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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Serie: Wiley - IEEE
- Sprache: Englisch
- Veröffentlichungsjahr: 2018
A practical guide to facilitate statistically well-founded decisions in the management of assets of an electricity grid Effective and economic electric grid asset management and incident management involve many complex decisions on inspection, maintenance, repair and replacement. This timely reference provides statistically well-founded, tried and tested analysis methodologies for improved decision making and asset management strategy for optimum grid reliability and availability. The techniques described are also sufficiently robust to apply to small data sets enabling asset managers to deal with early failures or testing with limited sample sets. The book describes the background, concepts and statistical techniques to evaluate failure distributions, probabilities, remaining lifetime, similarity and compliancy of observed data with specifications, asymptotic behavior of parameter estimators, effectiveness of network configurations and stocks of spare parts. It also shows how the graphical representation and parameter estimation from analysis of data can be made consistent, as well as explaining modern upcoming methodologies such as the Health Index and Risk Index. Key features: * Offers hands-on tools and techniques for data analysis, similarity index, failure forecasting, health and risk indices and the resulting maintenance strategies. * End-of-chapter problems and solutions to facilitate self-study via a book companion website. The book is essential reading for advanced undergraduate and graduate students in electrical engineering, quality engineers, utilities and industry strategists, transmission and distribution system planners, asset managers and risk managers.
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Reliability Analysis for Asset Management of Electric Power Grids
Robert Ross
IWO, Ede, the Netherlands TU Delft, Delft, the Netherlands
Copyright
This edition first published 2019
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The right of Robert Ross to be identified as the author of this work has been asserted in accordance with law.
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Library of Congress Cataloging‐in‐Publication Data
Names: Ross, Robert, author.
Title: Reliability analysis for asset management of electric power grids /
Prof Dr Robert Ross, IWO, Ede, the Netherlands; TU Delft, Delft, the
Netherlands.
Description: Hoboken, NJ : Wiley, [2019] | Includes bibliographical
references and index. |
Identifiers: LCCN 2018031845 (print) | LCCN 2018033843 (ebook) | ISBN
9781119125181 (Adobe PDF) | ISBN 9781119125198 (ePub) | ISBN 9781119125174
(hardcover)
Subjects: LCSH: Electric power distribution–Reliability.
Classification: LCC TK3001 (ebook) | LCC TK3001 .R65 2018 (print) | DDC
621.319–dc23
LC record available at https://lccn.loc.gov/2018031845
Cover Design: Wiley
Cover Image: © iStock.com/yangphoto; Equations courtesy of Robert Ross
Dedication
In remembrance of my parents and dedicated to my dear wife, children and sister.
R. R.
Preface
Reliability Analysis for Asset Management of Electric Power Grids aims to provide understanding and skills for analysing data in order to assess the reliability of components and systems. The understanding and skills support not only asset management and maintenance, but also incident management. The latter deals with unexpected failures that need to be evaluated to assist in decision‐making.
The structure of the book is presented in Table 1 below. After an introduction (Chapter 1) that pictures asset management and incident management in qualitative terms, seven chapters follow. The subjects of these chapters are: the basics of statistics (Chapter 2), measures to quantify (Chapter 3), a range of statistical distributions with their aims and properties (Chapter 4), graphical analysis of data (Chapter 5), distribution parameter estimation (Chapter 6), system and component reliability (Chapter 7) and, finally, system states with their reliability, availability and redundancy (Chapter 8). These provide an arsenal of techniques that form a foundation for statistical analysis in asset and incident management. These eight chapters form the core of the course in reliability analysis.
Table 1 Overview of the subjects treated in the book.
Qualitative introduction on:
asset management; maintenance styles
incident management
Chapter 1
↓
Basics of statistics, addressing:
concept outcomes, sample space, events, distribution, probability
statistical functions
F
,
R
,
f
,
h
,
H
; combinations of distributions and processes; two bath tub models depending on child mortality type
concept of ageing dose, power law and accelerated ageing
Chapter 2
↓
Measures in statistics:
expected values; conditional values and Bayes' theorem
moments; mean, median, mode, variance, standard deviation
covariance, correlation, similarity index and compliance
Chapter 3
↓
The purpose, characteristics and use of various specific distributions:
uniform, beta, Weibull, exponential, normal, lognormal, binomial, Poisson, hypergeometric and multinomial
Chapter 4
↓
Graphical data analysis and representations of distributions:
parameter‐free graphs, confidence intervals
parametric plots: Weibull, exponential, normal and lognormal, Duane and Crow/AMSAA
Chapter 5
↓
Parameter estimation:
bias, efficiency, consistency and small data sets
maximum likelihood, least squares and weighted least squares
application to Weibull, exponential and normal distributions
asymptotic behaviour, power function and unbiasing
beta distribution‐based and regression‐based confidence limits
Chapter 6
↓
System and component reliability:
block diagrams
series systems and competing processes, parallel systems and redundancy, combined systems and common‐cause failure
analysis of complex systems
Chapter 7
↓
System states in terms of working versus down:
states and transitions; failure and repair, absorbing down‐states
Markov chains and Laplace transforms
mean time to first failure and mean time between failures
availability and steady states
Chapter 8
↕↕
Practical applications to asset management and incident management:
period‐based, corrective, condition‐based, risk‐based maintenance
health index, risk index and combined health index
testing and quality with small test sets and accelerated ageing
failure cases and probability forecast of next failures
Chapter 9
↕↕↕
Miscellaneous background subjects:
combinatorics and the gamma function
power functions and asymptotic behaviour
regression analysis and regression‐based confidence intervals
sampling, Monte Carlo and random number generators
hypothesis testing
Chapter 10
Graph template and data tables
Appendices
The final two chapters (Chapters 9 and 10) aim to provide deeper insight and may be used in parallel with Chapters 1–8. Chapter 9 discusses a range of practical cases from asset management and incident management while using the techniques as explained in the previous chapters. Per case, it is indicated from which section the information is taken. Elements of the sections can be used for illustration during the course to complement the teaching from other chapters. Interested readers from the electric power industry may choose to start with Chapter 9 and select the aspects of reliability analysis that they would like to study more deeply, then follow the references.
Chapter 10 also aims at providing deeper insight, not so much by treating practical cases, but rather by studying a range of subjects in more depth. Depending on the courses given, the topics from this chapter may be added to lectures on Chapters 1–8.
The book covers some relatively new subjects and approaches, such as:
The difference in the meaning of statistical distribution and probability, as discussed and followed throughout the text.
Child mortality and the bath tub model, discussed with two different meanings. Statistically often associated with a declining hazard rate, in practice the meaning of child mortality is often encountered as a weak subpopulation which does not necessarily mean a declining hazard rate at all. The two models are discussed.
The power law concept associated with the ageing dose concept and used for discussing accelerated ageing and testing.
Asymptotic behaviour of bias and variances, described with a three‐parameter power function. This approach leads to an elegant unbiasing method in parameter estimation.
The power function also used to approximate the normal distribution.
The similarity index, introduced to compare distributions, which is useful for evaluating whether two distributions are the same and for estimating the number of failures yet to come. Determining the significance is discussed, and various examples are elaborated.
Consistency between graphical analysis and parameter estimation, which means that the best fit in a graph is identical to the best fit from parameter estimation.
Comparable views on the confidence limits based on random sampling (beta distribution) versus linear regression.
The relation between Monte Carlo simulations and sampling from the ranked cumulative distribution space. Numerical integration and mapping the ranked
F
‐space is discussed. The effect of quality control testing on the resulting ranked
F
‐space and confidence limits is demonstrated.
Much attention is paid to analysing small data sets. It is acknowledged that large data sets are necessary for accurate statistics. On the other hand, data are often scarce, with incident management and timely decision‐making required. While conclusions may not be very accurate, for decision‐making after unexpected failures they may be good enough and – more important – can support timely decision‐making.
The statistical models related to maintenance models, like corrective, period‐based, condition‐based and risk‐based maintenance, as well as models like the health index, risk index and combined health index.
As shown in Table 1, a range of distributions and plotting methods are discussed, including the Poisson distribution and Crow/AMSAA plots. Five distributions stand out in the discussions:
Weibull, because it is the asymptotic distribution for the weakest link in the chain, which applies to many failure incidents.
Exponential, because maintained components and systems tend to have a more or less constant hazard rate, which is a property of the exponential distribution.
Normal, because it is the asymptotic distribution for the mean and standard deviation. It helps in evaluating the accuracy of regression analysis.
Beta, for confidence limits and ranked sampling.
Uniform, which is fundamental to random sampling.
Additional distributions are discussed due to their peculiar properties.
The book is a considerable extension of the manuscript used for courses on system reliability at the Netherlands Royal Institute for the Navy. It is extended with experience gained during failure investigations and forensic studies, as well as asset management of grids and maintenance.
Finally, the book contains more than 30 years of experience and, moreover, stands on the shoulders of generations of experts before me. It is the sincerest wish of the author to match the needs in this field of students and colleagues in charge of strategic infrastructures, and the electric power industry in particular. This book also aims to contribute to the lively discussions that accompany the further development of asset management and incident management. The subject is not considered a finished topic with this book. Many experts contribute with questions, ideas and research daily, which makes this a still maturing field. The author welcomes any suggestions for improvements in order to maximize the contribution to the field of reliability analysis.
Robert Ross
IWO, Ede, the Netherlands
TU Delft, Delft, the Netherlands
Acknowledgements
I gratefully acknowledge the reviews, reflections and comments of Sonja Bouwman (AVANS), Remko Logemann (TenneT), Marijke Ross (New Perspective), Robbert Ross (Rijkswaterstaat) and Peter Ypma (IWO and TU Delft). In addition, I would like to thank Aart‐Jan de Graaf (HAN), René Janssen (NLDA) and various colleagues at TenneT for fruitful discussions on selected topics in this book.
List of Symbols and Abbreviations
∞
Infinity
±
Plus or minus; e.g.
x
= ±1 means
x
can be +1 or −1
≠
Is not equal to
≈
Is approximately equal to; e.g.
π
≈ 3.14159
∼
Is similar to
with quantities: same order of magnitude; e.g. 4∼6
with probability: has a similar probability distribution
with functions: is asymptotically equal to; definition
f
(
x
) ∼
g
(
x
) (
x
→
a
) means ; e.g.
≅
Isomorph; having the same measures or being congruent (identical measures)
let ABC and A′B′C′ be triangles. ABC and A′B′C′ are isomorphic (congruential) if and only if their shape (i.e. angles) and distances are equal. They may be oriented differently
*
Complex conjugate
if
z
= (
x
+
iy
) with
z
∈ ℂ then (
x
+
iy
)* = (
x
−
iy
) with
x,y
∈ ℝ and
i
the imaginary unit, i.e.
i
2
= −1
∝
Is proportional to
x
∝
y
means:
x
=
c
.
y
with
c
being a constant, i.e. a linear relationship
≡
Equal by definition; congruence
with definitions:
x
≡
y
means
x
by definition is equal to
y
with congruence:
a
≡
b
(mod
c
) means
a
−
b
is divisible by
c
; e.g. 7 ≡ 4 (mod 3)
√
Square root
√
x
means the non‐negative number whose square is equal to
x
!
Factorial; e.g.
n
! = 1·2·…·
n
, and
n
! = Γ(
n
+ 1)
, the gamma function
<
Is smaller than
≤
Is smaller than or equal to
«
Is much smaller than
>
Is larger than
≥
Is larger than or equal to
»
Is much larger than
<
x
>
Expected value of
x
. This is the mean value of
x
as the average of all possible
x
values
⇒
Implication
with statements: implies that; e.g. A ⇒ B: if A is true then B is true
⇐
Reverse implication
with statements: implies that; e.g. A ⇐ B: if B is true then A is true
⇔
Equivalence, mutual implication
with statements: is equivalent to; e.g. A ⇔ B: A is true if and only if B is true
∈
Is an element of
with set theory: is an element of; e.g.
n
∈
ℤ
means
n
is an integer
∉
Is not an element of
with set theory: is not an element of; e.g.
π
∉
ℤ
means
π
is not an integer
∅
Empty set; e.g.
A
∩
A
c
= ∅
means the intersection of a set
A
and its complement is an empty set
∩
Intersection; e.g.
x
∈
A
∩
B
means
x
is an element of both set
A
and set
B
∪
Union; e.g.
x
∈
A
∪
B
means
x
is an element of the union of set
A
and set
B
⊂
Is a proper subset of; i.e. not an empty set or equal set; e.g.
A
A
⊆
Is a subset of; includes proper sets and the complete set; e.g.
A
⊆
A
\
Excluding; e.g. ℕ\{0} means the set of natural numbers without 0, i.e. {1, 2, …}
∧
And; e.g.
x
> 0 ∧
x
< 1 means 0 <
x
< 1, i.e. both statements are true
∨
Or; e.g.
x
< 0 ∨
x
> 1 means
x
outside range [0, 1], i.e. either statement is true
|…|
Absolute value
with real number
x
: absolute value of
x
; i.e. if
x
≥ 0, |
x
| =
x
; if
x
< 0, |
x
| = −
x
with complex number
x
=
a
+
i
·
b
where
i
is imaginary number and
a
and
b
are real numbers: , with |
a
| and |
b
| absolute real numbers
with vector
X
= (
x
1
,…,
x
n
): the Euclidean length; i.e.
with matrix
A
: determinant of the matrix: |
A
|
with set
X
= {
x
1
,…,
x
n
}: the cardinality, i.e. number of elements. |
X
| =
n
‖…‖
Rounded off;
‖
x
‖
is the nearest integer to
x
; e.g.
‖1.5‖=2
;
‖1.4‖ = 1
Binomial coefficient, also called
n
‐over‐
k
; see also factorial
n
! ; e.g.
⌊…⌋
Floor function of real number
x
: largest integer
n
less than or equal to
x
; i.e.
n
+ 1 > ⌊
x
⌋ ≥
n
⌈…⌉
Ceiling function of real number
x
: smallest integer
n
greater than or equal to
x
; i.e.
n
− 1 < ⌈
x
⌉ ≤
n
A
(
t
)
Availability at time
t
A
∞
Availability long term, i.e. for infinite period of time or the limit
t
→ ∞
A
%
In context of distributions: failed fraction of population at specific reliability; e.g. 50% median where 50% of the population failed
In context of availability: up‐time as a percentage of total time (i.e. up‐time + down‐time)
A
c
Complement of
with set theory: the set
A
c
contains all elements of the total sample space Ω that are not in set
A
α
Lowercase alpha
with two‐ or three‐parameter Weibull distribution: scale parameter
∀
For all;
∀
t
:
t
·
t
=
t
2
, i.e. for all variables
t
the product of
t
and itself is the square of
t
a.u.
Arbitrary unit; i.e. unit not specified
AM
Asset management
B
(
i,j
)
Beta distribution
β
Lowercase beta
with two‐ or three‐parameter Weibull distribution: shape parameter
ℂ
The set of complex numbers
the sum of a real and an imaginary number; e.g.
z
=
x
+
iy
with
z
∈ ℂ and
x,y
∈ ℝ and
i
the imaginary unit, i.e.
i
2
= −1
CM
Corrective maintenance, i.e. action after failure (usually replacement)
CBM
Condition‐based maintenance
Cigré
Conseil International des Grands Réseaux Électriques, Eng.: International Council on Large Electric Systems
D
Ageing dose in terms of power law
D
%
Relative (ageing) dose; i.e.
D
%
= D/D
tot
D
tot
Total or fatal (ageing) dose
Δ
Delta; e.g. Δ
t
(usually small) step or variation in time
t
f
(
t
)
Distribution density; derivative of cumulative failure distribution
F
(
t
)
Cumulative failure distribution
ϕ
X
(
t
)
Characteristic function
G
X
(
t
)
Moment generating function
γ
Lowercase gamma
Euler's constant
γ
≈ 0.5772156649…
with three‐parameter Weibull distribution: threshold or location parameter
Γ
Uppercase gamma; e.g. used for gamma function (see below)
Γ(
n
)
Gamma function; if
n
is a natural number: factorial of (
n
− 1), i.e. Γ(
n
) = (
n
− 1)!
h
(
t
)
Hazard rate
H
(
t
)
Cumulative hazard rate over range [0,
t
]
H
(
t
,Δ
t
)
Cumulative hazard rate over range [
t
,
t
+ Δ
t
]
i
Lowercase
i
often used as index to distinguish an arbitrary element from a group; e.g.
x
i
(
i
= 1, …, 10) to represent any element from the group {
x
1
,
x
2
,
x
3
,
x
4
,
x
5
,
x
6
,
x
7
,
x
8
,
x
9
,
x
10
}
in case of imaginary number
i
: the square root of −1; i.e.
i
2
= − 1
IAM
Institute for Asset Management (UK)
IEC
International Electrotechnical Commission (CH)
IEEE
Institute for Electronic and Electrical Engineers (USA)
j
Lowercase
j
often used as index to distinguish an arbitrary element from a group; e.g.
x
j
(
j
= 1, …, 10) to represent any element from the group {
x
1
,
x
2
,
x
3
,
x
4
,
x
5
,
x
6
,
x
7
,
x
8
,
x
9
,
x
10
}
in case of imaginary number
j
: the square root of −1; i.e.
j
2
= − 1
λ
Lambda
with exponential distribution: inverse characteristic time
with systems: hazard rate
L
(
t
)
Likelihood function; product of failure probability densities (uncensored case)
M
As suffix: denoting median
μ
Lowercase mu
with distributed variable
X
: expected mean, i.e.
μ
= <
X
>
with systems: repair rate
Max(
a
,
b
)
Function that selects the largest of two values
a
and
b
Min(
a
,
b
)
Function that selects the smallest of two values
a
and
b
MTBF
Mean time between failure
MTTFF
Mean time to first failure
ℕ
The set of natural numbers; i.e. ℕ = {0, 1, …}
Ω
Uppercase omega
total sample space
π
Lowercase pi
fundamental constant
π
≈ 3.1415926536…
Π
Uppercase pi; symbol denoting product
means:
a
1
·
a
2
· … ·
a
n
PBM
Period‐based maintenance
PM
Periodic maintenance; also preventive maintenance; both used as synonym for PBM
ℚ
The set of rational numbers
numbers that can be derived from ratios of integer numbers; i.e. ℚ = {
p/q
:
p
,
q
∈ ℤ,
q
≠ 0}; e.g. , but
π
∉ ℚ
ℝ
The set of real numbers
the set of rational and irrational numbers; e.g. and
π
∈ ℝ
R
(
t
)
Reliability, also survivability
RBM
Risk‐based maintenance
RCM
Reliability‐centred maintenance
s
Lowercase
s
with averaging: scatter or standard deviation of (population) sample
with Laplace transform: variable of Laplace‐transformed function
σ
Lowercase sigma
with Gaussian or normal distribution: scatter or standard deviation
Σ
Uppercase sigma; symbol denoting summation
means:
a
1
+
a
2
+ … +
a
n
τ
Lowercase tau; measure of time
θ
Lowercase theta
mean lifetime
with exponential distribution: characteristic time, i.e. scale parameter
ℤ
The set of integer numbers; i.e. ℤ = {0, ±1, ±2, …}
About the Companion website
This book is accompanied by a companion website:
www.wiley.com/go/ross/reliabilityanalysis
The website includes:
Figures and tables from the book in PowerPoint format, answers to the questions and collected comments by readers.
Scan this QR code to visit the companion website.
1Introduction
This chapter provides an introduction to electric power grids as strategic infrastructures (Section 1.1); introduces asset management (AM) as an approach to optimize performance versus efforts (Section 1.2); provides an overview of maintenance concepts with their underlying philosophies and under which conditions they are preferred (Section 1.3); and finally discusses incident management (IM) (Section 1.4). The purpose of this chapter is to describe the context of the statistical methods discussed in this book.
1.1 Electric Power Grids
In 1880 Joseph Wilson Swan started to provide electricity to a residence in Rothbury, UK and in 1882 Thomas Edison's Pearl Street Station power plant generated electricity for homes in Manhattan. Edison envisaged: ‘After the electric light goes into general use, none but the extravagant will burn tallow candles’ [1]. This statement evolved into: ‘We will make electricity so cheap that only the rich will burn candles’. Interestingly, these statements refer to three themes that are important to AM: performance (electric light versus candles), general use (utility) and price (extravagant, rich).
In many countries electricity lived up to this expectation and became a dominant energy carrier, not just for lighting, but also for industrial machines, transportation, many household functionalities and more. Electric power networks grew, were integrated and expanded to continental grids and larger. Interconnections have been developed, such as the 580‐km NorNed submarine cable that enables hydropower reservoirs and consumption in Norway to act as energy storage for, for example, a surplus of wind power generated out of the German coast. That energy can be transmitted through the Netherlands' grid and transmitted through this NorNed cable or connected to other grids in Europe and beyond. The 130‐kW power plant of 1882 has been succeeded by GW power plants installed by State Grid, China. Nowadays, developed countries depend to a great extent on a reliable supply of electric power, which makes electric power grids very strategic infrastructures. The growing environmental awareness and focus on sustainable energy led to initiatives like the German ‘Energiewende’ (in English: energy transition, namely the shift from fossil and nuclear energy to sustainable energy), which only adds to the importance of electric power grids with electricity as a convenient energy carrier.
A power grid consists of many assets, but it is much more than a collection of components. The assets depend on each other to such an extent that they form clusters and connections which can be treated as units. For instance, a cable system consists of a cable termination and cable sections that are linked with cable joints. In addition, this cable system combined with primary components like switchgear, measuring transformers (current, voltage), a power transformer and a rail forms a circuit. Two parallel circuits can form a redundant connection, and so on. The components and connections are often controlled and protected by secondary systems. All those systems together form the network or grid.
