CN106383960A - Minimum cut set analysis method-based power system reliability analysis method for more-electric aircraft - Google Patents

Abstract

The invention relates to a minimum cut set analysis method-based power system reliability analysis method for a more-electric aircraft. The method comprises the steps of making a topological structure diagram equivalent to a network node diagram; calculating a minimum path set from a source node to a specified node by adopting an adjacent matrix method; calculating a minimum cut set of fault occurrence; performing non-cross processing on the minimum cut set by adopting a ''delete and leave'' method; and converting an actual fault rate of a part into an equivalent fault rate of a node or an arc, and further calculating equivalent reliability. According to the method, weak points in an aircraft energy flowing process can be definitely displayed, improved suggestions are provided for aircraft energy usage schemes, and pointed schemes are provided for aircraft overhaul in different running time segments.

Description

Many electricity aircraft electrical power system analysis method for reliability based on minimal cut set analytic process

Technical field

The present invention relates to the field such as the analysis of Aero-Space, power system security and graph theory parsing.Can be generally applicable In the fail-safe analysis of the poor power system of self-healing ability, the such as aircraft of mission phase, underway steamer, high-speed motion Electric automobile etc..It is suitable for reliability during analysis micro-grid system independent operating.

Background technology

How electric aircraft replaces original hydraulic pressure, air pressure and mechanical drive system with power system as much as possible, tries hard to make The secondary energy of aircraft as far as possible more uses electric energy.Power system occupies mastery reaction how in electric aircraft, its reliable and stable fortune Row is related to the normal work of whole aircraft system.The aircraft safety accident that aircraft electrical power system fault leads to happens occasionally, this Make power system reliability service caused many including aircraft manufacturing producer, airline, Ge great research institution Aspect is paid close attention to.

Open source literature for aircraft electrical power system reliability assessment is simultaneously few both at home and abroad now, and wherein most is studied into Fruit all employs Monte Carlo method or Fault Tree, and obtains relatively good achievement.

Monte Carlo method is a kind of probability analysis method, have application flexibly, realize simple feature.It can solve effectively Certainly fail-safe analysis problem.But during using Monte Carlo method to the fail-safe analysis of power system, its computation burden is big, efficiency The low always inevitable shortcoming of Monte Carlo method." the aviation based on model importance written by Lv Hong, Yuan Haiwen et al. The reliability of power-supply system " introduces the concept of model importance, partial extent solves and adopts merely DSMC Less efficient problem.Model importance is significance level in subsystem for the finger, and it passes through to calculate a certain component failure The subsystem failure number of times causing accounts for subsystem total Failure count ratio and obtains.Model importance has distinguished normal elements and weakness The impact effect to system reliability for the part.This also inspires us：In systems reliability analysis, treat different parts and want root According to physical unit action time and significance level, the reliability that it is converted is made with corresponding adjustment.But lack due to each in the method Interact between individual subsystem the correction to reliability for the impact producing.

Fault Tree is using the undesirable event of system as analysis purpose, by successively push down on trace back all Possible the reason, thus enumerate component failure that may be present, environmental effect, human error and various factors and thrashing Logical relation, and formed one handstand fault tree.Zhou Suying master thesis " electrical power system of more electric aircraft research " with The reliability of its distribution system is analyzed study by Fault Tree as a example advanced tactical fighter distribution system.In reality Set up the problem that huge amount of calculation during fault tree can bring multiple shot array in application, and fault tree is only once set up The reliability of some time point of system can be obtained, the relation of reliability and run time can not be reflected well.But such as By the thought of retrodicting in fault tree, the knowledge using graph theory can be analyzed effect under computer aided calculation to fruit well Really.The logical source of this namely fail-safe analysis method of the present invention.

General with present generation aircraft in " large aircraft power supply-distribution system reliability assessment and analysis " written by Cai Lin, Zhang Ling et al. Time power supply-distribution system structure give power supply-distribution system analysis method for reliability flow process, simultaneously by graph theory knowledge utilization Computer aided calculation obtains the reliability for run time for the aircraft components, and its analytical mathematics is perfectly in harmony with the present invention.

But this thinking also lifts improved space at several aspects：One side Shi road collection ask for during using depth Degree first search, leads to road collection matrix that energy Flow order can not be reflected very well so that excessive system information is hidden；Another Aspect does not deeply excavate the change of part reliability under different time dimensions.

Content of the invention

The purpose of the present invention is that prior art is improved, and provides one kind many electricity aircraft electrical power system fail-safe analysis side Method, shows the concrete path of energy Flow by adjacency Matrix Method, and different in length to aircraft overall operation reliability Further mining analysis are made under time dimension.This improved method more clearly show the aircraft energy flowing during thin Weakness, provides improvement idea for aircraft energy operational version simultaneously, and the also maintenance under the different operation periods to aircraft is provided with pin Scheme to property.Technical scheme is as follows：

A kind of many electricity aircraft electrical power system analysis method for reliability based on minimal cut set analytic process, comprise the steps：

Step one：According to following four principle, topology diagram is equivalent to network node figure：

1) network node figure needs to accurately reflect parts relationship and working condition；

2) by abstract for the components and parts such as chopper, wire, transformator, commutator and electromotor in former topological diagram for network node The arc of in figure, wherein control energy Flow direction for directed arc, otherwise for undirected arc；

3) by abstract to busbar in the power system and load node for network node in figure, a virtual source is set simultaneously Node, is equivalent to the source node of whole system power input；

4) network node figure is as far as possible succinct, and the components and parts connected when not having branch road are equivalent to an arc, type Load that is identical and deriving in a busbar is equivalent to a node；

Step 2：Source node is asked for using Adjacent Matrix Method and reaches specified node minimal path sets, method is as follows：

1) meshed network N=(V), wherein V={ v are set₁, v₂, v₃, v_nIt is set of node；

2) define adjacency matrix A₁=[a¹ _ij], the topology information of description network node in figure, the wherein subscript of element name 1 is expressed as the adjacency matrix that path is 1, also illustrates that first time Iterative Matrix in calculating process, i, j represent square respectively simultaneously The row and column of battle array：

3) define terminal matrix R=[r_jk], the terminal of every arc in reflection meshed network N：

4) define A₁A is tried to achieve in computing " ※ " with R₂=[a² _ij], its algorithm is as follows, a² _ij={ a¹ _ij※r_jk|i,j,k =1,2,3, n }：

5) ask for virtual source node to the minimal path sets specifying node using as above method, minimal path sets are transformed into simultaneously Entering minimal path sets matrix facilitates subsequent calculations machine auxiliary to calculate；

Step 3：Ask for the minimal cut set of fault generation：

Observe minimal path sets, if each paths are all through same node or same arc in minimal path sets, that This node or arc break down, and mean the situation specifying node or arc that dead electricity necessarily occurs, then this node or arc be just Belong to the single order minimal cut set in grid；Remove the node failure represented by single order minimal cut set, other paths all comprise Two components and parts or wherein any one components and parts, if two components and parts break down simultaneously, mean to specify node Or arc necessarily occurs the situation of dead electricity, then this two components and parts just belong to second order cut set；Found using the method by that analogy Source node is to specified node each rank minimal cut set；That is any one ranks vector in minimal path sets matrix is all 1, then this Column vector corresponding components and parts fault is exactly single order minimal cut set；Again by any two column vector in addition to single order minimal cut set Carry out inclusive-OR operation, if result all 1, obtain the minimum second order cut set of this two column vectors corresponding components and parts composition, with Such inquire into all minimum second order cut sets, obtain three rank cut sets, quadravalence cut set in the same manner；

Step 4：Non cross link process is carried out to minimal cut set using " leave out and stay " method：

1) a n-dimensional vector E is defined for each minimal cut set_i(x₁, x₂, x₃···x_n), x_iTake 0 or 1, work as x_iTake Represent that minimal cut set includes this node or arc when obtaining 1, work as x_iObtain and when 0, represent that minimal cut set does not include this section Point or arc；

2) define companion matrix T_i(t₁, t₂, t_n) and meet following condition：

3) set E₁For the minimal cut set of first non cross link, when i=2, compare E_iAnd T_iFind T_iMiddle all elements are 1 and corresponding E_iMiddle relevant position is 0 position, according to size order record position sequence number and number and obtain k₁, k₂, k_r；

4) by E_iTo k_iDecomposed successively, be decomposed into E_i(k₁) andWherein E_iMiddle k_iOn position 0 is by 1 or -1 Replace respectively；

5) compare E_i(k₁) and E_j, wherein j<I, if all of E_i(k₁) in have -1 position correspond to E_jPosition have one Individual 1 or several 1 corresponding, and these 1 number summations are equal to line number j, obtain E_i(k₁) and E_jIncompatible, become non cross link Minimal cut set, therefore by E_i(k₁) storage stay, simultaneously forDo identical mode to process；

6) for the E not having storage to stay_i(k₁) and T_i-1Relatively, if T_i-1In be not 0 position correspond to E_i(k₁) in phase The position answered is not 0 then it is assumed that E_i(k₁) by E_jAbsorbed, wherein j<i；Simultaneously forDo identical mode to process；

7) if E_i(k₁) do not stay and do not delete, it is continued decompose, by E_i(k₁) it is decomposed into E_i(k₁)(k₂) andWith respectively with 1, -1 replacement k₂On position 0, according to step 4,5,6 replicate analysis, until all of E_iStaying Or make a choice in deleting；

Step 5：Based on following principle, part physical fault rate is converted to the equivalent fault rate of node or arc, enters one Step tries to achieve equivalent reliability；

1) for single components and parts give tacit consent to its qualified dispatch from the factory after, its service life should obey exponential law distribution, that is, this Components and parts fault rate λ is steady state value, and assert that institute is faulty and be all on components and parts, and wire connects good, and junction point is not Situations such as there is rosin joint；

2) arc connected by some components and parts for network node in figure, its equivalent process is carried out using below equation Conversion：

${\mathrm{\&lambda;}}_e={\mathrm{\&Sigma;}}_{i=1}^n{\mathrm{\&lambda;}}_i$

In formula, λ_eThe fault rate of this arc, λ after equivalent_iRepresent the fault rate of each components and parts in series connection successively, n represents network In node diagram, the component counts of series connection in this arc；

3) individual node fault rate is made up of three parts, and first is fault rate λ of node or arc itself_b；Second not In the case of protection device, the expansion fault rate that fault that adjacent node or arc produce causes, all of in arc Non-protected device, quantity is designated as w, and the expansion fault rate of wherein certain components and parts k is designated as λ_ek；3rd through protection device situation Lower and when protection device occurs tripping, adjacent node or arc are delivered to the probability of this nodes break down after producing fault, In arc, all protection devices are designated as m, and record fault is isolated and have passed through several isolating devices or protection switch before Quantity is designated as u, then being calculated according to the following formula of the reliability of a node：

${\mathrm{\&lambda;}}_{eb}={\mathrm{\&lambda;}}_b+{\mathrm{\&Sigma;}}_1^w{\mathrm{\&lambda;}}_{ek}+{\mathrm{\&Sigma;}}_1^m\left(P_{fi}{\mathrm{\&Sigma;}}_1^u{\mathrm{\&lambda;}}_{ek}\right)$

λ in formula_ebFor the equivalent rear fault rate of individual node, P_fiFor protection device fault transport=relay fail number of times/guarantor Shield should move number of times；

4) fault rate of individual node and arc is brought in the minimal cut set after non cross link is processed, using obtaining respective objects section Point is for the fault rate change of time：

$P\left(S\right)=\underset{i=1}{\overset{m}{\mathrm{\&Sigma;}}}p\left(E_i\right)$

In formula, P (S) represents this destination node fault rate, is equivalent to the reliability of corresponding system, p (E_i) represent m cut set In each cut set fault rate.

Brief description

Fig. 1 Model in Reliability Evaluation of Power Systems algorithm flow chart

Fig. 2 cut set non cross link flow chart

The total electrical structure diagram of Fig. 3 A380

Fig. 4 A380 network node figure

Specific embodiment

The present invention will be described with reference to the accompanying drawings and examples.

The specific targets of fail-safe analysis are mainly reflected on the trouble-proof probability of its aircraft in the unit interval.By general Probability (abbreviation reliability) summation of fault rate and reliability service knowable to rate opinion knowledge is 1.

And the mathematical principle of path matrix will be asked for using adjacency matrix in the graph theory being mainly concerned with patent and delete Going or staying purgation is described below

Ask for path matrix mathematical principle using adjacency matrix：

Meshed network N=(V), wherein V={ v₁, v₂, v₃, v_nIt is set of node.Arc can have two sections Point represents, direction is that previous node points to a rear node.

Adjacency matrix A₁=[a¹ _ij], describe the topology information in network.Wherein 1 be expressed as path be 1 adjacent Matrix is it is also possible to be considered first time Iterative Matrix in calculating process.The row and column of i, j difference representing matrix.

Terminal matrix R=[r_jk], matrix R reflects the terminal of every arc in meshed network N.

Define A₁A is tried to achieve in a kind of computing " ※ " with R₂, its algorithm such as a² _ij={ a¹ _ij※r_jk| j=1,2, 3, n }.

Leave out and leave method：

1) a n-dimensional vector Ei (x1, x2, xn) is defined for each minimal cut set, xi can obtain 0,1.When Xi represents that minimal cut set includes this node or arc when obtaining 1, represents that minimal cut set does not include when xi obtains 0 This node or arc.

2) define companion matrix Ti (t1t2, tn) and meet following condition

3) assume the minimal cut set that E1 is first non cross link.When i=2, compare Ei and Ti and find all units in Ti Element corresponds to, for 1, the position that relevant position in Ei is 0, according to size order record position sequence number and number and obtain k1, K2, kr.

4) Ei is decomposed successively according to K1, be decomposed into Ei (k1) andReplace ki position in Ei with 1, -1 respectively On 0 obtain.

5) compare Ei (k1) and Ej (j<I), the position if all of the corresponding Ej in the position having -1 in Ei (k1) has one Individual 1 or several 1 corresponding, and these 1 number summations are equal to line number j, it is incompatible with Ej to obtain Ei (k1), becomes not The minimal cut set of friendshipization.Therefore Ei (k1) storage is stayed, simultaneously forDo identical mode to process.

6) Ei (k1) and Ti-1 not having storage to stay is compared, if in Ti-1 not being the 0 corresponding Ei (k1) in position In corresponding position be not 0, then it is considered that Ei (k1) is by Ej (j<I) absorbed.Simultaneously forDo identical side Formula is processed.

7) do not delete if Ei (k1) does not stay, it is continued decompose, by Ei (k1) be decomposed into Ei (k1) (k2) andWith 1, -1 replace on k2 position 0, according to step 4,5,6 replicate analysis, until all of Ei staying or Make a choice in deletion.

From obtain an aircraft system structural topology figure to calculate its operation the relation to the time for the reliability, need with Lower five steps.With how, electric aircraft A380 is for the exploitativeness of case verification fail-safe analysis method.As Fig. 3 for A380 is electrically always System structure chart, and five steps as example correspondence analysis, the exploitativeness of checking fail-safe analysis method.

1st, according to following four principle, by abstract for topological network figure for network node figure：

1) network needs to accurately reflect parts relationship and the working condition of former topological diagram.

2) the abstract arc in figure such as power system breaker in middle, wire, transformator, commutator, electromotor, wherein can control Energy Flow direction processed for directed arc.

3) busbar, the abstract node in figure of load in power system, arranges a virtual source node, is equivalent to whole The source node of system power output.

4) network is as far as possible succinct, and the components and parts connected when not having branch road can be equivalent to an arc, type Load that is identical and deriving in a busbar can be equivalent to a node.

Network node primitive is changed according to structure chart and then Fig. 3 is changed into Fig. 4.

2nd, ask for source node and reach given load node minimal path sets

Using a kind of particular algorithm of adjacency matrix and terminal matrix, directly ask for the minimal path sets of network.Through excessive A is asked in secondary calculating₃, A₄,···A_n-1, can get All Paths in network, i.e. required minimal path sets.And by calculating process In character matrix be converted to basic road collection matrix, to facilitate subsequent calculations.

3rd, ask for the minimal cut set of fault generation

Observe minimal path sets, if each paths are concentrated all through a node in road, then this nodes break down Then mean that destination node necessarily occurs the situation of dead electricity, then this node element fault is exactly the single order minimal cut in network Collection.Node failure represented by removing minimal cut set, two elements that other paths all comprise or wherein any one element, Then this two element faults are exactly then second order cut set.Source node can be found using the method to each rank of destination node by that analogy Minimal cut set.That is any one ranks vector in minimal path matrix is all 1, then this row for element be exactly single order Little cut set.Again any two column vector correspondence in addition to single order minimal cut set is carried out inclusive-OR operation, if result is all 1, the minimum second order cut set that this two column vectors are constituted can be obtained, obtain all minimum second order cut sets by that analogy, can obtain in the same manner Little three rank cut sets, quadravalence cut set.

4th, the non cross link of minimal cut set is processed

" leave out and stay " method using by the agency of above carries out non cross link process to minimal cut set.This method is mainly thought Road is passed through increase argument and is progressively melted into disjoint minimal cut set, facilitates the calculating of last reliability.Leave out as Fig. 2 flow process uses Stay method that minimal cut set is processed.

5th, the conversion of reliability index

Because different nodes and arc are equivalent by different devices, wherein each device has corresponding failure rate λ of oneself.And And when breaking down, each device has interaction, can lead to corresponding failure rate λ that change occurs.Then being based on following principle will Part physical fault rate is converted to the equivalent fault rate of node or arc, can try to achieve equivalent reliability further.

1) for discrete component give tacit consent to its qualified dispatch from the factory after, its service life should obey exponential law distribution, i.e. this yuan Part fault rate λ is steady state value.And assert that institute is faulty to be all on components and parts, wire connects well, and junction point does not exist Situations such as rosin joint.

2) for the arc connected by some components and parts in network, its equivalent process can be carried out using below equation Conversion.Wherein λ_iRepresent the fault rate of each components and parts in series connection successively, n represents in network, first device of series connection in this arc Part number.

${\mathrm{\&lambda;}}_e={\mathrm{\&Sigma;}}_{i=1}^n{\mathrm{\&lambda;}}_i$

3) individual node fault rate is made up of three parts, and first is fault rate λ of node or arc itself_b；Second not In the case of protection device, the expansion fault rate that fault that adjacent node or arc produce causes, all of in arc Non-protected device, quantity is designated as w, and the expansion fault rate of wherein certain element k is designated as λ_ek；3rd in the case of protection device And protection device is when occurring tripping, adjacent node or arc are delivered to the probability of this nodes break down, warp after producing fault Cross all protection devices in arc and be designated as m, and record fault is isolated and have passed through several isolating devices or protection switch number before Amount is designated as u.Then being calculated according to formula of the reliability of a node

${\mathrm{\&lambda;}}_{eb}={\mathrm{\&lambda;}}_b+{\mathrm{\&Sigma;}}_1^w{\mathrm{\&lambda;}}_{ek}+{\mathrm{\&Sigma;}}_1^m\left(P_{fi}{\mathrm{\&Sigma;}}_1^u{\mathrm{\&lambda;}}_{ek}\right)$

P in formula_fiNumber of times should be moved for protection device fault transport=relay fail number of times/protection.

Finally the fault rate of each node and arc is brought in the minimal cut set of non cross link process, by theory of probability knowledge The fault rate that respective objects node can be obtained for the time changes.

Write in USAF《AS aviation electronics plan》In to crucial airplane load require in two hours Interior uninterrupted power supply reliability reaches 0.999 975 25.Important load in two hours in the calculating with A380 as example Reliability reach 0.999 997 88, reach related request.

Through instance analysis, the circuit redundancy improving critical loads can significantly improve long playing reliability； Reduce the number of components passed through before electric energy reaches critical loads and can improve short time reliability.This is to many electricity aircraft early stage electricity Power design provides good reference and help, and gives a comparison feasible weak spot in prediction many electricity aircraft electrical power system Method, the optimization and upgrading to later stage whole power system simultaneously has corresponding directive function.

Based on following precondition：Disconnecting switch, passage paralleling switch have actuator and control according to grid weight Structure requires automatically to close, then dead electricity situation is the situation that still cannot make service restoration after failure reconfiguration.

Eventually pass step 5 and achieved reliability is switched to node reliability, its result of calculation such as table 1

The concrete node of table 1 and the reliability of arc

The conclusion that analysis can obtain correlation is calculated to the result obtaining.Its method is equally applicable to the warship under running status The Calculation of Reliability of the not strong independent microgrid system of ship, electric automobile and self-healing ability.Need the not homology according to practical application System, flexibly holds to the principle of network conversion；The physical significance of reasonable definition different rank minimal cut set；Suitably change coefficient Part physical fault rate is changed into node failure rate.And interpretation of result is obtained with corresponding conclusion, to microgrid operation maintenance meter Draw and advise it is also possible to suitable adjustable strategies are given to microgrid structure updating.

Claims (1)

1. a kind of many electricity aircraft electrical power system analysis method for reliability based on minimal cut set analytic process, comprise the steps：

Step one：According to following four principle, topology diagram is equivalent to network node figure：

1) network node figure needs to accurately reflect parts relationship and working condition；

2) by abstract for the components and parts such as chopper, wire, transformator, commutator and electromotor in former topological diagram for network node in figure Arc, wherein control energy Flow direction for directed arc, otherwise for undirected arc；

3) by abstract to busbar in the power system and load node for network node in figure, a virtual source section is set simultaneously Point, is equivalent to the source node of whole system power input；

4) network node figure is as far as possible succinct, and the components and parts connected when not having branch road are equivalent to an arc, and type is identical And the load in a busbar derivation is equivalent to a node；

Step 2：Source node is asked for using Adjacent Matrix Method and reaches specified node minimal path sets, method is as follows：

1) meshed network N=(V), wherein V={ v are set₁, v₂, v₃, v_nIt is set of node；

2) define adjacency matrix A₁=[a¹ _ij], the topology information of description network node in figure, the subscript 1 of wherein element name represents The adjacency matrix being 1 for path, also illustrates that first time Iterative Matrix in calculating process simultaneously, i, j difference representing matrix Row and column：

3) define terminal matrix R=[r_jk], the terminal of every arc in reflection meshed network N：

4) define A₁A is tried to achieve in computing " ※ " with R₂=[a² _ij], its algorithm is as follows, a² _ij={ a¹ _ij※r_jk| i, j, k=1, 2,3,···,n}：

5) ask for virtual source node to the minimal path sets specifying node using as above method, minimal path sets conversion is entered the most simultaneously Path collection matrix facilitates subsequent calculations machine auxiliary to calculate；

Step 3：Ask for the minimal cut set of fault generation：

Observe minimal path sets, if each paths are all through same node or same arc in minimal path sets, then this Individual node or arc break down, and mean to specify node or arc that the situation of dead electricity necessarily occurs, then this node or arc just belong to Single order minimal cut set in grid；Remove single order minimal cut set represented by node failure, other paths all comprise two Individual components and parts or wherein any one components and parts, if two components and parts break down simultaneously, mean to specify node or arc The situation of dead electricity necessarily occurs, then this two components and parts just belong to second order cut set；Find source section using the method by that analogy Point is to specified node each rank minimal cut set；That is any one ranks vector in minimal path sets matrix is all 1, then this arrange to Measuring corresponding components and parts fault is exactly single order minimal cut set；Again any two column vector in addition to single order minimal cut set is carried out Inclusive-OR operation, if result all 1, obtains the minimum second order cut set of this two column vectors corresponding components and parts composition, with such Inquire into and all minimum second order cut sets, obtain three rank cut sets, quadravalence cut set in the same manner；

Step 4：Non cross link process is carried out to minimal cut set using " leave out and stay " method：

1) a n-dimensional vector E is defined for each minimal cut set_i(x₁, x₂, x₃···x_n), x_iTake 0 or 1, work as x_iObtain 1 When represent that minimal cut set includes this node or arc, work as x_iObtain represent when 0 minimal cut set do not include this node or Person's arc；

2) define companion matrix T_i(t₁, t₂, t_n) and meet following condition：

3) set E₁For the minimal cut set of first non cross link, when i=2, compare E_iAnd T_iFind T_iMiddle all elements are right for 1 Answer E_iMiddle relevant position is 0 position, according to size order record position sequence number and number and obtain k₁, k₂, k_r；

4) by E_iTo k_iDecomposed successively, be decomposed into E_i(k₁) andWherein E_iMiddle k_iOn position 0 is by 1 or -1 respectively Replace；

5) compare E_i(k₁) and E_j, wherein j<I, if all of E_i(k₁) in have -1 position correspond to E_jPosition have one 1 Or several 1 is corresponding, and these 1 number summations are equal to line number j, obtain E_i(k₁) and E_jIncompatible, become non cross link Little cut set, therefore by E_i(k₁) storage stay, simultaneously forDo identical mode to process；

6) for the E not having storage to stay_i(k₁) and T_i-1Relatively, if T_i-1In be not 0 position correspond to E_i(k₁) in corresponding Position is not 0 then it is assumed that E_i(k₁) by E_jAbsorbed, wherein j<i；Simultaneously forDo identical mode to process；

7) if E_i(k₁) do not stay and do not delete, it is continued decompose, by E_i(k₁) it is decomposed into E_i(k₁)(k₂) andWith respectively with 1, -1 replacement k₂On position 0, according to step 4,5,6 replicate analysis, until all of E_iStaying Make a choice down or in deleting；

Step 5：Based on following principle, part physical fault rate is converted to the equivalent fault rate of node or arc, asks further Obtain equivalent reliability；

1) for single components and parts give tacit consent to its qualified dispatch from the factory after, its service life should obey exponential law distribution, i.e. this yuan of device Part fault rate λ is steady state value, and assert that institute is faulty and be all on components and parts, and wire connects good, and junction point does not exist Situations such as rosin joint；

2) arc connected by some components and parts for network node in figure, its equivalent process is rolled over using below equation Calculate：

${\mathrm{\&lambda;}}_e={\mathrm{\&Sigma;}}_{i=1}^n{\mathrm{\&lambda;}}_i$

In formula, λ_eThe fault rate of this arc, λ after equivalent_iRepresent the fault rate of each components and parts in series connection successively, n represents network node In figure, the component counts of series connection in this arc；

3) individual node fault rate is made up of three parts, and first is fault rate λ of node or arc itself_b；Second without guarantor In the case of shield device, the expansion fault rate that fault that adjacent node or arc produce causes, all of non-protected in arc Device, quantity is designated as w, and the expansion fault rate of wherein certain components and parts k is designated as λ_ek；3rd in the case of protection device and protect When shield device occurs tripping, it is delivered to the probability of this nodes break down after adjacent node or arc generation fault, through arc In all protection devices be designated as m, and record fault be isolated before have passed through several isolating devices or protection switch quantity note For u, then being calculated according to the following formula of the reliability of a node：

${\mathrm{\&lambda;}}_{eb}={\mathrm{\&lambda;}}_b+{\mathrm{\&Sigma;}}_1^w{\mathrm{\&lambda;}}_{ek}+{\mathrm{\&Sigma;}}_1^m\left(P_{fi}{\mathrm{\&Sigma;}}_1^u{\mathrm{\&lambda;}}_{ek}\right)$

λ in formula_ebFor the equivalent rear fault rate of individual node, P_fiShould for protection device fault transport=relay fail number of times/protection Dynamic number of times；

4) fault rate of individual node and arc is brought in the minimal cut set after non cross link is processed, using obtaining respective objects node pair Fault rate change in the time：

$P\left(S\right)=\underset{i=1}{\overset{m}{\&Sigma;}}p\left(E_i\right)$

In formula, P (S) represents this destination node fault rate, is equivalent to the reliability of corresponding system, p (E_i) represent each in m cut set The fault rate of cut set.