Classification existing methods reservation is shown in fig.
Reservation
Above we have described the essence of the types of redundancy. Note that at present in technical systems, structural redundancy is most widely used.
The essence of structural redundancy lies in the fact that one or more additional (reserve) elements are attached to the main element (i.e., the minimum necessary to perform the specified functions), designed to ensure the operability of the object in the event of failure of the main element).
According to the volume of reservation, the following types are distinguished;
- - general, providing for the reservation of the entire object
- - separate, in which a separate element or their groups are reserved
- - mixed, combining different types of reservation.
reserve as well as technical systems, can be recoverable or non-recoverable. The first of these will be used on serviced systems, and the strategy for its recovery is built in such a way that the security of the system does not decrease below a given level. On serviced systems (non-returnable spacecraft, automatic weather stations, etc.), the reserve, as a rule, is fully used and cannot be restored.
Reserving elements can be in different modes:
Loaded, light and unloaded.
In the unloaded mode, the redundant elements are in the same state as the main element, i.e. all elements work simultaneously under the same conditions.
The light standby mode means that the load of the redundant elements is less than that of the main element.
An unloaded reserve is reduced to a situation in which the redundant elements have no load until the main element fails.
By the nature of the connection, they distinguish:
- - permanent redundancy, in which reserve elements participate in the operation of the facility on an equal basis with the main ones:
- - substitution, when the function of the main element is transferred to the backup only after the failure of the main
- - sliding, in which any failed element can be replaced by a reserve one.
Considering a system consisting of n series-connected elements, we can offer several options for its redundancy.
General reservation(Fig. 6.9, a) assumes that if any element of the main circuit fails, a backup circuit is switched on, which completely replaces the main one.
Probability of no-failure operation of the j-th circuit
,
where 
- the probability of failure-free operation of the i-th element of the j-th circuit, referred to the considered time point t.
Probability of failure-free operation of a system of m parallel circuits (for simplicity of analysis, redundancy is assumed to be loaded)
.
(6.26)
.
(6.27)
Example 1 Probability of non-failure operation of the system with total redundancy at n=4; m=3; p=0.8 will be: P(t)=1–(1–0.8 4) 3 =0.7942. In the absence of a reserve, the probability of failure-free operation of a sequential system of n=4 elements at p=0.8 will be:
P(t)=pn = 0,8 4 =0,4096.
Separate reservation(Fig. 6.9, b) provides the ability to turn on the next backup element in case of failure of any element of the main circuit. A kind of separate reservation is rolling reservation, when the reserve element (elements) can replace any failed element of the main circuit.
With separate redundancy, the probability of failure-free operation of the i-th element, taking into account m - 1 reserve elements (we consider the redundancy to be loaded) will be:
.
Probability of failure-free operation of a system with separate redundancy
.
(6.28)
If all elements have the same reliability, i.e. P ij (t)=p, then
.
(6.29)
Example 2 Probability of failure-free operation of a system with separate redundancy at n=4; m=3; p=0.8 will be:
P(t)=4=0.9684.
Comparison of the results of calculations given in examples 1 and 2 shows that separate redundancy provides a higher level of reliability compared to total redundancy with the same number of redundant elements (redundancy ratio). It should be noted, however, that separate redundancy leads to the complication of the entire system, caused by the need to use a large number control and switching devices, which in practice reduces the effect of its use.
Also apply mixed reservation- a combination of general redundancy of individual circuits with separate redundancy of the most critical and least reliable elements. Comparison of redundancy options in this case can be made by similar methods.
6.3. Redundancy as a method of ensuring the reliability of technological systems at the stage of their creation
Reservation- application additional funds and (or) opportunities in order to maintain the operability (increase the reliability) of the object.
Reservation types:
1. Structural redundancy- redundancy using reserve elements of the object structure. Structural redundancy is implemented by introducing reserve (redundant) elements into the system, which, given the absolute reliability of the elements of the original system, are not functionally necessary. With structural redundancy of elements (or circuits) of the system, reliability indicators increase discretely (jumps). Various structural redundancy options are discussed in 6.2.2-6.2.3.
2. Functional redundancy- redundancy with the use of functional reserves. With this method of redundancy, the system is built in such a way that the specified function can be performed different ways and/or technical means. For example, in some CNC machines, the function of interpolating the trajectories of movement of the working bodies can be performed by software and hardware, using a special device - an interpolator (linear-circular, parabolic, etc.).
3. Temporary reservation- reservation with the use of time reserves. The slack can be used for troubleshooting, maintenance, etc. The slack in technological systems ah can be provided in various ways:
a) increase in operational time (by reducing the time for maintenance, planned downtime, increasing shift work, etc.);
b) creating a performance margin;
c) giving the system the property of functional inertia. Functional inertia- a property of the system that characterizes its ability to allow interruptions in work without loss of the output effect. The functional inertia of the technological system can be given by the use of inter-operational accumulators (buffering).
4. Information redundancy - redundancy using information reserves. It is implemented by introducing redundant codes and symbols during the transmission, processing and display of information (for example, additional units of information that allow you to detect and eliminate errors in the transmission of information: correction codes, checksums, parity checks, etc.).
5. Load redundancy- redundancy with the use of load reserves. The essence of the principle of load redundancy (parametric redundancy) is to expand the scope of the object's operability; in this case, the state area of the object is removed from the boundaries of the health area, determined by the maximum allowable values of the output parameters of the object. This is realized by creating margins of strength, wear resistance (increasing wear tolerances, increasing the area of bearing surfaces, using wear-resistant materials, etc.), rigidity, vibration resistance, heat resistance, etc. Load redundancy allows you to continuously improve system reliability up to required level by increasing the efficiency and resistance to failures of individual elements of the systems. In systems with a coupled or combined structure, to establish this level, it is necessary to consider the operation of the entire system, taking into account the interaction of its elements and subsystems and the participation of individual elements and subsystems in the formation of the output parameters of the system as a whole.
Classification of system redundancy methods
The level of reliability of the element base of electronics, radio engineering, mechanical elements, electrical engineering achieved at present is characterized by the values of the failure rate λ=10 -6 ...10 -7 1/h. In the near future, this level should be expected to rise to λ= 10 -8 1/h. This will make it possible to raise the time between failures of a system consisting of N = 10 6 elements up to 100 hours, which is clearly not enough. Required Reliability complex systems can only be achieved by using various kinds reservations .
Redundancy is one of the main means of ensuring a given level of reliability (especially reliability) of an object with insufficiently reliable elements.
In accordance with GOST 27.002-89 reservation called the use of additional tools and (or) capabilities in order to maintain the operable state of the object in case of failure of one or more of its elements. Thus, redundancy is a method of increasing the reliability of an object by introducing redundancy. In its turn, redundancy - these are additional means and (or) capabilities that are superminimal necessary for the object to perform the specified functions. The task of introducing redundancy is to ensure normal functioning object after a failure occurs in its elements.
There are various backup methods. It is advisable to divide them according to the following criteria (Figure 4.7): type of redundancy, method of connecting elements, multiplicity of redundancy, method of switching on the reserve, mode of operation of the reserve, recoverability of the reserve.
Figure 4.7 - Classification of redundancy methods
Structural redundancy, sometimes called hardware (element, circuit), provides for the use of reserve elements of the structure of the object. The essence of structural redundancy is that in a minimum required option object, additional elements are introduced. The elements of a redundant system have the following names. main element- element of the structure of the object, necessary for the object to perform the required functions in the absence of failures of its elements. Reserve element - element of the object, intended to perform the functions of the main element in case of failure of the latter.
The definition of the main element is not related to the concept of minimality of the main structure of the object, since the element, which is the main one in some modes of operation, can serve as a backup in other conditions.
Reserved element- the main element, in case of failure, which is provided in the facility as a backup element.
Figures 4.8 - 4.10 show the connection diagrams of the main and reserve elements, the so-called parallel connection of elements. A system with parallel connection of elements is a system that fails only if all its elements fail.
Figure 4.8 - Example of parallel connection of elements
a - circuit diagram, b – design scheme
Figure 4.9 - An example of a parallel-serial connection of the elements of the SUHTP
a - functional diagram, b – calculation scheme
Figure 4.10 - An example of a bridge connection of elements
Temporary reservation associated with the use of time reserves. It is assumed that for execution by the object necessary work allotted time is obviously more than the minimum required. Time reserves can be created by increasing the productivity of the object, the inertia of its elements, etc.
Information redundancy- this is a redundancy with the use of information redundancy. Examples of information redundancy are multiple transmission of the same message over a communication channel; the use of various codes in the transmission of information over communication channels that detect and correct errors that appear as a result of equipment failures and the influence of interference; the introduction of redundant information symbols in the processing, transmission and display of information. The excess of information makes it possible, to some extent, to compensate for the distortions of the transmitted information or to eliminate them.
Functional Redundancy- redundancy, in which a given function can be performed in various ways and technical means. For example, the function of transmitting information to the automated control system can be performed using radio channels, telegraph, telephone and other means of communication. Therefore, the usual average reliability indicators (mean time between failures, the probability of failure-free operation, etc.) become uninformative and not suitable enough for use in this case. The most appropriate indicators for assessing functional reliability are: the probability of performing a given function, the average time to complete a function, the availability factor for performing a given function.
Load redundancy- this is a redundancy with the use of load reserves. Load redundancy, first of all, consists in ensuring optimal reserves of the ability of elements to withstand the loads acting on them. With other methods of load redundancy, it is possible to introduce additional protective or unloading elements.
Listed species reservations can be applied either to the system as a whole, or to individual elements of the system or to their groups. In the first case, the reservation is called general, in the second - separate. The combination of different types of reservation in the same object is called mixed.
According to the method of including reserve elements, there are permanent, dynamic, replacement reservation, sliding and majority reservation. Permanent reservation- this is redundancy without restructuring the structure of the object in the event of a failure of its element. For permanent redundancy, it is essential that in the event of a failure of the main element, no special devices are required to put the reserve element into operation, and there is also no interruption in operation (Figures 4.11 - 4.13). Permanent redundancy in the simplest case is a parallel connection of elements without switching devices.
Figure 4.13 - Mixed redundancy with permanently switched on reserve
Dynamic Redundancy- this is a redundancy with the restructuring of the object structure in the event of a failure of its element. Dynamic redundancy has a number of varieties.
Reservation by replacement- This is a dynamic redundancy, in which the functions of the main element are transferred to the backup only after the failure of the main element. The inclusion of a reserve by replacement (Figures 4.14, 4.15) has the following advantages:
- does not violate the mode of operation of the reserve;
- retains the reliability of the backup elements to a greater extent, since during the operation of the main elements they are in a non-operating state;
- allows you to use a reserve element for several main elements.
A significant disadvantage of replacement redundancy is the need for switching devices. With separate redundancy, the number of switching devices is equal to the number of main elements, which can greatly reduce the reliability of the entire system. Therefore, it is beneficial to reserve large nodes or the entire system by replacement, and in all other cases - with high reliability of switching devices.
rolling reservation- this is redundancy by replacement, in which a group of the main elements of the object is backed up by one or more backup elements, each of which can replace any failed main element in this group (Figure 4.16).
Figure 4.16 - Sliding reservation of the same type (a) and heterogeneous (b) elements
Found in control systems wide application majority reservation(using "vote"). This method is based on the use of an additional element called the majority or logical element. The logical element allows you to compare the signals coming from the elements that perform the same function. If the results match, then they are transferred to the output of the device.
Figure 4.17 shows a 2 out of 3 redundancy, i.e. any two out of three matching results are considered true and passed to the output of the device. According to this principle, many schemes of subsystems of control and protection systems (CPS) are built. It is possible to apply the ratios "3 out of 5", etc. The main advantage of this method is to ensure an increase in reliability for any types of element failures and an increase in the reliability of information-logical objects.
Figure 4.17 - Majority reservation
The degree of redundancy is characterized by the multiplicity of redundancy. Reserve ratio- this is the ratio of the number of reserve elements of the object to the number of the main elements reserved by them, expressed as a non-reduced fraction. Integer redundancy occurs when one primary element is backed up by one or more reserve elements.
Fractional Redundancy – this is such a reservation, when two or more elements of the same type are reserved by one or more reserve elements. The most common redundancy with fractional multiplicity is when the number of main elements exceeds the number of reserve ones. Reservation, the multiplicity of which is equal to one, is called duplication.
Depending on the mode of operation of the reserve, loaded, light and unloaded reserves are distinguished. loaded reserve - it is a reserve that contains one or more standby elements that are in the mode of the main element. At the same time, it is assumed that the elements of the loaded reserve have the same level of reliability, durability and persistence as the main elements of the object reserved by them. Lightweight Reserve - this is a reserve that contains one or more reserve elements that are in a less loaded mode than the main one. Lightweight reserve elements usually have more high level reliability, durability and persistence than the main elements. Unloaded reserve- this is a reserve that contains one or more backup elements that are in an unloaded mode before they begin to perform the functions of the main element. For the elements of an unloaded reserve, it is conditionally assumed that they never fail and do not reach the limit state.
Redundancy, in which the operability of any one or more redundant elements in the event of failures is subject to restoration during operation, is called redundancy with recovery, otherwise there is redundancy without recovery. The recoverability of the reserve is ensured in the presence of monitoring the health of the elements. In the presence of redundancy, this is especially important, since in this case the number of hidden failures may be greater than in the absence of redundancy. AT ideal the failure of any element of the object is detected without delay, and the failed element is immediately replaced or repaired.
Classification of redundancy methods. One of the main means of ensuring the required level of reliability and, above all, the reliability of an object or ES with insufficiently reliable elements is redundancy.
Under reservation means the use of additional tools and capabilities in order to maintain a healthy state electrical system failure of one or more of its elements. Reservation is effective method creation of electrical systems, the reliability of which is higher than the reliability of the elements included in the system.
Reservation differs. main elements structures necessary for the system to perform the required functions in the absence of failures of its elements, and backup items, designed to perform the functions of the main elements in the event of their failure.
The ratio of the number of reserve elements etc systems to the number of basic elements they reserve on, expressed as a non-reduced fraction is called the reserve ratio
m p = n p / n o .
Reservation with a reserve ratio of one to one m p \u003d 1/1 is called duplication.
Additional tools and capabilities used in redundancy include elements introduced into the structure of the system as backup, the use of functional and information tools and capabilities, the use of excess time and stocks. load capacity. Accordingly, according to the type of additional funds, they distinguish structural redundancy using reserve elements of the object structure, functional using functional reserves, informational using information reserves, temporary with time reserves and load with the use of load reserves (Fig. 3.28).
In ES, structural redundancy is most often used, and other types of redundancy are also used. So, with functional redundancy, sometimes multifunctional elements of automation are used, and if they fail, they can be used in this system for other purposes, functional redundancy is also carried out for various modes of operation, for example, by transmitting information in various ways, depending on which elements of the system remained functional. Information redundancy is used in systems where the occurrence of a failure leads to the loss or distortion of some part of the processed or transmitted information. Temporary redundancy can be carried out by increasing the productivity of the object, the inertia of its elements, repetition with a shift in time of individual operations. Load redundancy is expressed in the provision of optimal margins for the ability of elements to withstand the loads acting on them or in the introduction of additional protective or unloading elements into the system to protect some of the main elements of the system from the loads acting on them.
According to the method of switching on the reserve, a distinction is made between permanent and dynamic redundancy. Permanent reservation is carried out without restructuring the system structure in the event of a failure of its element, and dynamic redundancy- with the restructuring of the system structure in the event of: the failure of its element.
In the simplest case, with permanent redundancy, elements are connected in parallel or in series without switching devices, and with dynamic redundancy, switching devices are required that respond to element failures.
Dynamic redundancy is often a redundancy substitution in which the functions of the main element are transferred to the backup only after the failure of the main element.
A common type of replacement redundancy is a sliding redundancy, in which a group of main system elements is backed up by one or more redundant elements, each of which can replace any failed main element in this group.
According to the mode of operation of the reserve elements before the failure of the main element, they differ loaded reserve(one or more standby elements are in primary element mode), light reserve(one or more backup elements are in a less loaded mode than the main element) and idle reserve(one or more reserve elements are in an unloaded mode until they start performing the functions of the main element).
The concepts of loaded light and unloaded reserve are used to distinguish between redundant elements in terms of their level of reliability. The elements of the loaded reserve have the same level of reliability (reliability, durability and persistence) as the main elements of the object reserved by them, since the resource of the reserve elements is consumed in the same way as the main elements. Lightweight reserve elements have a higher level of reliability, since the intensity of resource consumption of reserve elements until they are switched on instead of failed ones is much lower than that of the main ones. With an unloaded reserve, the resource of reserve elements begins to be consumed practically only from the moment they are switched on instead of failed elements.
Fig.3.28. Classification scheme of reservation types
According to the method of reserving an object (an element of an object), there are general and separate reservations. At general reservation the object as a whole is reserved, instead of one object, the simultaneous operation of two or more objects of the same type or similar in terms of their functions is provided. The method is simple and widely used in practice when backing up the most critical systems. At separate reservation reserved are individual elements of an object or their groups, which are usually built into an object, both individual elements of the system and its rather large parts (blocks) can be separately reserved.
Dynamic redundancy can be separate and common and allows the use of reserve elements not only in loaded, but also in light and unloaded reserve, which allows you to save the resource of reserve elements, increase the reliability of the electrical system as a whole and reduce energy consumption.
When redundant by substitution, sliding redundancy can be used, which makes it possible to ensure the required reliability of the system at low cost and a slight increase in its weight and dimensions.
The disadvantages of dynamic redundancy by replacement include the need for switching devices and the presence of interruptions in operation when switching to redundant elements, as well as a search system for a failed element or block, which reduces the reliability of the entire redundant system. Reservation by substitution is advisable to use for redundancy of sufficiently large functional units and blocks of complex electrical systems.
Permanent redundancy, which involves the permanent connection of elements to the main ones, is simple, and switching devices are not needed. If the main element fails, the system continues to operate normally without interruption and without switching. The disadvantages of permanent redundancy are the increased resource consumption of redundant elements and the change in the parameters of the redundant node in case of element failure.
Permanent redundancy is used in critical systems for which even a short interruption in operation is unacceptable, and when redundant relatively small elements - nodes, blocks and elements of ESA electronic equipment (resistors, capacitors, diodes, etc.).
The redundancy of the electrical radio elements included in the ESA, the failure of which can lead to especially dangerous consequences, is carried out taking into account the possibility of both short circuits and element breaks. Redundancy in case of element breaks is performed by parallel connection, and in case of short circuits - by series connection of elements, assuming that the element fails, but the electrical circuit of other elements connected in series with it is not violated. For example, permanent separate redundancy of a diode with a loaded reserve in case of failure due to a short circuit (short circuit), open circuit or short circuit and open circuit is carried out by switching on the reserve diodes, respectively, in series, in parallel and in series-parallel to the main one (Fig. 3.29, a, in).
Total permanent rectifier redundancy UD loaded reserve is performed by parallel connection of the reserve, and diodes are used to prevent the backup rectifier current from flowing through the output circuit of the failed one (Fig. 3.29, G). The general redundancy of the rectifier with an unloaded reserve is carried out using the device BUT switching, which receives a signal CO about the failure and sends a control signal US to the switch QW to turn off the failed rectifier and turn on the backup one (Fig. 3.29, d).
Permanent reservation. Such redundancy can be carried out by parallel or serial connection to the main element (system) of one or more redundant ones that perform the same functions as the main element (system). Such redundancy is performed, for example, during parallel operation of generators, computers, ESA units, resistors, etc., as well as when diodes, break contacts, capacitors, etc. are connected in series. d.
Electrical systems with a permanently switched on reserve are made in such a way that failed elements do not affect the operation of the system as a whole. The consequences of element failure with permanent redundancy in extreme cases can be: short circuit or breakage of one or more elements, which should be taken into account when designing the system. For this, limiting resistances are introduced, the
Rice. 3.29. Typical structural redundancy schemes:
a B C - diode VD respectively, in the event of a short circuit type failure, open circuit, short circuit and open circuit;
d, d - rectifier UD respectively with loaded and unloaded reserve
dividing transformers, as well as increase the tolerances of individual system parameters, etc.
Permanent redundancy provides for a loaded reserve and can be shared and separate; on the block diagram for calculating the reliability, the main and reserve elements are connected in parallel (Fig. 3.30).
Rice. 3.30. Schemes of general (a) and separate (b) permanent redundancy
An electrical system with general redundancy (Fig. 3.30, a) will function normally while maintaining the operability of at least one of t+1 parallel circuits consisting of series-connected elements. The probability of failure-free operation of each i-th chain With P elements connected in series, taking into account (3.68) in time t(For simplicity, no further time is given)
Pi =(3.95)
where P ij- probability of non-failure operation of the j-th i-th element chains. The probability of failure-free operation of a system with a common redundancy of m + 1 parallel circuits is found taking into account (3.72) and (3.95):
P s.o = 
(3.96)
With the same reliability of all elements Р ij = Р e formula (3.96) will take the form
R s.o \u003d 1 - (1 - P e n) m +1. (3.97)
For a given probability of non-failure operation of the electrical system s.o. on the basis of (3.97) it is possible to determine the necessary quantity t, under which the condition c.o = P c.o is satisfied, i.e.
t o =
With an exponential distribution law for the elements of the system P e = exp (- λ e t) the probability of failure-free operation (3.97) and the mean time to failure of the system are determined by the formulas
P c.o (t) = 1 - m +1;
where = pλ e - circuit failure rate of P elements; T cf = 1/ - mean time to failure of one circuit.
WPP with separate redundancy assumes the constant inclusion of backup elements in individual sections of the system (Fig. 3.30.6).
Probability of failure-free operation of an individual redundant system element
and the entire system with separate redundancy
(3.99)
With the same reliability of all elements (3.99) takes the form
Р с.р = n , (3.100)
whence, for a given probability of no-failure operation of the system, the corresponding value is determined
With an exponential law of distribution of equally reliable elements Р e = exp (-λ e t) the probability of no-failure operation
P s.p (t) = (1 - m +1 ) n (3.101)
and mean time to failure of the system
where v i = (i + 1) /(m + 1); λ = λ e.
The increase in ES reliability as a result of redundancy can be estimated by the ratio of the probability of failure of the main non-redundant system
and redundant system
With the same reliability of the main and backup systems
γ pe z \u003d l / Q i m \u003d l / Q o m.
An important conclusion follows from the ratio obtained: the greater the probability of a system failure (the less its failure-free operation), the less the redundancy effect. From this conclusion, sometimes called reservation paradox, one can conclude the following:
the possibility of redundancy does not remove the task of increasing the reliability of redundant elements and systems;
general redundancy of the system, other things being equal, is less profitable than separate redundancy, so the probability of failure of a part of the system is less than the probability of failure of the entire system.
With an exponential distribution of time to failure, the probability of failure of the redundant system
Q p (t)=Q o m+1 (t)= m+l ,
where λ o = const is the failure rate of one redundant system.
In practice, usually λ about t< 0,1 тогда
Q o (t)≈ λ o t = t/T cp and
Q P (t) ≈ (λ o t) m +1 = (t/T cp) m +1 ,
where T cf =1/λ o - mean time to failure of the redundant system.
Taking into account the above relations, the gain from reservation can be represented as
γ res ≈ (T cf / t) m.
It follows that the redundancy gain decreases as the required time increases. t system operation.
On the reliability of redundant ES big influence provides restoration of the main or backup systems (circuits) immediately after their failure. In steady state operation, the probability of circuit operability with an average recovery time T c. cf and mean time between failures That at an arbitrary point in time (except for planned periods during which its intended use is not provided) is the chain availability factor.
To r =
since in most practical problems T v.sr / T about<< 1.
Accordingly, the probability of circuit failure can be defined as the probability of inoperability
Q o (t) \u003d 1 - K T ≈ T in. cf /T o .
Then the increase in the reliability of the redundant ES with recovery immediately after the failure of the main or backup systems
γ pe z \u003d l / Q o m ≈ (T o / T in. with p) m ≈ const.
As can be seen, the qualitative difference between redundancy with restoration and redundancy without restoration lies in the fact that when restoring, y, in the first approximation, does not depend on the operating time t. Therefore, the benefits of redundant redundancy increase over non-renewable redundancy as the required operating time increases. t. At the same time, it should be borne in mind that recovery immediately after a failure can be implemented with constant monitoring, the technical means of which should have a probability of failure that is much less than that of a controlled system.
Separate redundancy is more efficient in terms of increasing the reliability of ES, especially for large n (Fig. 3.31). This is explained by the fact that for a system failure with a general redundancy, it is enough for one element from each circuit to fail, and for a separate one, for all elements in any group to fail.
Of practical interest is the question of choosing a rational way to improve the reliability of ES: with the help of redundancy or by choosing highly reliable elements. If from the point of view of mass, dimensions and cost, both ways are equivalent, then the most important thing in solving this issue is the required duration of continuous operation of the system. t.
The influence of time t for trouble-free operation P c . p(t) ES from two identical blocks, working and reserve, with a loaded reserve can be determined using formulas (3.98) with m = 1 and n = 1:
P s.p (t) = 2exp (-t/T cf.b)-exp (-2t/T cp . 6);
T cf = 1.5 T cf. b, (3.103)
Rice. 3.31. Dependences of the probability of failure-free operation of electrical systems with a common (1) and separate (2) redundancy from the number of reserve elements with a different number of consecutive elements
Rice. 3.32. Dependences of the probability of non-failure operation of the system on time with a loaded reserve (1) and with increased reliability of the unit (2)
where T cf.b = 1/λ 6 - mean time to failure of one block; λ b- failure rate of one unit of the redundant system.
For a non-redundant electrical system from a single block of increased reliability with the same mean time to failure T cf. as for a redundant system (3.103), the probability of failure-free operation will be
P sn (t) \u003d exp [- t / (1.5T cf. b)]. (3.104)
Dependencies (3.103) and (3.104) show that redundancy is more efficient than directly increasing the block reliability in the initial period of system operation t< 2Т ср.б, при t >> 2T c r.b, on the contrary, it is more effective to increase the reliability of the block (Fig. 3.32).
Constant series-parallel connection of mutually redundant elements is used in cases where failures of short circuit and breakage types are possible. For example, a capacitor may fail due to loss of capacitance due to an open circuit or due to breakdown due to a short circuit; Relay contacts may fail due to their oxidation (break) or due to their "welding" or "sticking" (short circuit), etc. (see Table 3.7).
Taking into account the possibility of failures such as open circuit and short circuit, in many cases a constant series-parallel connection of four mutually redundant elements is used (Fig. 3.33). When element failures of the short circuit type prevail
Q kz (t) > Q o 6 (t),
Rice. 3.33. Permanent serial-parallel connection of mutually redundant elements in case of failures mainly: short-circuit type (a) and break (b)
where Q kz (t) and Q o 6 (t) - the probability of failure of an element of the short circuit type and open circuit, respectively, series-parallel switching circuits without a jumper are used (Fig. 3.33, a), and when failures of the open circuit type prevail
Q kz (t)< Q об (t) -
Series-parallel circuits with a jumper (Fig. 3.33, b).
The probability of failure of the redundant circuit in case of failures of the open type Q r.ob (t) and the type of short circuit Q r.kz (t) for the required period of operation t is a function of element failure probabilities Q kz (t) and Q o b (t) and depends on the redundancy scheme used and the type of failure (Table 3.13).
From the table. 3.13 of the relations it follows that the efficiency γ res of serial-parallel redundancy decreases as the probability of failure of the circuit element increases. At a certain critical value Q kz (t) or Q about (t) the probability of failure of the redundant circuit becomes greater than the probability of failure of one element, then the use of series-parallel redundancy becomes inappropriate. Taking into account the reliability and accuracy of a priori information about the reliability of elements, it is usually recommended to use serial-parallel redundancy in cases where the probability of failure of the circuit element is Q kz ( t) 0,l and Q o 6 (t) 0,l.
Table 3.13.
Design ratios for series-parallel connection
four elements
Rice. 3.34. Schemes of general (a) and separate (b) dynamic redundancy
with switching devices
Dynamic redundancy. With such redundancy, it becomes possible to use a light or unloaded reserve, if interruptions in the operation of the ES necessary for switching on the reserve are acceptable, and it becomes necessary to use additional elements- switching devices for connecting the reserve. The inclusion of reserve elements can be done manually or automatically, switching devices can be separate or common for parallel-connected elements or circuits (blocks) of the electrical system (Fig. 3.34).
If we neglect the influence of switching devices and consider them absolutely reliable, then with a loaded reserve, the reliability of an ES with dynamic redundancy will be equal to the reliability of a system with a permanently switched on reserve. With light and unloaded redundancy, dynamic redundancy improves system reliability.
The influence of the reliability of switching devices on the reliability of a redundant system is quite simply taken into account for systems with a loaded reserve.
WPP with general redundancy and loaded reserve in normal mode, all circuit breakers are switched on and the main and backup circuits from P elements are under load. In the event of a main circuit failure, switch K . turns it off, in case of failure of the first backup circuit, it is turned off by the switch K1, etc.
System failure occurs when the main and all backup circuits, consisting of P elements and switch To each. Assuming that switches and elements of the system fail independently, one can find the probability of failure-free operation of one circuit from P elements
and the probability of failure-free operation of the entire system of m + 1 such parallel circuits
P s.o = 
,(3.105)
where P ki- probability of failure-free operation switch i-th chains.
With the same reliability of all P elements P e and the same reliability of switches P k formula (3.105) will take the form
P s.o \u003d 1 - (1 - P k P e n) m +1. (3.106)
From (3.106) for a given value P s.o = find the required value of the number of backup circuits
With an exponential distribution law for elements P e \u003d exp (- λ e t) and switches Р k = exp(- λkt) of the system, the mean time to failure and the probability of no-failure operation of the system are determined by formulas (3.98), in which in this case the failure rate of the circuit is calculated by the formula
WPP with separate redundancy and loaded reserve all circuit breakers To in the initial period of the system operation are switched on, in case of failure of any main or backup element, the corresponding switch disconnects this failed element. System failure occurs when any main element j (or its switch K) fails and all elements that reserve it i(or all their switches K i).
Probability of failure-free operation of the entire system with separate redundancy, taking into account the probability of failure-free operation of circuit breakers
(3.107)
For a system with equally reliable elements and switches, expression (3.107) takes the form
R s.r = n. (3.108)
With an exponential distribution law for elements λ e \u003d const and switches λ k \u003d const, the values \u200b\u200bof T cf.r and P c.r are calculated using formulas (3.101) and (3.102), in which in this case they take
λ \u003d λ e + λ k.
It can be seen from the obtained formulas that with dynamic redundancy with a loaded reserve due to the presence of switching devices K, the system reliability indicators are lower compared to permanent redundancy. It is advisable to use dynamic redundancy with a loaded reserve in cases where interruptions in the operation of the system are unacceptable and the failed element (system) must be turned off so that there is no abrupt change in the mode of operation of the redundant system.
Calculations according to formulas (3.106) and (3.108), which determine the probability of failure-free operation of the systems shown in Fig. 3.34, show that with the same reliability of the elements and the same sufficiently high reliability of the switches for the same values P and t the probability of failure-free operation of an ES with separate redundancy and a switch for each element is higher than that of an ES with a common redundancy and a switch in each circuit.
Thus, separate redundancy is more efficient than general redundancy in the case of dynamic redundancy.
The effectiveness of dynamic redundancy is enhanced when it is implemented as replacement redundancy with light or light redundancy. Below we consider redundancy by replacement with an unloaded reserve; it is obvious that the reliability indicators with a light reserve will have intermediate values between those with a loaded and unloaded reserve.
In a redundant system with general redundancy and an unloaded reserve, the main circuit with a circuit breaker operates first To(fig.3.34, a), if it fails, it is switched on by a switch instead K i one of the spare circuits. There can be no more such substitutions. t;(m + 1) - failure leads to the failure of the system as a whole.
To simplify the analysis, we consider a system with an exponential distribution law for the elements Р ij (t) = exp(-λ jt) and switches P ki (t)=exp(- λkit). Then the probability of failure-free operation of one circuit from P elements with a switch
P i (t) = (3.109)
where λ i = λ j n + λ k - failure rate of the i-th circuit of the redundant system.
Average running time up to i-th failure chain, taking into account (3.109), will be
T cf. i = 
At each of the intervals t i only one circuit is working and can fail, so the mean time to failure of the entire system will be
T cp . o = T cp . i(m+1). (3.110)
Probability of non-failure operation of a redundant ES with an unloaded reserve during the time t can be determined under the assumption that if one circuit fails, there is an instantaneous switching to one of the backup circuits, and the system will fail after the failure of the main circuit and all t backup circuits. Then the probability that one chain from P elements and switch TO, having a failure rate λ i over time t fails ztimes (taking into account the possibility of its replacement with reserve ones), can be determined by Poisson's law
P z (t) = (λ i t) z /z! exp(-λ i t), (3.111)
where λ i t is the average number of circuit failures over time t.
The entire redundant system over time t will work flawlessly if at least one of the following incompatible events takes place during this time: C o - all circuits of the system worked flawlessly, From 1 - one circuit failed Cz- failed z chains from (t+1); C t - refused t chains from (m+1).
Thus, the probability of failure-free operation of the entire redundant system is determined according to the probability addition theorem full group of incompatible events C with (3.111) taken into account
P s.o (t) = (3.112)
Comparing the obtained formulas (3.110) and (3.112) with the corresponding formulas for a loaded reserve, it follows that with an unloaded reserve, the probability of no-failure operation and the mean time to failure increase.
At the same time, it is practically impossible to achieve an increase in the mean time to failure by more than an order of magnitude due to such redundancy due to the presence of switching devices and auxiliary equipment. With an increase in the number of redundant elements (blocks, systems), the mass, dimensions and cost of auxiliary equipment significantly limit the achievable level of reliability in redundancy, allowing in practice to use redundancy with m ≤ 2 ... 3.
If the ES consists of groups identical elements, then it is advisable to use sliding reservation by replacement, when one or more reserve elements (blocks) t systems can replace any of the failed main elements (blocks) of the system (Fig. 3.35).
Rice. 3.35. Rolling Reservation Scheme
If the sliding redundancy is with an unloaded reserve, the failures of the elements are independent and have an exponential distribution, the device for searching for the failed element and turning on the backup instead of it (switch) is absolutely reliable, then the probability of the system fail-safe operation during the time t, i.e., the probability of failure during this time no more t elements, is determined according to Poisson's law similarly to (3.112)
P c . c(t) = 
(3.113)
where λ e - element failure rate.
Mean time to failure of the system, i.e. expected value the time of occurrence of the (m+1)-th failure is determined in the usual way:
T cf \u003d 1 / (pλ e) + t / (pλ e) \u003d (t + 1) (pλ e).(3.114)
The efficiency of the sliding redundancy of an electrical system can be estimated by comparing the dependencies (3.113) and (3.114) for a system with a sliding redundancy with the corresponding dependencies P c \u003d exp (- nλ e t) and T cf \u003d 1 / (pλ e) for non-redundant system
(t) = P c . c (t)/P c (t) = 1+ nλ e t + (nλ e t) 2 /2! + . . .+ (nλ e t) m /m!;
(t) = T cp . c/T cp = (m+1).(3.115)
From (3.115) it follows that from the point of view of increasing the probability of failure-free operation and the mean time to failure of the ES, the efficiency of sliding redundancy in comparison with the corresponding non-redundant system increases with an increase in the number of reserve elements, an increase in the system operation time and the number of redundant main elements (blocks) of the system.
Rolling redundancy can be economically more profitable, since it is implemented with a smaller number of reserve elements than the main ones.
Optimal redundancy. In the practical implementation of ES redundancy, the problem arises of optimal redundancy, i.e., ensuring the required system reliability at the lowest cost.
The number and range of reserve elements (blocks) of ES can be determined based on the following two formulations of the problem of optimal redundancy:
1) the given probability of non-failure operation of the system must be ensured at minimal cost With mi p on reserve elements, i.e. at C min ;
2) at given costs for reserve elements, it is necessary to ensure the maximum possible probability trouble-free operation of the system R s. m ah, i.e. at R s. m ah.
To solve both problems, first determine the number of elements (sections) of the system redundancy, calculate the probabilities of failure-free operation of each section and the system as a whole, and determine the cost of each section.
Then, to solve the first problem, the minimum of the function С = 
on condition P c \u003d 
where FROM - the cost of a redundant system, C i - the cost of one reserve element of the i-th section of the system; C 0 i - the initial cost of the i-th section of the system; m i - number of spare elements per i-th section; P i (m i) - the probability of failure-free operation of the i-th section of the system if it has m i -reserve elements.
The solution of the second problem of optimal redundancy is reduced to finding the maximum of the function P c = under the condition C = 
The calculation of the optimal redundant ES is a multi-step process. At the first step, such a redundancy section is found, the addition of one reserve section to which gives the greatest increase in the probability of failure-free operation of the system in terms of unit cost. At the second step, the next section is determined (including the previously reserved section), the addition of one reserve section to which gives the greatest increase in the probability of failure-free operation of the system, etc. Calculations are performed in tabular form; calculation stops at this step
M = , when the condition is met for the first task P c (M-1)< (М), а для второй задачи - С(М) 
Reservation is the most effective method achieve the most high performance systems reliability.
Redundancy is a method of increasing reliability by including a reserve. Redundancy allows you to create systems whose reliability may be higher than the reliability of its constituent elements. Reservations can be made various methods, which are characterized common feature- the principle of redundancy. This means that along with the main elements, nodes or blocks that perform the specified functions, the system must contain redundant (reserve) components that are not functionally necessary, but are intended only to maintain a certain level of system reliability. The application of the principle of redundancy leads to the complication of REA, an increase in weight, dimensions, and cost. The classification of redundancy methods is shown in fig. 3.5.
Rice. 3.5. Classification of reservation types
In redundant replacement systems, a failed element is replaced by a working one from among the redundant ones, and this replacement is most often carried out using a switch (automatically or manually).
The advantages of replacement reservations include:
no need to adjust the system parameters after replacing a failed element with a serviceable one;
· redundant elements can be located until they are included in the system in a light mode, which contributes to the preservation of their resource and reduces power consumption.
However, such systems have disadvantages:
· the need to use switches, which are the least reliable elements of REA;
the need to create additional devices, monitoring performance, looking for a failed element and replacing it with a serviceable one.
All these shortcomings lead to the fact that redundancy by replacement is mainly used for redundancy of relatively large functional units of complex systems.
In systems with a constant inclusion of a reserve, all elements (both main and backup) are electrically connected so that they are in the same modes. This type of redundancy is calculated taking into account the consequences of element failures and the types of these failures.
The advantages of such a reservation are:
Ease of implementation of redundancy, therefore, a slight increase in the weight, dimensions and cost of the system;
No interruptions in the operation of the system after the occurrence of failures. Permanent redundancy is the only possible one in those systems where even a short interruption in operation is unacceptable.
The disadvantages include:
Repaid resource consumption of reserve elements;
Failure of one of the elements leads to a change in the modes of operation of the others.
The use of permanent redundancy is limited by the fact that the simultaneous parallel operation of elements, nodes and blocks is possible only in some systems. Therefore, the permanent inclusion of the reserve is most convenient when reserving relatively small devices of the system (mainly elements).
General redundancy is the redundancy of the entire system. Separate redundancy consists in redundant system in parts, according to separate sections.
A general redundant system (Figure 3.6) functions normally until the last remaining good circuit fails. Let m- multiplicity of redundancy, that is, the number of redundant circuits. If each j-th circuit consists of n elements with the probability of correct operation P ij, then, using the probability multiplication theorem, we obtain that the probability of a complex event, which consists in the fact that in j-th circuit, no failure will occur, is equal to the product of the probabilities of proper operation of each element of the circuit, then:
Single Circuit Failure Probability
Then the probability of correct operation of the system
For the case when all elements of the system have the same reliability, i.e. Pij=P, we get
Rice. 3.6. General reservation
Rice. 3.7. Separate reservation
A system with separate redundancy (Fig. 3.7) will work normally while maintaining the operability of at least one element in each of n- links, probability of failure i-th link
where q ij- probability of failure j-th element i-th link.
Probability of correct operation of a system with separate redundancy P with is equal to the product of the probabilities of correct operation Pi all n- links
For the case of identical elements in terms of reliability Pij=P we have
Mixed redundancy (Fig. 3.8) is a combination of common and separate, and the calculation of reliability in mixed redundancy is made using formulas for common and separate redundancy.
Rice. 3.8. Mixed redundancy
Rice. 3.9. Efficiency of various types of redundancy
To compare the effectiveness of the application various types redundancy suppose that there is a system consisting of n elements connected in series, identical in reliability, having reliability P=0.9.
As follows from Fig. 3.9, on which the calculated values of the corresponding probabilities are plotted, separate redundancy has the greatest efficiency, moreover, than more quantity elements n, topics more advantage. However, it is necessary to remember the assumption that was used when deriving the formula for the reliability of redundant systems, namely, here the reliability of a system with a permanently switched on reserve was calculated.
Examples of such inclusion are:
systems consisting of several transmitters operating on a common antenna;
Radar stations containing several indicator devices operating in parallel;
· parallel electrical connection of several elements (resistors, capacitors, etc.).
Let's find the value of the average time of correct operation T s a system consisting of elements connected in parallel, one of which is the main one, and the second is a backup.
Let the failure rates of these elements be respectively equal to λ1 and λ2. Then, under the exponential law of reliability, the probability of their failure-free operation by the time t equal
; and 
For System
As is known,
After substituting the limits of integration, we obtain
If the elements are equally reliable, i.e. λ 1 = λ 2 = λ, then
where T0- the average time of correct operation of one element.
For a system consisting of three elements of the same type connected in parallel, we find
In the general case, with the multiplicity of redundancy m
From last expression it follows that an increase in the multiplicity leads to a decrease in the contribution of a new reserve element to the mean time of correct operation of the system. This phenomenon is explained by the fact that with constant switching on, the reserve circuits consume their working capacity simultaneously with the main circuit.
Redundancy by replacement involves the inclusion of backup circuits only after the failure of the main circuit. Switching on of reserve circuits can be carried out both manually and automatically. In either case, a fault indicator, a control device and a switch are required. As the latter, relays or electronic switches are usually used.
On fig. 3.10 shows a system where
B 1 ... B m- blocks of the main and reserve circuits,
n 11 …n m1– input circuit switches,
n 12 …n m2– output circuit switches,
U 1 ... B m- 1 - indicator and control devices.
Rice. 3.10. Reservation by replacement
When a unit fails B 1 failure indicator sends a signal to the controller 1 which disables B 1 by input and output by connecting the block B 2. After a block failure occurs B 2 the system behaves similarly.
The failure of any of the switches leads to the failure of the redundant circuit in which it is included (provided that the failure of the switch does not disable the entire redundant system). Therefore, when calculating the reliability, the switch is considered as an element connected in series with its block (in terms of reliability).
