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-
a
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
a
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
l
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
m
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
e
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,,)
n
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。.)
S
t
G
—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—
g
subsys蛐sinlowefIevel'suchasse【iesorparaIkl
ForiIlstance,
if也ereexlsts
a
seriesrelaⅡonamongaUsubsystemsmfirstkvel,thentheobjectiVe
mnc石on
of(Po)canbeexpressed
as
follows
R=兀尺L
q
lfthereexists
a
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)
s
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
a
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
a
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
2
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
i
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。.
l
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:
G
m一卜√》水,
C广
C
☆
C
C
‰∑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
I
扣1
(1,
。/塞l。If
口=1
J
‰。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
a
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
a
convex如nctionabout
ch
l≤A≤”1,
(P)isaconvexpro伊ar衄ingproblem
weknowthat
theK-Tpointof
a
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,
比
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
a
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
6
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-
a
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
a
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
l
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
m
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
e
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,,)
n
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。.)
S
t
G
—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—
g
subsys蛐sinlowefIevel'suchasse【iesorparaIkl
ForiIlstance,
if也ereexlsts
a
seriesrelaⅡonamongaUsubsystemsmfirstkvel,thentheobjectiVe
mnc石on
of(Po)canbeexpressed
as
follows
R=兀尺L
q
lfthereexists
a
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)
s
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
a
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
a
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
2
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
i
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。.
l
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:
G
m一卜√》水,
C广
C
☆
C
C
‰∑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
I
扣1
(1,
。/塞l。If
口=1
J
‰。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
a
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
a
convex如nctionabout
ch
l≤A≤”1,
(P)isaconvexpro伊ar衄ingproblem
weknowthat
theK-Tpointof
a
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,
比
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
a
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
6
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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