Electrical Actuators E-Book

Bernard de Fornel

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 553

Veröffentlichungsjahr: 2013

Beschreibung

The different chapters of this book cover a large range of information regarding electrical actuators, including: synchronous and asynchronous machine modeling in order to measure and identify offline and online parameters using modern optimization methods; identification in real time of parameters with Luenberger filter and the extended Kalman filter; estimation of non-measurable variables, first by linear estimates and observers, then by lower observers. Robustness is a very problematic issue, as well, which is fully explored in a chapter dedicated to the subject. Finally, the estimate of non-measurable mechanical variables is particularly dealt with: estimate of load moment, then observation of the positioning of a command without mechanical sensor. The conditions to measure variables and real implementation of numerical algorithms are also examined with particular attention.

Leseprobe

Table of Contents

Introduction

Part I. Measures and Identifications

Chapter 1. Identification of Induction Motor in Sinusoidal Mode

1.1. Introduction

1.2. The models

1.3. Traditional methods from a limited number of measurements

1.4. Estimation by minimization of a criteria based on admittance

1.5. Linear estimation

1.6. Conclusion

1.7. Appendix

1.8. Bibliography

Chapter 2. Modeling and Parameter Determination of the Saturated Synchronous Machine

2.1. Modeling of the synchronous machine: general theory

2.2. Classical models and tests

2.3. Advanced models: the synchronous machine in saturated mode

2.4. Bibliography

Chapter 3. Real-Time Estimation of the Induction Machine Parameters

3.1. Introduction

3.2. Objectives of parameter estimation

3.3. Fundamental problems

3.4. Least square methods

3.5. Extended Kalman filter

3.6. Extended Luenberger observer

3.7. Conclusion

3.8. Appendix: machine characteristics

3.9. Bibliography

Part II. Observer Examples

Chapter 4. Linear Estimators and Observers for the Induction Machine (IM)

4.1. Introduction

4.2. Estimation models for the induction machine

4.3. Flux estimation

4.4. Flux observation

4.5. Linear stochastic observers—Kalman-Bucy filters

4.6. Separate estimation and observation structures of the rotation speed

4.7. Adaptive observer

4.8. Variable structure mechanical observer (VSMO)

4.9. Conclusion

4.10. Bibliography

Chapter 5. Decomposition of a Determinist Flux Observer for the Induction Machine: Cartesian and Reduced Order Structures

5.1. Introduction

5.2. Estimation models for the induction machine

5.3. Cartesian observers

5.4. Reduced order observers

5.5. Conclusion on Cartesian and reduced order observers

5.6. Appendix: parameters of the study induction machine

5.7. Bibliography

Chapter 6. Observer Gain Determination Based on Parameter Sensitivity Analysis

6.1. Introduction

6.2. Flux observers

6.3. Analysis method of the parametric sensitivity

6.4. Choice of observer gains

6.5. Reduced order flux observer

6.6. Full order flux observer

6.7. Conclusion

6.8. Appendix: parameters of the squirrel-cage induction machine

6.9. Bibliography

Chapter 7. Observation of the Load Torque of an Electrical Machine

7.1. Introduction

7.2. Characterization of a load torque relative to an axis of rotation

7.3. Modal control of the actuator with load torque observation

7.4. Observation of load torque

7.5. Robustness of control law by state feedback with observation of the resistant torque

7.6. Experimental results

7.7. Conclusion

7.8. Bibliography

Chapter 8. Observation of the Rotor Position to Control the Synchronous Machine without Mechanical Sensor

8.1. State of the art

8.2. Reconstruction of the low-resolution position

8.3. Exact reconstruction by redundant observer

8.4. Exact reconstruction by Kalman filter

8.5. Comparison of reconstructions by Kalman filter or analytical redundancy observer

8.6. Bibliography

List of Authors

Index

First published 2010 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc. Adapted and updated from two volumes Identification et observation des actionneurs électriques published 2007 in France by Hermes Science/Lavoisier © LAVOISIER 2007

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

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© ISTE Ltd 2010

The rights of Bernard de Fornel and Jean-Paul Louis to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Cataloging-in-Publication Data

Identification et observation des actionneurs électriques.

English Electrical actuators : identification and observation / edited by Bernard de Fornel, Jean-Paul Louis.

p. cm.

Includes bibliographical references and index.

ISBN 978-1-84821-096-7

1. Actuators. 2. Electromechanical devices. I. Fornel, Bernard de. II. Louis, Jean-Paul, 1945-TJ223.A25I3413 2010

621.3--dc22

2009046733

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

ISBN 978-1-84821-096-7

Introduction1

Electric actuators, at least the most traditional ones (direct-current machine and alternating-current machines working under Park’s assumptions), have been the subject of a very large number of scientific studies and industrial realizations, and we can consider that they are currently well understood. The control structures use the machine’s decoupling properties in both axes (direct axis for the flux and quadrature axis for the torque), and the performance and robustness of the regulators are well adapted to the system specifications.

The implementation of overlapped regulations makes it possible to control the dynamics of the main variables, magnetic flux and rotation speed (via the torque), and to create “active safety features” (instant limitations of power amplitudes for example). These controls are even more efficient as long as the designer has precise models with known parameters. In fact, controllers most often use the innermost properties of actuators. The “direct model” is derived from the physical equations of the machine. From this model a reverse model is then obtained enabling direct access to the control architecture and allowing the selection of the control algorithms, the regulators or the controller best adapted to the original specifications. Knowledge of the physical laws and parameter values is therefore a requirement.

In addition, these controls involve variables whose direct measurements cannot always be achieved such as the magnetic flux and the electromagnetic torque of the induction machine; even if flux and torque sensors exist, they are very expensive and not often used. That is also the case with the rotation speed since controls without mechanical sensors are increasingly widespread. High performance controls require a very good knowledge of these variables.

This work is based on the expertise of the authors which have a threefold experience of research, teaching and industrial applications. This book is intended to provide the reader with a reference work on parameter identification, both “off-line” (in the background) and “on-line” (in real time, during the control operation) and of estimation or observation of the variables of alternating-current electric machines that cannot be directly measured.

The reader will observe that all chapters in this book devote an important part to modeling. From the one which identifies the speed of a machine without mechanical sensor to the one that enables the estimation of the parameters of a saturated induction machine, the variety of models is large. The goal of this book is to provide the user with the methods necessary to acquire the expertise that will enable him to choose the most appropriate model, and not necessarily the ideal model (since the perfect model that solves all problems does not exist).

In this book, many different approaches are explained in order to find the best compromise between two opposite constraints:

– the physical validity of models often quite complex to account for the large number of phenomena;

– the mathematical model that must be handled by real time computers and therefore must be simple enough. In fact, the calculation period is linked to the fastest time constants of the physical system. Some are very short and the electric machine control is very demanding in terms of execution time of the algorithms.

The measured variables, either online or offline, can also be submitted to a physical filtering or a numerical process treatment.

The chapters of the first part of the book (1, 2 and 3) are dedicated to measurement and parameter identification of the synchronous and induction machines. The authors have tried to give an overview of different aspects: steady state measurements of physical parameters, including some non-linearities (saturation, for example), and dynamic parameter estimation in order to gain a better understanding of the machine’s physics, as well as enabling the creation of the dynamic models necessary to develop the controllers. We had to find compromises between the “white box” and “black box” approaches. In order to do this, we had to use:

– off line measurement or identification of physical parameters required by the simulation models and necessary for controller implementation;

– the real-time identification of parameters for adaptive control that takes into account the parameter variations linked to conditions of operation, magnetic state, temperature, etc. This online identification uses filtering techniques, mainly the Kalman Bucy technique.

The chapter “Identification of Induction Motor in Sinusoidal Mode” by E. Laroche and J.-P. Louis is an extension of the classical methods for measuring induction machine parameters. These usually rely on an equivalent circuit where leakage fluxes are first divided between the rotor and the stator, then, for convenience, referred traditionally to the rotor. The steady-state model gives access to several parameters, which can be rightly used for transient state analysis or for control, such as for vector control. An equivalent circuit can be exact, but its parameters may not be measurable physically, or without unacceptable errors. Moreover, the well-known no-load and short circuit tests are not sufficient to obtain the required precision. In the end, the magnetic saturation must be taken into account by the high performance controllers, and introduced into the models. Optimized parameter identification methods are thus developed: which models should we use? Can we identify them? What measurements should we make? Modern methods go beyond the simple optimization method to estimate the best parameter values: there is a need to “optimize the optimization”.

In the chapter “Modeling and Parameters Determination of the Saturated Synchronous Machine” by E. Matagne and E. de Jaeger, the authors present a Park model but without considering the usual linearity hypothesis. This enables the authors to not only present the classical tests which make it possible to determine the numerical values of accessible parameters, but also to introduce the “cross saturation” phenomenon caused by the intrinsic non-linearity of magnetic materials. This is an appropriate model to harmonize the traditional and modern points of view. In particular, the “magnetic quadrature” condition requires the use of the “magnetic co-energy” concept (at the expense of the magnetic energy which is a state function of essential physical signification). In this chapter, the authors show that measurements must be performed with great precaution, and that the experimenter must know and understand the physical properties of the models (for which he is looking to identify the parameters) well: non-linearity effects, iron losses, etc. A good knowledge of the order of magnitude of the parameters is useful if not mandatory to carry out fine measurements and carefully make the necessary approximations. Clearly, the authors have sought to pass on their own experience in the domain.

In the chapter “Real-Time Estimation of the Induction Machine Parameters”, Luc Loron considers “on-line” or “real-time” processing for the determination of variable parameters (temperature dependent winding resistance, inductance, which depend with the magnetic state of the cores) and non-measurable variables (flow and velocity for controllers without mechanical sensors). In the preceding chapters, the models were close to the machine physics and the parameter identification tools were quite cumbersome necessitating an “off-line” process of the recorded data. On the other hand, on-line processing imposes real-time algorithms, which have short calculation times and robustness. They cannot destabilize the system, and they must provide trustworthy data at every moment. The reader will find in this chapter not only reliable information on the least square method, on the extended Kalman filter theory and the Luenberger observer theory, but also advice on their implantation and concerning the relevance of models (reparametrization), the validity of algorithms, the problem of monitoring (still very open to discussion), the influence of the sampling period, the analog filtering of measurements, the adaptation of algorithms when parameters or variables are no longer identifiable, etc. Again, the concrete experience of an expert is put at the disposition of future practitioners.

The second part of the book (Chapters 4, 5, 6, 7 and 8) focuses on several specific studies involving the control of these electric machines:

– study and implementation of reduced order observers and methods to determine the robustness of observers for the induction machine;

– estimation and observation approaches of the load torque and the rotor angular position of the synchronous machine.

The chapter “Linear Estimators and Observers for the Induction Machine (IM)” by Maria Pietrzak-David, Bernard de Fornel and Alain Bouscayrol concerns the estimation and observation of non-measurable variables. In an induction the magnetic flow, an essential value for the control of this machine, is not accessible through direct measurements. In fact, flux sensors in the air-gap greatly increase the cost of the machine; they constitute intrinsically fragile elements, and they often produce very noisy signals. They are only used for prototypes. A large part of this chapter is dedicated to the estimation or observation of the flux state. The estimation of the rotation speed and of the load torque is also studied. In this chapter, the modeling is critical since the “natural model” (referred to as the real windings) given by the physics of the system must be rewritten to meet the objectives. The numerous works carried out by the community of specialists have shown that the choice of four electric state variables (stator and rotor fluxes or currents) on the one hand, and the choice of the reference frame (link to the stator, rotor or to the rotating flux) on the other hand, play a very important role in the control structures of the speed controller (vector control or direct couple control). These particular choices for the state variables and the reference frame also influence the performances of the estimators and the observers. The authors present the estimation and observation of the flux of the induction machine with the help of automatic and signal processing tools: the complete order deterministic observer (synthesized with the help of pole locations) and the linear stochastic observer (Kalman-Bucy filters synthesized using optimization methods). These observers are then associated with structures dedicated to speed observation. Various non-linear observation methods (adaptable, of variable structure) are also presented. The authors of this chapter present several solutions based on their expertise and illustrated by examples. Researchers and engineers, facing these questions, now have access to a variety of solutions, which should help them not to waste their time with solutions badly adapted to their problem.

The chapter “Decomposition of a Determinist Flux Observer for the Induction Machine: Cartesian and Reduced Order Structures”, by Alain Bouscayrol, Maria Pietrzak-David and Bernard de Fornel focuses on very specific problems and solutions. The goal of this chapter is the study and creation of reduced order observers leading to smaller models and algorithms than those given by general theories (based on extended models). The authors also consider extended observers, but with a very interesting, even if but not very traditional, original solution of “Cartesian structures”. In this solution, the extended observer is broken down into (coupled) sub-observers, each one corresponding to the variables relating to an axis. We must clarify that this breakdown corresponds to an approximation justified by the time scale difference between speed of rotation and electric variables. In this way, the fourth order observer is replaced by the combination of two reduced order coupled observers for a simpler synthesis. It is well adapted for certain specific problems:

– robust estimation of the stator flux for “DTC” commands (Direct Torque Control);

– robust estimation of the rotor flux for traditional vector controls.

The authors use a Cartesian observer for stator and rotor flux to highlight the inability of observing the zero speed flux and provide a complete synthesis of this observer. The discretization of the Cartesian observer is simpler than the one resulting from an extended observer. The authors present several variations of reduced order observers and a number of synthesis examples. Since numerous studies on these subjects have been already published on this topic, the readers will appreciate suggestions for well adapted solutions given by experienced authors.

The chapter “Observer Gain Determination Based on Parameter Sensitivity Analysis” by Benoît Robyns proposes original tools for the resolution of a traditional but tricky problem. The rotor flux observer, for the vector control, is a real-time simulation algorithm of electric equations (either extended or reduced order) with matrix of undetermined gains. These gains are normally chosen by pole location techniques. However, since they depend on speed, they should be continuously recalculated at each sampling, which greatly increases calculation time. In practice, we often define speed ranges where chosen gains are constant. But this is in contradiction to the selection of “good poles” and to the robustness of this observer in terms of the various parameters. In order to resolve this contradiction, the author uses a very powerful, and not sufficiently known, tool called parametric sensitivity which provides access to observer errors made in the determination of the flux. With the sensitivity study, the observer’s gains are chosen to greatly reduce its parametric sensitivity, and maintain a satisfying dynamic. In this chapter, the author clarifies his models, algorithms and choice criteria. His theoretical results are supplemented by representative, precise and well explained examples. He shows the advantage of an extended observer, optimized through a sensitivity study, compared to a reduced order observer. The latter seems more sensitive to rotor resistance error, the most critical parameter in the vector control.

In the “Observation of the Load Torque of an Electrical Machine” chapter, the authors, Maurice Fadel and Bernard de Fornel, develop different load torque observation structures based on the mechanical quantities that can be measured. For electro-mechanical actuators, the requirements on speed and position control have a strong impact on the drive control loops. The electromagnetic torque must be perfectly mastered to obtain the most satisfying speed or position evolutions.

In addition, the mechanical loads often show ill-defined characteristics at low speed or in the vicinity of zero or quasi-zero speeds. The variable speed drive’s control in these specific operation zones can turn out to be problematic and the traditional control laws are often inadequate. The proposed study develops a detailed model aimed to improve the global behavior of the actuator.

The major contribution involves the load torque observer which, due to its structure and operation, can monitor the perturbations inherent to the internal structure of the electrical machine (cogging torque, electromotive force distortion, etc.) resulting from load parameters variations. This quantity, the torque that should be compensated, is then injected in the control law in order to smooth the machine’s effective torque. The system thus functions in disturbance rejection.

The solutions presented in this chapter are based on studies conducted at LEEI (Laboratoire d’Electrotechnique et d’Electronique Industrielle in Toulouse and now a team of LAPLACE laboratory) and resulting in experimental prototypes with digital controllers and extensive measurements sytems. The chapter details the problem associated with:

– noise and filtering of speed and/or position measurements;

– an erroneous identification of parameters;

– excessive dynamic response of the controller or of the observer.

Several results, most of them experimental, confirm the relevant character of the approach and the robust performances obtained in presence of load and/or machine parameter variations. One of the main objectives of this work is the search for the necessary compromise between control stability and the observer dynamic based on the disturbance rejection effectiveness and drive parameters variations. These questions result in a specific look at the choice of the observer dynamic in relation to controller settings and to the number of sensors used.

In the last chapter of the book, entitled “Observation of the Rotor Position to Control the Synchronous Machine without Mechanical Sensor”, the authors, Stéphane Caux and Maurice Fadel, review several position estimation approaches to get rid of this measure and thus suppress a mechanical sensor. For some applications, a “low resolution” reconstitution of the position is sufficient, mainly in the case of a synchronous trapezoid electromotive force machine. Two more precise methods are then presented by the authors using either the Kalman filter or an analytical redundancy approach called Matsui’s observer. Kalman’s method is very systematic and corresponds to a calculation intensive algorithm, with the usual problems of Kalman gains definition, initial covariances and statistic properties of noise. Moreover, the Kalman filter is sensitive to position initialization errors. The analytical redundancy algorithm is simpler than the Kalman filter and provides better estimation at low rotation speeds than the latter. Performances of both estimators were compared over different speed ranges and for parameter sensitivity and initialization errors. The studies found in this chapter offer practical suggestions for their implementation (filtering) and adjustment (choice of gains, observers selection, identification of noise sources, initialization and calibration). They show the feasibility of the rotor position estimation and provide fundamental information on the choice between the different approaches.

This second part of this book completes the first one dedicated to the definition of measures, to the key models and to the estimation and state observation tools. It presents the application of these methods and tools to control several actuators based on synchronous and induction machines. In particular, it describes:

– the study and implementation of reduced order observers and methods to determine the observers’ robustness for the induction machine;

– the estimation and observation approaches of the load torque and angular position of the rotor for the synchronous machine.

1 Introduction written by Bernard DE FORNEL and Jean-Paul LOUIS.

PART 1

Measures and Identifications

Chapter 1

Identification of Induction Motor in Sinusoidal Mode1

1.1. Introduction

Models generally used in electrical engineering are obtained from the laws of physics and are based on the knowledge of a certain number of parameters, they are called parametric models. As with the model structure, knowledge of parameters can also be obtained from laws of physics, if modeling is pushed far enough. Nevertheless, this approach has a few drawbacks: calculations are often fastidious; parameters depend on geometry and the materials used, which may not be well known. In addition, some parameters may vary during the life of a system and this is difficult to model by applying the laws of physics.

Another approach, that is more pragmatic, consists of estimating the numeric values of parameters so that the model has the same behavior as the experimental system. In this chapter, we will focus on the estimation procedures for the induction machine parameters from sinusoidal mode measurements. The methods we will present apply to all types of induction machines and are not based on usual stator measurements and rotation speed. First, we will develop the parametric models liable to use this type of estimation procedure. We will then present the most basic methods based on a limited number of measurements. In order to improve estimation precision, it may be necessary to use a larger number of measurements. We will present two methods with this possibility.

1.2. The models

The sinusoidal mode is not a steady-state mode strictly speaking because the electrical variables are not constant. In order to obtain the model rigorously, it is necessary to rely on the actuator’s dynamic model.

1.2.1. Dynamic model of the induction machine

For the establishment of the dynamic model, consider a wound rotor three-phase induction machine. We presume that the stator and rotor windings are perfectly symmetrical (hypothesis of concentricity). We use the hypothesis of the first space harmonic, i.e. we presume that magnetomotive forces created by windings are sinusoidal space functions. We ignore salience effects and teeth harmonics. Subsequently, we will presume that the model is valid for squirrel-cage machines, a more widespread technology.

First, take the dynamic model of the three-phase induction machine which, for the purpose of this report, is presumed to be star connected. We can detect the three phases by indices a, b, and c. We write i for currents, v voltage, and φ fluxes. The resistance of the stator winding is Rs, and the rotor circuit is Rr. We note Ls as the cyclic inductance of the stator, Lr for the rotor, and M is the cyclic mutual inductance between stator and rotor. The machine has p pole pairs.

A set of balanced three-phase stator quantities {xsa(t), xsb(t), xsc(t)} without homopolar can be represented by the complex phasor referred to the stator [LOU 04]:

[1.1]

with , corresponding to the first component of the Fortescue transform. This transformation is inversible with

[1.2]

For a three-phase rotor quantity {xra (t), xrb (t), xrc (t)}, we define the complex phasor referred to the stator, corresponding to the first component of the Ku transform:

[1.3]

where θ(t) is the angular position of the rotor.

The resulting model is expressed in two series of equations, equations to fluxes

[1.4]

and voltage equations

[1.5]

In the second equation, a zero voltage appears in the rotor corresponding to winding short circuit, and an electromotive force proportional to rotation speed Ω of the rotor, resulting from the change in reference frame of rotor variables.

This model depends on five parameters: two resistances Rs and Rr and three inductances: Ls, Lr, and M. The goal of an estimation procedure is to determine the numeric values of these parameters. And yet, this model cannot directly be used. In fact, an infinite number of parameter values correspond to an identical stator behavior [POL 67]. We say that this model is unidentifiable.

1.2.2. Establishment of the four parameter models

1.2.2.1. Total rotor leakage model

These equations are similar to those of a transformer. As with this other electromagnetic system, we can introduce the magnetizing current relative to stator flux:

[1.6]

implying that

[1.7]

By noting the stator/rotor transformation ratio and the rotor current referred to the stator, we can write the rotor flux in the following form:

[1.8]

We define the total rotor leakage inductance referred to the stator:

[1.9]

where is the dispersion coefficient, representative of the leakage part in the magnetic flux. The rotor flux referred to the stator is

[1.10]

and the rotor resistance referred to the stator is . The model, referred to as the stator, is in the following form:

[1.11]

1.2.2.2. Total stator leakage model

Another possibility of writing a model based on four parameters is to define the magnetizing current from the rotor flux:

[1.12]

Figure 1.1.Dynamic total rotor leakage model referred to the stator

Table 1.1.Parameters of the four-parameter dynamic models

Resulting in node law with . By defining the inductance of total leakages referred to the stator:

[1.13]

and the magnetizing inductance , the stator flux is written as

[1.14]

By noting the variables referred to the stator we rewrite the following flux equations:

[1.15]

and voltage equations

[1.16]

The model defined by equations [1.15] and [1.16] is presented in Figure 1.2. It depends on four parameters (Rs, R2r, Ns, and Lmr), illustrated in Table 1.1. Suppose that these parameters are estimated, the parameters of the initial model are then obtained by arbitrarily setting the mr transformation ratio:

[1.17]

Figure 1.2.Dynamic total stator leakage model referred to the stator

1.2.2.3. Equivalence of total leakage models

Since the total rotor or stator leakage models are equivalent to the initial model, they are therefore equivalent between each other, and we can go from one to the other by

[1.18]

The inverse of these relations results in

[1.19]

These different equations reveal a single factor, which is less than 1:

[1.20]

1.2.3. Magnetic circuit saturation

The increase in the magnetic field in certain parts of the machine’s magnetic circuit leads to an decrease in their permeability, creating a magnetic saturation phenomenon. We generally consider that this phenomenon only affects the mutual stator/rotor flux. In fact, leakage fluxes go through a large portion of air and because of that they are less sensitive to the saturation of magnetic parts. We now separate the fluxes in a main flux, noted φm and leakage fluxes:

[1.21]

where m is the transformation ratio, equal to the number of turn ratio, and ls and lr are the leakage inductances of the stator and rotor, respectively, presumed to be constant. We also define the magnetizing current relative to the air-gap flux:

[1.22]

In the absence of the rotor measurement, the transformation ratio is not available and can anyway be arbitrarily chosen with no effect on the behavior of the model referred to the stator. The model then depends on four constant parameters (Rs, Rr, ls, and lr) and a characteristic Lm (.).

Saturation can be taken into consideration in two ways in the model: by noting the different values of Lm based on the saturation variable in a table, or by attempting to interpolate this characteristic by a parametered function. In the last case, it is practical to determine the magnetizing current according to flux (im (φm)), which can be easily done from a polynomial development of the form:

[1.23]

[1.24]

Other authors prefer to use a development of Lm according to magnetizing current im. We can then use the same form of development. In any case, this consists in choosing a characteristic in the form:

[1.25]

where is the no-load inductance and . In practice, we choose a reduced number of non-zero factors αk, in order to limit the number of parameters to estimate.

When we consider saturation, leakage separation between stator and rotor is theoretically possible because of saturation. In practice, it is difficult to determine experimentally because the measurement errors produce estimation errors of high order1. We can then decide to work on similar models with total leakage at the stator or the rotor. The advantage of working with a model with better identifiability largely compensates slight loss in precision.

1.2.4. Iron losses

Field variations in the magnetic circuit of the machine lead to ferromagnetic losses. A first source of magnetic loss is caused by eddy currents, which are currents induced by field variations (also known as Foucault currents). Their power is proportional to the square of the field amplitude and the square of the frequency. The solution to decrease them is to use foliated material circuits.

Hysteresis losses are a second type of magnetic loss. They are connected to the depth of the hysteresis cycle of the magnetic material characteristic. Their power is proportional to the frequency and function of the surface of the cycle run. This surface increases in a non-linear way according to the field amplitude. Different approximations can be proposed to parameter this surface based on the field amplitude. One of them proposes that this surface is proportional to the square of the field amplitude. In any case, these losses can only be calculated for one period.

Eddy current losses are well modeled by a resistance added to the model in parallel to magnetizing inductance. For hysteresis losses, we generally use the same model, which has the advantage of making modeling of all magnetic losses by a single resistance possible. Nevertheless, in this last case, it is an approximation. To be more precise in writing the model, we could parameter the value of the resistance according to frequency, and possibly to the magnetic field amplitude. Nevertheless, in this chapter, we will only consider the case where iron losses are modeled by a single additional resistance, only adding one parameter to models previously presented.

Based on what we have just written, the resistance must be parallel to the magnetizing inductance, which corresponds to the main flux. If we ignore saturation, we then have a model with six parameters (resistances Rs and Rr , leakage inductances ls and lr, magnetization inductance Lm, and resistance of iron losses Rf ). This model, as we will see later, can theoretically be identified when Rf is not infinite. However, as in the saturation case, leaks are difficult to separate because of the high sensitivity with respect to measurement errors. We then use in practice five parameter models with a single leakage inductance.

1.2.5. Sinusoidal mode

In sinusoidal mode (SM), each three-phase system {xa (t), xb (t), xc (t)} can be written in the following form:

[1.26]

In a balanced mode, we can restrict to studying the first phase that is characterized by complex amplitude (equivalent to the Fresnel vector). The Fortescue component, obtained according to [1.1], is written as

[1.27]

Except for factor both notations are equivalent. In this way, all the models developed so far are equally valid for representing the equivalent diagram of the machine. We just have to replace the complex phasors noting the voltage, currents, and flux by the complex amplitudes by noting these same quantities for a phase of the machine.

By noting g as the slip, the rotor rotation speed is linked to the stator angular frequency by the p Ω (1 — g) relation. We will again use the equation of the rotor voltage [1.5.b]. It is now written as , where is the complex amplitude of the current of a rotor phase, written again by dividing by g:

[1.28]

As a result, the electromotive force behaves as a resistance with value . By adding the resistance Rr of the rotor winding, we then get global resistance corresponding to the sum of the converted power and rotor Joule losses. This general principle is valid for all the diagrams introduced earlier. In order to get the sinusoidal mode diagram, we just have to replace electromotive force by a resistance in the form .

Important note: we can observe that all the parameters of the dynamic model are present in the sinusoidal mode model. If we can estimate all these parameters by sinusoidal mode measurements, we then have a valid sinusoidal model as well as a valid dynamic model.

We showed that saturation can be taken into consideration by using magnetizing inductance Lm (or Ls in the case of total rotor leakage models) as a function of ξm equal to im or to φm; these variables represent modules of complex numbers defined by the transformation presented in section 1.2.1. In the case of SM, the rms value Ξm of ξma(t), relative to one phase, and module ξm (t) of vector are in a ratio of (see equation [1.27]). We can then adapt the characteristic of saturation [1.25] to obtain saturation characteristic depending on the rms value of the saturation variable:

[1.29]

[1.30]

1.2.6. Summary of the different models

The different models that we have obtained can all be put in a single form corresponding to the phase equivalent diagram represented in Figure 1.3. In the case of the total stator model (see Figure 1.2), we will consider that

[1.31]

Figure 1.3.Sinusoidal mode model

In the case of the total rotor leakage model (see Figure 1.1), we will consider that

[1.32]

The equivalent phase model impedance is written as

[1.33]

where Xm can be a function of the level of saturation. In the case of parametric saturation identification, we will from now on consider the model:

[1.34]

resulting in sinusoidal mode

[1.35]

with

[1.36]

There are 12 different models considered. One choice in three for leakage position (rotor, stator, or distributed), one in two for saturation (with or without), and one in two for iron losses (with or without). These different models are explained in Table 1.2. The first letter indicates if the leaks are totaled at the stator (S), rotor (R) or if the leaks are distributed between the stator and the rotor (D for Sales). The models that consider the saturation have an s as a second letter; and the models taking into account iron losses have letter f. Each model is in the general form of the diagram in Figure 1.3 as long as certain parameters are set at a zero value (Xs, Xr , or A) or infinite (Rf ). The asterisks represent the parameters to estimate; four to six according to the models. Model D cannot be identified. The other models can theoretically be identified but, we will see later, separate leak models are difficult to identify in practice.

Table 1.2.Different models considered

1.2.7. Measurements

The methods that we will present from now on help us to determine the numeric values of the parameters from sinusoidal measurements. Two types of measurements are necessary: electrical measurements for the stator (voltage, current, power, etc.) and a mechanical measurement: the rotor’s rotation speed.

Speed measurement Ω must be precise. In fact, it is used to determine slip form relation , with pΩ similar to ω. In this way, a relatively small speed error can generate a relatively important error on the slip.

The test bench for the estimation of parameters must enable the variation of the point of operation. In order to do this, we need a variable mechanical load for imposing a variable torque. If this load is passive (powder brake, direct current generator discharging in a rheostat), only the stable operation zones will be used (for slips of some percentage). A controlled speed load makes it possible to carry out measurements for all the values of speed (and thus slip)3. It is also interesting to have a reversible load, to reverse the direction of transfer of energy and to make the induction machine work as a generator, corresponding to faster speeds than the speed of synchronism.

If we want to identify the saturation characteristic, it is necessary to change the flux and consequently the voltage. We usually use a variable autotransformer in this case.

1.2.8. Use of the nameplates

The nameplates of induction motors vary from one manufacturer to another. Nevertheless, we can consider that they will at least provide the following information relative to the nominal working point: nominal slip gN, rms value UsN of the voltage between two phases, rms value Isn of the line current, power factor Fp, and output power Pu. It is not possible to determine the series of parameters of induction machine models only from the nameplate information. We will then introduce a “well designed” hypothesis; i.e., we will presume that the nominal machine operation corresponds to the optimal power factor. In addition, we will ignore stator resistance.

[1.37]

with . This equation has two real positive solutions. For each root R21 and R22, we can then determine the magnetizing reactance by replacing in the general admittance expression, resulting in

[1.38]

We will select the most plausible couple (R2k, Xmk).

Figure 1.4.Admittance diagram

1.3. Traditional methods from a limited number of measurements

1.3.1. Measurement of stator resistance

Resistance Rs of a stator winding is the only parameter that can be measured independently from the other parameters. We just have to power a phase of the motor with direct current, to sample the average values of voltage and current and to determine Rs as the voltage over current ratio.

Since the resistance values are sensitive to variations in temperature, it is recommended to bring the machine to its working temperature prior to making this measurement4.

1.3.2. Total rotor leakage model

Different procedures based on two tests help us to obtain the parameters of the total rotor leakage model5.

1.3.2.1. Usual method

The most usual method to determine the parameter values contains two tests:

– no-load test (without load torque) under nominal voltage with stator voltage Vs0, stator current Is0, power P0 and speed Ω06,

[1.39]

where φ0 is the phase difference of voltage in relation to the no-load current. From an active and reactive energy balance, we then determine the following:

[1.40]

with

1.3.2.1.1. Simplified calculation

[1.41]

with . The unknown parameters are then determined by

[1.42]

1.3.2.1.2. Precise calculation

In order to obtain more precise results, it is preferable not to ignore the saturation of the magnetic circuit8. An ingenious way of obtaining results is to write calculations in complex form. is the equivalent impedance calculated from measurements by the method described in section 1.2.7. The model gives the following relation:

[1.43]

written as

[1.44]

The real and imaginary parts are enough to isolate R2 and Xr, respectively:

[1.45]

1.3.2.2. Use of any two measurements

The previous method can be generalized in order to take any two measurement points into consideration9. Note the different measurements with indices a and b corresponding to two measurement points. To simplify calculations, note as complex admittances of circuits once stator Joule losses compensated. If is the equivalent impedance determined from experimental measurements as explained in section 1.2.7, we can determine them with relation . Admittance expressions:

[1.46]

Two of the unknown parameters are easily eliminated considering the difference term for term of both equations, which then results in:

[1.47]

The imaginary part of this last equation determines the value of Xr :

[1.48]

We then have to determine R2 as the only positive solution of the second-order equation obtained with the real section of relation [1.47]:

[1.49]

with , or

[1.50]

The values of Xm and Rf are then simply obtained by using one of the two equations of [1.46], the first one, for example:

[1.51]

We can use this method by choosing a no-load measurement and a nominal load measurement. We can then consider, when appropriate, the value of the no-load speed if it is slightly different from the speed of synchronism.

1.3.3. Total stator leakage models

In the case of total stator leakage models, the estimation of parameters with the help of two measurements is clearly more difficult to achieve. In fact, it is a four nonlinear equation system (if we consider the real and imaginary parts of both equations) with four unknowns. In the present case, the no-load test makes it possible to eliminate a single parameter instead of two in the previous case, the same applies to the technique previously used consisting of calculating the difference between both equations.

If, however, we wish to obtain a total stator leakage model with a simple method, the easiest way is to estimate the parameters of the total rotor leakage model and to find an equivalent total stator leakage model. When we ignore iron losses, we know that they are perfectly equivalent as long as we use the equivalence formulas from section 1.2.2.3. For the value of iron loss resistance, we propose to settle for the previously estimated value. It is obviously an approximation; a small difference will then remain between the values of impedances given by the two measurements and those given by the model.

1.3.4. Saturation characteristic

Once the values for the parameters in non-saturated mode are estimated, i.e. at low flux, we can try to extend the field of validity of the model by identifying the saturation characteristic10. We consider the measurements for a point of saturation with a level of saturation ξm:Vs, and Is rms value of the voltage and current relative to a phase, P the power absorbed and g the slip. It is then possible to determine the equivalent impedance per phase as explained in section 1.2.7; note to clarify that it comes from the measurements. This impedance is also written based on model [1.33] where the value of Lm must be recalculated to correspond to the measurement point as much as possible. We can then choose:

[1.52]

[1.53]

The magnetizing current module is then written as . After iteration of measurements, we have two saturation characteristics: Xm (Em) and Xm (Im) which can be chosen as needed.

When appropriate, we can then identify the characteristic as a parametric function of the saturation variable by estimating parameters by a least-square technique. This type of estimation method will be presented in detail and used in the following part of this chapter for the estimation of parameters from a high number of measurements.

1.3.5. Experimental results

1.3.5.1. Total rotor leakage model

The estimation method previously discussed was implemented in the MAS3 motor, and the characteristics are provided in Appendix 1.7.2. Two usual measurements were carried out: no-load test and nominal load test. Different variable voltage no-load tests were then carried out in order to identify the saturation characteristic.

1.3.5.2. Total stator leakage model

Because of iron losses, both models are not rigorously identical. In order to compare them, their complex admittances were calculated for different slip values between — 10% and +10% and are represented in Figure 1.5. We note that both models provide similar admittance values.

Figure 1.5.Comparison between models R.f (×)and Sf (+)

1.3.5.3. Saturation

We now focus on the effect of saturation on Xm value. For different measurement points, we calculate Xm with the help of relation [1.52]. The corresponding electromotive force Em is determined with the help of relation [1.39], making it possible to draw Xm variations according to Em. We can also calculate current .

We propose to handle the identification of the saturation characteristic in the following form:

[1.54]

where N is an integer to determine. The model depends linearly on parameters α1. and α2. We can then search for the values of parameters minimizing the gap quadratic criterion between the model and measurements:

[1.55]

[1.56]

where

[1.57]

and

[1.58]

1.4. Estimation by minimization of a criteria based on admittance

1.4.1. Estimation of parameters by minimization of a criterion

Figure 1.6.Saturation characteristics

1.4.2. Choice of criterion

Considering the circle diagram to characterize the operation of an induction machine is usual. Since this diagram represents complex admittance, it is legitimate to consider this quantity as relevant for identification . In this way, identifying the machine comes down to finding a model providing a circle diagram approximating measurements the best.

[1.59]

By noting and by grouping the different values in a vector: , we can write

[1.60]

where XH represents the Hermitian of X, i.e. the conjugate of its transpose.

1.4.3. Implementation

Since the model is non-linear according to parameters, we should use a minimization algorithm to determine the numerical values of parameters. In order to improve the speed of convergence, it is preferable to use methods based on a limited development of the criterion.

The first derivative of the criterion with respect to the vector of parameters, called gradient, is a vector noted ∇(Θ*,M) with dimension n (number of parameters), where the ith component is written . The gradient can be written as

[1.61]

where the term at line k and column l of is in addition, . The expression of the module sensitivities is given in Appendix 1.7.1.

The second derivative of the criterion in relation to the vector of parameters, called Hessian, is a n x n matrix where the term (i, j) is written .We generally use the following approximate expression, valid for low values of :

[1.62]

Around an arbitrary value Θ* of the vector of parameters, the second-order development is written as

[1.63]

The gradient has a first-order development written as

[1.64]

1.4.4. Analysis of estimation errors

A method of estimation able to determine the values of parameters in ideal conditions is not enough. It must also be able to deliver a relatively precise estimation of the parameters despite errors affecting the system. There are two types of errors: (i) the measurement errors linked to inaccuracies in the measurement device and the chain of acquisition; (ii) those linked to model imperfections (ignored phenomena, idealization of reality). In order to validate or invalidate estimators, in this section, we conduct a complete theoretical accuracy analysis of estimators of the different models. This study is based on raw values of parameters obtained from prior investigation (these values are referred to as “a priori values” in the sequel; they can be derived from the name plate). In section 1.4.5, we will see that this study helps us to clarify experimental results.

1.4.4.1. Method for error evaluation

A first method for evaluating estimation errors consists of creating the estimation procedure from simulated measurements in which we have introduced one or more sources of error. For stochastic measurement errors, the resulting estimation errors are random variables, and we will focus on their stochastic properties (bias and standard deviation), which will be evaluated in a panel containing a large enough number of samples12.

Another method consists of writing an approximate analytical development for the error in parameters according to measurement or model errors. The estimate of © verifies the first-order condition . From the development of gradient [1.64], we get the relation:

[1.65]

Now consider that Θ* is the “true” value of parameters, and that ≠ Θ*, because of measurement and model errors. The estimation error of parameters is then written as

[1.66]

It can be evaluated based on the a priori values of parameters Θ*.

1.4.4.2. Experiment design

In order to best identify the behavior of the machine, we have considered a measurement set that is as large as possible. In practice, the slip is limited to a value gmax for stability purposes; we then limited ourselves to a range of [—gmax;gmax]. In order to identify saturation, it is interesting to vary the level of saturation (and thus voltage) in a wide range. By noting Vmax as the maximum value that can be reached for voltage, we can choose to use as measurement range . It is not necessary to measure at very low voltage because the relative precision is then low and, in the absence of saturation, these measurements only bring redundant information. The experimental results used in this part were obtained on the MAS1 machine with characteristics available in Appendix 1.7.1. With a maximum slip of approximately 10% and maximum voltage of 120 V. We chose to carry out 66 different measurements for 11 Em values and 6 slip values.

In order to test the choice of measurement points on estimation precision, we also considered two other series of measurements containing the same number of measurements. In one, the measurements are done at positive slip, corresponding to motoring operation. In the other, the measurements are done at negative slip, i.e. in generating operation. Subsequently, we will see how to optimally design a set of measurement points.

1.4.4.3. Effect of measurement errors

We now present the evaluation of estimation errors, achieved with the help of analytical development presented in section 1.4.4.1 considering the measurement set presented in section 1.4.4.2. The series of results will be the subject of a discussion in section 1.4.4.5.

1.4.4.3.1. Offset and error of gain on sensors

We considered offsets on the different sensors corresponding to ±1% of the nominal value in voltage, current, phase difference, and speed (nominal values are given in Appendix 1.7.2), and we calculated the impact on the estimated value of the parameters of model Rsf (see Table 1.3). We provided the effect of each sensor offset and the worst-case conjugated effect of an offset in each sensor. We can observe that the most effective measurement is the phase measurement. The most sensitive parameters are Rf and Rs. The high level of these errors (117% and 43%, respectively) shows the necessity of an unbiased measurement of phase difference; it would also be useful to ensure that phase measurement has a lower offset by a tenth of what was considered in the evaluation, or less than 0.1% of 2π.

Table 1.3.Errors of parameter estimations caused by sensor offsets (in percentage of nominal parameter value)

We also evaluated the effect of a gain error of 1% in each sensor. The results are presented in Table 1.4 and show that the phase measurement is again the trickiest measurement that can lead to estimation errors in Rf and Rs.

Table 1.4.Errors of parameter estimations because of sensor gain errors (in percentage of the nominal value of parameters)

1.4.4.3.2. Stochastic measurement noise

The parameter estimation of the different models was done from measurements containing a random additive error with standard deviation equal to 1% of nominal values (see Table 1.5). We start with model R with four parameters that will serve as reference. We get accurate precision in R2 and Lm0 and a less accurate, but still satisfactory, precision in Rs and N2. When we consider iron losses in model Rf, the precisions in the four starting parameters are almost maintained and Rf has an acceptable precision. At this stage, model Rf is considered as usable for identification.

In model Df , we try to separate leaks, theoretically possible if we account for iron losses. However, the precisions obtained in leakage inductances are catastrophic. We conclude that this model is inappropriate for identification.

Model Rs is different from reference model R because of the introduction of saturation. We can observe that the precision of the four physical parameters remains correct; model Rs is therefore usable. That is also the case with model Rsf accounting for iron losses and saturation. However, with models Ds and Dsf attempting to separate leaks with the presence of saturation, we cannot obtain satisfying leakage inductance values. They should therefore be rejected for identification. Generally speaking, separate leakage models cannot be practically identified.

Table 1.5.Standard deviation of estimation errors caused by stochastic errors of measurements (in percentage of the nominal value of parameters) for a random additive error of 1% of nominal value

1.4.4.4. Effect of model error

Ignoring a phenomenon can lead to significant estimation errors. Validating an estimation procedure in relation to this problem is a vital step. In the case of sinusoidal mode operation, the main model errors emerge when we ignore saturation and iron losses. The reader will find a more detailed analysis of model errors of the dynamic model in [LAR 00, LAR 08].

1.4.4.4.1. Iron losses

Models Rs and Rsf only differ in iron losses. By simulating measurements with Rsf and by estimating Rs parameters, we can evaluate estimation errors caused by iron losses, reported in Table 1.6