大型复杂系统的可靠性与冗余度优化

第二9卷第!期二00:年4目

华北电力大学学报

JournalofNorthCbinaElectrjcP0werUniversny

VoI29.No2Apr,2002

文章编号:1007—269l(2002)02一0034—05

大型复杂系统的可靠性与冗余度优化

谭忠富

(华北电力走学工商管理学院,北京102206)

Reliabili毋andredundancyoptimizationforalarge-scalecompoundsystem

TANZhong—m

Instltutc。fBuslness

Management,NonhchlnaE1ectncPowerunlversity.Be蟮ing102206,chitla)

摘要:个系统可以分解为组r系统.而每个产系统叉

1arge—scalesystemwhichiscomposedofmanycom-ponemscaⅡbedecomposedintomulti-1evelsubsys-tems109ic

In

可以分解为一维于系统,如此下去,犬犁复杂系统可咀扶二到}舒解为绷多级于系统。为了获得系统叫靠忭与兀袅童的总体优化.将每叶、f系统的可靠胜及其走r成本的偏导数作为分解参数.每个子系统的成本作为协调参数.构

遗J’练台分解协调算法

关键词:可靠性:冗余度;分解:协调中图分类号:023l

eachlevel,thefeexistsomesubsystems,the

are

reIations锄ongthem

assumedtobeseries

or

parallelForeachbonomsubsystem,melogicrelationsamongits

componeⅡtsare船sumedtobeparallelor(1/

How

七)(G)orstand_by

文献标识码:A

ls

shouldcomponentsbealloc-

ated

decomposedmto

so

t11atitsreIiabili“rcachesitsmaximummlder

Abstract:Ala唱e-scalecompo岫dsystemmuⅢ一le、elsub黔s忙ms,{iomupcomposcdof

to

givencost?In血isp叩er,consideringcost

as

ofeachsub—

ma—

down,eachsⅡb5ystem堪inIower1evel,unt儿nleDeed

10

8印up

ofsubsystems

nol

systeminits10werleVelximizingthesubsystem。s

deslgn

v撕able,by

bono抛suhyslemswhlchdoCons】den“geachsub5ystem’sParameter疳omupden、an、e陀latedparameter

to

be

as

decomposedcoordlnatlon

1ts

reliabili吼the

sub-prog姗一

constmcnon

cost

mingproblemc卸becons廿1lcted,

can

itsoptimalsolution

down.eachsub5ystem’srellabilit,a11d

to

beusedforconstmcⅡngsub—progmmmingfor

constmctlon

to

costas

decomposltl蚰

subsystemsinitslowerlevel,untiltheoptimalcos忸forbouomsubsvstems

are

m)mdown

up.adecomposltion—coordinatlon

1s

attained

Hence.fbreach

algonthrn{brreIlabllltyaIldredundaIl。yoptlⅢlzatlonK。ywords:rellablli哆:

natlon

givencoordl-

bonomsubsystem,consideringfeliabilityandredun—dancyofeach

redundan。);decomposlnon,

c伽1ponent船designv鲥able,a

mixed

pro群ammingproblemispresented.Sothelarge—scalecompouDds”lem

is订韧sfo埘ed

bes01ved

iI】loa11bom)msub—

frnmdownto

Introductionsvstemswhich

can

easiIv,蛐d

up,血ereliabilityanditsderivativerelatedto

It

cost啪

isknownthatmanyengineeringsystems盯e

ones

calculatedforeachsubsvstemineach1eveltosoIvethesub—programmingproblemforitsuppersubsystem.。BytheaboVe,reliabilitycan’tbeimprovedundefthe

cost

complexandlarge-scale

whichconsistofsomealsobedecomposed

subsystems,eachsubsystemmto

can

group

subsystemsagaiⅡ,untilthebonomsub-

limltation

Now,theoptimalallocationforreha—

ofcornponents

caIl

systemswhichconsistofsomecomponents.Hence,a

bil时aⅡdredundaIlcy

be

anai∽d.

Received:2002—01一01

Sponsoredby:NauonaINatura}SclenceFoundatIon

ofChina(GramNo

79800002)

AuthorbriefintrodⅡction:TAN7hong—fu,Bomin1964,male.VIcepmfessor

万方数据 

第2期谭忠富:大型复杂系统的可靠性与冗余度优化

35

Decomposmonoptimiza60nmodelfora

largescalecompoundsystem

Fora

la玛escalecompoundsystem,沁multi-l剁-

eldecomposihonfon】qmaybegivenin行gure】,wherethemainsystemSconsistsof月lsubs弘temsSo:。In

f酏tlevel,1玉lsHlJlo。consistsofmlo.subsystems最Jj¨

iⅡsecoⅡd

level,l句≤m10f,where(2il,2f2,…,2加Io.)=

11.…,H2),le,there

aren2

subsystemsinsecondlevel,

是~consistsof小nsubsystemsS』M。Int11irdlevel,

2扩952immls七茎m:。where

(Ⅳl,可2,…,可埘zF)=

(1,2,…,月;),i

e,也ereare

subsyst锄sin

mird1evel,

and

soon,on

the

a呻logy

oftheabovestatement,there

areH,1

subsystems乱,in(,_1)tlllevel,S山。consists

of

m¨岫subsystems乱"inlthlevel,where最uFisabot-

tomsubsystcmwhichis

co唧osed

ofsomecompon-

ents

inparallelor(1/七)(G)ofstand-by

logicrelation,

1墨,墨m川"and(jn,阮,…,打mu_,hj:【1,2,…,一-),i

there

are月l

bottomsubsystemsin1th1evelDenoting

R。且sreliabilityof(i,,七)thsubsystem,C止asa110c砷甜

cost

of(f,^)也subsystem,whichisconsidered

as

de-

si2丑variable,whereiismeorderofnlelevelwhichiIl—cludessub—pmFa豳g

SJ√is血eorder

ofS。in(卜lⅫleVel

Hence,

forcoordinatingsubsystemsineach

level

can

beconsnlJcted

as

follows:

Thesub-programmmgforcoordinatiIlgsubsys.

temsinfirstkvel:

(P0)m确=石(RⅧt,R,nz,…,RL0,,)

st.

∑c。。=G

(1)

2二】

CⅧ。≥O,1Si茎”l

长■

乳川√矗/

。\/。卜卜

。兮夺~念滁

Fig.1

Mult■levetdecomposinonform

万 

方数据Where风Issystemreliabili吼c0isconstmction

cost

limitatioⅡgivenforthesystem.

Sub-pm芦ammingforcoordinatingsubsystems

insecond1evel:

(P。¨)ma斌¨《n。【尺2“。,R:。t,…,尺:一。.)

—C

(2)

‰∑一

c2¨。≥O,1巧smm

whefec¨..1s

given疔omtheoptimalresult

oflR),

l兰f≤Ⅳ..

Sub-programmingforcoordinatingsubsystemsintllirdlevel:

(尸2。,)maxR2广弧.,(只¨}J,R¨”.…,尺¨枷。)

卅‘

s。t.

∑Gm。=cz。

(3)

l=l

G,计≥0,1≤七sm:。

whereC.¨isgiven的mtheoptimalresult

of妒Io.),

l!f≤吼,2盯≤,s2i州m.

ontheanalogyoftlleaboVesub_programming,

抽I

sub—pm乒ammiIlgforcoordinatingsubsystemsm,

tlllevel:

(P『。,jmaxRr,,,=项一川(只^,批月¨w,‘‘‘,尺^,怖。)

mlElH

s.t.∑cf.。=c』“,

(4)

扣1

(1,¨≥O,ls七s聊。一1h,

、ⅣhereG_1叫isgjven丘omtheoptlmalresultofasub—

pr0尊aInIllillgfofcoordillanngsubsystemsin(,~1)th

level

Foreachsub-pmgr猢ing

above,

its

objectiVe

mnctioncan

becons仃1lctedbytlle109icrelation

aInon—

subsys蛐sinlowefIevel'suchasse【iesorparaIkl

ForiIlstance,

if也ereexlsts

seriesrelaⅡonamongaUsubsystemsmfirstkvel,thentheobjectiVe

mnc石on

of(Po)canbeexpressed

as

follows

R=兀尺L

lfthereexists

parallelrelatlonamongallsub—systemsin疗rstlevel,thenthe

objectiVe

mnctionof

华北电力人学学报

(R)caⅡbeexpressedby:

Ro_l一兀(1一月.¨1

Fl

For血eothersub—programmlng,也eir

objectivemnctionscan

alsobcexprcssed

on

theanalogyofthe

above.

Weassumemere

exlsts

only

one

classofl091c

re—

lanonamongallcomponentsineachbonomsubsys—tem,such

as

parallelor【^,卅^)(G)orstaⅡd-by

Hence.

foreachbottomsubsystem,1tssub—pro—

grammmgProblem

can

begiven

as

f0110ws:

【|D¨)max尼,t=∥“(^(G),卅t)

t.

m。G=G¨{5)

WhereG≥O,G¨1sgi ̄en

fomtheoptjmal

solution

of(,㈠…).意=1,2,…,月hand卅t≥^forthe(厶/州。)(G)

sub8ystem.There

are

t11reemncliontypes

f0听…for

instance。ifS,{ls

parallelsubsystem,then

只¨。=1一(1一“)“

(6)

IfS.}isa(^/埘I)(G)subsystem,men

,wl一^

R¨。=∑a.【l一^y一・。

(7)

F0

Where^istheminimumnumberofcomponents,

卅t≥^.c:.=埘t!/[口!(卅。一口)味

IfS.。is

stand-bysubsystem,

then

Ⅲ。』

月。。=∑(一1n“y,√q!

(8)

q卸

For

eachcomponent,

withoutlossof

general咄

thcmnctlonrela“onforitsfeliabilitynandits

con—

stn】ctloncost

GisassumedtobeⅡ1at:

“㈦)=exp[d√慨一G)]

(9)

whered≥0.口≥O

AnalgOrithmforsolvingthedecompOsi—

tiOnOptimiztionmodel

Analgorimmforsolv{ngtheaboVedecompos卜

tlon

optmlizationmodel

can

begiven

as

follows:

step1.Forthe91vencons仃1lctioncost

11ml僦ionG

forthesystem,厅omuptodown,giveinitialconstruc・

tlon

cost

allocationfofeach

subsystem

e,glve

万 

方数据c’1

o。,l≤f≤n1.to

satis矗(1),G,i,,i茎,s州m,to

satis印【2),cn”l≤七兰m叩tosatisfy(3),・・.c,M,

1≤置玉州㈨,to

satls母(4)Then士rom

downtoup,

solvesub—programmlngsln

eachIeVclbygradjentpm—

Jectionmelhod

as

follows

Step2.S01Vethesub-programingsforbonomsubsys—

tems,forinstance,to(尸㈡):lfSL,isparallelbonomsubsystemor

stand_bybot—

tomsubsystem,take

Ⅲ』_1.2.3.-..

1fs¨is(厶/州})(G)bonom

subsystem,take

州。=^,^+1,^+2,…

Foreach卅.,calculatee=G,。/mhn(c^Jln(9),andtheobjectlve

of(B¨),i

e.,R¨=^.-I^((j),,"I¨n

(6)or(7)or(8),unIil尼¨,stanstodecreaseStore

瓣妙

瓣m‰丧

“01

where曼挚is90nenfonn(6)。r(?)州乳and

害鲁=(仉假)exp[一吼(G伊。一川

Step3.SolveIhesub-programmingsforco(Irdmatingbottomsubsystems,forins协nce,to

solVe(F:“,)

calculate;鲁等andp眯cti。n可adient:

黠=舞羔器:面jii百万。。0e。。

Vm。-【麟,憋.…,謦等)T_

【l,卜。。川翟咎。‰。;,

glvenaccofdlngt。thespeci6cfunc-

lionfom。fthe。bjectivein(九。,瓣:ts

where%鲁蔫is

given

bySlep2,1et

(C,。,o。,….C。~)’

一lC¨,G_山…,C。,)1。^可.月I,,

WhereIisiteratestcplengtlldeterminedb)’one—di—

mensionalsearch.Ifitissalisnedthat

llV月.…川<£,wheIes

ls

a出ven

c。n㈣precision'thenst。re};等astk

foll。wmg:

玛:硝

谭忠富:犬犁复杂系统的可靠性与冗余J皇优化

3,

othenvise,g。tosteD

跨2跨Ⅲ翻。。~,,鼾印61凡“蹦≯1翼嚣??2瓦i两ij啊了

step

4.on暑eanal矗y。fStcp3’s0Ivesub-pr。盯am.

mlngfor∞ordinatingsubsystemsin(,一1)thleVel,sub—

pmgrammingforcoordinatlngsubsystems血(,.2)th

le、el,and

so

on,un州thesub。pmgrammingfor

coor-

diⅡatiⅡgsubsystemsin

mst

leVel,forinstance,

the

solVingprocessof(只¨)and(尸0),seet|lef0110wing

step

s.胁c‰^叫c慨暑鲁兰astn。如--。。,。。:囊塾:梁u未彗一,l哥辄。百石i百ij,j百瓦i11爿3%…

where}戋is

glvenatthe

optiml

point。f(‰^

恶。,can

be

given

accordingt。thespecincfonIl。f

mnction^o

Ca}culatethe

projection

gradieⅡI

of

【Pl。)anditeratedesignvariables

as

f0110wsF一‰:【氆

妇砭

':o

㈦卜...川蓦疆渺。

。…,o!。.)7一

(G二。,G。一…,G。。)1“V.R。.

ls

iteratest。Plen甜hdetelminedbyone—di—

search.

Ifitissatisfiedthat

ff

V月1

o,ff龟,

store导訾ev

挚-:嘉擎Ⅱ,i班‰。a瓦ii一石瓦_『j’1纠3”10‘

112)I“J

to

Step6.

o也envlsea110cateagainconstmctio“

ofsubsystemsinthird1evel,fonhlevel,and

so

untiI^hIevel

as

f0IIows:

m一卜√》水,

C广

‰∑一

Ste02

万 

方数据where

g慧-sgiven

forrnstc

p5’。粤ts

g-ver

ac—

cordlngtothespecl矗cfonnofmnctlon工

lfltissat—

isnedthat||V吼【】勺,then

outputthecostsofcompon。

ents

andthesyscemreliabiljcy尺0.Sf。p

()【hernise.

calculatethe

projectiongrddientoffP。)andlteratede—

signvanablesby

甲肛(袅i,悲,…、最卜

(1-卜一,w耋袅i/一,

(C-一,,C。!,….(1㈨.)1一

(G川,C“-.….C,¨)Lj守凰

whereiisIteratesteplengthdeterrnlnedbyonc_d卜mensionalsearchAllocateconstmc廿oncostsorsub.

systemsagain

1nsecondlevel’thirdlevel,and

h。on.

until^hJevel

as

f0”ows:

㈡。一k。/至。。k

扣1

(1,

。/塞l。If

口=1

‰。十。。,擎。十。

goto

Step2

Proposition1.Attheoptlmalpomtofeach)ub-

problem,onehastllat:

裳唔孤;参_碧.t螺n

謦声辩.-盥m.

鲁≯=器。、:.,,啦m。

ProoC

By

use

oftheK—TcondlLlonofeachsub—

call

have

《C.:。,C

wheremensional帆n

go

progmmmingcosts

on.

窖oto

programmjngprobIem,a11d(1)~(5).wc

华北电力夫学学报

2002年

(10)~(12).

P呷os埘oⅡ2.

Forttleabove

alg嘶恤,

its

iteratesequencesare

ofthemain

pr0伊鲫岫g

convergentto也eK_Tp0Intwhichisexpressed

as

follows:

0))maxJR。=

烈,t(c』),r2(c;),…,n(e),小。,珊:,…,卅。)

s.t.∑,Ⅲ.G=c0

^=l

m±≥厶,珊l+l!七兰肌1+小2

where

4k萌(^o.,(…),…,』n.(…,正.。,(…),…)…,

』o.(…)),

itis

compoundmnctionf研all

objecⅡve

如ncnonsoftlleabovesub—pmgra【n:札-mg.

ProoC

niseas订yprovedthatthecomposinonof

K_Tcondi廿onsf研allsub-programmingistheK—Tcondi恼onofthemain

pro铲町ming(P)since

nlepro—

jectiongradientmethodusedins01vingeachsub-pro-

掣铷nmingisconvergent,sotheiteratesequences

are

c蚰vergent

to

theK—Tpointof也emaillpr0可amming

(P).

Proposition3.

FortheabovealgoritIlI工l,

its

iteratesequencesare

convergemtotheoptimals01-

udonof山emain

progmmmiIlg(P).

PmoCBy也enmchonf0Ⅱnof尼aboutRln.,it

】sto

knowthat

器≥o

OntlleaIlalogyoftlleabove,t11efollowingcan

begonen

溉>o,溉她爨巩

Ffom(9)'

it

caIl

bee船ilypfovednlat

百西‘”铝≥o

瓯砥贰…切i忑]F砸“穗畿溉…瓣警凳≥o

万 

方数据ie铬茎o

Hence士is

convex如nctionabout

ch

l≤A≤”1,

(P)isaconvexpro伊ar衄ingproblem

weknowthat

theK-Tpointof

conVex

programmiⅡgproblemis

also池optimalsolution,sofbmProposition2,one

hasthatiteratesequencesoftlleabovealgorimm

afe

conve娼em

to

t11eoptimalsolutionofthemampm掣a-

mming(P).

Example.AssumingthatSiscomposedof6par_

allelsubsystemsin行rstlevel,i.e,HI=6,Sl州consistsof"。1parallelsubsystems

and"n(1服)(G)subsyste雌

and月astaIld_bysubsystems1Ⅱsecondlevel,

n,1(1茎f蔓6)as2,3,4,5,6,7,”口(1sjs6)as2.5,6,6,7.7,no(1s注6)as5,2,3,3,2,2.Somere

are

77redundant

subsystemstotallyinsecondlevel

Consideringt|Ie

pl锄e

positionofallredllIldantsubsystemsmsecond

level

as

Irla仃ix,we

give

血eorderrlotes

七=1,2,3,…,77to

them,仔omuptodown,斤om

1eftcolurnntonghtcolumn.Tak号夙(1』七兰77)as9+20,10+23。13’25,14+lO,15+17,16+14,d±(1≤七s77)

as

9+16,10+12,

13+l5,

14+10,15+17,16+19,厶

for

3≤七≤4,

13≤七517,

24≤七≤29,38≤七≤43,

50≤七156,61兰七兰67,as2+2,5+2,6+2,6+2,7+2,

7+2.

Bytheabovealgor主Ihm,theoptimairesultisob-

tained

as

fbllows:

m15m72ml32m19=4,m2523,m312ml-2m412Ⅲ4q-2,埘。=2for3≤丘茎4,13茎七兰17,24≤膏≤29,381^143,50≤七≤56.6l≤七≤67,州Flforother七,n=r,=O.7l,n]_r19=h5=n1=n,=rd3=h9=O.896,,、=004forother七,尺o

=0955

References

【1】Gu扎gyu卸Wangnleson如slgⅡtheory

for

en91Bee血g

systems(inchinese)【M1.Beqiog:sclencePress,1992【2】AIl00pKDhingraoptimalappo^ionmentofrellabil・ty

叩dred咖daⅡcyin

senes

3ystemsundermul【lpk

[门.IEEEl觚actio啊onReliabdi吼1992,4l(4)576—58l

objectIVes

【3】Jinhuac∞,YuedongwaⅡgopnmalallocatlonofa‘印alrabk

system[J]Microelec们nicsandRellabiIitM

1990,30(6)

109l,1093

大型复杂系统的可靠性与冗余度优化

作者:作者单位:刊名:英文刊名:年,卷(期):被引用次数:

谭忠富

华北电力大学工商管理学院,北京,102206

华北电力大学学报

JOURNAL OF NORTH CHINA ELECTRIC POWER UNIVERSITY2002,29(2)4次

参考文献(3条)

1. GuangyuanWang The soft design theory for engineering systems 1992

2. Anoop K Dhingra Optimal apportionment of reliability and redundancy in series systems undermultiple objectives 1992(04)

3. Jinhua Cao. Yuedong Wang Optimal allocation ofa repairable system 1990(06)

引证文献(4条)

1. 于晓东. 高会生. 郭爱玲 基于模糊理论的电力通信网络有效性研究[期刊论文]-华北电力大学学报 2008(5)2. 杨明顺. 李鹏阳. 李言. 袁启龙 考虑不同类型可靠性-成本函数下的可靠性优化设计[期刊论文]-中国机械工程2006(22)

3. 李蒙. 胡兆光 基于智能工程理论拓展的政策正向状态模拟新方法[期刊论文]-中国电机工程学报 2006(z1)4. 张海泉 集散控制系统中冗余通信设计及实现[学位论文]硕士 2006

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下载时间:2010年10月10日

第二9卷第!期二00:年4目

华北电力大学学报

JournalofNorthCbinaElectrjcP0werUniversny

VoI29.No2Apr,2002

文章编号:1007—269l(2002)02一0034—05

大型复杂系统的可靠性与冗余度优化

谭忠富

(华北电力走学工商管理学院,北京102206)

Reliabili毋andredundancyoptimizationforalarge-scalecompoundsystem

TANZhong—m

Instltutc。fBuslness

Management,NonhchlnaE1ectncPowerunlversity.Be蟮ing102206,chitla)

摘要:个系统可以分解为组r系统.而每个产系统叉

1arge—scalesystemwhichiscomposedofmanycom-ponemscaⅡbedecomposedintomulti-1evelsubsys-tems109ic

In

可以分解为一维于系统,如此下去,犬犁复杂系统可咀扶二到}舒解为绷多级于系统。为了获得系统叫靠忭与兀袅童的总体优化.将每叶、f系统的可靠胜及其走r成本的偏导数作为分解参数.每个子系统的成本作为协调参数.构

遗J’练台分解协调算法

关键词:可靠性:冗余度;分解:协调中图分类号:023l

eachlevel,thefeexistsomesubsystems,the

are

reIations锄ongthem

assumedtobeseries

or

parallelForeachbonomsubsystem,melogicrelationsamongits

componeⅡtsare船sumedtobeparallelor(1/

How

七)(G)orstand_by

文献标识码:A

ls

shouldcomponentsbealloc-

ated

decomposedmto

so

t11atitsreIiabili“rcachesitsmaximummlder

Abstract:Ala唱e-scalecompo岫dsystemmuⅢ一le、elsub黔s忙ms,{iomupcomposcdof

to

givencost?In血isp叩er,consideringcost

as

ofeachsub—

ma—

down,eachsⅡb5ystem堪inIower1evel,unt儿nleDeed

10

8印up

ofsubsystems

nol

systeminits10werleVelximizingthesubsystem。s

deslgn

v撕able,by

bono抛suhyslemswhlchdoCons】den“geachsub5ystem’sParameter疳omupden、an、e陀latedparameter

to

be

as

decomposedcoordlnatlon

1ts

reliabili吼the

sub-prog姗一

constmcnon

cost

mingproblemc卸becons廿1lcted,

can

itsoptimalsolution

down.eachsub5ystem’srellabilit,a11d

to

beusedforconstmcⅡngsub—progmmmingfor

constmctlon

to

costas

decomposltl蚰

subsystemsinitslowerlevel,untiltheoptimalcos忸forbouomsubsvstems

are

m)mdown

up.adecomposltion—coordinatlon

1s

attained

Hence.fbreach

algonthrn{brreIlabllltyaIldredundaIl。yoptlⅢlzatlonK。ywords:rellablli哆:

natlon

givencoordl-

bonomsubsystem,consideringfeliabilityandredun—dancyofeach

redundan。);decomposlnon,

c伽1ponent船designv鲥able,a

mixed

pro群ammingproblemispresented.Sothelarge—scalecompouDds”lem

is订韧sfo埘ed

bes01ved

iI】loa11bom)msub—

frnmdownto

Introductionsvstemswhich

can

easiIv,蛐d

up,血ereliabilityanditsderivativerelatedto

It

cost啪

isknownthatmanyengineeringsystems盯e

ones

calculatedforeachsubsvstemineach1eveltosoIvethesub—programmingproblemforitsuppersubsystem.。BytheaboVe,reliabilitycan’tbeimprovedundefthe

cost

complexandlarge-scale

whichconsistofsomealsobedecomposed

subsystems,eachsubsystemmto

can

group

subsystemsagaiⅡ,untilthebonomsub-

limltation

Now,theoptimalallocationforreha—

ofcornponents

caIl

systemswhichconsistofsomecomponents.Hence,a

bil时aⅡdredundaIlcy

be

anai∽d.

Received:2002—01一01

Sponsoredby:NauonaINatura}SclenceFoundatIon

ofChina(GramNo

79800002)

AuthorbriefintrodⅡction:TAN7hong—fu,Bomin1964,male.VIcepmfessor

万方数据 

第2期谭忠富:大型复杂系统的可靠性与冗余度优化

35

Decomposmonoptimiza60nmodelfora

largescalecompoundsystem

Fora

la玛escalecompoundsystem,沁multi-l剁-

eldecomposihonfon】qmaybegivenin行gure】,wherethemainsystemSconsistsof月lsubs弘temsSo:。In

f酏tlevel,1玉lsHlJlo。consistsofmlo.subsystems最Jj¨

iⅡsecoⅡd

level,l句≤m10f,where(2il,2f2,…,2加Io.)=

11.…,H2),le,there

aren2

subsystemsinsecondlevel,

是~consistsof小nsubsystemsS』M。Int11irdlevel,

2扩952immls七茎m:。where

(Ⅳl,可2,…,可埘zF)=

(1,2,…,月;),i

e,也ereare

subsyst锄sin

mird1evel,

and

soon,on

the

a呻logy

oftheabovestatement,there

areH,1

subsystems乱,in(,_1)tlllevel,S山。consists

of

m¨岫subsystems乱"inlthlevel,where最uFisabot-

tomsubsystcmwhichis

co唧osed

ofsomecompon-

ents

inparallelor(1/七)(G)ofstand-by

logicrelation,

1墨,墨m川"and(jn,阮,…,打mu_,hj:【1,2,…,一-),i

there

are月l

bottomsubsystemsin1th1evelDenoting

R。且sreliabilityof(i,,七)thsubsystem,C止asa110c砷甜

cost

of(f,^)也subsystem,whichisconsidered

as

de-

si2丑variable,whereiismeorderofnlelevelwhichiIl—cludessub—pmFa豳g

SJ√is血eorder

ofS。in(卜lⅫleVel

Hence,

forcoordinatingsubsystemsineach

level

can

beconsnlJcted

as

follows:

Thesub-programmmgforcoordinatiIlgsubsys.

temsinfirstkvel:

(P0)m确=石(RⅧt,R,nz,…,RL0,,)

st.

∑c。。=G

(1)

2二】

CⅧ。≥O,1Si茎”l

长■

乳川√矗/

。\/。卜卜

。兮夺~念滁

Fig.1

Mult■levetdecomposinonform

万 

方数据Where风Issystemreliabili吼c0isconstmction

cost

limitatioⅡgivenforthesystem.

Sub-pm芦ammingforcoordinatingsubsystems

insecond1evel:

(P。¨)ma斌¨《n。【尺2“。,R:。t,…,尺:一。.)

—C

(2)

‰∑一

c2¨。≥O,1巧smm

whefec¨..1s

given疔omtheoptimalresult

oflR),

l兰f≤Ⅳ..

Sub-programmingforcoordinatingsubsystemsintllirdlevel:

(尸2。,)maxR2广弧.,(只¨}J,R¨”.…,尺¨枷。)

卅‘

s。t.

∑Gm。=cz。

(3)

l=l

G,计≥0,1≤七sm:。

whereC.¨isgiven的mtheoptimalresult

of妒Io.),

l!f≤吼,2盯≤,s2i州m.

ontheanalogyoftlleaboVesub_programming,

抽I

sub—pm乒ammiIlgforcoordinatingsubsystemsm,

tlllevel:

(P『。,jmaxRr,,,=项一川(只^,批月¨w,‘‘‘,尺^,怖。)

mlElH

s.t.∑cf.。=c』“,

(4)

扣1

(1,¨≥O,ls七s聊。一1h,

、ⅣhereG_1叫isgjven丘omtheoptlmalresultofasub—

pr0尊aInIllillgfofcoordillanngsubsystemsin(,~1)th

level

Foreachsub-pmgr猢ing

above,

its

objectiVe

mnctioncan

becons仃1lctedbytlle109icrelation

aInon—

subsys蛐sinlowefIevel'suchasse【iesorparaIkl

ForiIlstance,

if也ereexlsts

seriesrelaⅡonamongaUsubsystemsmfirstkvel,thentheobjectiVe

mnc石on

of(Po)canbeexpressed

as

follows

R=兀尺L

lfthereexists

parallelrelatlonamongallsub—systemsin疗rstlevel,thenthe

objectiVe

mnctionof

华北电力人学学报

(R)caⅡbeexpressedby:

Ro_l一兀(1一月.¨1

Fl

For血eothersub—programmlng,也eir

objectivemnctionscan

alsobcexprcssed

on

theanalogyofthe

above.

Weassumemere

exlsts

only

one

classofl091c

re—

lanonamongallcomponentsineachbonomsubsys—tem,such

as

parallelor【^,卅^)(G)orstaⅡd-by

Hence.

foreachbottomsubsystem,1tssub—pro—

grammmgProblem

can

begiven

as

f0110ws:

【|D¨)max尼,t=∥“(^(G),卅t)

t.

m。G=G¨{5)

WhereG≥O,G¨1sgi ̄en

fomtheoptjmal

solution

of(,㈠…).意=1,2,…,月hand卅t≥^forthe(厶/州。)(G)

sub8ystem.There

are

t11reemncliontypes

f0听…for

instance。ifS,{ls

parallelsubsystem,then

只¨。=1一(1一“)“

(6)

IfS.}isa(^/埘I)(G)subsystem,men

,wl一^

R¨。=∑a.【l一^y一・。

(7)

F0

Where^istheminimumnumberofcomponents,

卅t≥^.c:.=埘t!/[口!(卅。一口)味

IfS.。is

stand-bysubsystem,

then

Ⅲ。』

月。。=∑(一1n“y,√q!

(8)

q卸

For

eachcomponent,

withoutlossof

general咄

thcmnctlonrela“onforitsfeliabilitynandits

con—

stn】ctloncost

GisassumedtobeⅡ1at:

“㈦)=exp[d√慨一G)]

(9)

whered≥0.口≥O

AnalgOrithmforsolvingthedecompOsi—

tiOnOptimiztionmodel

Analgorimmforsolv{ngtheaboVedecompos卜

tlon

optmlizationmodel

can

begiven

as

follows:

step1.Forthe91vencons仃1lctioncost

11ml僦ionG

forthesystem,厅omuptodown,giveinitialconstruc・

tlon

cost

allocationfofeach

subsystem

e,glve

万 

方数据c’1

o。,l≤f≤n1.to

satis矗(1),G,i,,i茎,s州m,to

satis印【2),cn”l≤七兰m叩tosatisfy(3),・・.c,M,

1≤置玉州㈨,to

satls母(4)Then士rom

downtoup,

solvesub—programmlngsln

eachIeVclbygradjentpm—

Jectionmelhod

as

follows

Step2.S01Vethesub-programingsforbonomsubsys—

tems,forinstance,to(尸㈡):lfSL,isparallelbonomsubsystemor

stand_bybot—

tomsubsystem,take

Ⅲ』_1.2.3.-..

1fs¨is(厶/州})(G)bonom

subsystem,take

州。=^,^+1,^+2,…

Foreach卅.,calculatee=G,。/mhn(c^Jln(9),andtheobjectlve

of(B¨),i

e.,R¨=^.-I^((j),,"I¨n

(6)or(7)or(8),unIil尼¨,stanstodecreaseStore

瓣妙

瓣m‰丧

“01

where曼挚is90nenfonn(6)。r(?)州乳and

害鲁=(仉假)exp[一吼(G伊。一川

Step3.SolveIhesub-programmingsforco(Irdmatingbottomsubsystems,forins协nce,to

solVe(F:“,)

calculate;鲁等andp眯cti。n可adient:

黠=舞羔器:面jii百万。。0e。。

Vm。-【麟,憋.…,謦等)T_

【l,卜。。川翟咎。‰。;,

glvenaccofdlngt。thespeci6cfunc-

lionfom。fthe。bjectivein(九。,瓣:ts

where%鲁蔫is

given

bySlep2,1et

(C,。,o。,….C。~)’

一lC¨,G_山…,C。,)1。^可.月I,,

WhereIisiteratestcplengtlldeterminedb)’one—di—

mensionalsearch.Ifitissalisnedthat

llV月.…川<£,wheIes

ls

a出ven

c。n㈣precision'thenst。re};等astk

foll。wmg:

玛:硝

谭忠富:犬犁复杂系统的可靠性与冗余J皇优化

3,

othenvise,g。tosteD

跨2跨Ⅲ翻。。~,,鼾印61凡“蹦≯1翼嚣??2瓦i两ij啊了

step

4.on暑eanal矗y。fStcp3’s0Ivesub-pr。盯am.

mlngfor∞ordinatingsubsystemsin(,一1)thleVel,sub—

pmgrammingforcoordinatlngsubsystems血(,.2)th

le、el,and

so

on,un州thesub。pmgrammingfor

coor-

diⅡatiⅡgsubsystemsin

mst

leVel,forinstance,

the

solVingprocessof(只¨)and(尸0),seet|lef0110wing

step

s.胁c‰^叫c慨暑鲁兰astn。如--。。,。。:囊塾:梁u未彗一,l哥辄。百石i百ij,j百瓦i11爿3%…

where}戋is

glvenatthe

optiml

point。f(‰^

恶。,can

be

given

accordingt。thespecincfonIl。f

mnction^o

Ca}culatethe

projection

gradieⅡI

of

【Pl。)anditeratedesignvariables

as

f0110wsF一‰:【氆

妇砭

':o

㈦卜...川蓦疆渺。

。…,o!。.)7一

(G二。,G。一…,G。。)1“V.R。.

ls

iteratest。Plen甜hdetelminedbyone—di—

search.

Ifitissatisfiedthat

ff

V月1

o,ff龟,

store导訾ev

挚-:嘉擎Ⅱ,i班‰。a瓦ii一石瓦_『j’1纠3”10‘

112)I“J

to

Step6.

o也envlsea110cateagainconstmctio“

ofsubsystemsinthird1evel,fonhlevel,and

so

untiI^hIevel

as

f0IIows:

m一卜√》水,

C广

‰∑一

Ste02

万 

方数据where

g慧-sgiven

forrnstc

p5’。粤ts

g-ver

ac—

cordlngtothespecl矗cfonnofmnctlon工

lfltissat—

isnedthat||V吼【】勺,then

outputthecostsofcompon。

ents

andthesyscemreliabiljcy尺0.Sf。p

()【hernise.

calculatethe

projectiongrddientoffP。)andlteratede—

signvanablesby

甲肛(袅i,悲,…、最卜

(1-卜一,w耋袅i/一,

(C-一,,C。!,….(1㈨.)1一

(G川,C“-.….C,¨)Lj守凰

whereiisIteratesteplengthdeterrnlnedbyonc_d卜mensionalsearchAllocateconstmc廿oncostsorsub.

systemsagain

1nsecondlevel’thirdlevel,and

h。on.

until^hJevel

as

f0”ows:

㈡。一k。/至。。k

扣1

(1,

。/塞l。If

口=1

‰。十。。,擎。十。

goto

Step2

Proposition1.Attheoptlmalpomtofeach)ub-

problem,onehastllat:

裳唔孤;参_碧.t螺n

謦声辩.-盥m.

鲁≯=器。、:.,,啦m。

ProoC

By

use

oftheK—TcondlLlonofeachsub—

call

have

《C.:。,C

wheremensional帆n

go

progmmmingcosts

on.

窖oto

programmjngprobIem,a11d(1)~(5).wc

华北电力夫学学报

2002年

(10)~(12).

P呷os埘oⅡ2.

Forttleabove

alg嘶恤,

its

iteratesequencesare

ofthemain

pr0伊鲫岫g

convergentto也eK_Tp0Intwhichisexpressed

as

follows:

0))maxJR。=

烈,t(c』),r2(c;),…,n(e),小。,珊:,…,卅。)

s.t.∑,Ⅲ.G=c0

^=l

m±≥厶,珊l+l!七兰肌1+小2

where

4k萌(^o.,(…),…,』n.(…,正.。,(…),…)…,

』o.(…)),

itis

compoundmnctionf研all

objecⅡve

如ncnonsoftlleabovesub—pmgra【n:札-mg.

ProoC

niseas订yprovedthatthecomposinonof

K_Tcondi廿onsf研allsub-programmingistheK—Tcondi恼onofthemain

pro铲町ming(P)since

nlepro—

jectiongradientmethodusedins01vingeachsub-pro-

掣铷nmingisconvergent,sotheiteratesequences

are

c蚰vergent

to

theK—Tpointof也emaillpr0可amming

(P).

Proposition3.

FortheabovealgoritIlI工l,

its

iteratesequencesare

convergemtotheoptimals01-

udonof山emain

progmmmiIlg(P).

PmoCBy也enmchonf0Ⅱnof尼aboutRln.,it

】sto

knowthat

器≥o

OntlleaIlalogyoftlleabove,t11efollowingcan

begonen

溉>o,溉她爨巩

Ffom(9)'

it

caIl

bee船ilypfovednlat

百西‘”铝≥o

瓯砥贰…切i忑]F砸“穗畿溉…瓣警凳≥o

万 

方数据ie铬茎o

Hence士is

convex如nctionabout

ch

l≤A≤”1,

(P)isaconvexpro伊ar衄ingproblem

weknowthat

theK-Tpointof

conVex

programmiⅡgproblemis

also池optimalsolution,sofbmProposition2,one

hasthatiteratesequencesoftlleabovealgorimm

afe

conve娼em

to

t11eoptimalsolutionofthemampm掣a-

mming(P).

Example.AssumingthatSiscomposedof6par_

allelsubsystemsin行rstlevel,i.e,HI=6,Sl州consistsof"。1parallelsubsystems

and"n(1服)(G)subsyste雌

and月astaIld_bysubsystems1Ⅱsecondlevel,

n,1(1茎f蔓6)as2,3,4,5,6,7,”口(1sjs6)as2.5,6,6,7.7,no(1s注6)as5,2,3,3,2,2.Somere

are

77redundant

subsystemstotallyinsecondlevel

Consideringt|Ie

pl锄e

positionofallredllIldantsubsystemsmsecond

level

as

Irla仃ix,we

give

血eorderrlotes

七=1,2,3,…,77to

them,仔omuptodown,斤om

1eftcolurnntonghtcolumn.Tak号夙(1』七兰77)as9+20,10+23。13’25,14+lO,15+17,16+14,d±(1≤七s77)

as

9+16,10+12,

13+l5,

14+10,15+17,16+19,厶

for

3≤七≤4,

13≤七517,

24≤七≤29,38≤七≤43,

50≤七156,61兰七兰67,as2+2,5+2,6+2,6+2,7+2,

7+2.

Bytheabovealgor主Ihm,theoptimairesultisob-

tained

as

fbllows:

m15m72ml32m19=4,m2523,m312ml-2m412Ⅲ4q-2,埘。=2for3≤丘茎4,13茎七兰17,24≤膏≤29,381^143,50≤七≤56.6l≤七≤67,州Flforother七,n=r,=O.7l,n]_r19=h5=n1=n,=rd3=h9=O.896,,、=004forother七,尺o

=0955

References

【1】Gu扎gyu卸Wangnleson如slgⅡtheory

for

en91Bee血g

systems(inchinese)【M1.Beqiog:sclencePress,1992【2】AIl00pKDhingraoptimalappo^ionmentofrellabil・ty

叩dred咖daⅡcyin

senes

3ystemsundermul【lpk

[门.IEEEl觚actio啊onReliabdi吼1992,4l(4)576—58l

objectIVes

【3】Jinhuac∞,YuedongwaⅡgopnmalallocatlonofa‘印alrabk

system[J]Microelec们nicsandRellabiIitM

1990,30(6)

109l,1093

大型复杂系统的可靠性与冗余度优化

作者:作者单位:刊名:英文刊名:年,卷(期):被引用次数:

谭忠富

华北电力大学工商管理学院,北京,102206

华北电力大学学报

JOURNAL OF NORTH CHINA ELECTRIC POWER UNIVERSITY2002,29(2)4次

参考文献(3条)

1. GuangyuanWang The soft design theory for engineering systems 1992

2. Anoop K Dhingra Optimal apportionment of reliability and redundancy in series systems undermultiple objectives 1992(04)

3. Jinhua Cao. Yuedong Wang Optimal allocation ofa repairable system 1990(06)

引证文献(4条)

1. 于晓东. 高会生. 郭爱玲 基于模糊理论的电力通信网络有效性研究[期刊论文]-华北电力大学学报 2008(5)2. 杨明顺. 李鹏阳. 李言. 袁启龙 考虑不同类型可靠性-成本函数下的可靠性优化设计[期刊论文]-中国机械工程2006(22)

3. 李蒙. 胡兆光 基于智能工程理论拓展的政策正向状态模拟新方法[期刊论文]-中国电机工程学报 2006(z1)4. 张海泉 集散控制系统中冗余通信设计及实现[学位论文]硕士 2006

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