Mathematical and Physical Properties of Reliability Models in View of their Application to Modern Power System Components

Chapter
Part of the Springer Series in Reliability Engineering book series (RELIABILITY)

Keywords

Weibull Model Reliability Function Life Model Hazard Rate Function Accelerate Life Test

Abbreviations and list of symbols

ALT

Accelerated life test

a.s.

Almost surely

BS

Birnbaum–Saunders (distribution)

cdf

Cumulative distribution function

CLT

Central limit theorem

CV

Coefficient of variation

CRF

Conditional reliability function

DHR

Decreasing hazard rate

DRA

Direct reliability assessment

D[Y]

Standard deviation of the RV Y

E[Y]

Expectation of the RV Y

EV

Extreme value (distribution)

F(x)

Generic cdf

f(x)

Generic pdf

f(x), F(x)

pdf and cdf of stress

g(y), G(y)

pdf and cdf of strength

G(r, ϕ)

Gamma distribution with parameters (r, ϕ)

H( )

Cumulative hrf

hrf

Hazard rate function

HRM

Hyperbolic reliability model

HV

High voltage

IDHR

First increasing, then decreasing hazard rate

IG

Inverse Gaussian (distribution)

IHR

Increasing hazard rate

IID

Independent and identically distributed (random variables)

IPM

Inverse power model

IRA

Indirect reliability assessment

IW

Inverse Weibull (distribution)

LL

Log-logistic (distribution)

LN

Lognormal (distribution)

LT

MRL

Mean residual life

MV

Medium voltage

N(α, β)

Normal (Gaussian) random variable with mean α and standard deviation β

pdf

Probability density function

r(s)

Mean residual life function at age s

RF

Reliability function

R(t)

Reliability function at mission time t

R(t|s)

Conditional reliability function at mission time t, after age s

RV

Random variable

SD, σ

Standard deviation

s-independent

Statistically independent

SP

Stochastic process

SS

Stress-strength

Var, σ2

Variance

W(t)

Wear process at time t acting on a device

W(a, b)

Weibull model with RF: R(x) = exp(−ax b )

W′(α, β)

Alternative form of Weibull model, with RF: R(x) = exp[−(x/α)β]

δ(·)

Dirac delta function

X, X(t)

Stress (RV or SP)

Y, Y(t)

Strength (RV or SP)

Γ( )

Euler–Gamma function

Γ(,)

Incomplete gamma function

μ

Mean value (expectation)

Φ(z)

Standard normal cdf

φ(z)

Standard normal pdf

Notes

Acknowledgments

The authors wish to express their thankful acknowledgments to some friends and colleagues of the University of Naples “Federico II”, for their very useful advices and significant contributions to the improvement of the present chapter. In particular, we express our thanks to Profs. Ernesto Conte, Pasquale Erto, Biagio Palumbo. We are also grateful to Profs. Erto and Palumbo and Dr. Giuliana Pallotta for bringing to our attention the main preliminary results of their last paper, while it was still in progress [75]. We also thank Dr. Claudia Battistelli for her help in revising the manuscript.

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Authors and Affiliations

1. 1.Department of Electrical EngineeringUniversity of NaplesNaplesItaly
2. 2.Department of Electrical EngineeringUniversity of BolognaBolognaItaly