APPLICATIONS OF RELIABILITY ANALYSIS TO POWER ELECTRONICS SYSTEMS

APPLICATIONS OF RELIABILITY ANALYSIS TO POWER ELECTRONICS SYSTEMS Chanan Sngh, Fellow IEEE Praad Enjet, Fellow IEEE Department o Electrcal Engneerng Texa A&M Unverty College Staton, Texa USA Joydeep Mtra, Sr. Member IEEE Department o Electrcal & Computer Engneerng North Dakota State Unverty Fargo, North Dakota USA ABSTRACT Relablty and cot conderaton play an mportant role n the choce o varou alternatve. Thee could be alternate degn to meet an objectve or could be alternate plannng cenaro. To be able to make tradeo between cot and relablty, relablty need to be quanted and there need to be method or predctng the relablty. Th paper decrbe three technque o relablty analy, a technque baed on Markov procee, cut et method and Monte Carlo mulaton. Thee technque are llutrated ung two example. The Markov proce llutrated by applcaton to tudy ault tolerant eature n PEBB baed converter conguraton. Smlarly the Monte Carlo mulaton and cut et technque are llutrated by applcaton to tandby power ytem ncludng unnterruptble power upply ytem. BACKGROUND Th ecton gve a bre ntroducton o the relablty evaluaton method. There are many method avalable n the lterature; here three method wll be revewed. The two analytcal method are the Markov procee and cut et and one orm o Monte Carlo Smulaton alo decrbed. ANALYTICAL METHOD The analytcal method decrbed here ue the concept o Markov chan and Markov cut-et. A detaled treatment o thee technque avalable n [1]; a ummary o the method preented here. Markov Chan A Markov chan a equence o event contng o the tranton o a component or ytem o component amongt a et o tate uch that the tate to whch the component or ytem trant n the uture depend only on the current tate o the component or ytem and not on the tate t ha tranted 1

through n the pat. I a ytem conorm to (or, more realtcally, approxmate) th behavor, then the theory o Markov chan may be appled to t. In th knd o analy, normaton o ntertate tranton rate ued to determne the rate o trantng to alure tate. The tranton rate rom tate to tate j the mean rate o the ytem pang rom tate to tate j. In th paper, the duraton o component tate are aumed exponentally dtrbuted, whch mean that the ntertate tranton rate are contant. Th explaned n the next ecton. The tate probablte can be obtaned by olvng BP = C (1) where B matrx obtaned rom A by replacng the element o an arbtrarly elected row k by 1; A matrx o tranton rate uch that the element a j = j and a = a ; j contant tranton rate rom tate to j; P column vector whoe th term p the teady-tate probablty o the ytem beng n tate ; C column vector wth kth element equal to one and other element et to zero. j j Once the teady-tate probablte have been calculated, the relablty ndce [1] can be computed ung the ollowng relatonhp. Frequency o ytem alure: The relatonhp or the requency o ytem alure [1] gven by Eq. (2) or Eq. (3) = p ( S F) = F j F j p (3) j j ( S F) (2) where the requency o the ytem alure, S the ytem tate pace and F the ubet o aled tate. Mean Down Tme: The expected tme o tay n F n one cycle where r P / = (4) 2

P = (5) p F Cut Set A cut et a et o component that, removed rom the ytem, reult n lo o power to the load pont. I th et doe not contan any cut et a a ubet, then t called a mnmal cut et. In th denton, component ued n a wder ene o hardware or a partcular ytem condton. In general cut et contanng m component are called m-order cut et. Generally up to econd order cut et are condered and the contrbuton rom hgher order cut et condered neglgble. The ollowng relatonhp are mportant n cut et calculaton. 1) Frequency and duraton o a cut et (a) Frt order cut et [6] : c = (6) r = r (7) c where, r = requency and mean duraton o cut et c c, r = alure rate and mean duraton o component comprng cut et (b) Second order cut et [6] = r + r) (8) c j( j r c = rr j/( r + rj) (9) where, = alure rate o component and j comprng cut et j r, r= mean alure duraton o component and j comprng cut et j (c) Condtonal econd order cut et [6]: Aumng that component al gven component j ha aled, c = r) (10) j ( j r c = rr j/( r + rj) (11) 3

The term alure requency and alure rate are oten ued nterchangeably. Strctly peakng alure rate the mean number o alure per unt o the up tme and alure requency the mean number o alure per unt o the total tme. I the probablty o ytem beng up cloe to unty, then thee two quantte are very cloe. In general, alure requency le than alure rate. 2) Frequency and duraton o nterrupton [6]: = requency o load nterrupton = (12) c r = mean duraton o load nterrupton = r / (13) c c The theory concernng Markov chan and mnmal cut-et dealt wth n greater detal n (4). Thee concept are ued a ollow to analyze the ample ytem. Monte Carlo mulaton Th another method o determnng relablty ndce. Th method decrbed n detal n [1], whle an applcaton o th method to the ample ytem gven n [9]. A ummary o the method preented here. The relablty ndce o an actual phycal ytem can be etmated by collectng data on the occurrence o alure and the duraton o repar. The Monte Carlo method mmc the alure and repar htory o the component and the ytem by ung the probablty dtrbuton o the component tate duraton. Stattc are then collected and ndce etmated ung tattcal nerence. Though there are derent way o mplementng Monte Carlo mulaton, the technque decrbed here the next event method and capable o mulatng dependent event. Th a equental mulaton method whch proceed by generatng a equence o event ung random number and probablty dtrbuton o random varable repreentng component tate duraton. A lowchart gven n Fg. 1. The nput data cont o the alure rate () and mean down tme (r) o every component. The alure rate the recprocal o the mean up tme. The mean down tme the recprocal o the repar rate (µ). The alure and repar rate, and µ, o a component wll be ued to determne how long the component wll reman n the UP tate and the DOWN tate. Smulaton could be tarted rom any ytem tate, but t cutomary to begn mulaton wth all the component n the UP tate. 4

START READ FAILURE RATE AND DURATION DATA FOR ALL COMPONENTS SET INITIAL STATE OF ALL COMPONENTS AS UP FOR EACH COMPONENT DRAW A RANDOM NUMBER AND COMPUTE THE TIME TO THE NEXT EVENT FIND THE MINIMUM TIME AND CHANGE THE STATE OF THE CORRESPONDING COMPONENT; UPDATE TOTAL TIME NO IS THERE A CHANGE IN SYSTEM STATUS? YES UPDATE INDICES SIMULATION CONVERGED? NO YES PRINT OUTPUT STOP Fg. 1. Flowchart or next-event mulaton. The tme to the next event generated by ung the nvere o probablty dtrbuton method. Th explaned a ollow. I the tranton tme o the component are aumed to be exponentally dtrbuted: t ( t) = ρe ρ (14) where ρ the tranton rate. The mean tranton tme, thereore, 0 1 ( t) dt= (15) ρ 5

Th mean that, or ntance, a component UP, then, regardle o how long t ha been n the UP tate, the expected tme to the next alure 1/,.e., the mean up tme. The probablty dtrbuton uncton o the tranton tme T would be F( t) t ρt = P( t T) = ρe dτ = 1 0 e ρt (16) Now F(t) can be regarded a a random varable, unormly dtrbuted between 0 and 1. Th mean that the urvvor uncton S ρt ( T) = 1 F( T) = e (17) alo unormly dtrbuted between 0 and 1. So a random number R generated, n 0 R 1, t can be aocated wth the event that the next tranton occur ater tme t r, gven by n ρt r R n = e, that, ln( Rn) tr = (18) ρ Th method ued to determne the tme to the next tranton or every component, ung or µ or ρ, dependng on whether the component UP or DOWN. At the end o any mulated tme nterval [0, t], where t = total up tme n [0, t] + total down tme n [0, t], the etmate o the relablty ndce are gven a ollow. alure rate: number o alure n [0, t] t = (19) totaluptmen [0, t] mean down tme: totaldown tmen [0, t] r t = (20) number o alure n [0, t] The value o and r at the ntant the mulaton converge are the relablty ndce or the ytem a obtaned rom the Monte Carlo method. The mulaton ad to have converged when the ndce attan table value. Th tablzaton o the value o an ndex meaured by t tandard error, dened a: σ η = (21) n c 6

where σ = tandard devaton o the ndex n = number o cycle mulated c Convergence ad to occur when the tandard error drop below a prepeced racton, ε, o the ndex,.e., when η ε (22) I, or ntance, the mean down tme r choen a the ndex to converge upon, then, ater every ytem retoraton mulated, the ollowng relaton teted or valdty: σ r ε r (23) n c I th crteron ated, the mulaton ad to have converged. Smulaton advantageou n that t not only allow the computaton o ndce at varou pont n the ytem, but alo permt the accumulaton o data pertanng to the dtrbuton o thee ndce, thereby aordng a better undertandng o the ytem behavor. APPLICATIONS The technque decrbed n the paper are llutrated by two applcaton. The rt applcaton to model ault tolerant eature n the PEBB- baed converter propoed n re. [3]. The econd applcaton the tudy o relablty mprovement by ung UPS a tandby power MODELING FAULT TOLERANT FEATURES IN THE PEBB-BASED CONVERTERS [3] In th ecton a modelng approach preented to analyze the ault tolerant behavor o PEBB baed converter propoed n [3]. Fg. 2 (a) how a detaled connecton dagram o combnng three, 3-phae nverter block wth 18 IGBT wtche. The nterconnecton nductance between the module lmt the hgh requency crculatng current. Fg. 2 (b) how the vector dagram o the undamental voltage generated by IGBT wtch par. 1. Fault Tolerant Feature & Relablty Analy: 7

From Fg. 2 (a) and (b) t clear that alure (open crcut) o IGBT wtche 3,6 and 7,10 or 2,5 and 13,16 or 15,18 and 8,11 reult n an open delta tuaton and the modular nverter are tll capable o upplyng the load at a reduced power level. However, the alure o IGBT wtch par 1,4 or 9,12 or 14,17 wll caue a crtcal alure and reult n a ytem hut down. The trade o between relablty and cot can be acheved by the knowledge o cot o nterrupton and contructng relablty model and computng the relablty ndce. The two ndce that are ueul n th tuaton are: 1. Frequency o alure: Th the mean number o alure per year. 2. Mean duraton o alure: Th the mean duraton o a alure event. The cot o nterrupton n ome applcaton depend on the total down tme wherea n other applcaton the requency o nterrupton alo mportant. A central problem n the optmzaton o the overall cot developng a relablty model that relect the actor that eect the relablty. Some approache or contructng th model are dcued n [8-10]. A utable approach would be to ue o Markov Procee to model thee conguraton. In th approach, varou tate reultng rom the alure and repar or replacement o component are dented. Then the ntertate tranton rate are agned that are uncton o the alure rate and repar tme o the component. The tate probablte and other ndce are then calculated ung the mathematcal technque decrbed n [1-2]. 2. State Tranton Dagram: The tate tranton dagram or the topology hown n Fg. 2 (a) hown n Fg. 3. The varou parameter are: = Falure rate o one leg o the nverter. µ = Repar rate when only one leg aled. 1 µ = Repar rate when two leg are aled. 2 µ = Repar rate when three leg are aled. 3 µ = Repar rate when alure detected a a reult o npecton beore the conguraton alure. I In developng th tate tranton model, ollowng aumpton are made: 1. Once the conguraton aled, there are no more component alure a the conguraton wll not be powered. 2. There ome proce o npecton or detecton o alure even beore the conguraton completely down. 3. The conguraton ether up or down. It, however, poble to model degraded mode o alure. 8

Vdc/2 Vdc/2 + - + - o 1 3 5 4 6 2 A Vdc/2 + - o + Vdc/2 ' - + Vdc/2 - o'' + Vdc/2-7 9 11 10 12 8 13 15 17 16 18 14 1,4 o B 2, 3,6 5 13,1 7,10 o'' 6 o 14,17 8,11 ' 15,18 9,12 b) C A B C a) Fg. 2. (a) Detaled connecton dagram o the propoed modular nverter topology; (b) Vector dagram wth wtch par number The tate equaton can be wrtten ung the concept o requency balance [1, 2]: 9P1 µ 1P2 µ I P3 µ IP4 µ 2P5 µ 3P6 0 3 P 1+ µ 1P2 0 ( µ ) 3 6 P 1 + I + 8 P 0 ( µ ) 4 = (24) = (25) = (26) P 3 + I + 7 P = 0 (27) 7 P 3+ µ 2P5 0 7 P 4+ µ 3P6 0 = (28) = (29) Where P the probablty o tate. Any ve o the above equaton wth the ollowng equaton can be olved to nd the tate probablte, P 1 + P2 + P3 + P4 + P5 + P6 = 1 (30) The varou relablty ndce can now be ound ung the ollowng relatonhp [1]: Sytem Unavalablty = P 2 + P5 + P6 9

Frequency o ytem alure = 3 P1 + 7P3 + 7P4 Mean Down Tme = Sytem unavalablty / Frequency o ytem alure. µ 2 µ 3 1 µ I ALL COMPONENTS UP 3 (1,4) OR (9,12) OR (14,17) DOWN 2 µ 1 µ I 6 3 (2,5) OR (3,6) OR (7,10) OR (8,11) OR (13,16) OR (15,18) DOWN 5 ANY COMBINATION OF TWO LEGS OTHER THAN IN STATE 4 7 4 [(3,6)&(7,10)] OR [(2,5)& (13,16)] OR [(8,11)&(15,18) DOWN 6 ONE MORE LEG THAN IN STATE 4, DOWN 7 CONFIGURATION FAILURE Fg. 3. State tranton dagram o conguraton decrbed n Fg. 2 (a) STANDBY POWER SYSTEM INCLUDING UPS Sytem Decrpton The tet ytem hown n Fg. 4. Th ytem the ame a the one analyzed n [8, 9]. The power normally uppled rom utlty nput power through the UPS. A ynchronzed bypa and tatc wtch protect the crtcal load n the event o an nverter alure. I voltage lot to the crtcal load, the STS reetablhe voltage n le than one quarter o a cycle. Th condered contnuou power or mot load. When the generator are n tandby mode, ther alure reman undetected except durng perodc npecton. Thereore, whle tartng, there probablty p that a generator may al to tart. Only one generatng unt taken out or planned mantenance. I a generator al whle the other on planned mantenance, t poble to accelerate the mantenance on the econd generator by a actor oα. I power al at bu A, the battery can utan the load or upto 4 hour. 10

UTILITY INPUT POWER SYNCHRONIZED BYPASS ATS BUS A RECTIFIER INVERTER STS BATTERY CRITICAL LOAD BUS G G Fg. 4. Parallel uppled non-redundant unnterruptble power upply TABLE I lt the data pertanng to the ytem. Smulaton Model The lowchart hown n Fg. 1 wa mplemented. In computng the tme to the next event, the method decrbed earler wa ued, but the ollowng dependence were alo ncluded: 1. A generator cannot al whle the utlty upply up 2. A generator cannot be taken out or mantenance the utlty down or the other generator down or under mantenance 3. The UPS cannot be taken out or mantenance unle power avalable at bu A TABLE I: SYSTEM DATA Equpment/Supply (/y) r (h/) Utlty Supply, ngle crcut 0.53700 5.66 Generator (per hour o ue) 0.00536 478.00 Inverter 1.25400 107.00 Recter 0.03800 39.00 ATS 0.00600 5.00 STS 0.08760 24.00 Battery 0.03130 24.00 Equpment Mantenance req (/y) dur (h) Generator 1.00 10.00 UPS 1.00 4.00 Other data: Battery can upply load or 4.0 h Common mode alure o generator ( cm ) 0.0 Acceleraton actor or planned mantenance o (α) 2.0 generator Probablty o alure to tart a generator (p ) 0.015 11

The alure o a generator to tart wa modeled a ollow. A random number z wa generated, 0 z 1. I z p, the generator wa aumed to al. Reult In th ecton the ndce obtaned rom mulaton are compared wth thoe obtaned analytcally, ung cut et analy, n [8]. Cut Set Analy The relablty ndce o the tet ytem have been analytcally evaluated, ung the cut et approach, n [5]. A ummary o the oluton tep and the reult wll be preented here, to provde a ba or valdaton o the mulaton technque. Frt, the combnaton o the utlty upply and the two generator analyzed or alure mode. Fg. 5 how the poble tate th combnaton can aume, and the tranton rate between thee tate. Baed on thee tranton rate, the probablte and requence o occurrence o the aled tate are determned. Th enable computaton o the alure rate and duraton o the utlty-generator ubytem, whch are determned to be p = 0.001576/ y and rp = 5. 443h The next tep nvolve computaton o the rate and duraton o power alure at bu A: A = p+ ATS = 0.007576/ y prp+ ATSrATS ra = = 5. 092h p+ ATS The remander o the cut et analy perormed a hown n TABLE II. The ytem ndce,.e., rate and duraton o power lo at the Crtcal Load Bu (CLB) are determned to be = CLB = 0.005225/ cut et y r r CLB = = 9. 648h 12

µ 9 S, 1 GEN MT 1 GEN READY TO START 2 g 1 r mg 4 S, BOTH GENS READY TO START µ p p _ 1 S, BOTH GENS UP 1 r mg µ g 10 S, 1 GEN MT OTHER DN µ 1 r mg µ g S, 1 GEN DN 5 2 µ g µ _ 2 p p _ p µ g 2 g _ 2 S, 1 GEN DN OTHER UP 2 µ g α r mg _ p _ 7 S, 1 GEN MT OTHER GEN UP µ g S, BOTH GENS DN 6 µ p p p g _ 3 S, BOTH GENS DN cm DOWN g _ 8 S, 1 GEN MT OTHER GEN DN p Fg. 5. State tranton dagram or utlty upply and generator I all dtrbuton are aumed exponental, then the tandard devaton o all the up tme and down tme would equal the correpondng mean up tme and down tme. Th mple that the tandard devaton o the alure rate would alo equal the correpondng mean computed. TABLE II: FREQUENCY AND DURATION OF POWER LOSS AT CRITICAL LOAD BUS (CLB) Cut Set (/y) r (h/) r Power lo at bu A > 4 h 0.003454 5.092 0.017588 Power lo at bu A (0.007576, 5.092) and Falure o [Inverter or battery or STS] (1.3729, 99.812) 0.000125 4.845 0.000606 Mantenance on UPS (1.0, 4) and Power lo at bu A (0.007576, 5.092) Inverter alure(1.254, 107.0) and STS alure(0.0876, 24.0) 0.000003 2.240 0.000007 0.001643 19.603 0.032208 Σ 0.005225 0.050409 Smulaton Reult The mulaton method decrbed earler wa ued to compute the ollowng tattc or the tet ytem: 1. Frequency p and duraton r p o alure o the utlty-generator ubytem (Fg. 4); the tandard devaton o p and r p. 13

2. Frequency A and duraton r A o power alure at bu A; the tandard devaton o A and r A. 3. Frequency CLB and duraton r CLB o power alure at the crtcal load bu; the tandard devaton o CLB and r CLB. 4. Data or the Probablty Ma Functon o the number o ytem alure per year, N. 5. Data or the Probablty Dtrbuton Functon o the ytem down tme, T. TABLE III compare the ndce obtaned rom mulaton wth thoe obtaned analytcally. TABLE III: COMPARISON OF SIMULATED AND CALCULATED INDICES Smulated Calculated Index Mean SD Mean SD p (/y) 0.001352 0.001497 0.001576 0.001576 r p (h/) 5.028 5.503 5.443 5.443 A (/y) 0.007193 0.006851 0.007576 0.007576 r A (h/) 5.216 5.201 5.092 5.092 CLB (/y) 0.004868 0.004888 0.005225 0.005225 r CLB (h/) 9.7135 13.3312 9.648 9.648 TABLE IV compare the PMF o the number o ytem alure per year. Now or exponentally dtrbuted up tme the alure are Poon dtrbuted,.e., P( N k CLB CLBe = k) = (31) k! Th equaton ued to generate the calculated data or the PMF o N, n TABLE IV. Fg. 6 compare the PDF o the ytem down tme. For exponentally dtrbuted down tme T, the PDF gven by t/ r CLB F( t) = P( T t) = 1 e (32) Th equaton ued to generate the calculated data or the PDF o T, or Fg 5. 14

TABLE IV: PMF OF NUMBER OF FAILURES PER YEAR P( N = k) k 0 1 2 3 4 Smulated 0.9952 0.4825 10-2 0.2167 10-2 0.0000 0.0000 Calculated 0.9947 0.5198 10-2 0.1358 10-2 0.2365 10-7 0.3089 10-10 CONCLUDING REMARKS Th paper ha decrbed how a gven conguraton can be analyzed or producng a quanttatve relablty ndex. The technque decrbed can help to quanty relablty o alternate degn conguraton or the ame degn but wth derent qualty o component, relectng derent alure rate.. Then the relablty can be traded o wth cot to produce conguraton whch have relablty that that conumer prepared to pay or. 1.00 0.80 probablty 0.60 0.40 0.20 0.00 0 20 40 60 80 100 120 down tme (hour) mulated calculated Fg. 6. Probablty dtrbuton uncton o down tme 15

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