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Abstract

Содержание

Introduction

Using the concept of the cross section, minimum cross section, method for estimating the reliability of structurally complex systems whose elements can be in three inconsistent states: operable, failed (type failure open circuit and failure of type short circuit) is proposed.

An example of calculation is given.

Keywords: system, minimum cross section, complex scheme, probability, load node.

1. Formulation of the problem

To assess the reliability of complex non-renewable systems, the elements of which can be in three incompatible states, the minimum cross section method is using.[1–2].

This method helps to evaluate the reliability of systems whose elements can be in three inconsistent states (operable, inoperative – type failure: open circuit, inoperative – type failure: short circuit), which allow to take into account failures in the operation of protective switching devices and significantly improve the accuracy of calculations.

2. Purpose, analysis of studies and publications

Objective: improvement of the method for assessing the reliability of complex non-renewable systems, the elements of which can be found in two incompatible states.

To assess the reliability of complex non-renewable systems, the elements of which can be in three incompatible states, three methods are known based on the use of: decomposition of a complex structure with respect to a base element, applying the triangle-star and star-triangle transformations, and using the algebra of tuples.[3–6] All proposed methods are rather difficult to use for engineering calculations, so the improvement of the minimum cross section method will allow to obtain a lower evaluation of the probability for failure-free operation of the systemwith considering two types of incompatible failures of its elements.

3. Research results

We denote by Pi the probability of failure-free operation of the i-th element of the system, qOi – is the probability of failure in the i-th element of the open circuit type, and by qSi – is the probability of failures in the i-th element of the short circuit type. These three states constitute a complete group of incompatible events.

2 (3.1)

The index O и S n the formula indicate that element faults that lead to faults such as open circuit, or faults of the type short circuit, respectively.

In that case, if the elements of the system are subject to two types of incompatible failures: failure such as an open circuit or short circuit failure then the probability of its failure during the time t can be determined as follows [4]:

1 (3.2)

where λOi,λSi – are the failure rate constants for the i-th element, taking into account its failures such as open circuit and short circuit type, respectively;

t – current time of the i-th element of the system.

For non-renewable systems, the elements of which can be in three incompatible states, the formula:

3 (3.3)

where RH – is the lower estimate of the probability that such random events will not occur that result in a disconnection or an end-to-end fault current between the Input and Output nodes of the replacement circuit;

QOH,QSH – is the lower estimate of the probability that a disconnection will occur or an end-to-end fault current flows between the Input and Output nodes of the replacement circuit, respectively.

By a simple scheme of substitution of a system, we will take a one whose elements can be connected: sequentially, parallel, sequentially parallel, or parallel-sequential.

For a simple scheme of substitution of a system consisting of n logical connection of elements, failure of the type open circuit any of the n elements (i = 1...n), leads to disconnection between the Input and Output.

If the probability of failures of the elements of the substitution circuit qOi is given, then the probability of disconnection between the nodes Input and Output QO is defined as follows:

4 (3.4)

If the probability of failure of the elements of the substitution circuit qSiis given, then the probability QO of that between the Input and Output nodes of the replacement circuit will pass the through fault current, we find using the formula:

5 (3.5)

Using the formulas (3-5), we find for the system that consists of n logically sequential connection of the elements of the probability Rn that such random events will not pass, resulting in a disconnection or passing through fault current between the nodes Input and Output of the replacement circuit:

6 (3.6)

For a simple scheme for replacing a system that consists of m ogically parallel elements, a failure of the type open circuit in each of the j elements (j = 1...m), causes the connection between the Input and Output.

If the probability of failure of the elements of the substitution circuit qOiis given, then the probability QO that the connection between the Input and Output nodes will be broken is found using the formula:

`
7(3.7)

If the probability of failures qSj, (j = 1...m) of the elements of the substitution circuit is given, which consists of m logically parallel elements, then QS – is the probability that an end-to-end fault current flows between the Input and Output terminals of the replacement circuit, formula:

8 (3.8)

Substituting formulas (7-8) into formula (3), we find that for a system that consists of m logically parallel elements, the probability Rm is that such random events do not pass, resulting in a break connection or an end-to-end fault current between the nodes Input and Output of the replacement circuit:

9 (3.9)

Formulas (7) and (10) are valid if the following conditions are satisfied: qOi  +  qSi < 1, qOj  +  qSj < 1 and obtained in various ways [4–5].

Under a complex scheme of substitution of the system, we mean a one that includes at least one group of elements connected in the form of a bridge structure [11].

4. Example

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Figure 1  – System replacement scheme

For the scheme of substitution of the system in Fig. 1 the following initial data are given :

11

Define the lower bound RH of the fact that there is no disconnect or an end-to-end fault current between the point Input and Output of the substitution circuit.

Using the notion minimum section, the circuit of Fig. 1, the minimum section scheme, taking into account failures of type open circuit/q> will take the form:

20

Figure 2  – The scheme minimum section when accounting for the failure of elements of the type open circuit

Using the replacement circuit of Fig. 2, formulas (4),(7) we find:

13

We substitute in the obtained formula the values: qOi (i = 1...7), given in the example condition, we obtain:

14

Using the notion minimum section, the circuit of Fig. 1, the minimum section scheme, taking into account failures of type short circuit/q> will take the form:

21

Figure 3  – The scheme minimum section when accounting for the failure of elements of the type short circuit

Using the replacement scheme obtained in Fig. 3, formulas (5), (8), we find the lower bound QSH of the probability that an event will occur in which an end-to-end fault current passes between node Input и Output of the replacement circuit:

16

Using the initial data of the example, we substitute them in the obtained formula:

17

The probability that there will be no disconnection or the through fault current between node Input и Output of the substitution circuit will be found using formula (3):

18

We substitute the obtained values QOH and QSH in the formula and define:

19

A similar result was obtained in solving this problem using an exact calculation method using the method of decomposition of a complex structure by the basis element [3].

Conclusions

1. An engineering technique for calculating the reliability of complex non-renewable systems is proposed, the elements of which can be in three incompatible states.

2. The accuracy of the proposed methodology for highly reliable systems (the probability of failure of elements of the substitution scheme is less than or equal to 0.1) is not inferior to the exact, tried and tested methods.

References

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