Modeling reliability of power systems substations by using stochastic automata networks

https://doi.org/10.1016/j.ress.2016.08.006Get rights and content

Highlights

  • We present the methodology to apply stochastic automata network formalism to create Markov chain models of power systems.

  • The stochastic automata network approach is combined with minimal path sets and structural functions.

  • Two models of substation configurations with different model assumptions are presented to illustrate the proposed methodology.

  • Modeling results of system with independent automata and functional transition rates are similar.

  • The conditions when total independence of automata can be assumed are addressed.

Abstract

In this paper, stochastic automata networks (SANs) formalism to model reliability of power systems substations is applied. The proposed strategy allows reducing the size of state space of Markov chain model and simplifying system specification. Two case studies of standard configurations of substations are considered in detail. SAN models with different assumptions were created. SAN approach is compared with exact reliability calculation by using a minimal path set method. Modeling results showed that total independence of automata can be assumed for relatively small power systems substations with reliable equipment. In this case, the implementation of Markov chain model by a using SAN method is a relatively easy task.

Introduction

Markov chain modeling has been applied in power system reliability modeling for a long time [1], [2]. It has a well-developed mathematical apparatus and can describe complex behavior of a system [3]. Two alternatives to Markov models are common in power system reliability modeling: methods based on the independence of system components (e.g., fault tree analysis, reliability block diagrams, etc.) and Monte Carlo simulation.

Methods which assumes total independence of system components are simpler, but even these techniques can be cumbersome, and classical methods, such as fault tree analysis [4] or generation of minimal set paths [5], require computer assistance and use of special algorithms. These methods can also be combined in modeling of Markov chains [6], [7].

A Monte Carlo simulation method is often applied in reliability modeling and has been used to simulate various types of objects, such as substations [8], power plants [9] or standard and composite generation and transmission systems [10], [11]. However, simulation of rare events (which is common in reliability modeling) requires special attention [12], [13], and Markov chain models tend to be more efficient if a system is not too large and highly reliable [14]. Moreover, since by nature Monte Carlo method is based on random experiments, it is beneficial to be able to compare its results to other methods, for which Markov chain modeling can be used [15], [16].

Markov chain modeling allows evaluating complex system behavior [17], [18]; however, the main drawback is rapid growth of system states and, consequentially, rising complexity. Therefore, most examples in Markov chain reliability modeling deal with relatively small state space [17], [19]. For a more complex system, special software can be applied [20], [21], but there is still a need for systematic approach, which would allow for simplified description of complex systems. Some proposed strategies are based on system decomposition into smaller independent subsystems [22], [23], but the problems might arise if subsystems are interdependent.

One of the methods suitable for complex Markov chain model creation is stochastic automata networks (SANs) formalism [24]. SAN formalism applies Kronecker algebra operations, which enables to store infinitesimal generator matrix in compact format; therefore, it is especially suitable for solving the problem of dimensionality. It also allows for systematic description of interaction between smaller subsystems which provides an exact solution. The SAN method was applied to different areas of research. For example, SAN formalism was used to evaluate availability of large-scale computer networks [25], in system theory, SANs were applied in creating the influence model [26]; in cell biology, SANs can be used to model ion channels [27], etc.

Kronecker algebra approach is not new in system reliability, but most examples deal with systems of independent components [28]. A more sophisticated use of Kronecker algebra with functional transitions in respect to system reliability is considered in [29], though it uses different terminologies than that of SAN formalism. In [30], theoretical k out of n system was specified by using SAN formalism. However, practical application of SANs in solving real-world reliability problems is still uncommon.

In [31], SAN formalism is used to specify reliability of cogeneration power plant substation. In this paper, it has been elaborated on these ideas to propose a methodology to model system reliability of power systems substations. System division into different automata and the use of arrowhead matrices are addressed in detail. We also consider the use of SANs together with classic reliability modeling techniques like minimal path sets, failure modes and effect analysis (FMEA) and structure functions. Reliability models of two standard configurations of substation under different model assumptions are created using the proposed methodology. The calculation of system measures and evaluation of independence of individual automata are also considered.

The main principles of Markov chain numerical modeling and SANs are introduced in Section 2, while the theoretical background on SAN formalism is presented in Section 3.

In Section 4, the application of SANs of real world power systems is addressed. In this chapter, the formation of individual automaton is considered due to preventative circuit breaker actions of power system. The proposed technique leads to automata, whose infinitesimal generators are arrowhead matrices. We demonstrated that arrowhead matrices and SAN formalism allow specifying various scenarios, which are common in reliability modeling.

This methodology is applied in two case studies. In Section 5 we present the model of sectionalized bus and in Section 6 – a ring bus substation configuration. Both reliability models are considered in detail under different assumptions, which can be easily implemented by the use of functional transition rates. Functional transition rates are used to specify preventative circuit breaker operation, shared load and repair capacity.

In Section 7, it is shown how SAN reliability modeling can be combined with minimal path set methods and structure functions for estimation of system availability.

In Section 8, the modeling results are presented. In Section 9, we discuss the implication of modeling results and the conditions, under which the independence of automata can be assumed. In this case, the implementation of SAN modeling becomes a much easier task.

Section snippets

Development of Markov chain model

In this paper we assume stationary analysis of homogenous irreducible continuous time Markov chain reliability models, with finite number of system states. Development of Markov chain model can be divided into three main stages:

  • 1)

    Defining the set of states of the system and possible transitions amongst them.

  • 2)

    Computation of steady-state probabilities.

  • 3)

    Computation of necessary probabilistic characteristics of the system, using steady-state probabilities.

The first step is model specification. For

Stochastic automata networks

One of the methods which allow creating large Markov chain models is stochastic automata networks (SANs) formalism. SANs allows storing infinitesimal generator Q in a compact form by using tensor (Kronecker) algebra operations.

Using SAN formalism the system is described as a few different automata which can interact among themselves. Each automaton is represented by a Markov chain, i.e., a set of states and possible transitions among them. If two automata interact, transition in one automaton

Power system reliability modeling by the use of SANs

The reliability models are created under the following three assumptions: 1) each item can be in one of two possible states – operating or failed; 2) a failed item is detected and repair is initiated immediately; 3) the repaired item is as good as new.

Reliability modeling of sectionalized bus configuration

In this chapter we present the reliability model of substation with sectionalized bus, presented in Fig. 1. We use the proposed methodology for system division into different automata, when each automaton is described by an arrowhead matrix. Three different models were considered: one with independent automata and two models with preventative failures and shared load. In both later cases functional transition rates will be used.

Reliability modeling of ring bus substation configuration

In this chapter we consider a reliability model of the ring bus configuration (see Fig. 2). We use the same approach to create SAN reliability model – system division into individual automata is performed according to failure modes and effect analysis, while each automaton is described by an arrowhead matrix with functional transition rates.

The circuit breaker operation under the repair is presented in Table 3.

Estimating system measures of SAN reliability models

Stochastic automata networks approach can be easily implemented together with classical reliability methods such as minimal set paths and structure functions.

Modeling results

We calculated probabilities Pr(A1) and Pr(A2) for sectionalized bus and ring bus substation configurations with two different sets of failure and repair rates. The first set of estimated model parameters (see Table 4) is statistical data collected by Lithuanian Energy Institute. The second set of parameters was chosen from [38]. In this case we chose higher failure rates and average repair times, in order to get more visible difference between different reliability modeling methods.

Evaluating independence of automata

Modeling results showed a relatively small difference between SANs of independent automata and SANs with functional transition rates.

It is obvious that modeling system as a SAN of independent automata is a much easier task than SAN with functional transition rates. Assuming automata independence means not only an easier model solution by (7). System specification is also much easier, because in this case the definition of functional transition rates can be discarded. Therefore it would be

Discussion

The paper presents the use of SAN formalism in system reliability modeling. The proposed strategy of system division into individual automata allows for simplification of model specification. The use of decomposition techniques in Markov chain reliability modeling is not new and is probably unavoidable for a large system. The applications differ in types of modeled systems (e.g., nuclear power plants [23] or electrical bus networks [22]) and used methodology (Markov models can be specified

Acknowledgments

This work was supported by Grant (ATE-no. 04/2012) from the Research Council of Lithuania.

References (42)

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