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Abstract

Content

Introduction

Elements with three incompatible states: healthy, a failure type open circuit and a failure type short circuit can be distinguished in the systems: power supply, gas supply, water supply, etc. Analogs of elements with three incompatible states in such systems can be protective switching devices, taps, valves of various types, stop valves, plugs and other types of flow interrupters (electric power, gas, liquid, information, etc.), for which are inoperable the flow is not interrupted (short circuit) or is not transmitted (open circuit) [3].

In those cases, if it is necessary to increase the reliability of the designed system without changing the reliability of its component parts, it is usual to introduce redundant (backup) elements or change the block diagram of the network. For systems consisting of elements that can be in three incompatible states, the introduction of redundant elements with three states can not only not increase its reliability, but even reduce it. Everything will depend on the ratio between the different types of failure of the circuit element, its configuration and the number of backup elements or their groups.

1. Theme urgency

The development of new and improved existing methods for assessing the reliability of nonrecoverable systems, whose elements may be in three incompatible states, is an important scientific problem, the solution of which will significantly improve the accuracy of prediction of their reliability, both at the design stage and at the stage of reconstruction and operation.

2. The purpose and objectives of the study, the planned results

The purpose of the master's work is to review one of the existing methods for assessing the reliability of unrecoverable, complex in structure systems, elements of which can be in three incompatible states – the Minimum Sections method.

The main objectives of the study:

  1. A review of the literature and the definition of existing methods for assessing the reliability of unrecoverable, complex in structure systems, whose elements may be in three incompatible states.
  2. Parsing the Minimum Sections method.
  3. The application of this method of calculation on a specific example.

Planned results:

  1. Determination of the accuracy of the calculation of the reliability of unrecoverable structurally complex schemes for t ≤ 0.1 year, the proposed method using schemes minimum sections.
  2. Comparison of calculation accuracy with other known methods for calculating the assessment of the reliability of nonrecoverable systems, whose elements may be in three incompatible states.

3. Practical section

The equivalent circuit of a complex system, to assess its reliability is shown in Fig. one:

Схема замещения системы

Figure 1 – System replacement scheme
(animation: 8 frames, 3 reiteration cycles, 75 kilobytes)

For the equivalent circuit elements (Fig. 1), the following failure rates are set: λo1 = 0,58 year−1; λs1 = 0,34 year−1; λo2 = 0,42 year−1; λs2 = 0,25 year−1; λo3 = 0,75 year−1; λs3 = 0,25 year−1; λo4 = 0,5 year−1; λs4 = 0,34 year−1; λo5 = 0,75 year−1; λs5 = 0,25 year−1; λo6 = 0,34 year−1; λs6 = 0,25 year−1; λo7 = 0,67 year−1; λs7 = 0,5 year−1; λo8 = 0,5 year−1; λs8 = 0,42 year−1.

Determine the lower estimate Rн of the fact that during the time t = 0.1 year there will be no disconnection or pass-through emergency current between the nodes input and output of the equivalent circuit.

Using formulas (2) and (3), the initial data of the example, for t = 0.1 year we find qoi, qsi (i = (1÷8)): qo1 = 0,0554; qs1 = 0,0325; qo2 = 0,0406; qs2 = 0,0242; qo3 = 0,0714; qs3 = 0,0238; qo4 = 0,0479; qs4 = 0,0326; qo5 = 0,0714; qs5 = 0,0238; qo6 = 0,033; qs6 = 0,0243; qo7 = 0,0632; qs7 = 0,0472; qo8 = 0,0478; qs8 = 0,0401.

To obtain the minimum section scheme (the failures of elements of the open circuit type are taken into account), we use the scheme in Fig. 1 and table is filled. 1 in accordance with the algorithm described above.


Table 1 – Algorithm for obtaining minimum sections when accounting for failures of a open circuit type

№ Substitution pattern node Elements of communication between nodes Section Minimum section
1 2 3 4
1 Ао 1,2,3 1.2.3 1.2.3
2 Ао+А1 1,2,3
2,4,7		 1.3.4.7 1.3.4.7
3 Ао+А2 1,2,3
3,7,6,8		 1.2.6.7.8 1.2.6.7.8
4 Ао+А3 1,2,3
1,4,5,6		 2.3.4.5.6 2.3.4.5.6
5 АоА1+А3 1,3,4,7
1,4,5,6		 3.5.6.7 3.5.6.7
6 АоА1+А2 1,3,4,7
3,6,7,8		 1.4.6.8 1.4.6.8
7 АоА2+А3 1,2,6,7,8
1,4,5,6		 2.4.5.7.8 —
8 АоА1А3+А2 3,5,6,7
3,6,7,8		 5.8 5.8


Section 7 (2.4.5.7.8) is not minimal, since it already includes the minimum section (5.8), therefore column 4 in this case remains empty.

Using the resulting fourth column (Table 1), we construct the minimum cross section scheme, arranging their cross sections as follows:

рисунок

Figure 2 – The scheme of minimum cross sections with regard to failures of elements open circuit

To obtain the minimum cross-section scheme (the refusal in the triggering element is taken into account), you should use the diagram in fig. 1 and fill in the table. 2 in accordance with the rule described above.


Table 2 – Algorithm for obtaining minimum sections when taking into account element failures in triggering

№ Circuit nodes through which emergency current flows Elements of communication between nodes Section Minimum section
1 2 3 4
1 Ао; А2; А4 3,7 3.7 3.7
2 Ао; А3; А4 1,6 1.6 1.6
3 Ао; А3; А2; А4 1,6,8 1.6.8 1.6.8
4 Ао; А1; А3; А4 2,4,5 2.4.5 2.4.5
5 Ао; А1; А2; А4 2,7,8 2.7.8 2.7.8
6 Ао; А2; А3; А4 3,6,5 3.6.5 3.6.5
7 Ао; А1; А3; А2; А4 2,4,6,8 2.4.6.8 2.4.6.8
8 Ао; А2; А1; А3; А4 3,7,4,5 3.7.4.5 3.7.4.5
9 Ао; А3; А1; А2; А4 1,4,7,8 1.4.7.8 1.4.7.8
10 Ао; А1; А2; А3; А4 2,7,6,5 2.7.6.5 2.7.6.5


Using the data table. 2, column 4, we construct the scheme minimum sections (the failures in the triggering of its elements are taken into account):

рисунок

Figure 3 – The scheme of minimum cross sections when accounting for failures in triggering elements

Using substitution schemes for minimum sections, fig. (2 and 3), formulas (6), (8), (7), (9) we find:



Qoн = 1−(1−qo5qo8)(1−qo1qo2qo3)(1−qo1qo3qo4qo7)(1−qo1qo4qo6qo8)(1−qo3qo5qo6qo7)(1−qo2qo3qo4qo5qo6)(1−qo1qo2qo6qo7qo8);


Qsн = 1−(1−qs1qs5)(1−qs3qs8)(1−qs1qs6qs8)(1−qs2qs4qs5)(1−qs2qs7qs8)(1−qs3qs6qs5)(1−qs2qs4qs6qs8)(1−qs3qs7qs4qs5)(1−qs1qs4qs7qs8)(1−qs2qs7qs6qs5).


Substitute the values qoi(0,1), qsi(0,1), (i = (1÷8)) in the resulting formulas, we find:



Qoн = 0,0036; Qsн = 0,0018.


Using the formula (10) we find:



R = 1−Qoн−Qsн = 1−0,0036−0,0018 = 0,9946.


A similar result was obtained when solving the same problem using the triangle–star and star–triangle method [9], i.e формула = 0,9946. A similar result was obtained if the problem was solved by exact methods based on the use of the notion of the algebra of tuples [10] and using the logical-probabilistic method [7], that is, in these cases: R = 0,9946.

Conclusion

  1. One of the possible ways to assess the reliability of structurally complex non-recoverable systems, whose elements may be in three incompatible states, is proposed.
  2. Unless the reliability of non-recoverable systems is taken into account, failures in the actuation of the protective switching devices, i.e. Qsн = 0, then, for this particular example, the reliability of the system is overestimated by 2 times.
  3. Accuracy of calculations of reliability of non-recoverable structurally complex schemes for t ≤ 0.1 year, the proposed method using schemes minimum sections does not differ from the known.
  4. The proposed method for assessing the reliability of complex nonrecoverable systems, whose elements may be in three incompatible states, is based on simple and understandable to the engineer concepts and can be used to train electrical engineers of any specialties.

At the time of writing this essay, the master's work is not yet completed. Estimated completion date of master's work: June 2019. The full text of the work and materials on the topic can be obtained from the author or his manager after the specified date.

References

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