We will now proceed by first finding a Kω making the term Y0Ystab passive inSubsection 9.4. Thereafter the second term in the modified admittance isconsidered in Section 9.5. First however some useful approximations arestated in Subsection 9.3.


9.3 ApproximationsIn order to derive expressions for the stabilization gain in the followingsections we need explicit expressions for the closed loop transfer function andthe sensitivity function for the torque. The sensitivity function appears in theexpression for Y0, see (7.30) and the closed loop torque transfer function isneeded in (9.2). For simplicity we will neglect time delays and average whencalculating the sensitivity function, hence we make the followingapproximation (c.f. (7.24))

(9.6)

(Note that (9.6) would be the exact expression for the sensitivity function ifthe torque would be predicted). With the controller gains (8.24) the sensitivityfunction and the closed loop transfer function can now be approximated as

TORQUE

CTORQUE

TORQUE TORQUE

T

T

(9.7)

The sensitivity function S11 is shown in Figure 9-9.a for the given example,controller tuning and zero torque. The effect of the approximation on thesensitivity function is shown in Figure 9-9.b where the Bode plot of thequotient between the approximation and the original sensitivity function fortorque control is plotted.

100

101

102

100

Frequency [Hz]

Gai

n

Sensitivity function

100

101

102

-90

0

90

180

Frequency [Hz]

Pha

se [d

eg]

100

101

102

0.5

0.75

1

1.5

Frequency [Hz]

Gai

n

Bode Plot of Sapprox

/S

100

101

102

-30-20-10

0102030

Frequency [Hz]

Pha

se [d

eg]

Figure 9-9: Bode plot of (a) the sensitivity function S11 and (b) the error whenneglecting the time delays.


With the approximation (8.9) the first term of the input admittance Y0, definedby (7.30), can be written

ADS4

R M)-

R D

TORQUE

N

T

(9.8)

For small frequencies the exponential in (9.8) can be approximated by a firstorder Taylor expansion, i.e.

ADS4

AD (9.9)

which means that

R M )- M SLIPAD )-

R D

TORQUE

T

(9.10)

Hence, as long as

SLIP )- M (9.11)

the phase of (9.10) is given by

TORQUE (9.12)

The requirement (9.11) only limits the validity of (9.12) for non-zero torque.Even at maximum torque it is however not a practical problem as the slipfrequency is still small. The absolute limit of the slip frequency is given by(4.28), which in our case is 4.8 Hz. Expression (9.12) is hence not valid formotor speeds below 2.4 Hz in the worst case. For such speeds the magnitudeof the input admittance is very small and consequently the loop gain in Figure3-2 is also small.

9.4 Stabilization at Zero TorqueThe analysis in Subsection 9.1.1 showed that the drive may become unstablealso with zero operating point torque. The problems were mainly due to thenon-passiveness of the inverter input admittance for low frequencies. At highfrequencies the Nyquist trajectories of the loop gain did pass the point –1 onthe right side although the stability margin was not very large. In order to


stabilize the system we must hence decrease the phase of the loop gain for lowfrequencies, without introducing new problems at high frequencies. It wouldeven be desirable to improve behavior at high frequencies.

In Subsection 9.4.1 an expression for the gain Kω is derived making theinverter input admittance passive at low frequencies. The resulting propertiesat high frequencies are then examined in Subsection 9.4.2. The section endsby briefly investigating the effects of the band-pass filter and torque controllertuning on stability in Subsection 9.4.3 and Subsection 9.4.4, respectively.

9.4.1 Low FrequenciesThe goal is to derive an expression for Kω making the modified inverter inputadmittance passive for low frequencies at zero torque. Using (9.4) and (9.5)the admittance at zero torque can be written Y0Ystab and from (9.12) we seethat the phase of Y0 is close to 180° for low frequencies. Making theadmittance passive hence means that the phase of the modified admittancemust be reduced from 180° to less than 90°, i.e.

STAB

o (9.13)

With the approximation (9.12), (9.13) is equivalent to

STAB TORQUE o (9.14)

The requirement on Kω may hence be expressed in terms of the phase of Ystab.With the controller parameters (8.24) and the following Taylor approximation

D

DA

S4

S4

DA

(9.15)

we see that the transfer function Ystab can be written as

TORQUEDSTAB

)- R AD TORQUE

T X

N

(9.16)

For small frequencies, ω << k1, we further obtain

TORQUE

STAB

X

(9.17)


where the constant K’ω was introduced as

D

D

)- R AD TORQUE )- R AD TORQUE

T X

T X

X

N N

(9.18)

The approximation in (9.18) follows as k1 >> k2. Now let Kω = Pωωµ∗ , where

Pω > 0 so that K’ω is independent of ωµ∗ and is larger than zero. It is shown in

Appendix A that this implies that

STAB

X

(9.19)

as long as

TORQUE

(9.20)

With the estimation (9.19) it now follows that (9.14) is satisfied if

TORQUEX

(9.21)

For small frequencies ω, the arctan on both sides in the equation above can beneglected, which means that the requirement on K’ω can be written

TORQUE

X

(9.22)

To obtain some margin, the gain K’ω is set to twice the limit in (9.22) and itthen follows from (9.18) that

)- R

AD

D

X N

T

(9.23)

Note that this expression is independent of the torque controller bandwidth,set through Ttorque (as long as (9.20) is satisfied). Note also that even thoughthe stabilization actually is turned off at low frequencies by the band-passfilter B, it was still possible to reduce the phase of the inverter inputadmittance at low frequencies. This follows as also the gain of the admittanceY0 decreases with the frequency, see (9.10).


The corresponding plots to Figure 9-1 and Figure 9-2 are shown below whenapplying stabilization according to (9.23).

-1 0 1 2 3 4 5-3

-2

-1

0

1

2


Re

Im

100

101

102

10-5

100

Frequency [Hz]

Gai

n


100

101

102

-90

0

90

180

Frequency [Hz]

Pha

se [d

eg]

Figure 9-10: Nyquist (a) and Bode (b) plots of the modified inverter inputadmittance Y’.

-5 -4 -3 -2 -1 0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

2

3

4


Real(YZDC)

Imag

(YZ D

C)

Figure 9-11: Nyquist plots of the modified loop gain Y’ZDC.

We see that the problems at low frequencies are solved and hence the systemis stable. However, the high frequency performance is made slightly worse,


which is seen by comparing the loop gains in Figure 9-2 and Figure 9-11. Wewill show in the following subsection how performance may be improved athigh frequencies.

9.4.2 High FrequenciesThe expression for Kω was derived to make the input admittance passive forlow frequencies and from Figure 9-10 we see this was achieved. However, thehigh frequency properties were made slightly worse and in order to analyzethe properties at high frequencies more in detail we examine Ystab with thestabilization gain set according to (9.23). In the pass band of B (set k1 to alarge number) it then follows that

D

DA

S4TORQUED

STAB S4TORQUE

(9.24)

Assuming that the torque controller is tuned to give a constant phase marginof 45°, it then follows from (8.28) that Ttorque ≈ 2Tda. For small sampling timesTs we may further approximate Td ≈ 0.5Tad, which follows by combining(5.31) and (8.13). Hence, Ystab may be written as

DA DA

DA DA

4 S 4 SDA

DA

STAB 4 S 4 SDA

(9.25)

Note that the expression for Ystab only depends on the quantity Tdas. With theassumption of the controller tuning, we may also write the phase of Y0 as

DAJ 4

DA

X

(9.26)

which follows from (9.8), subject to (9.11). The expressions (9.25) and (9.26)can now be used to plot the phases of Ystab and Y0 as functions of thenormalized frequency Tdaω, see Figure 9-12. As the filter B has beenneglected, the phase of Y0Ystab for low frequencies does not approach 90°, asin Figure 9-10.b. In Figure 9-12 we see that the phase of Ystab is negative forlow frequencies in order to compensate the large phase shift of Y0. As thephase of Y0 decreases, the phase of Ystab approaches zero. It however stays a


little negative, which means that the phase of the modified admittance will beslightly less than the phase of the original admittance for higher frequencies.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-90

0

90

arg(Y0), arg(Ystab) and arg(Y0Ystab)

Tdaω

Pha

se [

deg]

arg(Y0)

arg(Ystab)

arg(Y0Ystab)

Figure 9-12: Phase of the modified inverter input admittance.

To improve behavior at high frequencies, we hence have either to increase thephase of Ystab or decrease the magnitude of Ystab around the frequency wherethe phase of Y0 crosses -90°. One way of doing this is to add a phase shift tothe gain Kω, as shown in Figure 9-13.

Note that the phase shift of half a pulse period could be implemented byforming the average of the DC-link voltage over the pulse period, i.e.changing the band-pass filter (9.1) to

(9.27)

Note also that the derivation of the stabilization gain for low frequencies is notaffected by the extra time delay.

The resulting loop gains, when introducing the additional time delays, areshown in Figure 9-14. By comparing with Figure 9-11 we see an improvementof stability margins at high frequencies.


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0.8

1

1.2

1.4

Gain and phase of Ystab

Gai

n

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-40

-30

-20

-10

0

10

Pha

se [

deg]

Tdaω

Figure 9-13: Gain and phase of Ystab (solid), Ystab with 0.5Tp additional timedelay (dashed) and Ystab with 1.0Tp additional time delay (dotted).

-5 -4 -3 -2 -1 0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

2

3

4

5Nyquist plot of modified loop gain

Re

Im

-5 -4 -3 -2 -1 0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

2

3

4


Re

Im

Figure 9-14: Nyquist plots of modified loop gain with additional time delaysof (a) 0.5Tp and (b) Tp.

The benefit of an extra phase shift may also be understood in the followingmore intuitive way. Consider the expression for the modified admittance (9.2).With zero torque this equation becomes


DM S4

C

D

(9.28)

The Nyquist and Bode plots of Y0 are shown in Figure 9-1 and thecorresponding plots for the second term in (9.28) are shown in Figure 9-15.

-1 0 1 2 3 4 5-3

-2

-1

0

1

2

3Nyquist plot of stabilization contribution

Re

Im

100

101

102

10-5

100

Frequency [Hz]

Gai

n

Nyquist plot of stabilization contribution

100

101

102

-270

-180

-90

0

90

Frequency [Hz]

Pha

se [d

eg]

Figure 9-15: Nyquist (a) and Bode (b) plots of DC-link stabilizationcontribution.

The ideal stabilization would add positive real part to the original admittancefor all frequencies. Due to the phase shift of the torque control this is howevernot possible and one would intuitively like to “shut off” the stabilization forfrequencies where the stabilization adds negative real part to the admittance.The functionality for doing this is the upper cut-off frequency of the band-pass filter, which is examined in Subsection 9.4.3. On the other hand, fromFigure 9-1 it is clear that we actually only need to add positive real part to theinput admittance for low and high frequencies. For middle frequencies theadmittance stays in the right half plane and we could even add a smallnegative real part without causing a non-passive admittance. This is the keyobservation to understanding the positive effect with the extra time delay. Theinevitable negative real part of the contribution from the stabilization shouldbe generated for frequencies where the original admittance is passive itself.For higher frequencies where the stabilization is needed, i.e. where the phaseof the original admittance approaches -90°, we need the phase shift of thecontribution to be -270° or less. This way the contribution again adds positivereal part to the admittance. In order for the stabilization to have effect here,the gain of the contribution must not be too small, which might be amotivation for setting the upper cut-off frequency of the band-pass filter high.


To further illustrate the effect of the time delay we note that the phase of thecontribution from the stabilization in (9.28) (without additional phase shift)may be written

DAM

C DA

DDA

X

(9.29)

The phase of (9.29) is plotted in Figure 9-16 as the upper solid line where thedotted line is the phase of Y0. As the phase of Y0 reaches -90°, the phase of thecontribution is more or less -180°. Here the contribution adds pure negativereal part to the original admittance, which is bad from the stability point ofview. On the other hand, by adding extra time delay the phase of thecontribution is decreased below -270° for the same frequency. This is shownby the other two solid lines being the phases of the contribution withadditional time delay of half a pulse period and an entire pulse period,respectively. A phase between –450° and -270° makes the real part of thecontribution positive.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-270

-180

-90

0

90

arg(Y0) and the phase of the stabilization added to Y0

Normalized frequency Tdaω

Pha

se [

deg]

Figure 9-16: The solid lines show the phase of DC-link voltage contributionwith (1) no additional time delays (2) an additional time delay of half a pulseperiod and (3) an additional time delay of an entire pulse period. The dashedcurve is the phase of the original inverter input admittance Y0.


9.4.3 Choice of Band-Pass FilterWhen motivating the existence of the band-pass filter in the equation for thestabilization (2.16), it was said that there was no reason in trying to damp veryhigh frequent oscillations on the DC-link due to limited torque controllerbandwidth. As a matter of fact, the limited controller bandwidth also causesphase shift. A phase shift between -270° and -90° actually makes thestabilization amplify oscillations. This was also noted in Subsection 9.4.2where the contribution from the stabilization added negative real part to theadmittance. To avoid this destabilization, the idea is to turn the stabilizationoff at high frequencies and for that purpose there is an upper cut-off frequencyof the filter B.

The effect of modifying the upper cut-off frequency is seen in Figure 9-22showing the resulting Nyquist trajectories of the loop gain with k1 = 5000 andk1 = 500 respectively.

-5 -4 -3 -2 -1 0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

2

3

4


Re

Im

-5 -4 -3 -2 -1 0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

2

3

4


Re

Im

Figure 9-17: Nyquist plots of loop gain with (a) k1 = 5000 and (b) k1 = 500.

Reducing the upper cut-off frequency shows a very small improvement.Clearly the effect with the additional time delay shown in Figure 9-14 islarger.

9.4.4 Torque Controller Gain SchedulingIn Subsection 8.1.1 the question of torque controller gain scheduling wastouched upon. Below two examples of different types of gain scheduling areshown.


-5 -4 -3 -2 -1 0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

2

3

4


Im

Re-5 -4 -3 -2 -1 0 1 2 3 4 5

-5

-4

-3

-2

-1

0

1

2

3

4


Re

Im

Figure 9-18: Nyquist plots of loop gains with (a) constant phase margin of60° and with (b) a constant bandwidth with Ttorque = 5ms.

From the plots we see that the torque controller tuning is not critical for thelow frequency behavior of the loop gains (remember that the expression forthe stabilization gain Kω was independent of Ttorque). The high frequencyproperties however depend on the value of Ttorque. We see that a smallerbandwidth gives a better stability margin at high frequencies.

9.5 Stabilization at Non-Zero TorqueIn the previous section an expression for Kω was derived making the inverterinput admittance passive for zero torque. With a non-zero torque a secondterm appears in the original input admittance, which we compensate for withthe gain KT, see (9.4). In Subsection 9.5.1 it is shown how KT could be chosento maintain the passivity of the input admittance with a non-zero torque. InSubsection 9.5.2 the effects of the band-pass filter is investigated and theinfluence of the torque controller tuning on stability is examined in Subsection9.5.3. Finally Subsection 9.5.4 specifically treats stability in braking.

9.5.1 Choice of Stabilization GainIn order not to destroy the passive phase of the input admittance with a non-zero torque, we try to avoid adding negative real part to the input admittance,i.e. we aim to satisfy (see (9.4))


D

M MJ 4

C 4

D D

X

(9.30)

For low frequencies this is apparently not possible due to the filter B. As ZDC

normally is small for low frequencies, this is not a practical problem and wecan limit the scope to the pass band of B, where the expression above isapproximately equivalent to

D

M D 4 J 4

C

D

X

(9.31)

From this expression we conclude that the additional stabilization is actuallyonly needed in case of positive power flow and it is sufficient to set KT

according to

STAB

M

D4

M

(9.32)

where the parameter Nstab should be chosen as

D

STAB J 4

C

X

(9.33)

Within the torque controller bandwidth, this means that Nstab should be greateror equal to 1, as we could use the phase of the band-pass filter to somewhatcompensate for the phase lag of the time delay.

Remark: In the remainder we will only use Nstab = 1.

The input admittance achieved with stabilization according to (9.23) and(9.32) is plotted in Figure 9-19. Due to the band-pass filter it is not possible tomake the admittance passive for low frequencies but as the gain of the inputfilter is small at those frequencies the Nyquist curves of the loop gain do notencircle the point –1, see Figure 9-20.


-1 0 1 2 3 4 5-3

-2

-1

0

1

2


Re

Im

100

101

102

10-5

100

Frequency [Hz]

Gai

n


100

101

102

-90

0

90

180

Frequency [Hz]

Pha

se [d

eg]

Figure 9-19: Nyquist (a) and Bode (b) plots of modified input admittance withpositive torque.

-5 -4 -3 -2 -1 0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

2

3

4


Re

Im

Figure 9-20: Nyquist plots of modified loop gain with positive torque.

With positive torque we see the negative effects of the stabilization at highfrequencies more clearly compared to zero torque. The Nyquist curves passvery close to the point –1 and the stability margins are hence small. By addinga time delay to the band-pass filter, the high frequency properties may again


be improved, which is seen in Figure 9-21 where the extra time delay is set to0.5Tp and Tp respectively.

-5 -4 -3 -2 -1 0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

2

3

4


Re

Im

-5 -4 -3 -2 -1 0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

2

3

4


Re

Im

Figure 9-21: Nyquist plots of modified loop gain with additional time delay of(a) 0.5Tp and (b) Tp.

9.5.2 Choice of Band-Pass FilterAn alternative way to improve the high frequency properties, compared to thetime delays introduced in Subsection 9.4.2, is to use the upper cut-offfrequency of the band-pass filter. In Figure 9-17 the resulting Nyquisttrajectories of the loop gain are shown when using k1 = 5000 and k1 = 500,respectively.

-5 -4 -3 -2 -1 0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

2

3

4


Re

Im

-5 -4 -3 -2 -1 0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

2

3

4


Re

Im

Figure 9-22: Nyquist plots of loop gain with (a) k1 = 5000 and (b) k1 = 500.


Reducing the upper cut-off frequency shows a small improvement for someNyquist trajectories (higher motor speeds) but has bad influence for otheroperating points (lower motor speeds). Clearly the effect with the additionaltime delay shown in Figure 9-21 is larger.

9.5.3 Torque Controller Gain SchedulingGenerally the controller tuning affect the properties of the closed-loop system.This of course also holds for the input admittance but if we assume that theflux reference is constant, we only have to consider the torque controller, seeFigure 7-13. Some different possibilities to tune the torque controller werediscussed in Subsection 8.1.1 and in this subsection we investigate the effectsof some of these on the inverter input admittance. Figure 9-20 shows theadmittance with the torque controller tuned to give a constant phase margin of45°. The effects of increasing the phase margin, which also means decreasingthe bandwidth, are shown in Figure 9-23.a. For operating points with highmotor speeds (the trajectories closest to –1), the decrease of the bandwidth hasa positive effect on the stability margins at high frequencies. The trajectoriescorresponding to lower motor speeds however move closer to the point –1. InFigure 9-23.b the bandwidth is further reduced at operating points with lowstator frequencies by setting the bandwidth to 32 Hz for all operating points(Ttorque = 5ms). Now the effects are bad for all operating points and it isdemonstrated that high bandwidth is needed to stabilize the system.

-5 -4 -3 -2 -1 0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

2

3

4


Re

Im

-5 -4 -3 -2 -1 0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

2

3

4


Re

Im

Figure 9-23: Nyquist plots of loop gains with (a) constant phase margin of60° and (b) constant bandwidth with Ttorque = 5ms.


9.5.4 BrakingIn the beginning of this section we concluded that no additional stabilizationKT was needed in braking. The loop gain in braking without DC-linkstabilization in Figure 9-6 shows that the system is stable but the Nyquisttrajectories pass close to the point –1. The behavior is improved by addingstabilization with Kω. The Nyquist and Bode plots of the resulting admittanceare shown in Figure 9-24.

-1 0 1 2 3 4 5-3

-2

-1

0

1

2


Re

Im

100

101

102

10-5

100

Frequency [Hz]

Gai

n


100

101

102

-90

0

90

180

Frequency [Hz]P

hase

[deg

]

Figure 9-24: Nyquist (a) and Bode (b) plots of modified admittance inbraking.

From the figures we see that the phase of the admittance is improvedcompared to Figure 9-5. This also implies a better damped system whichfollows from the Nyquist plots of the loop gains in Figure 9-25.


-5 -4 -3 -2 -1 0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

2

3

4


Re

Im

Figure 9-25: Nyquist plots of modified loop gain in braking.

The effect of adding extra time delay is shown in Figure 9-26.

-5 -4 -3 -2 -1 0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

2

3

4


Re

Im

-5 -4 -3 -2 -1 0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

2

3

4


Re

Im

Figure 9-26: Nyquist plots of loop gain with additional time delay of (a) 0.5Tp

and (b) 1.0Tp.

By comparing Figure 9-26 with Figure 9-25 we see that extra time delaysslightly improve the high frequency properties but the stability marginswithout the additional time delays are probably good enough.


9.6 SummaryFrom investigating linearized models of the process, an expression for theadditive stabilization gain was derived in terms of motor data and conditionsof operation. The goal was to stabilize the system by making the inverter inputadmittance passive. The resulting gain is given by equations (9.3), (9.23) and(9.32), which are summarized by

MR

DAD

D

M

N

T

(9.34)

The high frequency performance may further be improved by adding a timedelay to the gain, for example by forming the average of the DC-link voltagebefore applying the band-pass filter in Figure 9-8.

Further, stabilization is most important for longer time delays, where theapproximation Tad ≈ Tp, given by (8.13) holds. Using this approximation,equation (9.34) may be written

MR

DP

D

M

N

T

(9.35)

For the example in Appendix C, the gain at zero torque as a function of statorfrequency is shown in Figure 9-27. As described in Appendix C, the switchingfrequency in the example is proportional to the stator frequency for all pulsepatterns but the first. This means that the pulse periods are inverselyproportional to the stator frequency and consequently the product of the pulseperiod and the stator frequency is constant. This is evident in Figure 9-27where the gain jumps between constant values. For low stator frequencies thepulse period is constant and hence the gain increases with the statorfrequency. The gain finally decreases for high stator frequencies due tofieldweakening.


0 10 20 30 40 50 60 70 80 900

0.5

1

1.5

2

2.5

3

Stator Frequency [Hz]

DC-link stabilization gain

Figure 9-27: DC-link stabilization gain Kω as a function of stator frequency.

An alternative stabilization gain was proposed by (2.15). Comparing thisexpression with (9.35) we see that the terms proportional to the torque areequal. This however does not hold for the other terms in the two expressions.By selecting Rd

inv to 1.45, the values of (2.15) and (9.35) match at thefrequency where fieldweakening is entered. By also using this value as theupper limit of (2.15), the two gains are plotted in Figure 9-28.a.

0 20 40 60 800

0.5

1

1.5

2

2.5

3

Stator frequency [Hz]


-5 -4 -3 -2 -1 0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

2

3

4


Re

Im

Figure 9-28: (a) Comparison between additive stabilization gains and (b)Nyquist curves of the loop gain with stabilization according to (2.15).


The resulting loop gain Nyquist plots with the stabilization gain (2.15) forzero torque are shown in Figure 9-28.b. The system has indeed been stabilizedbut with a much larger gain for low stator frequencies. Large gains causetorque jerks and the gain should therefore be kept as small as possible.Expression (9.35) gives the same stability margins with much less gain. Notealso that we actually used (9.35) to set the equivalent resistance and the upperbound of (2.15). Without having had this information some on-line tuningwould still be required.

Chapter 10

Linear Stability AnalysisThe purpose of this chapter is twofold. First we want to verify the linearmodel of the inverter input admittance shown in Figure 9-8. What is moreimportant however, is to evaluate the proposed choice of stabilization gain,given by (9.35). Even if the models used to derive the expression for thestabilization gain in Chapter 9 are not entirely correct, the gain may stillstabilize the drive.

The evaluation is done by comparing the linearized inverter input admittancewith optimal linear models obtained through identification frommeasurements. The measurements are generated by a Hardware-in-the-Loopsimulator, where the control is implemented and run on real hardware,whereas the process is simulated in real time. This is of course asimplification but experience has shown that the results from simulations wellmatch measurements from power labs. The simulated process is run on asampling time of 40µs, which should be fast enough to capture all relevantprocess dynamics from the stability point of view.

For coasting, driving and braking, linear models are identified for 100 motorspeeds each. The motor speeds are selected between 6 and 48 Hz. Allsimulations are done for the example defined in Appendix C.

Section 10.1 briefly discusses how to identify the inverter input admittancefrom measurements and details of the implementation of the real timesoftware are given in Section 10.2. The evaluation without and with DC-linkstabilization are presented in Section 10.3 and Section 10.4, respectively.

Remark: In the field-weakening region the stator voltage saturates, which hastwo effects on the control. First, the stator flux must be decreased as the motorspeed increases and second, the stator voltage magnitude saturates and maynot be modified by the control. This was discussed in Section 7.6. The linearmodels only consider fieldweakening by adjusting the operating point statorflux.

120 Linear Stability Analysis

10.1 Identification of Inverter Input AdmittanceThe theory and practice of system identification of linear time invariant (LTI)models are well developed and advanced methods and tools are available. Theproblem gets more difficult when considering non-linear systems. Whenapplying standard identification methods for estimating linear models to datafrom a nonlinear system, the best linear approximation in a certain sense isobtained. Consider a nonlinear dynamical system with input sequence u(tn)and corresponding output sequence y(tn), tn=nTs, n=1,2,…,N, where Ts is thesampling time. The best linear model of this input-output relation, in the sensethat it produces the same second order statistics, is given by the frequencyresponse

S

S

S

J 4YUJ 4

J 4UU CAUSAL

X

X

X

(10.1)

where subscript causal denotes the causal part, SJ 4

UU X is the power

spectral density of the input and SJ 4

YU X is the cross power spectral

density of the input and output signals. This linear approximation depends ofcourse on the specific input signal. This is closely related to linearizationusing Taylor expansion of the nonlinear system, which often is done around agiven operating point.

We will consider variations around a specific operating point u(tn) = u0 andy(tn) = y0. It is often easier to remove the DC-components before forming themodel, i.e. to work with the signals ∆u(tn)=u(tn)-u0 and ∆y(tn)=y(t)-y0. We willassume that the DC-components have been removed, but still use thenotations u(tn) and y(tn).

A common choice of input signal is u(tn) = C1 sin(ω1tn), i.e. a sinusoidalsignal with frequency ω1 and amplitude C1. Let y(tn) be the correspondingoutput. From observations y(tn),u(tn), the (finite) Fourier transforms of thesignals can be calculated as

S S

.

J 4 J 4 N. N

N

X X

(10.2)

S S

.

J 4 J 4 N. N

N

X X

(10.3)

Linear Stability Analysis 121

The Fourier transform of u(tn), is then approximately equal to NC12/4 for ω =

ω1 and almost zero otherwise. The empirical transfer function estimate equals

S

S

S

J 4.J 4

. J 4.

X

X

X (10.4)

It is possible to show [14], that SJ 4

. X converges to

S

J 4 X , under

certain assumptions on the system and the disturbances, as N tends to infinity.In this case

S

J 4 X is the best linear frequency response approximation of

the nonlinear system for a sinusoidal input signal with amplitude C1 andfrequency ω1 (around the operating point u0).

The continuous time Fourier transform can be approximated by

SJ 4

. .S

X

(10.5)

The linear model of the inverter input admittance was identified in thefrequency domain by measuring the gain and phase shift of the system forsingle sinusoids of varying frequencies. 100 frequencies between 8 and 208Hz were used.

10.2 Software ImplementationThis section describes some relevant details of the software implementation ofthe torque controller and the DC-link stabilization.

10.2.1 Time DelaysIn Section 5.3.2 the total time delay was estimated by expression (5.31). Thisexpression contains the sample time Ts, which is generated in the software bydividing the pulse periods into a number of equally long time intervals. Theseintervals are lower limited by 150µs. As the pulse periods vary with theoperating points, also Ts varies with the operating points. More precisely, Ts

takes values between 150µs and 300µs. This effect is however not consideredby the theoretical models, where Ts is set to the fixed value of 200µs.


10.2.2 Torque ControllerGain scheduling of the torque controller parameters in the softwareimplementation is done according to

P P

PI PI I I

P P

(10.6)

Here Kpi0 = 12 and Ki

0 = 3.3 are constant parameters and Tp1= 0.91ms is the

constant pulse period in pulse pattern number 1, see Appendix C. In this pulsepattern, the proportional gain gives Ttorque = 2.6ms according to expression(8.24).

10.2.3 DC-Link StabilizationIn the software implementation, the first term of (9.35) is approximated by thedashed line in Figure 10-1.

0 10 20 30 40 50 60 70 80 900

0.5

1

1.5

2

2.5

3



Figure 10-1: Approximation (dashed curve) of the first term of the DC-linkstabilization gain (9.35) (solid curve) as a function of stator frequency.

The two band-pass filter coefficients k1 and k2 are set to 2000 and 20,respectively. Hence, the pass-band of the filter is between 3 and 300 Hz.


Remark: The additional time delay discussed in Subsection 9.4.2 andSubsection 9.5.1, is not implemented.

10.3 Closed Loop without StabilizationThis section evaluates the model of the closed-loop inverter input admittancewithout DC-link stabilization for the three driving scenarios coasting, drivingand braking, respectively. Nyquist curves of the loop gain in Figure 3-2 arecompared and in Subsection 10.3.1 also Bode plots of the simulated andtheoretic models of the input admittance are shown.

Remark: Fieldweakening is modeled only by adjusting the operating pointflux. During the evaluation it turned out that the theoretical models slightlydeviated from the models obtained through identification in thefieldweakening region, denoted pulse pattern number 7 in Appendix C. Thiswas actually the case also for pulse pattern no 6, see Appendix C. Thetheoretical results showed a less damped system at large frequenciescompared to the simulation results. The shape of the simulated loop-gains infieldweakening indicated a lower bandwidth than what was modeled. Thetheoretical models were therefore adjusted ad hoc by multiplying the torquecontroller parameters by 0.4 in pulse pattern number 7 and by 0.5 in pulsepattern no 6. Results are shown both for the original model and the modelwith reduced controller gains, referred to as the modified theoretic model.

10.3.1 CoastingThe magnitude of the inverter input admittance achieved throughidentification is shown in Figure 10-2 for all 100 motor speeds. Thecorresponding plots with the theoretic linear models are given by Figure 10-3.The theoretic models only differ for higher motor speeds and the modifiedmodel shows a better agreement with the identified model.


Figure 10-2: Simulated magnitude response of the inverter input admittanceY.

Figure 10-3: Original (a) and modified (b) theoretic magnitude responses ofthe inverter input admittance Y.

The phase of the identified inverter input admittance and the phases of thetheoretic models are shown in Figure 10-4 and Figure 10-5, respectively.


Figure 10-4: Simulated phase response of the inverter input admittance Y.

Figure 10-5: Original (a) and modified (b) theoretic phase responses of theinverter input admittance Y.

Finally the loop gains YZDC are plotted in Figure 10-6 and Figure 10-7. Thetheoretical and simulated results match rather well. We also see that with thetuning used in the example, the system is stable in coasting also withoutstabilization. The stability margins are however small and result in a poorlydamped system.


-5 -4 -3 -2 -1 0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

2

3

4


Re

Im

Figure 10-6: Nyquist plots of the simulated loop gain YZDC.

-5 -4 -3 -2 -1 0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

2

3

4


Re

Im

-5 -4 -3 -2 -1 0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

2

3

4


Re

Im

Figure 10-7: Nyquist plots of the original (a) and modified (b) theoretic loopgains YZDC.

10.3.2 DrivingAs noted in Section 9.1, driving is the most critical case. We see from theNyquist plots of the loop gain in Figure 10-8 that the drive is not stable


without damping. We also see that the theoretical and simulated results matchfairly well.

-5 -4 -3 -2 -1 0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

2

3

4


Re

Im

-5 -4 -3 -2 -1 0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

2

3

4


Re

Im

Figure 10-8: Nyquist plots of the simulated (a) and modified theoretic (b)loop gains YZDC.

10.3.3 BrakingComparing with the Nyquist plots in coasting, we see that the stabilitymargins increase in braking, see Figure 10-9. We also see that the theoreticaland simulated results match well.

-5 -4 -3 -2 -1 0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

2

3

4


Re

Im

-5 -4 -3 -2 -1 0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

2

3

4


Re

Im



10.4 Closed Loop with StabilizationThis section evaluates the model of the closed-loop inverter input admittancewhen stabilization has been activated. Again the three driving scenarioscoasting, driving and braking are examined by showing simulated andtheoretic Nyquist curves of the loop gain in Figure 3-2.

Remark: Fieldweakening is not accurately modeled, see discussion in Section10.3. In order to make the plots fit better the controller gains were multipliedby a factor 0.4 in pulse pattern number 7 and by a factor of 0.5 in pulse patternnumber 6. Only results with the modified theoretical models are shown.

10.4.1 CoastingFrom Figure 10-10 we see that the stabilization improves damping of thedrive in coasting.

-5 -4 -3 -2 -1 0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

2

3

4


Re

Im

-5 -4 -3 -2 -1 0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

2

3

4


Re

Im


10.4.2 DrivingWithout stabilization the drive was unstable in driving, see Subsection 10.3.2.The derived stabilization however modifies the Nyquist curves of the loopgain such that the point –1 is not encircled, see Figure 10-11. Hence, the drivehas been stabilized.


-5 -4 -3 -2 -1 0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

2

3

4


Re

Im

-5 -4 -3 -2 -1 0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

2

3

4


Re

Im


Note that the properties at high frequencies look better for the loop gainsobtained from measurements compared to the theoretical models.

10.4.3 BrakingIn braking the drive was stable also without stabilization. From Figure 10-12we see that the stabilization turns the trajectories clock-wise. This waystability margins increase at low frequencies but at least for one operatingpoint the distance to the point –1 at high frequencies is decreased.

-5 -4 -3 -2 -1 0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

2

3

4


Re

Im

-5 -4 -3 -2 -1 0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

2

3

4


Re

Im


Chapter 11

Input-Output StabilityAnalysisIn Chapter 10 it was shown that the stabilization made the linear dynamics ofthe closed-loop drive stable for all modes of operation. However, as the realsystem in fact is non-linear, this may not be enough to guarantee input-outputstability. A small change of the input may still result in large changes of theoutput. In this chapter an attempt is made to estimate the errors made by thelinearization and the effect these errors have on stability.

In Section 11.1 it is shown how the errors introduced by the linearapproximations may be incorporated into the stability analysis and theexperimental results are presented in Section 11.2.

11.1 Robust Stability Analysis using MeasurementsIn Section 3.1 it was shown how the induction machine drive could berepresented as a non-linear feedback system, see Figure 3-1. Generally, input-output L2 stability of two interconnected systems can be studied using thesmall-gain theorem. Suppose both systems are L2 stable, that is

(11.1)

(11.2)

where

d

(11.3)

Then the feedback connection u1(t)=y2(t) and u2(t)=y1(t) is L2 stable if β1β2≤1,see [12]. Since no phase information is used, this test is quite conservative.However, by combining this result with stability tests for linear systems, less

132 Input-Output Stability Analysis

conservative conditions can be obtained. We will now show one example ofhow to do this for the system in Figure 3-1, inspired by the work ofSchoukens et al [22].

Let us represent the relation between DC-link voltage and DC-link current in(3.4) as

D D (11.4)

Here Y is a linear model and the additive error term v(t) accounts for modelerrors as well as noise. We will consider the following model error model

., (11.5)

< > ., D (11.6)

., V D V V V (11.7)

The error term has been decomposed into one non-linear input signalcontribution vNL(t) and additive noise e(t). The factor βv is the gain of the non-linear system and the constant αv can be used to model off-sets andexternal L2 signals, see [15]. The L2-gain of the nonlinear system isformally defined as

D

.,

5D

(11.8)

The problem is of course to find the input, which gives the maximal gain. Forlinear systems this corresponds to a sinusoidal input signal and the problemsimplifies to finding the correct frequency.

With the representation (11.4) and (11.5) it follows that Figure 3-1 isequivalent to Figure 11-1, if we neglect the additive noise.

+Y+

id

-

E

ZDC

UdZE

++ +

g(.)~

vNL

Figure 11-1: Equivalent representation of the induction machine drive.

Input-Output Stability Analysis 133

If the linear part of Figure 11-1 is stable, i.e. the system obtained with vNL(t) =0, we may close the linear feedback loop and stability of Figure 11-1 isequivalent to stability of the system in Figure 11-2.

g(.)~

-ZDC

1+YZDC

vNLUd

Figure 11-2: Non-linear feedback loop.

If we assume that

$#

:

$#

(11.9)

V (11.10)

the small gain criterion then implies that the feedback system is input-outputstable if βZβv≤1. Linear models of the inverter input admittance wereidentified in Chapter 10, which can be used to determine βZ in (11.9). It thenremains to determine βv appearing in (11.10). If we neglect the additive noise,the model error vNL(t) can be calculated as

., D D (11.11)

using measurements of Ud(t) and id(t). By forming (11.11) for a large numberof inputs Ud(t), an estimate of βv defined through (11.8) can be achieved. Aswe use a hardware-in-the-loop simulator to generate the measurements, theloop in Figure 11-1 can be broken and the model errors can be measured inopen-loop. This means that the loop in Figure 11-1 is opened such that theDC-link current does not affect the DC-link voltage through ZDC.

From Section 4.1 we know that due to the inverter, the stator voltages containhigh frequency components related to the switching frequency. This howeveralso holds for the DC-link current. Hence, feeding the inverter with smoothDC-link voltages still generates high frequency components in the DC-linkcurrent. These signals are not generated by the linear model and consequentlyappear in the model error vNL(t). This makes the gain βv large and it is notpossible to prove stability using βZβv<1. In the figures below an example of


the estimation error in open-loop is shown, where the input to the block Y inFigure 11-1 consists of 12 sinusoids.

0 0.01 0.02 0.03 0.04 0.05 0.06-400

-300

-200

-100

0

100

200

300

400Measured and estimated DC-link current

Time [s]

Cu

rren

t [A

]

0 0.01 0.02 0.03 0.04 0.05 0.06-400

-300

-200

-100

0

100

200

300

400DC-link current estimation error, v

NL(t)

Time [s]

Cu

rren

t [A

]

Figure 11-3: DC-link currents (a) and estimation error (b).

An alternative approach to prove input-output stability is to directly estimatethe total open loop gain in Figure 11-2

., D (11.12)

$#

.,

$#

(11.13)

i.e. the gain from the input Ud(t) directly to the signal w(t). If

W D W (11.14)

where 0 ≤ βw <1, we know from the small-gain theorem that the closed loopsystem is stable.

The signal w(t) is obtained by letting the error signal vNL(t) through the filterin (11.13), whose magnitude response is shown in Figure 11-4. For low statorfrequencies the magnitude of Y is small and ZDC /(1+YZDC) is approximatelyequal to ZDC. This explains the large resonance peak in Figure 11-4.


Figure 11-4: Magnitude of ZDC /(1+YZDC).

By applying the filter to the error current vNL(t) the high frequencycomponents dominating in Figure 11-3 are suppressed. The resulting filteredcurrents and the current errors are shown in Figure 11-5.

0 0.01 0.02 0.03 0.04 0.05 0.06-400

-300

-200

-100

0

100

200

300

400Measured and estimated filtered DC-link current

Time [s]

Cu

rren

t [A

]

0 0.01 0.02 0.03 0.04 0.05 0.06-400

-300

-200

-100

0

100

200

300

400Filtered DC-link current estimation error, w(t)

Time [s]

Cu

rren

t [A

]

Figure 11-5: Filtered DC-link currents (a) and filtered estimation error (b).

The calculation of vNL(t) involves the linear model of the inverter inputadmittance. As only a frequency domain model was identified in Chapter 10,


all calculations are done in the frequency domain. This can be done due toParsevals relation, which in discrete time says

< >

. .

J N .N .

N N

Q

(11.15)

where SJ 4

. X is the Fourier transform of w(t), as defined by (10.2).

11.2 Experimental ResultA most difficult problem is to find the input signal which gives the maximumgain. By studying the magnitude plot of ZDC /(1+YZDC) in Figure 11-4 we canhowever predict that the worst case input signal will have most of its energyaround the resonant frequency of this transfer function. The choice ofamplitude is important to obtain a good signal to noise ratio, but also to takethe non-linear effects into account. The gain from Ud(t) to w(t) has beencalculated for a large range of input signals. For example 100 test signals withrandom frequencies and phases of 12 sinusoids were used to excite thesystem. The conclusion is that a sinusoidal signal with frequency equal to theresonant frequency of ZDC /(1+YZDC) gives a reasonable estimate of themaximum gain of the non-linear system. This is in particular true for lowmotor speeds, while a more wide band signal gives slightly better results forhigher motor speeds.

Figure 11-6 shows the experimental results of the estimated stability gains forzero torque as a function of motor speed. This result shows that the gains arewell below one, and that we thus have quite good stability margin for mostmotor speeds. The main problem is for the case when the resonance frequencyof ZDC and the stator frequency are close. Here the signal to noise ratio is low,which implies larger estimation errors.

Figure 11-7 gives the experimental results for driving and braking, showinggains smaller than one for all operating points.


5 10 15 20 25 30 35 40 450

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Motor speed [Hz]

βw

Figure 11-6: Nonlinear gain β in coasting as a function of motor speed.

10 20 30 400

0.2

0.4

0.6

0.8

1βw

Motor speed [Hz]10 20 30 40

0

0.2

0.4

0.6

0.8

1βw

Motor speed [Hz]

Figure 11-7: Nonlinear gain β in (a) driving and (b) braking as functions ofmotor speed.

Chapter 12

Conclusions and FutureWork

12.1 ConclusionsThe intention with this thesis was to derive a structured way of designingadditive DC-link stabilization. For that purpose linear models of the closed-loop drive were obtained, including realistic descriptions of the control. It wasshown that stability problems arise as the inverter input admittance is non-passive for low frequencies. This also holds for operation with zero torque dueto imperfections in control. A critical parameter in this respect is the effectivetime delay in rejection of variations in the DC-link voltage.

For the example studied in this contribution, the non-passive inverter inputadmittance made the closed-loop drive unstable at zero torque. This meansthat so called multiplicative stabilization cannot stabilize the drive as itdoesn’t modify the properties of the inverter input admittance at zero torque.

To stabilize the system, an expression for the additive stabilization gain wasderived by appropriately shaping the linear model of the inverter inputadmittance. Important parameters to consider are the motor speed (statorfrequency), operating point torque and flux as well the actual DC-link voltageand time delays of the torque control. Also the leakage inductance of theinduction machines affect the stabilization. It was shown that with largepositive power, successful stabilization requires fast torque control. As controlbandwidth is limited by time delays in the control loops, small time delays arerequired.

For the given example it was shown how adding a time delay to thestabilization actually could increase the stability margins at high frequencies.This way the negative influence of large phase shifts of the torque control at

140 Conclusions and Future Work

high frequencies could be put at frequencies where the inverter inputadmittance is passive itself.

The achieved stability margins with the proposed method were validatedusing models obtained from frequency domain system identification togetherwith simple estimates of the gain of the nonlinear model error. The resultsshowed successful stabilization for a large number of motor speeds and torquelevels.

Expressions for the torque and flux controller parameters were also derived togive desired closed loop bandwidth and stability margins.

12.2 Suggestions for Future WorkSuccessful stabilization of the drive critically depends on the time delays inthe control loops, where large time delays make stabilization more difficult.When using synchronized pulse patterns the longest time delays appear infieldweakening and are inversely proportional to the stator frequency. Thelower the stator frequency where fieldweakening is entered, the longer thepulse periods get. As the fieldweakening region is determined by the DC-linkvoltage, it follows that operation with small DC-link voltages is especiallydifficult (assuming the power is not reduced). This thesis has only treated onenominal DC-link voltage level. The proposed stabilization should therefore beevaluated also at lower DC-link voltages.

With low DC-link voltages one may also have to further study the propertiesof the control in the fieldweakening region. It was noted that by neglectingsaturation of the stator voltage magnitude, the derived linear models showedslight deviations from the models obtained by identification infieldweakening.

The design of stabilization started with a given structure and the task was toproperly select the stabilization gain. What is really needed for stability of theclosed-loop drive is certain properties of the inverter input admittance.Feeding DC-link voltage oscillations to the torque reference is only one wayof influencing the inverter input admittance. As a matter of fact, due to thelimited bandwidth of the closed-loop torque control, better results may bereached for high frequency disturbances by for example tuning the rejection ofDC-link voltage disturbances. Note that the trend of reducing the capacitanceof the input filter to save space and weight increases the resonance frequencyand shifts stability problems to higher frequencies.

Conclusions and Future Work 141

All results were derived using the ISC for torque control. It would beinteresting to adapt the stabilization to also fit with other types of torquecontrollers.

Finally, the verification of input-output stability was used from measurementsof a partly simulated system. A rigorous treatment of the problem would usemeasurements from a power lab. Methods to verify stability of nonlinearsystems could also be further investigated.

Appendix A

Approximation ofAdmittance PhaseIn this appendix we show that

STAB

X

(A.1)

as long as

TORQUE

(A.2)

and

X X N X (A.3)

This inequality is used in Subsection 9.4.1 to derive an expression for Kωmaking the input admittance passive for low frequencies. The expressions forYstab and K’ω are given by (9.17) and (9.18), i.e.

TORQUE DSTAB

R AD TORQUE

T X

X X

N

(A.4)

First we note that (A.1) is equivalent to

STAB STAB

(A.5)

where Y’stab is given by

STAB

X (A.6)

144 Approximation of Admittance Phase

Together with (A.4), the assumptions on Kω and Pω in (A.3) imply that K’ω >0. From (A.6) it therefore follows that

STAB

TORQUE TORQUE

X

X

(A.7)

STAB

X

(A.8)

The real parts of the numerators in (A.7) and (A.8) are positive, andconsequently

STAB

(A.9)

and

STAB

(A.10)

where

TORQUE

TORQUE

X

X

X

(A.11)

We have thus shown that (A.5) is equivalent to

(A.12)

or

(A.13)

as arctan(.) is an increasing function of its argument. By inserting (A.11) into(A.13) we consequently have to show that

TORQUE TORQUE

X

(A.14)

Now, as K’ω is greater or equal to zero, the requirement on k2 in (A.2) issufficient for (A.14) to hold, which is shown in (A.15).

Approximation of Admittance Phase 145

TORQUE TORQUE

X

(A.15)

Appendix B

Linear Model of Field-OrientationIn Section 4.5 notation for the linearized induction machine was introducedthrough equation (4.37). The actual expressions for the transfer functions werehowever deliberately omitted as they get very complicated. As only thedynamics of the closed-loop system are considered here, transfer functionexpressions for the induction machine are not really needed. We may as wellstudy the closed-loop system directly. Equation (4.37) contains severaloutputs although only the first one, y(t), is to be controlled. The remainingoutputs are all needed for the non-linear feedback compensation, which iscalled field-orientation and is defined by equation (5.23). Hence, field-orientation is the transformation from the outputs of the torque and fluxcontrollers to the stator voltage reference, or with the terminology introducedin Subsection 5.2.2, the mapping from ucNL(t) to us(t). To keep the property ofthe field-orientation of being the mapping from the controller outputs to thestator voltage, the linear field-orientation was modified to also include thefeedback of the rotor flux magnitude mr(t), see (6.31), where the variable infield-coordinates is called uc(t). The dependency of the rotor flux magnitudeoriginates from the transformation of slip frequencies to torque, whichactually is done in stator flux coordinates. The linear field-orientation isvisualized in Figure 7-1, where uc(t) is the input and the output is the controlvariable y(t). The system is also affected by the measurable disturbances Ud(t)and ωm(t). In Section 7.1 the transfer functions from the input and thedisturbances to the output were derived and denoted P, Q and Gω respectively,see Figure 7-2 and expressions (7.7), (7.4) and (7.5).

This appendix evaluates explicit expressions for the transfer function matricesP, Q and Gω. Instead of directly using the expressions (7.7), (7.4) and (7.5),requiring a direct linearization of the induction machine equations,linearization is done first after applying parts of the field-orientation.

148 Linear Model of Field-Orientation

Alternative Expression for Field-OrientationThis section shows how the transfer function matrices P, Q and Gω in Figure7-2 can be calculated without explicitly having to work with G, the linearizedmodel of the induction machine. The idea is to rewrite the diagram in Figure7-1 as shown in Figure B-1.

TFO

ucNL +us

Ψµ

+

Fmr

uc +

+G

ωm

ymr

Ψµ

mr

+is

Ks

+ +

Kv-1GUd

Ud

+

+Kv

vs

G’

vsref

Figure B-1: Linear model of field-orientation.

By completing the stator current feedback, the system shown in Figure B-1can be described in terms of the shaded system, which is denoted G’. Thebenefit of proceeding this way is that the system G’ is easier to work with thanthe original system G. In Figure B-1 also the notation vs(t) was introduced forthe fictitious input signal to G’.

From Figure B-1 it follows that the stator current feedback affects the statorvoltage as

S V S S S

V S S I S I M

X

(B.1)

which means that

S S I V S S I M

X (B.2)

Applying the input (B.2) means that the system G’ is described by

Linear Model of Field-Orientation 149

Y

R MR S I V S

Y Y

MR S I S I MR M

N Z

X

X X

Z XZ

(B.3)

We will introduce the following notation for the transfer functions in (B.3)

Y Y

R MR S MR M

X

X

N Z XZ

(B.4)

that is

X X S I V (B.5)

X X S I S I X X X X

(B.6)

where x = y, mr, and Ψ. The system in Figure B-1 can then be represented asshown in Figure B-2.

TFO

ucNL

Ψµ

+

Fmr

uc +

+

ωm

ymr

Ψµ

mr

++ +

Kv-1GUd

Ud

vsvsref G’

Figure B-2: Alternative model of field-orientation.


The transfer function matrices P, Q and Gω may now be derived in terms ofthe system G’ from Figure B-2. The resulting expressions become

Y C MR MR Z Z

(B.7)

C MR MR Y

X X Z XZ X (B.8)

and P = QTc. In order to explicitly calculate (B.7) and (B.8), we need thelinear model G’, which is derived in the following sections.

Non-Linear State Space RepresentationThe previous section showed that the matrices P, Q and Gω describing thefield-orientation in Figure 7-2 can be expressed in terms of the system G’ inFigure B-2. Here G’ represents the linear system with vs(t) as input and thesignals y(t), mr(t), and Ψ(t) as outputs. Instead of deriving the linear model G’directly from (B.5) and (B.6), we will first form the corresponding nonlinearequations, which then are linearized to reach G’. The benefit of doing this isthat we avoid deriving the transfer functions Kv and Ks and performing thematrix calculations in (B.5) and (B.6). So how do we obtain these non-linearequations? As the (linearized) effect of the DC-link voltage has been movedout of the linear system G’ in Figure B-1, we are only interested in the non-linear equations relating the space vector vs

s(t) with the outputs of theinduction machine. We may consequently neglect the effects of the DC-linkvoltage, which means that

S S S

S S S S (B.9)

The equations for the induction machine in terms of space vectors were givenby (4.11) and (4.12). By inserting equation (B.9), the sought equations for theinduction machine with vs

s(t) as input become

S S

S N (B.10)

R RS S S

R )- M R

N

T T

(B.11)

The system G’ may hence be achieved by linearizing equations (B.10) and(B.11). The linearization will however be done on an equivalentrepresentation of the equations in terms of magnitudes and angles using (4.14), (4.15) and


V

S J T

S V D (B.12)

With the polar notation, the derivative of the stator flux space vector can bewritten as

J T J TS

N ND D

N N N N N (B.13)

With (B.13) it follows that equation (B.10) can be written

VJ T

V NE

N N N (B.14)

where

V V N N (B.15)

Similarly, by using the load angle (4.18), equation (B.11) may be written

R R R

R RJ T

)- M R

E

NT T

(B.16)

Separating real and imaginary parts of (B.14) and (B.16), assuming the fluxesare non-zero, leads to

V V

N N (B.17)

R R

R R

N

T T

(B.18)

V

V

N N

N

(B.19)

R

R )- M

R

N

T

(B.20)

By forming the derivatives of δvµ(t) and δ(t), equations (B.17)-(B.20) can alsobe written

V V

N N (B.21)

R R

N

T T

(B.22)


V

V V V

N N

N

(B.23)

V

V )- M

R

N

N

N T

(B.24)

where Tσ is given by (4.19) and ωv(t) in (B.23) is the derivative of χv(t) in(B.12), i.e.

V V

(B.25)

Note that with the input vector uIM (t) and the state vector xIM(t) given by

V

R

)- V )-

V

M

N

N

(B.26)

the equations (B.21)-(B.24) form a non-linear state space description, i.e.

)- )- )- )- (B.27)

Outputs of the system G’ in Figure B-2 are torque, stator flux magnitude,stator flux frequency (stator frequency) and rotor flux magnitude. Theequation for the torque is given by (4.17) and the equation for the statorfrequency is given by (B.19). The magnitudes of the stator and rotor fluxes aresimply states in the state space description (B.21)-(B.24). The outputs cantherefore be written in terms of the states xIM as

)-

R

N

T

(B.28)

N N (B.29)

V

V

N N

N

(B.30)

R R

(B.31)

By defining the output vector yIM(t) as


)-

R

N

N

(B.32)

it follows that equations (B.28)-(B.31) are non-linear functions of the statesxIM(t), i.e.

)- )- )- )- (B.33)

Stationary Operating PointsIn the previous section the induction machine was put on a non-linear statespace form after the feedback of the stator current was applied. The idea is tolinearize these equations around stationary operating points to obtain thesystem G’ in Figure B-2. Stationary operating points are characterized byuIM(t) = uIM

*, xIM (t) = xIM*, satisfying

)- )- (B.34)

The operating points (uIM*, xIM

*) will be specified through constant values oftorque, stator flux magnitude and motor speed. These operating point valuesare denoted T*, mµ

* and ωm* respectively.

The steady state relation between the stator and rotor flux amplitudes isachieved by setting the right hand side of equation (B.22) to zero. This implies

R

N (B.35)

To calculate the stationary load angle δ*, we use the torque equation (B.28) insteady state together with (B.35), which gives

)- )-

R

N N

T T

(B.36)

The load angle can now be solved for as

)-

T

N

(B.37)

where it was assumed that –π/2 ≤ δ *≤ π /2. It is shown later that this must bethe case.


As the angles δ* and δ*vµ are constant at steady state, the different electrical

quantities must rotate with the same frequency, which we may denote ωµ∗ .

Equation (B.20) together with (B.35), and the definition of Tσ in (4.19) thengive

R )- M

N

T

(B.38)

Equation (B.38) also means that the steady state slip frequency (see (4.23))can be written as

SLIP

T

(B.39)

For the steady state slip frequency to be finite, it hence follows that –π/2 ≤ δ*≤π /2. As a matter of fact, if we assume the steady state slip frequency to beless than the limit (4.28), the limits on δ* are tightened to –π/4 ≤ δ*≤ π /4.

Further, in steady state equation (B.21) equals

V V

N (B.40)

As the magnitude mv* is strictly positive, the angle δ*

vµ must be either ±π/2.Which sign to be used follows from equation (B.23), which implies that

V

V

N N

N

(B.41)

Again, the magnitudes are positive so the sign of the angle δ*vµ is determined

by the sign of the stator frequency, i.e.

V N N

(B.42)

(This expression could also be directly determined from equation (B.10)).Combining (B.41) and (B.42) also gives that

V N N (B.43)

Now all steady state inputs and states have been expressed in terms of thetorque, flux and mechanical speed at the operating point. The expressions forthe inputs and states are summarized below


VV

)- V )- M

MM

N

T

(B.44)

R

V

V

REF

)-

N

N

N

N

T

N

(B.45)

LinearizationThrough equations (B.21) - (B.24) and (B.28) - (B.31) the induction machinewas represented as a non-linear state space model of the form

)- )- )- )- (B.46)

)- )- )- )- (B.47)

The input vector uIM(t) and the state vector xIM(t) are given by (B.26) and theoutput vector yIM(t) by (B.32). In the previous section, stationary operatingpoints uIM

* and xIM* were defined by (B.44) and (B.45) such that

)- )- (B.48)

Equations (B.46) and (B.47) may now be linearized around the stationaryoperating points to give the following linear state space representation

)- )- )- )- )- (B.49)

)- )- )- )- )- (B.50)

The deltas denote deviations from the operating points and the state spacematrices are given by

)- )-

)- )- )- )-

)- X U

(B.51)


)- )-

)- )- )- )-

)- X U

(B.52)

)- )-

)- )- )- )-

)- X U

(B.53)

)- )-

)- )- )- )-

)- X U

(B.54)

The state space matrices (B.51)-(B.54) can be evaluated to

)-

N N

N

T T T

N

N

N

T TN T N

(B.55)

)-

)-

N

N

N

N

(B.56)

)-

)- )- )-

U U

U U

N N N

T T T

N N

N N

(B.57)


)-

U

N

N

(B.58)

From the state space representation (B.49) and (B.50) we may also form thetransfer function

)- )- )- (B.59)

where

)- )- )- )- )- (B.60)

From the transfer function GIM(s), the transfer functions (B.5) and (B.6) canbe extracted. These expressions are shown below for zero operating pointtorque, where they become particularly simple.

)- )-

R RY

N N N N

T T

N

N N N

(B.61)

N N N

Z N

N N

N

(B.62)

MR

N N N

T TN

(B.63)

)-

RY

N

TX

(B.64)

XZ

(B.65)


MR

X (B.66)

The expressions for the transfer functions (B.61) - (B.66) may also beevaluated for non-zero torque but they become a little more complicated andare therefore omitted here.

Field-OrientationIn the previous section a linear state space description was derived for theinduction machine equations after the voltage drop across the stator resistancewas cancelled. Also transfer function expressions were shown for the specialcase of zero torque. Allowing torque makes the expressions a little morecomplicated but it turns out that the transfer functions we really need, i.e. P, Qand Gω become tractable. The expression for Q is given by (B.7) and isrepeated below

Y C MR MR Z Z

(B.67)

The transfer functions G’y, G’mr and G’y can be derived from (B.60) and Tc, TΨand Fmr are given by (6.10), (6.11) and (6.18), respectively. Using theseexpressions, Q can be evaluated to

)-

R

T

N N N T T

T T

N N

(B.68)

where

T

T

(B.69)

From (7.7) it then follows that


)- R

R

T

T

T

T T

N

(B.70)

Finally, it remains to determine Gω. By evaluating (B.8) it follows that

)- R

R

TX

(B.71)

DC-Link Voltage DisturbanceThe previous sections showed that the system in Figure B-1 could be writtenas

C V 5D D M

X (B.72)

where explicit expressions have been given for all the transfer functionsexcept Kv. In this section it will however be shown that under certainconditions

V 5D 5D (B.73)

which means that (B.72) may be approximated by

C 5D D M

X (B.74)

The approximation is valid if (c.f. (6.2))

S S

VREF SREF S S S I (B.75)

where the following polar notation for the stator current was used

IS J T

S I D (B.76)

Condition (B.75) may also be expressed as

S I

VREF

(B.77)


and this quotient is plotted in Figure B-3 for the three driving scenarios,coasting, driving and breaking. As seen in the figure, condition (B.75) is validfor all but very low motor speeds.

0 5 10 15 20 25 30 35 40 450

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Rs*mi/mref

Motor speed [Hz]

CoastingDriving Braking

Figure B-3: Justification of relation (B.75).

To prove the validity of the approximation, we need an expression for Kv.That is, the linear relation between vsref(t) and usref(t), see (6.3). We thereforestart from space vector equation (6.2)

VREF I

I VREF VREF

VREF

S S S

SREF SREF S S

J T J T

VREF S I

J T T J T

VREF S I

VREF S I I VREFJ T

S I I VREF

D D

D D D

D

(B.78)

The real-valued vector representation introduced in this contribution relatesmagnitudes and frequencies of the space vectors. Hence, we need to rewritethe space vector equation in terms of magnitudes and angles. From theequation above it follows that


S

SREF

VREF VREF S I I VREF S I

(B.79)

and

S

SREF

S I I VREF

VREF

VREF S I I VREF

(B.80)

It is now straightforward to differentiate the magnitude and angle of ussref(t)

with respect to the magnitude and angle of vssref, which gives

S I

I VREF

VREFS

SREF

VREF

S I S I

I VREF

VREF VREF

(B.81)

S I I VREFS

SREF

VREF

S I S I

I VREF

VREF VREF

(B.82)

S I

I VREF

VREFS

SREF

VREFS I S I

I VREF

VREF VREF

(B.83)

S I S I

I VREF

VREF VREFS

SREF

VREFS I S I

I VREF

VREF VREF

(B.84)

Hence, if we assume that (B.77) holds then

SREF VREFS I I VREF

SREF VREF

(B.85)


which can be rewritten in terms of angular frequencies instead of angles

SREF VREFS I I VREF

SREF VREF

(B.86)

This means that

S I I VREF

V

(B.87)

and consequently

S I I VREF

V

(B.88)

This also gives that

V 5D 5D (B.89)

as the second element of GUd is zero, see (4.39).

Appendix C

Drive DataThis appendix gives all relevant information about the example needed for theillustrations and numerical evaluation of the obtained results. The data aretaken from the Guangzhou metro in China.

Drive ParametersThis section states the parameters of the induction machine and the input filterof the drive in Figure 1-2.

Motor data (Γ-ECD):

Stator resistance: Rs = 89.4 mΩRotor resistance: Rr = 65.8 mΩStator inductance: Lµ = 43.8 mHLeakage inductance: Lσ = 2.2 mHNumber of pole pairs: pIM = 2Number of motors in parallel: n = 4

Note that the resistances are given for 20° but all evaluation is done for 100°.The temperature coefficient is 0.0039.

Nominal values:

Nominal stator flux: mµ∗ = 2.59 Vs

Maximum torque per motor: Tmax = 1780 NmNominal DC-link voltage: Ud

* = 1700 V

Input filter data:

Filter inductance: L = 5 mHFilter capacitance: C = 4 mFFilter resistance: R = 40 mΩ

164 Drive Data

The resonance frequency of the input filter is 35.6 Hz and the damping factoris 0.02, see (1.1).

Tractive EffortThree different driving scenarios are considered in this contribution:

• Coasting (zero torque)

• Driving (maximum torque)

• Braking (minimum torque)

During driving and braking, the torque reference and the power vary withspeed as shown in Figure C-1.

0 10 20 30 40-2000

-1500

-1000

-500

0

500

1000

1500

2000Driving and braking torque as functions of speed

Motor speed [Hz]

Tor

que

[Nm

]

0 10 20 30 40-500

-400

-300

-200

-100

0

100

200

300

400

500Driving and braking power as functions of speed

Motor speed [Hz]

Pow

er [

kW]

Figure C-1: Torque (a) and power (b) as functions of speed in driving andbraking.

Switching Frequencies and Pulse PeriodsAs described in Section 4.1, the motor voltages are generated by switching thephases of the voltage source inverter. An important parameter when it comesto inverters is the switching frequency, see Section 4.1. The switchingfrequency will be denoted fsw and influences the generated harmonics, whichis clear from Figure 4-4 and Figure 4-14. Generally, high switchingfrequencies generate high harmonics that have little influence on theelectromechanical system. From this point of view, it is advantageous to use

Drive Data 165

as high switching frequency as possible. On the other hand, to generate largefundamental stator voltages, the switching frequency must be reduced. Thisalso follows by comparing Figure 4-4 and Figure 4-14. From equation (4.32)it further follows that large stator voltage are needed at high statorfrequencies. Hence, to efficiently use the inverter, the switching frequency isvaried with the stator frequency. At low stator frequencies a large switchingfrequency is used and as the stator frequency increases, the switchingfrequency has to be decreased. In our example, 7 different pulse patterns areapplied, with switching frequencies according to Table C-1. Pulse patternnumber 7 is so-called block mode, where maximum voltage is used, seeSection 4.3.

Pulse Pattern Switching frequency

No 1 fsw = 550 Hz

No 2 fsw = 21*ωµ/2/π





No 7 fsw = ωµ/2/π

Table C-1: Switching frequencies.

The reason for synchronizing the switching frequency to the stator frequency,is to avoid low frequency oscillations of the torque. Synchronized pulsepatterns are recommended when the quotient between switching frequencyand stator frequency is lower than 10 [24] (fsw/ωµ*2π<10). The switchingfrequency is shown as a function of stator frequency in Figure C-2.

166 Drive Data

0 10 20 30 40 50 60 70 80 900

100

200

300

400

500

600

700

800

900

1000

1100


Sw

itchi

ng f

requ

ency

[H

z]

Switching frequency as a function of stator frequency

21*fs →

15*fs →

7*fs → 5*fs → 3*fs →

fs →

Figure C-2: Switching frequency as a function of stator frequency.

As only the low frequency components of the stator voltages were modeled inSection 4.1, the only property of the actual pulse pattern that is needed is thedelay due to the pulse period. For the used pulse patterns, the pulse periodsare given by Table C-2.

Pulse Pattern Pulse Period

No 1 Tp = 1/(2*fsw)

No 2 Tp = 2π/(42*ωµ)

No 3 Tp = 1/(30*ωµ)

No 4 & No 5 Tp = 1/(12*ωµ)

No 6 & No 7 Tp = 1/(6*ωµ)

Table C-2: Pulse periods.

The pulse period as a function of stator frequency is shown in Figure C-3.

Drive Data 167

0 10 20 30 40 50 60 70 80 900

0.5

1

1.5

2

2.5

3


Pul

se p

erio

d [m

s]

Pulse period as a function of stator frequency

Figure C-3: Pulse period as a function of stator frequency.

Operating Points used for EvaluationEvaluation of results in all chapters except Chapter 10 and Chapter 11 aredone for 102 motor speeds between 0.16 and 48.4 Hz, with increment 0.48 Hz.

Appendix D

Space VectorsFor electrical three-phase quantities Q(t)=[QA(t) QB(t) QC(t)]T, the complex-valued space vector is defined as [11]

J JS

Q ! " # Q Q

(D.1)

where Kq is a scaling factor. In this contribution we will use Kq = 1, whichmeans that the magnitude of the space vector equal the amplitude of the threephase signals.

Three-phase quantities where the components add to zero, i.e.

! " # (D.2)

are said to have no zero sequence. For such signals the space vector operationis invertible, i.e. the space vector contains the same information as the originalthree-phase quantity. For general three-phase quantities the space vectorrepresentation however looses the information about the zero sequence.

One may interpret the space vector transformation (D.1) as a projection of athree-phase quantity on the plane defined by (D.2), followed by a scaling. Tosee this, we pick the following two orthonormal (orthogonal and normalizedto length 1) vectors v1 and v2 in the plane, i.e. two vectors satisfying (D.2), as

(D.3)

By adding the normal to the plane

170 Space Vectors

(D.4)

the transformation between the original coordinate system, with coordinatesdenoted QA(t), QB(t) and QC(t), to the coordinate system with the basis vectorsv1, v2, v3 and coordinates Q’A(t), Q’B(t) and Q’C(t) becomes

! !

" Q "

##

(D.5)

where

Q

(D.6)

By separating the real and imaginary parts, the space vector transformation(D.1) can be written on vector form as

!S

Q "S

#

B

C

(D.7)

Comparing (D.5) and (D.6) with (D.7), it follows that the space vectortransformation may be interpreted as a change of coordinates followed by aprojection onto the plane and a scaling.

Consider the following space vector with magnitude m(t) and angle χ(t)

S J T D (D.8)

With Kq = 1, this space vector may be transformed to a three-phase quantitywithout zero-sequence through the inverse transformation of (D.7), whichgives

Space Vectors 171

! S

" S

#

B

C

(D.9)

From (D.9) it follows that the magnitudes of the space vectors and the three-phase signals are equal with Kq = 1.

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