System Dependability Evaluation Including S-dependency and Uncertainty: Model-Driven Dependability Analyses

398 Pages·2018·7.55 MB·English

by  Hans-Dieter Kochs (auth.)

Hans-Dieter Kochs
System Dependability
Evaluation Including
S-dependency and
Uncertainty
Model-Driven Dependability Analyses
System Dependability Evaluation Including
S-dependency and Uncertainty
Hans-Dieter Kochs
System Dependability
Evaluation Including
S-dependency and
Uncertainty
Model-Driven Dependability Analyses
1 3
Hans-Dieter Kochs
Lehrstuhl für Informationslogistik
Universität Duisburg-Essen
Duisburg
Germany
ISBN 978-3-319-64990-0 ISBN 978-3-319-64991-7 (eBook)
DOI 10.1007/978-3-319-64991-7
Library of Congress Control Number: 2017950274
© Springer International Publishing AG 2018
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Preface
The introduction of new technologies and the increasing complexity of systems make
dependability (reliability and availability, defined in IEC 60050-192:2015) analyses in-
dispensable in avoiding economic disaster and huge penalties in the case of unreli-
able systems. Poor dependability can cause expensive rectification and loss of trust,
reputation, and market share. Thus, dependability modeling and evaluation should be
basic tasks of every systems engineer. Their results should be fixed in technical
specifications and contracts by manufacturers and customers of industrial systems.
Stochastic dependency (s-dependency) between components and the influence of
uncertainty can have significant impact on system dependability. In practice, s-de-
pendency and uncertainty are often not taken into account.
The book focuses on system dependability modeling and calculation, considering the
impact of s-dependency and uncertainty. The best suited approaches for practical sys-
tem dependability modeling and calculation, (1) the minimal cut approach, (2) the
Markov process approach, and (3) the Markov minimal cut approach as a combina-
tion of (1) and (2) are described in detail and applied to several examples. The strin-
gently used Boolean logic during the whole development process of the approaches
is the key for the combination of the approaches on a common basis. For large and
complex systems, efficient approximation approaches, e.g. the probable Markov path
approach, have been developed, which can take into account s-dependencies be-
tween components of complex system structures. A comprehensive analysis of alea-
tory uncertainty (due to randomness) and epistemic uncertainty (due to lack of knowl-
edge), and their combination, developed on the basis of basic reliability indices and
evaluated with the Monte Carlo simulation method, has been carried out. The uncer-
tainty impact on system dependability is investigated and discussed using several ex-
amples at different levels of difficulty. The applications cover a wide variety of large
and complex (real-world) systems. Actual state-of-the-art definitions of terms of the
IEC 60050-192:2015 standard, as well as the dependability indices, are used uniformly
in all six chapters of the book.
Pre-knowledge: Mathematical interest, basic knowledge of Boolean algebra, probabil-
ity theory, and theory of stochastic processes.
V
Preface VI
Why this book? The vast majority of current books and publications on dependability
is highly mathematical and often only for small systems. The intention of this
book is to bridge the gap between theory and practice, and to concentrate on easy
and effective approaches for dependability analyses of systems including s-depend-
ency and uncertainty, which have been proved to be applicable to industrial systems.
The developed modeling and calculation approaches are embedded in a framework
consisting of 8 steps, based on the author’s theoretical and industrial dependability
experience and application over several decades. A further aim of the book is also to
emphasize the close relationship between network models and Markov models,
based on the Boolean logic, which easily (and clearly) enables their combination. The
developed approaches are applicable to all large and complex systems that can be
structured as illustrated in Fig. 1.1, 3.7, and 5.1, which apply to all industrial systems
within the scope of this book. The aspiration of the author is to describe depend-
ability theory and its application in an understandable and applicable way. The depend-
ability approaches are compatible for all systems.
What the book is not? The book is not a summary or a collection of the wide variety of
pure theoretical dependability evaluation approaches.
Acknowledgements: I am greatly indebted and wish to thank all my colleagues and
my former research assistants at my chair of Computer Engineering and Information
Logistics at the University of Duisburg-Essen, Germany, for their innovative contribu-
tion and cooperation around the scope of dependability. I am also grateful to my col-
leagues from the Cooperative Institute of Mechatronik (imech) and the Collaborative
Research Centre 291 (Speaker Prof. M. Hiller) of the German Research Foundation
DFG, which have enabled extensive applied research work on system dependability.
Furthermore, the periodical meetings of the Fault Tolerant Discussion Panel (FTDP),
which took place alternately at different universities, have provided continuous stimulus
over the last 25 years. Representative for the FTDP, I particularly would like to thank
Prof. K. Echtle and Prof. W. G. Schneeweiss (initiators of FTDP) for their valuable
contributions and substantial discussions. I would like to thank all industrial coopera-
tion partners. The research cooperation with industry, especially the cooperation with
ABB Ladenburg, Germany, and ABB Basel, Switzerland, on areas such as automa-

tion and control systems, was very productive and stimulates new ideas concerning
the applicability of the developed dependability approaches. Research on the topic
uncertainty was carried out by Dr. Ph. Limbourg and Dr. P. Kongniratsaikul in co-
operation with Dr. F. Lutz (IPL technology). The close combination of theory and
Preface VII

practice in different application areas gave valuable impulses for the improvement of

the approaches, which are described in this book. Furthermore, I would like to thank
Dr. J. Petersen for the continual cooperation and discussions as well as for the
technical support. I am very grateful to Ms. S. Heidtmann for a large number of rele-
vant remarks and the correction of the manuscript.
The author thanks the International Electrotechnical Commission (IEC) for permis-
sion to reproduce Information from its International Standards. All such extracts are
copyright of IEC, Geneva, Switzerland. All rights reserved. Further information on the
IEC is available from www.iec.ch. IEC has no responsibility for the placement and
context in which the extracts and contents are reproduced by the author, nor is IEC in
any way responsible for the other content or accuracy therein (IEC 60050-192 ed.
1.0, Copyright © 2015 IEC Geneva, Switzerland. www.iec.ch).
The author thanks the Management of the Museum for Communication Berlin and
Mr. St. Sous (artist) for the permission to take photos for analyzing the stagecoach,
which is exhibited as an art object (slogan “Berliner Luft Post“) in an exploded view. It
offers a unique and clear insight into its construction details, which is used favorably
for the system dependability analysis (Chapter 3.9).
Finally, but by no means least, I would like to particularly highlight and cordially thank
my wife Anne for her persevarence and encouragement of the work. Without it, it
would not have been possible to produce the book.
Professional career: Hans-Dieter Kochs was head of the Chair of Computer Engi-
neering and Information Logistics at the University Duisburg-Essen, Germany (retired
2009). He received a Diploma-Degree in Electrical Engineering (1972) and a Dr.-Ing.
Degree (1976) from the Technical University (RWTH) Aachen, Germany. From 1972
to 1979 he was a member of the Institute of Power Systems and Power Economics
(IAEW) at the RWTH Aachen (Prof. K.W. Edwin) as a research assistant. From 1979
to 1991 he had leading positions in industry (AEG/Daimler Frankfurt, FAG Kugel-
fischer Erlangen, and ESWE Wiesbaden, Germany). Since 1991, he has been a full
Professor. From 1972 up till now he has been engaged in scientific and industrial de-
pendability analysis and studies. (e-mail: [e-mail])
Special thanks are also due to the Springer staff, especially Dr. J.-Ph. Schmidt and
Ms. P. Jantzen as well as the Springer production team of Mr. Jayanthan Veeraraghavan
for their editorial support.
Content
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V
List of definitions.....................................................................................................................................XV
List of figures.........................................................................................................................................XVII
List of tables..........................................................................................................................................XXV
List of symbols and abbreviations..........................................................................................XXVII
1 Definitions and objective ..................................................................................... 1
1.1 Definition of basic terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Objective of system dependability evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2 Brief review of system dependability approaches........................................ 23
2.1 Application area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Assessment criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3 Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27
2.4 Framework for system dependability modeling and evaluation . . . . . . . . . . . . . . . . . . . 32
2.5 Notes on guarantee declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3 Network approaches............................................................................................ 39
3.1 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2 Input data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3 Basic network models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3.1 Series system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3.2 Parallels ystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.4 Minimal cut (MC) approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4.1 Definitions and preconditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.4.3 Calculation of the objective indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.4.4 Calculation of the MC indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.5 Minimal path (MP) approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.5.1 Definitions and preconditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.5.2 Examples .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.5.3 Calculation of the objective indices ........................................................... 54
3.5.4 Calculation of the MP indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.6 Approximation: Probable minimal cut (pMC) approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.6.1 Mathematical basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.6.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.6.3 Reduction of system model complexity by MC segmentation . . . . . . . . . 62
IX
Content Page 2 of X4
3.6.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.6.5 Conclusive remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.7 Interrelation between combination approach and MC/MP approach . . . . . . . . 66
3.7.1 Example: Series structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.7.1.1 Combination approach (Truth table) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.7.1.2 MC approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.7.1.3 MC/MP approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.7.2 Example: Parallel structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.7.2.1 Combination approach (Truth table) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.7.2.2 MC approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.7.2.3 MC/MP approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.7.3 Combination approach( Trutht able)v ersusM C/MPa pproach . . . . . . . 74
3.8 Historical example 1: Communication chain in ancient Persia 500 BC . . . . . 76
3.9 Historical example 2: Horse-drawn stagecoach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.10Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
3.10.1 Derivation of Eq. 1.137 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
3.10.2 Derivation of VFC2DF, VFC2FD,and VMC2DF . . . . . . . . . . . . . . . . . . . . . . . . . .1 17
4 State-space approach ........................................................................................... 125
4.1 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
4.2 Input data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4.3 Definition of different types of stochastic processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4.3.1 2-state process model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
4.3.2 Multi-state process model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 33
4.4.Markov modeling and calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 39
4.4.1 Markov equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
4.4.2 Modeling of components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 40
4.4.3 Modeling and calculation of systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
4.4.3.1 Analytical approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
4.4.3.2 Numerical iteration approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 44
4.4.3.3 Objective indices of a parallel structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
4.4.3.4 Objective indices of a series structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
4.5 Approximation: Probable Markov path (pMp) approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
4.5.1 Mathematical basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
4.5.2 System with two s-independent components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
4.5.2.1 pMp calculation of the parallel system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
4.5.2.2 pMp calculation of the series system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
4.5.3 r-out-of-n system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
4.5.4 System of 4.5.2.1 with limited repair capacity and repair priority . . . . . . . 161
4.5.5 System of 4.5.4 with common cause failures (CCF) . . . . . . . . . . . . . . . . . . . . . . . . . .1 64
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4.5.6 System of 4.5.4 with scheduled maintenance ..................................168
4.5.7 Segmentation of the Markov model of 4.5.6 and aggregation of
the partial Markov models ........................................................................170
4.5.8 System with redundancy switching ......................................................172
4.5.8.1 pMp approach ............................................................................173
4.5.8.2 Numerical iteration approach ...............................................175
4.5.8.3 Examples ......................................................................................176
4.5.9 System excluding repair during system operation ..........................177
4.5.9.1 Long-term process behavior ................................................178
4.5.9.2 Short-term process behavior ................................................180
4.5.10 Item with periodic fault diagnosis ...........................................................181
4.5.11 Paradox of the periodic inspection and the short-term behavior. .....188
4.6 Appendix .........................................................................................................................190
4.6.1 Modeling and calculation of the alternating 2-state renewal
process in Fig. 4.2 .......................................................................................190
4.6.2 Decision trees of the processes [Z(t),t>0] graphically high-
lighted in Fig. 4.6-8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
5 Markov minimal cut (MMC) approach..............................................................203
5.1 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
5.2 S-dependency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
5.3 Integration of Markov process models into minimal cuts - MMC approach . . . 207
5.4 Definition of various types of s-dependency and their impact . . . . . . . . . . . . . . . . . . . 209
5.4.1 S-dependency of type 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
5.4.2 S-dependency of type 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
5.5 Theoretical study example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
5.6 Set of examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
5.7 Theoretical study example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
5.8 General conclusions concerning MMC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
5.9 Application example 1: Process automation and control system . . . . . . . . . . . . . . 231
5.10 Application example 2: Mechatronic system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
5.11 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
5.11.1 Derivation of the c term of Eq. 5.45 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .266
5.11.2 Steady state of the MMC model, Fig. 5.19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .267
5.11.3 Steady state of the MMC model, Fig. 5.20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .269
5.11.4 Steady state of the MMC model, Fig. 5.21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .271
5.11.5 Transient state of the MMC model, Fig. 5.19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .272
5.11.6 Transient state of the MMC model, Fig. 5.20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .275
5.11.7 Transient state of the MMC model, Fig. 5.21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .277