Redundancy analysis for repairable multi-state system by using combined stochastic processes methods and universal generating function technique

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Abstract

This paper discusses a type of redundancy that is typical in a multi-state system. It considers two interconnected multi-state systems where one multi-state system can satisfy its own stochastic demand and also can provide abundant resource (performance) to another system in order to improve the assisted system reliability. Traditional methods are usually not effective enough for reliability analysis for such multi-state systems because of the “dimensional curse” problem. This paper presents a new method for reliability evaluation for the repairable multi-state system considering such kind of redundancy. The proposed method is based on the combination of the universal generating function technique and random processes methods. The numerical example is presented to illustrate the proposed method.

Introduction

Redundancy problem in a multi-state system (MSS) is much more complex than that in a binary-state system. Some redundancy problems for MSSs were investigated in [1], [2], where typical parallel connections of multi-state state components, k-out-of-n multi-state system and corresponding extensions were discussed and summarized. Recent research has focused on the reliability evaluation and optimization of MSS [3], [4], [5], [6]. However, for the MSS, there is an important type of redundancy that has not existed for binary-state systems and has also not been investigated till now in the framework of MSS reliability analysis.

For MSS, it is typical that after satisfying its own demand one MSS can provide its abundant resource (performance) to another MSS directly or through the interconnection system (that can also be multi-state). In this case, the first MSS can be called as the reserve MSS and the second one as the main MSS. In general case demand for the reserve and the main MSS can also be described by two different independent stochastic processes. Typical examples of such kinds of MSS include power generating systems where one power station can assist another power station to satisfy demands, oil and gas production and transportation systems, computing systems with distributed computation resources, etc. Such multi-state structure with redundancy may be treated as MSSs with mutual aid or structure with interconnected MSSs. This type of redundancy is common enough for MSS. However, by using existing methods, it is very difficult to build the reliability model for a complex repairable MSS considering redundancy and to solve it for obtaining corresponding reliability indices.

In practice each multi-state component in a MSS can have different numbers of performance levels. This number may be relatively large – up to ten and more [7]. Even for relatively small MSSs consisting of 3–5 repairable components the number of the entire system states will be significantly great (10,000 or more). In general, for a MSS consisting of n repairable components, where each component i has ki different capacity levels, there are K=i=1nki system states. This number may be very large and increase dramatically with the increased number of components. For interconnected MSSs, the problem can be more serious. For such MSS, enormous efforts have to be performed to develop a stochastic process model and solve it (in order to obtain corresponding reliability indices) by using traditional straightforward methods. However, it is difficult to develop the stochastic process model for such a complex interconnected MSS. Determining all system states and transitions correctly is an arduous job. Moreover, it can challenge the available computing resources. If the random process is identified as a Markov process, the system state probabilities can be obtained by solving K=i=1nki differential equations. Therefore, in practice only long-term reliability analysis is performed to assess reliability of such systems, which is based on steady-state probabilities. In such case instead of differential equations only algebraic equations will be solved. Therefore, short-term transient dynamic behavior of a MSS is out of consideration. In general case, such an approach will lead to decreased accuracy.

In order to use multi-state models for all components and to avoid decreased accuracy for reliability analysis, a special technique is proposed in this paper. The technique is based on the combination of universal generating function (UGF) and random processes methods. The UGF was primarily introduced in [8] in order to reduce the MSS's computational complexity. Ref. [9] was the first book where a special chapter was devoted to the analysis of multi-state systems. The mathematical foundations of the technique were extended in [10], [11]. An updated comprehensive description of the UGF technique with many technical applications can be found in [1], [12]. Some recent research includes [13], [14]. In the presented paper, the combined UGF and random process method is presented for the reliability assessment of interconnected repairable MSSs. This combined method was suggested in [1], and extended in [15], [16], where it was applied for dynamic analysis of single MSS with constant demand. In the presented paper, the method is developed for MSS redundancy analysis and applied for interconnected MSSs with independent Markov stochastic demand processes considering mutual aid.

In this paper, the problem formulation is considered in Section 2. In Section 3, the UGF-based reliability evaluation technique for an interconnected generating system is developed. The basic steps of the proposed approach and corresponding algorithm are presented in Section 4. The reliability measures for the system are discussed in Section 5. An illustrative example is shown in Section 6.

Section snippets

Problem formulation

According to the generic MSS model any system component i in MSS can have ki different states corresponding to the performance levels, represented by the set gi={gi1,,giki}. The current state of the component i and the corresponding value of the component performance level Gi(t) at any instant t, are random variables. Gi(t) takes values from gi: Gi(t)∈gi. Therefore, for the time interval [0,T], where T is the MSS operation period, the performance level of component i is defined as a

Model descriptions

The suggested method is based on combined UGF and random processes technique. As it was mentioned in the introduction of this paper, UGF technique was primarily introduced by Ushakov in [8] and then was described by Ushakov in [9], [10], [11]. Recently, UGF technique is widely presented in the literature and the technique fundamentals – main definitions and properties – will not be described here. If it is necessary, one also can find more detailed description in [1], [12].

Algorithm of UGF computation for entire MSS

The procedure of the UGF computation for the entire MSS considering redundancy is graphically presented in Fig. 3.

The procedure consists of the following steps:

  • 1.

    Based on reliability data (failure and repair rates) for all components in MSSm and MSSr, individual UGFs (3), (10) for all components can be obtained by solving the corresponding systems of differential equation (1).

  • 2.

    Based on structure functions ψm, ψr and individual UGFs for all components in MSSm and MSSr UGFs Um(z, t), Ur(z, t)

Reliability measures computation for entire MSS

When the UGF (expression (23)) of the entire interconnected MSS is obtained, the reliability measures for the system can be easily evaluated.

The entire MSS availability A(t) at instant t>0 can be evaluated asA(t)=j=1MMSSpj(MSS)(t)1(gj(MSS)0),where 1(True)≡1, 1(False)≡0.

The expected performance deficiency at instant t>0:PD(t)=j=1MMSSpj(MSS)(t)·(-1)min(gj(MSS),0).

For a given period T, the expected accumulated performance deficiency can be calculated asPD=0TPD(t)dt,where PD(t) can be

Illustrative example

The presented technique is used to evaluate an interconnected electric generating system, which consists of two electric generating systems connected by a tie line as shown in Fig. 4. System 1 consists of two 360 MW coal units, one 220 MW gas unit, and one 220 MW oil unit respectively. System 2 consists of one 360 MW coal unit and one 220 MW gas unit, respectively. The corresponding parameters for these units [7] are shown in Table 1, Table 2, Table 3 in the appendix. The coal unit, gas unit, and

Conclusions

In the paper was considered an important type of redundancy in MSS that has not existed in binary-state systems. Traditional methods applied to reliability computation for such systems are usually not effective enough because of dimension curse. A new approach to evaluate the dynamic reliability of MSS with such redundancy is suggested. The approach is based on a combination of the UGF technique and the random processes method and takes into account multi-state models for all system components.

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