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Abstract

Content

Introduction

Using the concept of cross section, minimum cross section, a method is proposed for assessing the reliability of complex systems whose elements can be in three incompatible states: healthy, failed: open circuit failure and short circuit failure.

The calculation example is given.

Keywords: system, minimum section, complex scheme, triangle, star, probability of failures, failure type break, failure to operate, load node.

1. The urgency of the problem

Non-recoverable will include such recoverable systems that, for whatever reason, cannot be restored during the period in question.[1]

Under the system in this case, we mean a set of interrelated, structurally independent products that have the property to rebuild their structure as a result of accidental damage to its elements, without disturbing (or partial violation) the task performed before it.[2]

Methods for assessing the reliability of non-recoverable systems, the elements of which may be in two incompatible states – workable and failing (open circuit type failure) have been fully developed.[3–6]

In the above calculation methods it is assumed that the means of network protection are absolutely reliable. In real cable networks of 10–0.22 kV, failures in the operation of short–circuit protection devices do not rarely lead to ignition of the insulation of conductors and combustible materials near the laying of cable products, which often leads to large–scale fires at production sites. [3–5]

Therefore, the development of engineering techniques that allow you to take into account failures in the operation of protective switching devices operated in networks of different voltage classes, is an important scientific and technical problem.

2. The purpose of the master's work

Using well–known approximate methods for converting complex in the structure of non-recoverable systems, whose elements may be in three incompatible states, develop a methodology for calculating complex in the structure of non-recoverable power systems of industrial enterprises.

3. The main research material

A failure of the type of short circuit in power supply systems will be understood as a random event when a short circuit occurs in the area of the relay protection (RZ), and the system for switching off the protective switching device is in a failed state, i.e. refused to work..

An analogue of an element with three incompatible states: operational, failure of an open circuit and failure to operate in power supply systems can be a protective switching device, a protective switching device and electrical equipment that is within the range of its automatic protection can be considered as an element of the system that can be located in three incompatible states.

For example, if there is a fault in a network element that is in the area of the RZ of the switching apparatus, it is disconnected from the power supply source, then such damage in the network will be referred to as failures.

Protective switching apparatus and electrical equipment that is included in the zone of its automatic protection will be considered as an element of the system, which can be in three incompatible states.

By failures in response, we will attribute such damage (short-circuit in the network element) in which a through–current emergency passes through the switching device closest to the short–circuit location, and its RE system does not work.

To assess the reliability of technical systems, elemental methods are widely used. [6–8].

In these methods it is proposed that the electrical equipment in the equivalent circuits of power supply systems consists of independent (in the sense of reliability analysis) elements.

Substitution scheme nodes are physical points that are directly related to at least three directions of energy transfer, i.e. These are usually busbars or sections of distribution points. [8].

Using the schematic diagram of the power supply system, an equivalent circuit is drawn up to assess the reliability of consumers who receive electricity from the load node in question.

All independent sources of power supply systems are combined into one point, and it is considered absolutely reliable and is the input of the equivalent circuit. [8–10].

All damages in the power supply circuit above the selected entry point are not taken into account in the calculations.

The output of the power supply system's replacement circuit is busbars, from which consumers receive electricity.

Suppose that all the considered elements that make up the system can fail independently of each other; each element of the system may be in three incompatible states: healthy, inoperable – a type of open circuit failure, unworkable – a short circuit type failure; failure flows of elements (such as open circuit and short circuit type) are the simplest; the capacity of the elements is not limited in the same way as the ability to restrain the flow of energy (liquid, gas, electric current) regardless of the number of short–circuited elements; after failure, the element is not restored (not replaced with a new one) in the considered period of time [10].

Denote by Pi the probability of failure–free operation of the i–th element of the system, qOi – is the probability of occurrence of failures in the i–th element of the open circuit type, and через qSi – the probability of the occurrence of failures in the it short–type element. These three states make up a complete group of incompatible events.

1 (3.1)

In the event that the system elements are subject to two types of incompatible failures: open circuit type failure and short circuit type failure, then the probability of its failure during the time t can be determined as follows [11]:

2 (3.2)

Under the probability of failure–free operation by an unrecoverable system, whose elements may be in three incompatible states, we will take a measure of its reliability, which is characterized by the probability that during a given time interval such random events will not occur, as a result of which the connection is broken or the through current passes the input and output of the equivalent circuit, provided that at the initial moment of time all its elements are in a healthy state.

For unrecoverable systems, whose elements may be in three incompatible states, the formula is valid:

2 (3.3)

By simple, by definition, a system equivalent circuit will be understood as one whose elements can be connected: in series, in parallel, in series in parallel, or in parallel in series.

If the probabilities of failure of the elements of the equivalent circuit qOi, are given, then the probability of breaking the connection between the Input and Output nodes QO is defined as follows:

2 (3.4)

In the event that each of the n logical connection of elements fails like a short circuit in each of the i elements, this will lead to a pass–through emergency current between the Input and Output nodes of the equivalent circuit.

If the probabilities of failure of the equivalent circuit elements qSi, are given, then the probability QS of the fact that between the point of entry and output of the replacement circuit will pass through the emergency current, we will find using:

2 (3.5)

If the probabilities of failure of the equivalent circuit elements qOi, тare given, then the probability of breaking the connection between the Input and Output QO nodes is defined as follows:

2 (3.6)

The probability that the gap between the input and output of the equivalent circuit does not occur will be defined as follows:

2 (3.7)

Under the complex structure of the system replacement scheme, we will take one that includes at least one group of elements connected in the form of a logical star or triangle.(fig. 1)

In order to bring a complex replacement system of the power supply system to a simple one by definition, one should use the transition method from a logical triangle to an equivalent star in reliability, or use the transition method from a logical star to an equivalent in reliability logical triangle. (fig. 1 а,в).

Figure 1 –A logical connection of elements in the form: a) triangle; b) stars

Exact formulas for transitions from a logical triangle to a logical star equivalent in reliability (Fig. 1 a, б) and from a logical star to a logical triangle equivalent in reliability (Fig. 1 б, a) are given in [6–11]

Under the probability of failure–free operation of an unrecoverable system, the elements of which can be in three incompatible states, we mean the measure of its reliability, which is characterized by the probability that during a given time interval such random events will not occur, as a result of which the connection is broken or the emergency current passes through the entry and exit points of the replacement scheme, provided that at the initial moment of time all its elements were in a healthy state.

For a non-recoverable system, the relationship is valid[11]:

2 (3.8)

where R – is the probability that such random events will not occur, as a result of which the connection will break or pass through the emergency current between the input and output of the equivalent circuit;

QO,QS – the probability that a break in communication will occur or a pass–through emergency current between the "input" and "output" of the equivalent circuits, respectively, passes.

RO – the probability that such a random event will not happen as a result of which the link between the input and the output of the equivalent circuit will be broken.

4. Example

22

Figure 2  – Complex structure

For the equivalent circuit of the system shown in Figure 2, the following failure probabilities are given:

2

Determine the probability that the connection will not break and the emergency current will not pass through between the input and output of the equivalent circuit.

21

Figure 3  – The method of transforming a complex system structure into a simple one, using the star–triangle transformation method.

Elements 4, 7, 8 and 5, 6 connected in series (Fig. 3, a) are replaced with elements of equivalent reliability and 9 and 10, respectively (Fig. 3, b).

Using formulas (6), (7) we find:

2 (4.1)

In the equivalent circuit (Fig. 3b), the logical triangle ABC is replaced by the equivalent in reliability star AEHB (Fig. 3c). Using the equivalent circuit of fig. 3b, the exact formulas for transitions from a logical triangle to an equivalent in reliability star, taking into account failures of elements of a triangle of the open circuit type, we find:

2

Using the obtained equivalent circuit (Fig. 3c), formulas (7) - (10) we find: RO и QS .

2 (4.2)

Substituting the source data of the example and the previously calculated values of q011, qS11; q012, qS12; q013q, qS13, q09, qS9 и q010, qS10 in the resulting formula we find:

2

Using formula (26) we find:

2

The same problem can be solved using the minimal sections method for the case when an element of the system can be in three incompatible states. [14–15]

Using Fig. 3b we build the scheme of minimal sections for this task, which will take the form:

18

Figure 4  – The substitution patterns of the minimum sections for the case when an element of the system can be in three incompatible states.

Using the substitution patterns of fig. 4a, b, formulas (6) – (11) we find the lower estimates: Q0H и QSH.

2

Using formula (26) we find: RH=0,970275 .

In our case, the value of R and RH almost coincide.

Conclusions

  • 1. A method is proposed for assessing the reliability of power supply systems that are complex in structure, whose elements may be in three incompatible states.
  • 2. The accuracy of the proposed methodology for highly reliable systems (the probability of failure of the elements of the replacement circuit is less than or equal to 0.1) is not inferior to accurate, proven methods.

References

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