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In general, photovoltaic (PV) plants do not include components with moving parts and, as a consequence, they are wrongly considered maintenance-free. Thus, this article presents a maintenance, reliability, and availability analysis of ten PV systems, with different inverter configurations, in the context of the intermittent renewable energy source systems, including the wind farms. The first part of the analysis consists of the evaluation of reliability using failure rates from the literature. In the second part, these results are compared with data obtained from industrial maintenance reports in the years 2016–2018. Finally, the yearly energy losses and the availability of each PV plant are assessed.

The estimation of lithium-ion batteries degradation plays an important role for the correct operation of smart grid and electric vehicle applications. In fact, only healthy battery energy storage systems meet minimum performance in terms of supplied voltage and power. Health prognostic is mandatory to ensure safe and reliable operation of batteries, as unsuccessful operation may cause technical/economic detriments or complete failures. The battery remaining useful life (RUL) depends on several random factors and thus it should be probabilistically characterized to allow the application of decision-making processes. In this article, a hybrid methodology is developed to characterize the RUL even with small amount of data, in view of the typical scarcity of data for adequately fitting the RUL probability distributions. Several distributions are considered in the methodology and compared for such purpose: the inverse Burr (IB) distribution, and mixtures of IB with inverse Gaussian and with inverse Weibull distributions. Expectation-maximization algorithms are specifically developed for the parameter estimation of the considered mixture distributions in the proposed methodology. The IB-based distributions are applied on a RUL dataset created by Monte Carlo sampling on an electrochemical battery model fitted upon given charge/discharge cycles. Numerical experiments are reported to assess the proposal with respect to benchmark distributions.

This paper aims at refining an experimentally based reliability model for the insulation of power components subjected to the randomly varying harmonics generated by power electronic converters. Compared to previous papers of the same authors and to the existing literature, here the model is re-formulated from the theoretical viewpoint focusing on the foremost role played by low percentiles of time to failure&mdash, in particular by the 1st percentile&mdash, selected as the rated life in the framework of modern probabilistic design of components. This is not only more correct from the viewpoint of component design, but also on the safe side as for the reliability of devices. Moreover, the application of the model is broadened to treat the whole sequence of odd voltage harmonics from the 5th to the 25th, i.e., those taken as the most significant in power systems according to international standards. The limits to voltage distortion set in Standard EN50160 are the reference for establishing parametrically a series of typical distorted voltage waveshape analyzed in the applicative part, which account for the possible phase-shift angles between voltage harmonics. The effect of current harmonics is also considered, from both the theoretical and applicative viewpoint. As a last, but not least novelty, the reliability model is used here for life and reliability estimates not only of Medium Voltage (MV)/Low Voltage (LV) capacitors and cables&mdash, already studied in the previous stages of this investigation&mdash, but also of induction motors and transformers in the presence of harmonics from power converters.

This paper presents a novel approach for the optimal design of the control system for battery energy storage systems adopted in power systems in the framework of the primary frequency control. The challenge issued by the use of batteries for frequency control in modern power systems is related to the need of the repeatedly deep charge and discharge phases. The presence of intermittent loads and generators, in fact, imposes very stressful power profiles to the battery which imply high rate of degradation to its lifetime. Based on the stress model of the battery, the proposed approach aims to design a control system which allows for containing the state of charge time variation within intervals able to match the requirements of frequency profile improvements and lifetime preservation. At this aim, the typical control scheme for batteries adopted for primary frequency control is analyzed and elaborated to address stable dynamic behavior of the regulator. Moreover, starting from the observation of the statistical nature of the power request for the pure frequency regulation service, a statistical study is conducted to perform an optimal design of the regulator in terms of both lifetime duration and primary frequency regulation service.

Large-scale penetration of renewable energy sources in power systems is essentially related to the need of reducing the environmental impact caused by the fossil-fuel. As well known, the interface between the power grid and this kind of energy resource is achieved by power converters, with a consequent dynamic behavior quite different from the synchronous generators. This matter involves negative impacts on the operating conditions of power systems. In this context, it is crucial to individuate estimation techniques able to predict promptly critical conditions which could, in extreme cases, compromise the stability of whole system. In this paper the authors employ an auto-regressive model which can describe the dynamic evolution of the power system inertia. The core of the procedure relies on an inertia model conceived as the sum of a periodic component and a noise stochastic process distributed according a Logistic model. The robustness of a novel estimation procedure, able to capture the dynamic evolution of the inertia, is investigated by testing two scenarios of Italian Transmission Network. The assumptions in terms of the obtained numerical results show the validity of the estimation technique and of the probabilistic characterization of the noise.

Extreme Wind Speed modeling, i.e. the probabilistic characterization of extreme values of wind speed, is a key tool for properly understanding the destructive wind forces which may affect mechanical safety and reliability of wind power systems, but it is also extremely useful for the purpose of achieving accurate wind energy production estimations. Indeed, the need of more accurate wind estimations has been often highlighted in the literature, especially for Extreme Wind Speed, since classical adopted models, such as the Weibull distribution, behave poorly in the range of Extreme Wind Speed values. In the paper, a new model, generated by a proper mixture of the established Inverse Weibull distribution, is proposed and illustrated. The proposal is the “Inverse Log-logistic” distribution, whose adequacy in interpreting some sets of real Extreme Wind Speed data is shown after giving some hints to its identification. This study also develops a peculiar Bayesian statistical inference approach for the estimation of the above model from available data, using different prior distributions, i.e. the Lognormal, the Generalized Gamma and the Uniform distribution. Extensive numerical simulations confirm that the proposed estimation technique constitutes a very fast, efficient and robust method for the Extreme Wind Speed modeling.

It is well known that, especially in the field of electronic components reliability studies and applications, the Exponential reliability model is by far the most adopted, although the data fostering it are few. This appears to be due partly to its simplicity (also in view of estimation, since it is characterized by a unique parameter), and partly because most components seem to be well represented, at least in their “useful life” time interval, by the Exponential model. This adoption is basically due to its peculiar “memory-less” property, i.e., the fact that such model possesses a constant hazard rate function, meaning that stochastic “accidents” cause the failure of the component, independently of its service time. This theoretical reason behind the choice of the Exponential model is largely prevailing over the classical statistical “goodness of fit” tests, since the high-reliability values attained by such devices does not allow the availability of an adequate number of lifetime values to be observed and analyzed in a statistical data analysis procedure. A second model also widely adopted is the Weibull model, especially if characterized by a shape parameter greater than unity, so implying an increasing hazard rate function. However, there are many cases—which can be also justified on a theoretical basis, as reviewed in this paper—in which a decreasing hazard rate function (at least for relatively large mission times) may be the best suited to describe the true model behind a given failure mechanism. The afore-mentioned theoretical basis of these apparently peculiar models is the main core of the present review article, whose aim also includes the illustration of the basic features of the main reliability models featuring an hazard rate function diminishing with time. The paper also discusses, resorting to graphical and numerical case-studies relevant to both field and simulated data, the consequences of mistaken model identification in terms of the hazard rate function behaviour, which may imply wrong maintenance actions.

The majority of Photovoltaic (PV) plants do not include components with moving parts and often they are wrongly considered maintenance-free. The present work consists of a Reliability Analysis (RA) of five PV systems with different size from 50 kW to 1 MW. The exponential distribution is used to calculate the reliability of electrical components, and failure rates are derived from literature. The results are compared to experimental data, obtained by the analysis of maintenance activity in two years of life of the PV plants. In addition, the calculation of the related energy losses of the PV plants is presented. From these results, it can be argued that, in some cases, energy losses are not negligible.

This study is devoted to a proper modeling and estimation method of battery capacity fading under the widely adopted exponential decay model, in view of an efficient lifetime assessment of battery. The method is based upon recursive online Bayes estimation of capacity fading. It is also shown how a degradation-based modeling of battery reliability can be adopted, leading to a Bernstein lifetime distribution. The performances of the proposed method are successfully evaluated by means of extensive Monte Carlo simulations based upon available literature and experimental data. A brief account is also given of a robustness analysis of the proposed methodology with respect to departures from the assumption of Gaussian noise.

This paper proposes a new methodological approach in the field of studies devoted to proper and accurate selection of a reliability model for battery systems. The study is performed with particular emphasis on the modeling and estimation of the Conditional Reliability Function, conceived as a key analytical tool in predicting the "Remaining useful life" of the battery, which is in turn an important information in order to identify the best maintenance strategy selection, or for inspection optimization, and also spare parts provision. Estimation of the Conditional Reliability Function is developed by means of a method based on the Inverse Gaussian Distribution and its Bayes Estimation. The performances of this estimation are developed and validated by means of extensive simulations and available experimental data. A brief account is reported of robustness analyses of the method with respect to the assumed prior Distribution.

Power electronic devices are expected to play an ever more fundamental role in unlocking the potentialities of smart power systems and in developing more electric ground and air transportation systems. The reliability of power electronic devices at different hierarchical levels (single component, single device, installation and system) becomes a crucial point in this framework, as failures may determine technical, economical and safety issues that should be carefully addressed at the design and maintenance stages. Power electronic devices are subject to thermal, electrical and mechanical stresses, which can be assessed through consolidated, traditional techniques [1,2,3,4]. However, today these devices are expected to operate under challenging environmental conditions (e.g., high altitudes in more electric aircrafts or high temperatures on photovoltaic (PV) installations), undermining the effectiveness of traditional approaches that are typically based on historical failure data, fault rates or past observed scenarios. In fact, the rapid evolution of power electronic technologies and the ever more challenging operating frameworks pose severe limitations on the trustworthiness of available reliability data, as they are typically related to incoherent operating conditions [1,2,3,4].

Inverse Rayleigh probability distribution is shown in this paper to constitute a valid model for characterization and estimation of extreme values of wind speed, thus constituting a useful tool of wind power production evaluation and mechanical safety of installations. The first part of this paper illustrates such a model and its validity to interpret real wind speed field data. The inverse Rayleigh model is then adopted as the parent distribution for assessment of a dynamical &ldquo, risk index&rdquo, defined in terms of a stochastic Poisson process, based upon crossing a given value with part of the maximum value of wind speed on a certain time horizon. Then, a novel Bayes approach for the estimation of such an index under the above model is proposed. The method is based upon assessment of prior information in a novel way which should be easily feasible for a system engineer, being based upon a model quantile (e.g., the median value) or, alternatively, on the probability that the wind speed is greater than a given value. The results of a large set of numerical simulation&mdash, based upon typical values of wind-speed parameters&mdash, are reported to illustrate the efficiency and the precision of the proposed method, also with hints to its robustness. The validity of the approach is also verified with respect to the two different methods of assessing the prior information.

Battery energy storage systems are widely recognized as viable means for providing frequency regulation in electrical systems characterized by the massive deployment of renewable resources. In this case, the estimation of the battery lifetime is very complex since its response depends strongly on the stochastic nature of various contingencies. By starting from the characterization of the power system frequency in terms of the stochastic process, in this paper, tools needed for the battery lifetime estimation are detailed. On the basis of a realistic dynamic model, the state of charge profile requested to a battery providing primary frequency regulation is derived, with the aim of characterizing statistically its degradation. In this way, it is expected to obtain an efficient battery lifetime estimation compared to the heuristic methods. Numerical simulations are carried out in the last part of the paper, illustrating a simulation tool for the estimation of the battery working cycle. © 2018 IEEE.

In the companion paper [1], the first part of this technical contribution, the way Renewable Energy Sources (wind turbines and photovoltaic units) affect power system's rotational inertia was thoroughly discussed. It is well understood that this relationship, becoming obviously more and more important in the last years, has implications for frequency dynamics and power system stability. Frequency dynamics are indeed faster in power systems with low rotational inertia, making frequency control and power system stability more challenging. In the first paper, after discussing the impact of low rotational inertia on power system stability, a new Bayesian statistical inference approach has been proposed for the on-line estimation of the “Renewable Energy Source Share” and thus the rotational inertia of a given system. This was discussed from the point of view of developing a theoretical methodological contribution. In this second part, extensive numerical simulations confirm that the proposed estimation technique constitutes a very fast, efficient and, especially, a robust method for an efficient assessment of rotational inertia.

The growing diffusion of electric vehicles connected to distribution networks for charging purposes is an ongoing problem that utilities must deal with. Direct current networks and storage devices have emerged as a feasible means of satisfying the expected increases in the numbers of vehicles while preserving the effective operation of the network. In this paper, an innovative probabilistic methodology is proposed for the optimal sizing of electrical storage devices with the aim of maximizing the penetration of plug-in electric vehicles while preserving efficient and effective operation of the network. The proposed methodology is based on an analytical solution of the problem concerning the power losses minimization in distribution networks equipped with storage devices. The closed-form expression that was obtained is included in a Monte Carlo simulation procedure aimed at handling the uncertainties in loads and renewable generation units. The results of several numerical applications are reported and discussed to demonstrate the validity of the proposed solution. Also, different penetration levels of generation units were analyzed in order to focus on the importance of renewable generation.

A key role for the reliability model identification of the electrical insulation of power system components is played by adequate ageing models. This is the case of the popular “Stress- Strength” models, where “Strength” is the electrical endurance of the insulation and “Stress” is the applied stress magnitude. It is well known that reliability estimation may be better achieved, instead that using limited lifetime data, by the knowledge of the degradation mechanisms. The stress considered is the voltages surges which act on the insulation of a high voltage cable used in power systems. In order to exploit the flexibility of the Inverse Burr distribution as an Extreme value Distribution, the paper has the purpose of setting up a Stress-Strength insulation reliability model when both Stress and Strength follow Inverse distributions. Then, a practical Bayesian approach is proposed for the estimation of the model. The efficiency of the proposed Bayesian approach is verified by means of numerical simulations referred to real insulation data.

Physical/mathematical laws describing electrical insulation aging play a key role for the reliability model identification of the insulation itself. This holds for the popular Inverse Power Model, too. The paper first discusses the deduction of the Inverse Power Model from reasonable physical and mathematical models of ageing, described via proper characterization of the random variables or the stochastic processes involved. Then, some analytical aids are given in order to perform its identification and Bayes Estimation, also by means of numerical applications with reference to in-service electrical failure data.

In the last years, large-scale deployment of renewable energy sources has led to the use of significant shares of generation from intermittent sources. Renewable energy sources units are notably inverter-connected wind turbines and photovoltaic that as such do not provide rotational inertia: this has implications for frequency dynamics and power system stability and operation. Frequency dynamics are faster in case of systems with low rotational inertia, thus making frequency control and power system operation more challenging. While the impact of low rotational inertia on power system stability and operation have been widely investigated in the recent literature, in this paper a new Bayesian statistical inference approach is proposed for the on-line estimation of the rotational inertia of a given system. The need for estimation is based upon the observation that the amount of renewable energy affecting the system is a random process. The authors' researches of this paper is devoted to the Bayesian on-line recursive estimation of the amount of the so-called “Renewable Energy Source Share”. The numerical application reported in the companion paper [1] confirms that the proposed estimation technique constitutes a very fast, efficient and, especially, a robust method for ‘tracking’ the above share in view of an efficient assessment of rotational inertia.

The paper presents an advanced Bayesian technique to carry out availability analyses of photovoltaic inverters in presence of uncertain reliability data. In order to proper take into account the above uncertainty, photovoltaic inverter hazard rate and repair rate are considered as random variables characterized by proper Log-normal and/or Gamma distributions. These assumptions are adequately motivated by well known and established physical and mathematical considerations. In this manner the photovoltaic availability uncertainty can be expressed as a function of the component uncertain data. Afterwards, thanks to the tailored Bayesian technique we demonstrate how, even in presence of scarce data, an efficient updating of the system performance can be accomplished during the photovoltaic inverter operating life. The methodology is really useful in view of an optimal allocation of the components’ reliabilities at the design stage and in order to establish the actual expected payback of the whole photovoltaic system. Extensive numerical simulations confirm the effectiveness and the validity of the developed Bayesian technique by also verifying its inherent robustness and efficiency through a rigorous analysis.

A probabilistic battery design method is discussed in the paper, with reference to lithium-ion batteries, based upon battery lifetime and related economical aspects. The method takes into account the dependence of battery lifetime on different parameters, such as design maximum specific power and operating environment. The methods relies on a large set of experimental measurements, in particular accelerated life tests. In addition, the randomness of the above parameters in real operating conditions is considered in the paper. Based on available experimental data, the lifetime probability distribution of these batteries has been estimated by means of a Weibull model. More specifically, after a thorough statistical data analysis, a probabilistic method for battery design is proposed which ensures the minimization of a suitable cost function which has the expected lifetime as a basic input. The method allows to guarantee the required robustness design against the random variations in specific power experienced by a lithium ion battery designed for a small electric bus (public transport service).

A novel Inverse Burr stress-strength probabilistic model is proposed for reliability assessment of electrical components in the presence of overstresses, such as extreme wind-speed values for wind towers reliability modeling or extreme voltage surge amplitudes for insulation reliability modeling. These are kinds of stresses which can be modeled by means of adequate, well-known extreme value distributions, such as Gumbel and Inverse Weibull distributions. However, Inverse Burr models have also been recently proven to be efficient and flexible extreme values models. In this paper, also the “strength” of electrical components is characterized through an Inverse Burr distribution. The estimation of the overall reliability of the component is performed through classical Maximum Likelihood Estimation procedure and through a new proposed Bayesian methodology, based upon the assessment of prior distributions on given parameters of the stress and strength distributions. Indeed, usually only a limited amount of lifetime data are available for high-reliability electrical components, while data on strengths or stresses are easier to get. Therefore, the application of a Bayesian approach is particularly appropriate in situations characterized, on one hand, by lack of experimental data, and, on the other hand, by some degree of technical knowledge. The validation of the stress-strength model and the comparison between both reliability estimation methods is confirmed through a numerical application, considering typical values of reliability of electrical components. The results proved the usefulness of the Bayesian reliability estimation procedure and the feasibility of the Inverse Burr stress-strength model.

In this paper, the problem of the optimal rating of transformers in smart grids is properly discussed with respect to the specific load characteristics. The design is based on the accurate prediction of the lifetime degradation of mineral-oil-immersed transformers subject to highly intermittent loads. In fact, by investigating the nature of the loads in the smart grid scenario, it clearly appears that the intermittent nature of the power demand increases drastically. In this context, specific tools for the characterization of the lifetime duration are required, since severe reduction of the transformer’s life can be observed due to overloads, even in the case of short-duration overloads. The classical approaches based on using the equivalent thermal current to predict the transformer’s lifetime might result in incorrect estimates, thus requiring advanced models that can deal with the time variability of the load. In this paper, the randomness of the load powers is addressed in terms of stochastic processes. In particular, the Wiener process is demonstrated to provide robust modeling of load variability. By starting from this assumption, it was demonstrated analytically that the hot-spot temperature, which is a major contributor to the degradation of the transformer’s lifetime, also is a stochastic process. Then, in spite of the nonlinearity of the thermal model, the hot-spot temperature can be represented as a Wiener process, the robustness of which has been verified adequately. By taking into account the nonlinear relationship between the hot-spot temperature and the lifetime, the authors verified that the transformer’s lifetime is modeled as a lognormal, stochastic process. Hence, a novel, closed-form relationship was derived between the transformer’s lifetime and the distributional properties of the stochastic load. The usefulness of the closed-form expression is discussed for sake of design, even if a few of the considerations also are performed with respect to operating conditions. The aim of the numerical application was to demonstrate the feasibility and the easy applicability of the analytical methodology.

The paper investigates the performances of automotive lithium-ion battery systems, which are considered as one of the most promising candidates for the use on electric and hybrid vehicles. Indeed, lithium batteries show technical properties and features particularly suitable for these applications in which high energy and power densities are required. In order to investigate lithium-ion battery major technical limitations, a set of experimental testing activities has been carried on battery packs and systems, whose behaviours are expected to be significantly different from the test results on single cells. One of the most crucial challenge is the problem of the battery lifetime. Many approaches have been proposed in the relevant literature, but a lot of difficulties persist, being related to the incidence of many factors which do not allow to derive a quite general model able to describe in an exhaustive way battery performances under different operating conditions. In the paper, a method is proposed which takes into account the randomness of the battery parameters, such as design maximum specific power and operating environment, in real operating condition, with reference to lithium ion batteries designed for a small electric bus (public transport service). Based on available experimental data, the lifetime probability distribution of these batteries has been estimated by means of a Weibull model. © 2016 IEEE.

This article illustrates a Bayesian inference methodology for the parametric evaluation of the reliability of power system components in the presence of voltage harmonics. The methodology starts from an experimentally-based life model for solid insulation under distorted voltage, which belongs to the class of the popular Inverse Power Model. Thereafter, a novel approach is developed for building the prior distribution of the unknown life model parameters, described by a joint Normal distribution, that implies a Lognormal model for the hazard rate and the Mean Time To Failure. The applicability and efficiency of the proposed Bayesian methodology is checked by means of a broad amount of Monte Carlo simulations performed for the insulation of power capacitors and cables subjected to harmonic voltages. The results prove the satisfactory behavior of the Bayesian methodology especially for quite small samples; this is an important outcome, since – as well known in the case of real power systems - small samples are a common practical case.

The randomness of the wind source is a concerning issue for managing power plants in reliable conditions. High values of wind speed are undesirable since wind farms provide zero power for values greater than their cut-off thresholds. Also, the mechanical safety of the installations can be seriously compromised by extreme values of wind speed. Therefore, a reliable estimation of extreme values of wind speed is mandatory. An Inverse Burr distribution is proposed as an useful alternative for the probabilistic modeling of extreme values of wind speed. Distribution parameters were estimated through maximum likelihood and moment estimation procedures, and through a new proposal, the quantile estimation procedure. The proposed model is validated on several real wind datasets, comparing the proposed model with commonly-used extreme value models. Numerical applications showed that the proposed model is a valid and feasible alternative to the classical extreme value distributions for extreme values of wind speed.

This paper deals with the “physical reliability models” assessment and estimation for electrical insulation components. It is well known that the reliability model identification and estimation of most of the modern power system components, such as insulation components, may be better achieved, instead that using limited lifetime data, by the knowledge of the degradation mechanisms. Such mechanisms, which are responsible for component aging and failure, are indeed well established in the field of electrical insulation: this is also the case of the so called “Stress-Strength” models. In particular, the “Log-logistic” model, deduced by a suitable Weibull stress-strength probabilistic model, has found valid applications to the reliability assessment of the insulation components. In the framework of the estimation of such reliability model, a new Bayesian approach, based upon the “Odds Ratio” of the Log-logistic model is developed in this paper, based upon the properties that such information, being proportional to the reliability function, is available to the engineer on the basis of past data; moreover, being proportional to the Weibull scale parameter, allows to exploit known features of its conjugate prior Inverse Gamma distribution. Numerical examples and the results of extensive Monte Carlo simulations demonstrate the feasibility and efficiency of the proposed procedure.

Renewable energy sources provide power in a form not directly deliverable to the grid, thereby requiring power electronic converters. Parallel inverters are generally used to achieve high reliability and reduce inverter rating. In this paper the inherent limitations of conventional droop control method for low voltage parallel inverters with mainly resistive tie line are shown, and multi-loop control strategy is proposed. Such strategy uses proportional plus multi-resonant controllers to achieve high quality of the voltage through harmonic distortion rejection. The design procedure of this controller is investigated thoroughly and it is shown that both controller resonant gain and bandwidth affect the stability of the system. Also a new configuration for virtual resistive impedance loop is proposed to improve the accuracy of both linear and non-linear load sharing without degrading voltage quality. Simulation results with two 4-kVA inverters are provided, showing proper stability margin, having sinusoidal voltage in the presence of nonlinear load, power sharing with a good accuracy and proper dynamic. Furthermore, some highlights on system reliability, and its possible improvements by proper redundancy, are briefly illustrated at the end of the paper.

The estimation of wind speed extreme values is a topic assuming increasing importance in wind energy studies. Two different methods are compared here for that purpose, in view of safety applications. The first model is a parametric one, based upon a classical extreme value model, such as the Gum-bel or the Inverse Weibull distribution. The alternative model is a "non parametric" one, based upon a stochastic characterization of the wind speed by means of a Poisson distribution. For both methods, estimates are carried out by means of Bayes estimation approach. The two approaches are compared in terms of robustness of the estimates of a proper safety index, with respect to departures from the assumed wind speed model. A large set of simulations results are discussed, as a first step towards a deeper insight to wind speed estimation methods, taking into account model uncertainty.

Prediction of extreme values of wind speed is a key issue for both wind energy and wind tower safety assessment. The paper proposes a new method for such estimation, in the framework of safety assessment under extreme wind speed, based upon an adequate probabilistic model. The method assumes an Inverse Weibull probability distribution for the characterization of extreme wind speeds, and is developed by means of a novel Bayes estimation method. Such method uses a prior assessment of a given quantile of the wind speed by means of a “Negative Log- Lognormal" distribution. In the paper, by means of large set of numerical simulations relevant to typical wind speed data, the efficiency of the Bayes methods is discussed. Attention is focused in particular on the robustness of the estimates with respect to departures from the assumed wind speed probability distributions, assuming the Gumbel distribution as an alternative extreme value model.

The paper discusses some common reliability architectures, such as "parallel" and "k-out-of-n" systems, adopted to add redun-dancy in many modern industrial systems, such as parallel-inverter systems. The focus is on some crucial properties of the failure rate of such systems, motivated by the fact that, in applied literature, the system failure rate is often simply evaluated as the reciprocal of the "Mean Time To Failure" of the system. Howev-er, this relationship is valid if, and only if, the system has a "se-ries" reliability architecture. This is indeed the only case in which also the system has a constant failure rate, i.e. an Exponential lifetime distribution. Instead, the system failure rate of redundant systems is a func-tion of time, which can never be constant. It is simply shown in-deed that the failure rate of a parallel system with constant fail-ure rate components is an increasing, or "first increasing, then decreasing" function of time, eventually reaching the value of the smallest failure rate. These results are extended to "k-out-of-n" reliability systems, and also to more general reliability models with non-constant failure rate, such as the Weibull or the "bath-tub" model.

An advanced stage has been reached in the research and development of alternative vehicle propulsion systems, aiming at assuring acceptable levels of both consumption and emissions. Many different criteria, often expressed in incommensurable units, such as weight, reliability or availability, cost, etc., must be taken into account in order to obtain an “optimal” design for such systems. The design of such vehicles is the result of a complex synthesis, requiring high degree of methodologies involving tradeoffs among multiple criteria. This paper proposes a probabilistic multicriteria decision analysis applied to the design of a hybrid electric vehicle. In particular, a Bayesian estimation approach is proposed and numerically illustrated, with the purpose of determining the optimal decision for the systems’ design under uncertainty. In the course of the numerical application, it is shown how such methodology is able to integrate and update available information, even if poor and uncertain, on the basis of new experimental data.

PurposeThe purpose of this paper is the investigation of the main aspects of optimal reliability allocation with respect to the design of hybrid electric vehicles. In particular, with reference to the hybrid electric vehicle propulsion system, the problem of data uncertainty, due to a scarce knowledge of the components' reliabilities, is taken into account. This problem is crucial for new technology systems and it is faced with a Bayesian approach: components' reliabilities are considered as random variables, characterised in the paper by negative log‐gamma distributions.Design/methodology/approachThe main aspects of optimal reliability allocation with the design of hybrid electric vehicles are presented, pointing out the opportunity of a reliability evaluation in the planning stage.FindingsThe topic of a series hybrid vehicle reliability is addressed, nevertheless results can be easily extended to the parallel configuration. In particular, the opportunity of a reliability evaluation of the propulsion system in the design stage is highlighted, mainly when new technology components are involved.Originality/valueThe value of the paper consists in the methodology allowing to express the system reliability uncertainty as a function of component uncertain data. Then, as far as concern the practical implications, the optimal allocation of the components' reliabilities can be efficiently performed, in order to minimise the system total cost respecting a prefixed value of the system reliability. In the final part of the paper, a numerical application, related to a series hybrid electric vehicle propulsion system, is presented to show the feasibility of the approach.

The aim of this paper is to show the connections among uncertainty, information and human knowledge to develop methodologies able to support actions for measure and control of complex processes, and to propose new model to represent human hazard rate. The interest to human reliability analyses (HRA) arose for nuclear applications, observing that 50‐70 per cent of reported failures on operating systems were human‐induced. Since the middle of 1980, methods and tools of HRA have been transferred former to military weapons systems, latter to aviation designs and operations. At present, HRA, which consider human performance and human reliability knowledge, must be an integral element of complex system design and development. In this paper, system reliability function is carried out as a function of technological, information and human components, evidencing how human element affects the whole system reliability. On the basis of consideration that human errors are often the most unexpected and then the least protected, and subject to many random factors, an analytical model is proposed, based on a conditional Weibull hazard rate with a random scale parameter, for whose characterization the log‐normal, gamma and the inverse Gaussian distributions are considered. The aim of this model is to take into account random variability of human performances by introducing a random hazard rate.

In the paper, a general analytical method for the probabilistic evaluation of power system transient stability is discussed and a new statistical inference approach for this evaluation is proposed. In particular, the transient stability probability (TSP) is defined and evaluated by taking into account the random nature of both the system loads and the fault clearing times (FCT). The paper is focused upon the aspect of statistical estimation of the TSP—a topic generally neglected in literature—on the basis of the obvious consideration that the parameters affecting the TSP (e.g. mean value and variance of loads, FCTs, etc.) are not known, but must be estimated. New properties of point and interval estimations of the TSP are derived and, in particular, an efficient “lower confidence bound” for the TSP estimation is proposed, based upon a suitable Beta probability distribution. In order to show the feasibility of the proposed approach, a numerical application to the Cigre test network is illustrated. Moreover, extensive Monte Carlo simulations to evaluate the estimator efficiency are performed. In the final part of the paper, also a practical example of possible application to the optimization of system design is illustrated. The application of the method is illustrated and performed by using the potential energy boundary surface method, but the estimation results hold their validity irrespective of the method adopted for the transient stability problem formulation and resolution.

Series capacitive compensation in electrical power systems is generally recognized as a very economical and powerful means for increasing the transmission capability of long‐distance transmission lines, resulting in relevant technical advantages in power system behavior: increased steady‐state and transient stability margins, reduced voltage drop in receiving systems during occurrence of severe contingencies and reduction of transmission losses. In this paper, a general method for choosing the series compensation degree is proposed, focusing the attention on the transient stability aspect. The approach, based upon a probabilistic framework, allows to properly select – at the design stage – the optimal degree of series compensation in order to contain the instability risk at an acceptable value. The transient stability problem is formulated by using the transient energy function method. In order to show the feasibility of the proposed approach, a numerical application to the Cigre test network is performed in the final part of the paper.

Estimating the useful life of power system components is of considerable importance both in planning, to properly assess the costs, and in the management of such systems, to effectively schedule maintenance programs. The paper discusses the reliability of the insulation of power system components on the basis of well-known life models of such devices, with the aim of parametric statistical estimation of such models. With particular reference to insulated cables for traction systems under distorted regime, an experimentally-based and commonly adopted mathematical relationship between the lifetime of a given insulation affected by non-sinusoidal voltage and the levels of the harmonic voltages applied to such insulation is adopted, that is of the kind of the well-known Inverse Power Model (IPM). A Bayesian inference method for the estimation of the above model is illustrated, when the Gamma Exponential prior distribution - a new model here proposed - holds for the basic parameters of the above Inverse Power Model. The performance of these estimators are empirically analyzed through extensive numerical simulations under a wide range of parameter values. All the results show the feasibility and efficiency of such Bayes estimation, especially for very small sample sizes, as requested for the above applications.

An accurate wind power generation evaluation requires an adequate probabilistic description of the wind source which takes also into account the correlation among wind speeds at different wind generators installation sites. The problem has been addressed through the employment of several models for wind speed stochastic variable, i.e. Weibull, Lognormal, Gamma, Burr, Rayleigh, extending a previous work. Indeed different model assumptions, although all able to provide a global good fitting of the wind speeds data, may lead to quite different estimates of producibility. This is theoretically crucial both at planning and operating stage when the electrical infrastructures (e.g. transmission lines, transformers, etc.) devoted to transfer green energy have to be designed and managed. Indeed, the paper will demonstrate that while at planning stage a different probabilistic description of the wind resource does not affect in significant manner the transmission line design, at the operating stage instead different assumptions on WS distribution lead to different measures, in probabilistic terms, of the their loadability. This is accomplished by comparing the distribution quantiles of the entire wind farm producibility, estimated through a proper methodology of wind speeds simulation based upon an extensive Monte Carlo procedure.

The rapid growth in the use of wind energy calls for a more and more careful representation of wind speed probability distribution, both for identification and estimation purpose. In particular, a key point of the above identification and estimation aspects is the one of representing the extreme values of wind speed probability distributions, which in the "static" case, as defined in the paper, may be expressed in terms of "extreme upper quantiles". This topic has indeed brought about an increasing number of studies in the last years, both for wind energy production assessment and also in risk and reliability analysis. Concerning the aspect of energy production, it is well known that a great sensitivity exists in the relationship between wind speed extreme upper quantiles and the corresponding wind energy quantiles. Concerning risk and reliability analysis, the extreme wind speed values characterization is useful for a proper understanding of the destructive wind forces which may affect mechanical tower reliability, and consequently a proper choice of the “cut off” wind speed value. The above modeling is however difficult due to uncertainty in wind speed probability distributions. For this purpose, the paper proposes a novel Bayes approach for the estimation of the probability that wind speed is lower than a prefixed extreme value. The basic feature of the method is its being not dependent on the underlying wind speed model, be it the classical Weibull distribution, or alternative models such as the recently adopted Log-logistic or Burr distributions. A large set of numerical simulations are performed in the last part of the paper, in order to illustrate the feasibility and efficiency of the above method of estimation, especially when compared to the classical Maximum Likelihood method.

In the paper, the problem of uncertain data in reliability analysis of complex systems is examined. The analysis is addressed to system reliability assessment with imprecise knowledge of component reliabilities, an item becoming more and more important for systems affected by considerable technological change. Starting from component uncertain data, a new method for the whole system reliability uncertainty description, based upon a Bayesian approach and not depending on the reliability model of each component, is proposed. The reliability value of each component is considered as a random variable described by a Negative Log‐Gamma distribution. The proposed methodology makes it possible to compute the features of system reliability uncertainty (i.e. reliability distribution, confidence intervals, etc.) as functions of component uncertain data, thus characterizing the propagation of uncertainty from the components to the system. Numerical applications, related to a test system, are presented to show the validity of the method and its “robustness”, i.e. it is shown that it yields satisfactory results also when component reliabilities are not Negative Log‐Gamma but Beta distributed.

In recent years, the wide utilization of renewable energy sources has brought some problems for the dispatching of generated energy. In recent technical papers are proposed some solutions to improve the renewable energy dispatching: i. e. it is possible to set up an optimal strategy to recharge electric vehicles, or use batteries storage systems, supercapacitors or fuel cells. The aim of this papers is the application of multicriteria analysis to design batteries energy storage systems. By means of Bayesian inference is possible to take into account some uncertainties of batteries characteristics and estimate the best choice for the design of the storage system.

The reliability of electrical energy storage systems, related to the various mobile applications, needs to be carefully measured. The Italian research centre ENEA's "Low Environmental Impact Vehicles" Laboratory has started experimental testing activities on storage systems, as illustrated here with reference to acid batteries reliability performances. In this paper, starting from the results of the campaign of laboratory tests, a probabilistic method for describing the distributions of battery lifetime is presented. The approach is based upon the Rakhmatov and Virudhula battery model. In the final part of the paper, numerical tests show that the battery expected lifetime can be well described by models, such as the Inverse Weibull or the Lognormal, with decreasing hazard rate for large mission times, a fact which has been sometimes observed in literature as a paradox. Some considerations with respect to parameter estimation of the reliability models close the paper. © 2013 IEEE.

Wind speed (WS) probability distribution identification and estimation are the object of an increasing number of studies, especially related to the need of wind energy production evaluation. In this framework, the paper highlights the characterization of extreme WS quantiles, whose values and estimates are very sensitive to the assumed distributional form. This is a crucial issue not only for wind energy production assessment, but also in risk and reliability analysis. For the above purposes, the Lomax model is theoretically deduced and analysed: this model, indeed, well represents the typical “heavy tails” in WS probabilistic distributions arising from field data. A proper Bayes approach for the estimation of both the Lomax survivor function and of the above quantiles is analyzed. A large set of numerical simulations has been performed, and some typical subsets of them are shown to illustrate the efficiency of the estimates, showing excellent results.