A Method to Simplify the Markov Model of Protective Systems

Protection system in a power system domain plays a vital role as watchmen to safe guard power system apparatus in the event of faults. The integrity and effectiveness of these watchmen is heavily dependent on the reliability of the components that makes up the system. One of the main components in a protection system is protective relays that provides decision making element for correct protection and fault clearing operations. Failure of the protection devices may reduce the reliability of the power system protection that will impact the overall performance of the power system. Hence it is imperative for power utilities, such as Tenaga Nasional Berhad (TNB) to assess the reliability of protective relays to assure it will perform its intended function without failure. This paper will discuss the application of reliability block diagram (RBD) to assess the reliability and availability of a typical 132kV transmission overhead line protection system using historical failure data of protective relays.

This paper shows the span of results which can be obtained by modeling a system's behavior by stochastic processes and demonstrates practical rules for employing Markov and semi-Markov models. The introduction summarizes several methods for reliability analysis and gives the advantages and drawbacks of four methods: Markov processes, semi-Markov processes, supplementary variables, the method of stages. The remainder deals with reliability and availability modeling of a 2-unit redundant computer system. There are a) two types of maintenance: corrective (c. m.) and preventive (p. m.), and b) two system parameters: coverage, and an increased failure rate when one unit is under repair or inspection. Approximate expressions for reliability, mean time to failure, and asymptotic availability show the effects of the system parameters as well as of the shapes of the Cdf's of the times related to maintenance actions. For c.m., Markov modeling is a good approximation. For p.m., Markov modeling is a rough approximation; one can go to semi-Markov models or to the method of stages. Lastly, an approximate expression is given for the mean inspection interval which maximizes reliability and availability for p.m.

Traditional time-based reliability assessment methods evaluate the reliability of a multi-state system (MSS) from a population or a statistical perspective that the reliability of a system is computed purely based upon historical time-to-failure data collected from a large population of identical components or systems. These methods, however, fail to characterize the stochastic behaviors of a specific individual system. In this paper, by utilizing system-level observation history, a dynamic reliability assessment method for MSSs is put forth. The proposed recursive Bayesian formula is able to dynamically update the reliability function of a specific MSS over time by incorporating system-level inspection data. The dynamic reliability function, state probabilities, and remaining useful life distribution of an MSS in residual lifetime are derived for two common cases: the degradation of components follows a homogeneous continuous time Markov process, and a non-homogeneous continuous time Markov process. The effectiveness and accuracy of the proposed method are demonstrated via two numerical examples.

Due to the growing use of photovoltaic (PV) systems in recent years, the reliability of these systems as one of the most important issues regarding durability and correct performance is important. This paper presents a comprehensive assessment of the reliability of Residential PV System (RPVS) in various conditions based on the Markov model. The systems taken into considerations consist of PV modules, a DC-DC converter, an inverter, an inverter controller, and a maximum power point tracking (MPPT) controller. These systems are considered repairable. To evaluate the effect of the three-phase boost converter and the PV modules on reliability, these systems have been considered towards this specific perspective. Furthermore, the impact of various factors on the reliability of the system components and the entire system is evaluated. Simulation results demonstrate the capacity of the proposed method to assess and comparison of the reliability of different RPVS. Also, the results indicate the impact of various factors such as temperature on it.

The contribution first describes the implementation of Continuous-Time Markov Chains (CTMC) in the language Modelica; this enables the simulation by powerful solvers making typical Functional Safety applications numerically more efficient and reliable than with classically used Discrete-Time Markov Chains (DTMC). Secondly, the CTMC are extended to Multi-Phase Markov Chains (CTMC MP ) enabling the inclusion of maintenance (restoration) actions as well as online diagnostics, which is finally elaborated to the online prediction of an individual system's safety quantities.

Dependability is a subject of great importance in the development of critical systems and may be quantified in terms of various factors, such as reliability, maintainability and availability, whose significance may vary between different applications. A generic technique based on the Markov model is proposed in this paper, using fuzzy theory, for the reliability and safety assessment of fault-tolerant computational systems, composed of replicated modules. As the Markov model requires some parameters which may not be precisely known during the calculation, this method allows uncertainty-based parameters that are represented as fuzzy numbers. As a consequence, the reliability and the safety time response are composed by several curves which have associated degrees of confidence. This result may be used to estimate the reliability and safety value range at a given instant of time where each value has an associated possibility degree in which the developer believes the system has.

The Markov method is a common reliability assessment method. It is often used to describe the dynamic characteristics of a system, such as its repairability, fault sequence and multiple degradation states. However, the "curse of dimensionality", which refers to the exponential growth of the system state space with the increase in system complexity, presents a challenge to reliability assessments for complex systems based on the Markov method. In response to this challenge, a novel reliability assessment method for complex systems based on non-homogeneous Markov processes is proposed. This method entails the decomposition of a complex system into multilevel subsystems, each with a relatively small state space, in accordance with the system function. The homogeneous Markov model or the non-homogeneous Markov model is established for each subsystem/system from bottom to top. In order to utilize the outcomes of the lower-level subsystem models as inputs to the upper-level subsystem model, an algorithm is proposed for converting the unavailability curve of a subsystem into its corresponding 2×2 dynamic state transition probability matrix (STPM). The STPM is then employed as an input to the upper-level system's non-homogeneous Markov model. A case study is presented using the reliability assessment of the Reactor Protection System (RPS) based on the proposed method, which is then compared with the models based on the other two contrast methods. This comparison verifies the effectiveness and accuracy of the proposed method.

Existing methods of automatic creation for Markov models were limited to systems with identical channels. This paper focuses on how to automatically generate the Markov models of systems with non-identical channels. Each channel is characterized by its own failure rate. The system state matrix is defined and the failure criterion of the state is given. The automatic creation framework for system with non-identical channels is introduced in flow diagram. A computer program based on the automatic creation framework is developed. Many safety related factors, such as failure modes, restorations, common cause failures, self-diagnostic and repair rates are included in this framework. Finally, an example of 1oo2D system is studied, which shows that the average probability of failure on demand (PFDavg) of the system is closely related to the diversity of the channel’s failure rate. The results show that the diversity of the channel should not be neglected when building the precise Markov model of the safety system.

SummarySleep is a complex biological process regulated by networks of neurons and environmental factors. As one falls asleep, neurotransmitters from sleep–wake regulating neurones work in synergy to control the switching of different sleep states throughout the night. As sleep disorders or underlying neuropathology can manifest as irregular switching, analysing these patterns is crucial in sleep medicine and neuroscience. While hypnograms represent the switching of sleep states well, current analyses of hypnograms often rely on oversimplified temporal descriptive statistics (TDS, e.g., total time spent in a sleep state), which miss the opportunity to study the sleep state switching by overlooking the complex structures of hypnograms. In this paper, we propose analysing sleep hypnograms using a seven‐state continuous‐time Markov model (CTMM). This proposed model leverages the CTMM to depict the time‐varying sleep‐state transitions, and probes three types of insomnia by distinguishing three types of wake states. Fitting the proposed model to data from 2056 ageing adults in the Multi‐Ethnic Study of Atherosclerosis (MESA) Sleep study, we profiled sleep architectures in this population and identified the various associations between the sleep state transitions and demographic factors and subjective sleep questions. Ageing, sex, and race all show distinctive patterns of sleep state transitions. Furthermore, we also found that the perception of insomnia and restless sleep are significantly associated with critical transitions in the sleep architecture. By incorporating three wake states in a continuous‐time Markov model, our proposed method reveals interesting insights into the relationships between objective hypnogram data and subjective sleep quality assessments.

Markov models are commonly used for unavailability analysis of redundant systems. However, due to the exploding states of Markov models for redundant systems, the states need to be merged to simplify the computation, which is called micro-Markov models. However, how to derive the failure rates and repair rates of the newly developed micro-Markov models has not been studied thoroughly. Therefore, this paper proposes detailed explanations and rules to derive the static unavailability by the micro-Markov models for thek-out-of-n:G systems with multiple failure modes. Firstly, two properties about applying the Markov models to the repairable system with independent multiple failure modes are presented. Based on these two properties, two rules are proposed for implementing the micro-Markov models. The micro-Markov models provide the exact same results for the repairablek-out-of-n:G system with multiple independent failure modes and repair mechanisms and approximate results for systems with multiple hybrid failure modes. A case study of safety integrity verification for safety instrumented systems is provided to illustrate the application of the proposed method. The conceptual comparison and numerical examples demonstrate the reasonability and usefulness of the proposed micro-Markov models.

This chapter concentrates primarily on the general aspects of occupancy times when studying the reliability of systems that are described by finite-state continuous-time Markov or semi-Markov processes. Occupancy times for Markov models for system reliability are discussed together with a brief comment on reward processes. Similar problems, focused on joint distributions and cumulative distributions, are considered for semi-Markov models. Similar distribution problems for time interval omission situations are given briefly in the chapter. Some numerical examples are presented to illustrate the results obtained. The chapter also presents some general results for certain probability distributions of occupancy times in terms of Laplace-Stieltjes transforms. The advantages are that the results are elegant and moments can usually be easily obtained by differentiation. The chapter points out the similarity of some Markov and semi-Markov models for reliability and ion-channel theory. Controlled Vocabulary Terms laplace transforms

One of the main tasks in organizing the protection of computer networks from cyber attacks is to ensure the management of the protection system. Management includes obtaining information about the state of the protection system, analyzing the information received, developing and executing a solution, as well as implementing control actions on all network protective components. The paper discusses possible variants of models of the control system for the computer network protection system in the context of cyber attacks. Depending on the peculiarities for functioning of the control system of the computer network protection system, one or another controllability function can be selected. The parameters of the model should be determined in advance, so that the model adequately describes the activities of the control system of this class in solving problems in conditions of cyber attacks. The indicator for the effectiveness of the control system of the protection system is substantiated - the probability of an error-free and timely decision-making in time, which determines the cyber resilience of the network as a whole. The adequacy of the developed models has been experimentally confirmed.

Pricing American Drawdown Options under Markov Models

Aggregated Markov processes related by similarity transformation are equivalent in that they cannot be distinguished by steady-state experiments. We derive an explicit formula for the set of all detailed-balance preserving similarity transformations between such continuous time Markov chains with N states. The matrices that define the allowed similarity transformations are found to be a simple non-linear function applied to almost any element of the special orthogonal group in N dimensions. Since a model is identifiable only if there is no similarity transformations to an equivalent model, we expect this result to prove useful in the theory of identification of aggregated Markov chains, an enterprise of growing importance as more and more single molecules yield to observation.

Hardware systems are present in many fields of human activity. Markov models are sometimes used in hardware reliability, availability, and maintainability (RAM) modeling. They are especially useful in situations in which the system we want to analyze may be modeled with several states through which the system evolves, some of them corresponding to ON states, the rest to OFF states. We provide here RAM analyses of such systems within a Bayesian framework, addressing both short-term and long-term performance.