Optimal System for Warm Standby Components in the Presence of Standby Switching Failures, Two Types of Failures and General Repair Time

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1 Intenatonal Jounal of ompute Applcatons (5 ) Volume 44 No, Apl Optmal System fo Wam Standby omponents n the esence of Standby Swtchng Falues, Two Types of Falues and Geneal Repa Tme Mohamed Salah EL-Shebeny Depatment of Mathematcs, Faculty of Scence, Helwan Unvesty. O. Bo 5, ao, Egypt ABSTRAT Two dffeent system confguatons wth wam standby components, standby swtchng falues, two types of falues "common cause falue and hadwae falue" and geneal epa ae compaed based on the avalablty. The tme-tofalue fo each of the pmay and wam standby components ae assumed to follow the eponental dstbuton. Laplace tansfoms of state pobablty equatons ae developed by usng the supplementay vaable technque. We develop the eplct epessons fo the steady-state avalablty,, fo two confguatons. Fo all confguatons, compasons ae made fo specfc values of dstbuton paametes and of the cost of the components. The confguatons ae aned based on and cost/beneft, fo thee vaous epa tme dstbutons: Gamma (G), Webull (W) and Lognomal (L), beneft s. Keywods alablty; Standby swtchng falues; ommon cause falue "F"; Supplementay vaable; Geneal epa tmes.. INTRODUTION In ths pape we use a supplementay vaable technque to study the avalablty analyss of two dffeent sees system confguatons wth wam standby unts, mpefect swtch and two types of falues. The steady-state avalablty has wdely been analyzed n the lteatue because of ts pevalence n powe plants, manufactung systems, and ndustal systems. Mantanng a hgh o equed level of avalablty s often an essental equste. El-Sad and El- Shebeny [] studed two systems, each system wth two paallel components. The second system dffes fom the fst system due to the addtonal featue of peventve mantenance. The two systems ae analyzed unde the assumpton that the falue, eplacement and peventve mantenance tmes of the unts ae assumed to be abtaly dstbuted. Wang and Kuo [] dealt the elablty, the avalablty, and the cost/beneft analyss of fou dffeent sees system confguatons wth med standby (nclude cold standby and wam standby) components. The stochastc analyss of a non-dentcal two-unt paallel system wth common-cause falue by gaphcal evaluaton and evew technque (GERT) consdeed by Sdhaan and Kalyan [5]. Gaowsy, et al. [4] and Wang and ean [] analyzed the sees systems wth cold standby components and wam standby components, espectvely, the epa tme dstbuton of the seve s assumed to be eponentally dstbuted. Shen, et al. [] dscussed eponental asymptotc popety of the soluton of a paallel epaable system wth wam standby unde common cause falue. Dhllon and Anude [] studed ommon-cause falue analyss of a nondentcal unt paallel system wth abtaly dstbuted epa tmes. A standby component s called a "wam standby" f ts falue ate s nonzeo and s less than the falue ate of a pmay component. may and wam standby components can be consdeed to be epaable. A F s defned as the falue of sngle unt o multple unts due to a sngle common-cause. Some of the F may occu due to the followng easons. abnomal envonmental condtons, e.g. tempeatue, pessue; defectve desgn; and natual catastophe le fe,... etc. The poblem consdeed n ths pape s moe dffeent than the wos of M. Salah EL-Shebeny []. We fst pesent a ecusve method, usng the supplementay vaable technque and teatng the supplementay vaable as the elapsed epa tme, to develop the, fo confguaton, =,. Net, fo each confguaton, the eplct epessons fo the fo thee dffeent epa tme dstbutons such as Gamma (G), Webull (W) and Lognomal (L) ae povded. Fnally, we an two confguatons fo the based on assumed numecal values gven to the system paametes.. ASSUMTIONS AND MODEL DESRITION We consde a powe plant of MW satsfyng the followng assumptons:. The system compses of opeatve components and wam standby components.. The geneatos ae avalable n both and 5 MW.. Standby geneatos ae always necessay n case of falue. 4. In case of falue of an opeatve component, t s mmedately eplaced by a standby component f t s avalable. 5. The swtch fom standby component to opeatve component s mpefect swtch, wth pobablty.. The system suffes two types of falues, namely, hadwae and common-cause falue.. The falues ae statstcally ndependent.. Faled system epa tmes ae assumed to be abtaly dstbuted.. The common-cause and othe falue ates ae constant.. A unt's epa ate s constant.. A epaed unt (o system) s assumed to be as good as new. The above assumptons ae common to all of the followng two confguatons. be justfed, not agged.

2 Intenatonal Jounal of ompute Applcatons (5 ) Volume 44 No, Apl. onfguaton descptons The fst confguaton composes of one opeatve MW component and one standby MW component. The second confguaton contans two pmay 5 MW components and one standby 5 MW component. Notaton The followng symbols ae used n ths pape: onstant common-cause falue ate fom state to state ; fo =,. onstant hadwae falue ate of opeatve unt. onstant falue ate of the wam standby. onstant epa ate of a faled unt n state. The steady state pobablty that the system s n state ; =,,. ( ) The pobablty that the system s n state at tme t; fo =,,,. [swtchng devce s found to be good when needed]. E ( ) The mean tme to system epa that the faled system s n state and has an elapsed epa tme of ; fo =,. ( ) Tme-dependent system epa ate when the system s n state and has an elapsed epa tme of ; fo =,. (, t ) The pobablty that the faled system s n state and has an elapsed epa tme of ; fo =,. f ( ) obablty densty functon (pdf) of the system epa tme when the system s n state and has an elapsed epa tme of ; =,.. ost-beneft facto We assume that the sze-popotonal costs fo the pmay components and wam standby components ae gven n Table. Wth ths, we calculate the costs fo each confguaton, shown n Table. Let be the cost of the confguaton, and B be the beneft of the confguaton, B s the. Table.The cost fo the pmay and wam standby components omponent ost(n $) may MW may 5 MW Wam standby MW Wam standby 5 MW 5 5 Table. The costs fo each Model, =,. Models ost(n $) Model Model 4 5. alablty analyss of the system. alablty fo confguaton Descptve Equatons fo confguaton : d ( ) ( ) ( ) (, ) t t t d dt, () d ( t ) ( t ) dt, () ( ) (, t ), () t ( ) (, t ). (4) t The assocated bounday condtons ae as follows: (, t ) ( t ) ( t ), (5) (, t ) ( t ) ( t ). () When t then () and () (,) ; fo,. Steady-state avalablty analyss as t, the equatons (-4) educe to equaton (-), espectvely. ( ) (, t ) d, (), () ( ) ( ) ( ).,. () Smlaly, the bounday condtons become: (), () (). () s the steady-state pobablty that the system s n state, fo,...,. and ( ) d, fo,. () Also. () Solvng dffeental equaton (), we get ( ) ()ep ( w ) dw.fo,. Thus, fom equaton () and (4), we have ( ) d ()ep ( w ) dw d (4) (5) E ( ) Smlaly, ( ) d ()ep ( w ) dw d () ( ) E E ( ) ep ( w ) dw d E ( ) and E ( ) ae the mean tmes to epa fom state to state, fom state to state, espectvely. Solvng the set equatons (), (), (5) and (), we get the followng steady state pobabltes:

3 Intenatonal Jounal of ompute Applcatons (5 ) Volume 44 No, Apl, () a a a a E, () ( ), () a a E ( ). () aa ( ) ( ), a E ( ) E ( ) a E E and. The system steady state avalablty s. () aa Smlaly, the system steady state unavalablty s gven by E ( ) E ( ) U. aa ().. Specal cases ase I If the system epa tme s Gamma dstbuted and pobablty densty functon (pdf) of the epa tme s gven by f ( ) ep ( ). fo,;,, j. () and j ae two paametes of Gamma dstbuton. Thus, the mean tme to epa E ( ) s E ( ).,. (4) Substtutng Equaton (4) nto equatons (), we get the followng esultng system steady state avalablty fo the Gamma epa tme dstbuton: ( G ). (5) a4a5 a4, and a5. ase II If the system epa tme s Webull dstbuted and pobablty densty functon (pdf) of the epa tme s gven by f. ( ) ep fo,;,, j. () and j ae two paametes of Webull dstbuton. Thus, the mean tme to epa E ( ) s E ( ).,. (). Substtutng Equaton () nto equaton (), we get the followng esultng system steady state avalablty fo the Webull epa tme dstbuton: ( W ). () a a a, and a. ase III If the system epa tme s lognomal dstbuted, the pdf of the epa tme s defned by ln f ( ) ep. fo,;. () and ae the dstbuton paametes (mean value and standad devaton of ln espectvely). Thus, the mean tme to epa E ( ) s E ( ).,. () ep Substtutng equaton () nto equaton (), we get the followng esultng system steady state avalablty fo the lognomal epa tme dstbuton: ( L). () a a, and a a. alablty fo confguaton Descptve Equatons fo confguaton: d ( ) ( ) ( ) (, ) t t t d dt () d ( t ) ( t ) dt, () ( ) (, t ). fo,. (4) t The assocated bounday condtons ae as follows:, (, t ) ( t ) ( t ), (5) (, t ) ( t ) ( t ). () At t then () and () (,) ; fo,. Steady-state avalablty analyss When t, then the equatons (-4) educe to equaton (-), espectvely.

4 Intenatonal Jounal of ompute Applcatons (5 ) Volume 44 No, Apl ( ) (, t ) d, (), () ( ) ( ) ( ).,. () Smlaly, the bounday condtons become: (), (4) (). (4) s the steady-state pobablty that the system s n state, fo,...,. and ( ) d, fo,. (4) Also. (4) We get the followng steady state pobabltes:, (44) a a a a, (45) E ( ), (4) a a ( ) E. (4) a a a E ( ) E ( ), a E ( ) E ( ) and. The system steady state avalablty s. (4) a a Smlaly, the system steady state unavalablty s gven by E ( ) E ( ) U. (4) a a.. Specal cases Fo confguaton, we also consde thee specal cases fo dffeent epa tme dstbutons such as Gamma (G), Webull (W), and lognomal (L). We povde the followng eplct epessons fo the ( G ), ( W ) and ( ) L fo thee dffeent epa tme dstbutons: Gamma, Webull, and lognomal, espectvely. ( G ). (5) a a a, and a. ( W ). (5) a a a and a. ( L). (5) a4 a5,and a5 a ompason between the two confguatons The pupose of ths secton s to compae (, ) fo thee dffeent epa tme dstbutons: Gamma, Webull, and lognomal. Bascally, we consde the followng values:.,.,.5,.4,.,.5,.,.,. and ompasons fo the S cases ae llustated n Tables - ase. We f.,.5,.4,.,.5,.,.,.,.5 and vay the values of fom. to.. ase. We f.,.,.4,.,.5,.,.,.,.5 and vay the values of fom.5 to.. ase. We f.5,.,.4,.,.5,.,.,.,.5 and vay the values of fom. to.. ase 4. We f.,.,.5,.,.5,.,.,.,.5 and vay the values of fom.4 to.. ase 5. We f.,.,.5,.,.4,.,.,.,.5 and vay the values of fom.5 to.. ase. We f.,.,.5,.5,.4,.,.,.,.5 and vay the values of fom. to Table. ompason of the avalablty models, fo (case ) ( ) ( ) ( )

5 Intenatonal Jounal of ompute Applcatons (5 ) Volume 44 No, Apl ( ) ( ) ( ) Table 4. ompason of the avalablty models, fo (case ) ( G ) ( W ) ( L ) ( ) ( ) ( ) Table 5.ompason of the avalablty models, fo (case ) ( G ) ( W ) ( L ) ( ) ( ) ( ) Table. ompason of the avalablty models, fo (case 4) ( G ) ( W ) ( L ) ( ) ( ) ( )

6 Intenatonal Jounal of ompute Applcatons (5 ) Volume 44 No, Apl Table. ompason of the avalablty models, fo (case 5) ( G ) ( W ) ( L ) ( ) Table.ompason of the avalablty models, fo (case ) ( G ) ( W ) ( L ) ( ) ( ) ( ) cost/beneft ato compasons Let b be the cost of the confguaton, fo, whch ae lsted n Table. We ompae b,, n s cases as follow: Table. ompason of the cost/beneft models, fo (case ) b ( ) bw ( ) b ( ) b ( ) b ( ) b ( )

7 Intenatonal Jounal of ompute Applcatons (5 ) Volume 44 No, Apl Table. ompason of the cost/beneft models, fo (case ) b ( G ) bw ( ) b ( L ) b ( ) b ( ) b ( ) Table. ompason of the cost/beneft models, fo (case ) b ( ) bw ( ) b ( ) b ( ) b ( ) b ( ) Table. ompason of the cost/beneft models, fo (case 4) b ( G ) bw ( ) b ( L ) b ( ) b ( ) b ( )

8 Intenatonal Jounal of ompute Applcatons (5 ) Volume 44 No, Apl Table. ompason of the cost/beneft models, fo (case 5) b ( G ) bw ( ) b ( L ) b ( ) b ( ) b ( ) Table 4. ompason of the cost/beneft models, fo (case ) b ( G ) bw ( ) b ( L ) b ( ) b ( ) b ( ) Numecal esults of the b fo confguaton, ae calculated n Tables -4 fo s cases, espectvely. Fom these Tables -4, we deduce the optmal confguaton as follow: cases (,, 4) The optmal confguaton s confguaton, fo thee dffeent epa tme dstbutons: Gamma, Webull, and lognomal. case ) Gamma dstbuton {.5,...,.} the optmal confguaton s confguaton, {.,...,.} the optmal confguaton s confguaton. ) Webull dstbuton and lognomal dstbuton {.5,...,.} the optmal confguaton s confguaton, {.,...,.} the optmal confguaton s confguaton. case 5 ) Gamma dstbuton {.5,...,.} the optmal confguaton s confguaton, {,...,.} the optmal confguaton s confguaton. ) Webull dstbuton {.5,...,.} the optmal confguaton s confguaton, {.,...,.} the optmal confguaton s confguaton. ) lognomal dstbuton {.5,...,.} the optmal confguaton s confguaton. case ) Gamma dstbuton and Webull dstbuton {.,...,} the optmal confguaton s confguaton ) lognomal dstbuton {.,...,.} the optmal confguaton s confguaton, {.4,...,} the optmal confguaton s confguaton. 5. ONLUSIONS In ths pape we have fst utlzed the supplementay vaable technque to develop the steady-state avalablty,, of two 4

9 Intenatonal Jounal of ompute Applcatons (5 ) Volume 44 No, Apl dffeent sees system confguatons wth wam standby components, swtchng falues, two types of falues and geneal epa tmes. Net, fo each confguaton, we pesent the eplct epessons fo the fo thee vaous epa tme dstbutons such as Gamma (G), Webull (W) and Lognomal (L). Fnally, we an two confguatons based on the and the cost/ fo thee vaous epa tme dstbutons. System due to ommon ause Falues S ( ) One Unt Opeatve, One Unt Wam Standby ( ) S S ( ) One Unt Opeatve, One Unt Faled due to Hadwae S Two Unts Faled due to Hadwae Fg : State-space dagam fo confguaton. System Fals due to ommon ause Falues ( ) S Two Unts Opeatve, One Unts Wam Standby S S Two Unts Opeatve, One Unt Faled due to Hadwae ( ) S Two Unts Faled due to a Hadwae, One Unt Opeatve Fg : State-space dagam fo confguaton. 5

10 Intenatonal Jounal of ompute Applcatons (5 ) Volume 44 No, Apl 5. REFERENES [] Dhllon, B. S., Anude, O... ommon-cause falue analyss of a non-dentcal unt paallel system wth abtaly dstbuted epa tmes. Mcoelectoncs and Relablty, Vol., No., pp. - [] El-Sad, K.M., El-Shebeny, M.S.. ompang of elablty chaactestc between two dffeent systems. Appled Mathematcs and omputaton, Vol., pp. -. [] El-Shebeny, M.S.. The optmal system fo sees systems wth med standby components. Jounal of Qualty n Mantenance Engneeng. Vol. No., - 4 [4] Galowsy,., Svazlan, B. D., haovaltwongse,.. Optmal edundances fo elablty and avalablty of sees systems. Mcoelectoncs and Relablty,, 5 54 [5] Sdhaan, V., & Kalyan, T. V.. Stochastc analyss of a non-dentcal two-unt paallel system wth common-cause falue usng GERT technque. Infomaton and Management Scences,, 4 5 [] Shen, Z., Xaoao, Hu., Wefeng, Fan.. Eponental asymptotc popety of a paallel epaable system wth wam standby unde common-cause falue. J. Math. Anal. Appl., 4, 45 4 [] Wang, K. H., Kuo,... ost and pobablstc analyss of sees systems wth med standby components. Appled Mathematcal Modellng, 4, 5 [] Wang, K. H., ean, W. L.. ost beneft analyss of sees systems wth wam standby components. Mathematcal Methods of Opeatons Reseach, 5, 4 5