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

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


This chapter has a twofold purpose. The first is to present an up-to-date review of the basic theoretical and practical aspects of the main reliability models, and of some models that are rarely adopted in literature, although being useful in the authors’ opinion; some very new models, or new ways to justify their adequacy, are also presented. The above aspects are illustrated from a general, methodological, viewpoint, but with an outlook to their application to power system component characterization, aiming at contributing to a rational model selection. Such selection should be based upon a full insight into the basic consequences of assuming—sometimes with insufficient information—a given model. The second purpose of this chapter, closely related to the first, is to highlight the rationale behind a proper and accurate selection of a reliability model for the above devices, namely a selection which is based on phenomenological and physical models of aging, i.e., on the probabilistic laws governing the process of stress and degradation acting on the device. This “technological” approach, which is also denoted in the recent literature as an “indirect reliability assessment” (IRA), might be in practice the only feasible in the presence of a limited amount of data, as typically occurs in the field of modern power system. Although the present contribution does not address, for reasons of brevity, the topic of model or parameter statistical estimation, which is well covered in reliability literature, we believe that the development of the IRA is perfectly coherent—from a “philosophical” point of view—with the recent success and fast-growing adoption of the Bayesian estimation methodology in reliability. This success is proved by the ever-increasing number of papers devoted to such methodology, in particular, in the field of electric and electronic engineering. Indeed, the Bayesian approach makes use of prior information, which in such kind of analyses is provided by technological information available to the engineer, and—as well known—proves to be very efficient in the presence of data scarcity. Loosely speaking, IRA is a way of using prior information not (only) for random parameter assessment, but for a rational “model assessment”. In the framework of the investigation of innovations in reliability analyses regarding modern power systems, the present chapter takes its stimulus from the observation that the modern, deregulated, electrical energy market, striving toward higher system availability at lower costs, requires an accurate reliability estimation of electrical components. As witnessed by many papers appearing on the subject in literature, this is becoming an increasingly important, as well as difficult, task. Indeed, utilities have to face on one hand the progressive aging of many power system devices and on the other hand the high-reliability of such devices, for which only a small number of lifetime values are observed. This chapter gives theoretical and practical aids for the proper selection of reliability models for power system components. First, the most adopted reliability models in the literature about electrical components are synthetically reviewed from the viewpoint of the classical “direct reliability assessment”, i.e., a reliability assessment via statistical fitting directly from in-service failure data of components. The properties of these models, as well as their practical consequences, are discussed and it is shown that direct fitting of failure data may result poor or uncertain due to the limited number of data. Thus, practical aids for reliability assessment can be given by the knowledge of the degradation mechanisms responsible for component aging and failure. Such aging and life models, when inserted in a probabilistic framework, lead to “physical reliability models” that are employed for the above-mentioned IRA: in this respect, a key role is played by “Stress-Strength” models, whose properties are discussed in detail in the chapter. While the above part is essentially methodological and might be of interest even for non-electrical devices (e.g., Stress-Strength models were originally derived in mechanical engineering), a wide range of models can be deduced in the framework of IRA, that are useful for describing the reliability of electrical components such as switchgears, insulators, cables, capacitors, transformers and rotating electrical machines. Then, since insulation is the weakest part of most electrical devices—particularly in medium voltage and high voltage systems—phenomenological and physical models are developed over the years for the estimation of insulation aging and life is illustrated in this framework. Actually, in this kind of application the prior knowledge could be very fruitfully exploited within a “Stress-Strength” model, since Stress and Strength are clearly identifiable (mostly being applied voltage and dielectric Strength, respectively) and often measurable. By means of this approach, new derivations of the log-logistic distribution and of the “Inverse power model”, widely adopted for insulation applications, are shown among the others. Finally, the chapter shows by means of numerical and graphical examples that seemingly similar reliability models can possess very different lifetime percentiles, hazard rates and conditional (or “Residual”) reliability function values (and, thus, mean residual lives). This is a very practical consequence of the model selection which is generally neglected, but should be carefully accounted for, since it involves completely different maintenance actions and costs.


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

Abbreviations and list of symbols


Accelerated life test


Almost surely


Birnbaum–Saunders (distribution)


Cumulative distribution function


Central limit theorem


Coefficient of variation


Conditional reliability function


Decreasing hazard rate


Direct reliability assessment


Standard deviation of the RV Y


Expectation of the RV Y


Extreme value (distribution)


Generic cdf


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


Hazard rate function


Hyperbolic reliability model


High voltage


First increasing, then decreasing hazard rate


Inverse Gaussian (distribution)


Increasing hazard rate


Independent and identically distributed (random variables)


Inverse power model


Indirect reliability assessment


Inverse Weibull (distribution)


Log-logistic (distribution)


Lognormal (distribution)




Mean residual life


Medium voltage

N(α, β)

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


Probability density function


Mean residual life function at age s


Reliability function


Reliability function at mission time t


Conditional reliability function at mission time t, after age s


Random variable

SD, σ

Standard deviation


Statistically independent


Stochastic process



Var, σ2



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]


Dirac delta function

X, X(t)

Stress (RV or SP)

Y, Y(t)

Strength (RV or SP)

Γ( )

Euler–Gamma function


Incomplete gamma function


Mean value (expectation)


Standard normal cdf


Standard normal pdf



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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Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

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

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