Cost- Benefit Analysis of Two Similar Cold Standby Satellite System subject to Failure due to leak in the Russian cryogenic engine and Failure caused by an anomaly on the Fuel Booster Turbo Pump (FBTP) on the third stage of the vehicle Abstract- FAILURE: Indian GSLV fails during launch with GSAT-5P satellite December 25, 2010 The Indian Space Research Organization (ISRO) GSLV- F06 launch vehicle has suffered another failure, when it exploded around 60 seconds into ascent. The vehicle was carrying the GSAT-5P telecommunications satellite on Christmas Day, lifting off at 10:34am GMT from the Satish Dhawan Space Centre, Shriharikota. The launch was originally rescheduled due to a leak in the Russian cryogenic engine on the third stage of the vehicle. Indian Launch: The Geosynchronous Satellite Launch Vehicle (GSLV) is capable of placing the INSAT-II class of satellites (2000 2,500 kg) into Geosynchronous Transfer Orbit (GTO). The standard GSLV is a three stage vehicle GSLV is 49 m tall, with 414 t lift off weight. It has a maximum diameter of 3.4 m at the payload fairing. First stage comprises S125 solid booster with four liquid (L40) strap-ons. The second stage (GS2) is liquid engine and the third stage (GS3) is a cryo stage. The vehicle develops a lift off thrust of 6573 kn. The first flight of GSLV took place from SHAR on April 18, 2001 by launching 1540 kg GSAT-1. It was followed by four more launches; GSLV-D2 on May 8, 2003 (GSAT-2 1825 kg), GSLV-F01 on September 20, 2004 (EDUSAT 1950 kg), GSLV-F02 on July 10, 2006, GSLV-F04 on September 2, 2007 (INSAT-4CR 2130 kg) and GSLV-D3 on April 15, 2010. Two of its last three flights have ended unsuccessfully, most recently with the April 15 launch of GSAT-4 caused by an anomaly on the Fuel Booster Turbo Pump (FBTP) of the third stage. Saturday s failure leaves the ISRO with three failures from their last four launches, an extended record of four failures, one partial success and two successes from their previous seven launches. The vehicle launching GSAT-5P was taller by two metres and heavier by four tonnes as compared to its standard configuration. The Russian made cryogenic engine was powered with 15.2 tonnes of fuel (liquid hydrogen as fuel and liquid oxygen as oxidizer), an increase of around Ashok Kumar Saini B.L.J.S. College, Tosham (Bhiwani) Haryana Email ID drashokksaini2009@gmail.com The rocket had a larger fairing four-metres in diameter and made of fibre reinforced plastic (FRP) as compared to the standard configuration of 3.4-metre diameter fairing made from aluminium alloy metal. Although the current GSLVs are mixed in their fortunes, ISRO are still pressing ahead with the development of the GSLV Mk III which is scheduled to debut in 2012. This vehicle is designed to make ISRO fully self reliant in launching heavier communication satellites of INSAT-4 class, which weigh 4500 to 5000 kg, via mission launch capability for GTO, LEO, Polar and intermediate circular orbits. GSLV-Mk III is designed to be a three stage vehicle, with 42.4 m tall with a lift off weight of 630 tonnes. First stage comprises two identical S200 Large Solid Booster (LSB) with 200 tonne solid propellant, that are strapped on to the second stage, the L110 re-startable liquid stage. The third stage is the C25 LOX/LH2 cryo stage. The large payload fairing measures 5 m in diameter and can accommodate a payload volume of 100 cu m. GSAT-5P with 24 C-band transponders and 12 extended C-band transponders was meant for augmenting communication services currently provided by Indian National Satellite System (INSAT). It is meant to boost TV, telemedicine and tele-education, and telephone services KEYWORDS: Cold Standby, FLRCE-Failure due to leak in the Russian cryogenic engine on the third stage of the vehicle, FAFBTP- Failure caused by an anomaly on the Fuel Booster Turbo Pump (FBTP) of the third stage, first come first serve, MTSF, Availability, Busy period, Cost- Benefit analysis INTRODUCTION BANGALORE - A team that analyzed data from the failed launch of India's Geostationary Satellite Launch Vehicle (GSLV) has found the primary cause of the Dec. 25 mishap was the "untimely and inadvertent snapping" of a group of 10 connectors located at the bottom of the rocket's Russian supplied upper stage. The Indian Space Research Organisation (ISRO) said in a Dec. 31 statement that the vehicle's performance was normal up to 47.5 seconds after liftoff. It then began to three tonnes, and the engine s length also increased. 27
stray from its planned orientation angle, which resulted in higher structural loads than the vehicle was designed to handle. The rocket, which was carrying a telecommunications satellite, began to break apart about 58 seconds into the flight and was destroyed by command at 64 seconds. [8 Biggest Space Misfires of 2010] ISRO said some of the connectors that snapped carried command signals from the onboard computer near the top of the vehicle to the control systems for the four L40 strap-on boosters that augment the rocket's first stage. These connectors are intended to be separated by command 292 seconds after liftoff but their "premature" rupture stopped the flow of control commands to the boosters, resulting in the loss of control of the vehicle, ISRO said. "The exact cause of snapping of the set of connectors? whether due to external forces like vibration or dynamic pressure? is to be analyzed further and pin-pointed," ISRO said. The statement said ISRO has formed a committee to carry out an in-depth analysis of the failed flight as well as of the previous six GSLV missions and recommend corrective actions. The committee is chaired by G. Madhavan Nair, former chairman of ISRO, and has 11 experts drawn from within and outside ISRO. ISRO also has created a panel led by K. Kasturirangan, also a former ISRO chairman, to look into the future of the GSLV program and the vehicle's reliability for upcoming missions including the Chandrayaan-2 mission to the moon, which is slated for launch in 2013. ISRO plans to complete these reviews by the end of February, the statement said. All but one of the GSLV?s flights to date have used a Russian-built cryogenic upper stage, but only one of those engines remains. ISRO has developed its own cryogenic upper stage, but that engine failed in its debut mission earlier in 2010. Stochastic behavior of systems operating under changing environments has widely been studied. Dhillon, B.S. and Natesan, J. (1983) studied an outdoor power systems in fluctuating environment. Kan Cheng 1985) has studied reliability analysis of a system in a randomly changing environment. Jinhua Cao (1989) has studied a man machine system operating under changing environment subject to a Markov process with two states. The change in operating conditions viz. Fluctuations of voltage, corrosive atmosphere, very low gravity etc. may make a system completely inoperative. Severe environmental conditions can make the actual mission duration longer than the ideal mission duration. In this paper we have taken two types of failures (1) FLRCE-Failure due to leak in the Russian cryogenic engine on the third stage of the vehicle, (2)FAFBTP- Failure caused by an anomaly on the Fuel Booster Turbo Pump (FBTP) of the third stage When the main operative unit fails then cold standby system becomes operative. After failure the unit undergoes repair facility of very high cost in case of FLRCE- Failure due to leak in the Russian cryogenic engine on the third stage of the vehicle immediately. The repair is done on the basis of first fail first repaired. ASSUMPTIONS 1. F 1 (t) and F 2 (t) are general failure time distributions due to FLRCE-Failure due to leak in the Russian cryogenic engine on the third stage of the vehicle and FAFBTP- Failure caused by an anomaly on the Fuel Booster Turbo Pump (FBTP) of the third stage The repair is of two types -Type -I, Type-II with repair time distributions as G 1 (t) and G 2 (t) respectively. 2. The FLRCE-Failure due to leak in the Russian cryogenic engine on the third stage of the vehicle is non-instantaneous and it cannot be available simultaneously in both the units. 3. Whenever there is no failure caused by an anomaly on the Fuel Booster Turbo Pump (FBTP) of the third stage, it works as normal as before. But as soon as there is failure due to leak in the Russian cryogenic engine on the third stage of the vehicle, the operation of the unit stops automatically. 4. The repair starts immediately after detecting the failure due to FLRCE-Failure due to leak in the Russian cryogenic engine on the third stage of the vehicle, and works on the principle first fail first repaired basis. 5. The repair facility does no damage to the units and after repair units are as good as new. 6. The switches are perfect and instantaneous. 7. All random variables are mutually independent. 8. When both the units fail, we give priority to operative unit for repair. 9. Repairs are perfect and failure of a unit is detected immediately and perfectly. 10. The system is down when both the units are nonoperative. SYMBOLS FOR STATES OF THE SYSTEM F 1 (t) and F 2 (t) FLRCE-Failure due to leak in the Russian cryogenic engine on the third stage of the vehicle, FAFBTP- Failure caused by an anomaly on the Fuel Booster Turbo Pump (FBTP) of the third stage respectively G 1 (t), G 2 (t) repair time distribution Type -I, Type-II due to FLRCE-Failure due to leak in the Russian cryogenic engine on the third stage of the vehicle, FAFBTP- Failure caused by an anomaly on the Fuel Booster Turbo Pump (FBTP) of the third stage respectively 28
SUPERSCRIPTS O, CS, FLRCE, FAFBTP Operative, Cold Standby, FLRCE-Failure due to leak in the Russian cryogenic engine on the third stage of the vehicle, FAFBTP- Failure caused by an anomaly on the Fuel Booster Turbo Pump (FBTP) of the third stage respectively SUBSCRIPTS nflrce, flrce, nfafbtp, ur, wr, ur No Failure due to leak in the Russian cryogenic engine on the third stage of the vehicle, Failure due to leak in the Russian cryogenic engine on the third stage of the vehicle, no Failure caused by an anomaly on the Fuel Booster Turbo Pump (FBTP) of the third stage, under repair, waiting for repair, under repair continued from previous state respectively Up states 0, 1, 2; Down states 3, 4 regeneration point 0, 1, 2 NOTATIONS M i (t) System having started from state i is up at time t without visiting any other regenerative state A i (t) state is up state as instant t R i (t) System having started from state i is busy for repair at time t without visiting any other regenerative state. B i (t) the server is busy for repair at time t. H i (t) Expected number of visits by the server for repairing given that the system initially starts from regenerative state i STATES OF THE SYSTEM 0(O nflrce, CS nflrce ) One unit is operative and the other unit is cold standby and there is no failure due to leak in the Russian cryogenic engine on the third stage of the vehicle in both the units. 1(SOFLRCE flrce, ur, O nflrce ) The operating unit fails due to leak in the Russian cryogenic engine on the third stage of the vehicle and is under repair immediately of very costly Type- I and standby unit starts operating with no failure due to leak in the Russian cryogenic engine on the third stage of the vehicle. 2(FAFBTP fafbtp, nflrce, ur, O nflrce ) The operative unit fails caused by an anomaly on the Fuel Booster Turbo Pump (FBTP) of the third stage and undergoes repair of type II and the standby unit becomes operative with no failure due to leak in the Russian cryogenic engine on the third stage of the vehicle. 3(FLRCE nfafbtp,ur, FAFBTP fafbtp,nflrce, wr ) The first unit fails due to leak in the Russian cryogenic engine on the third stage of the vehicle and undergo very costly Type-I repair is continued from state 1 and the other unit failure caused by an anomaly on the Fuel Booster Turbo Pump (FBTP) of the third stage and is waiting for repair of Type -II. 4(FLRCE flrce,ur, FLRCE flrce,wr ) The failure due to leak in the Russian cryogenic engine on the third stage of the vehicle is continues and undergo very costly repair of Type - I from state 1 and there is failure due to leak in the Russian cryogenic engine on the third stage of the vehicle in another unit is waiting for very costly Type -I repair. 5(FAFBTP nflrce,fafbtp,ur,flrce flrce, wr ) The operating unit failure caused by an anomaly on the Fuel Booster Turbo Pump (FBTP) of the third stage and undergo repair of Type - II continues from the state 2 and in the other unit fails due leak in the Russian cryogenic engine on the third stage of the vehicle is waiting for repair of very costly Type- I. 6(FAFBTP fafbtp,nflrce,ur, FAFBTP fafbtp,nflrce,wr ) The operative unit Failure caused by an anomaly on the Fuel Booster Turbo Pump (FBTP) of the third stage and under repair continues from state 2 of Type II and the other unit is also Failure caused by an anomaly on the Fuel Booster Turbo Pump (FBTP) of the third stage is waiting for repair of Type-II and there is no to leak in the Russian cryogenic engine on the third stage of the vehicle Fig.1 the State Transition Diagram regeneration point -states 0, 1, 2 Up State Down State 29
Transition Probabilities Simple probabilistic considerations yield the following expressions: p 01 =, p 02 = p 10 =, p 13 =p 11 (3) = p 11 (4) = p 25 = p 22 (5) = p 22 (6) = clearly p 01 + p 02 = 1, p 10 + p 13 =(p 11 (3) ) + p 14 = ( p 11 (4) ) = 1, p 20 + p 25 = (p 22 (5) ) + p 26 =(p 22 (6) ) = 1 (1) And mean sojourn time is µ 0 = E(T) = (2) MEAN TIME TO SYSTEM FAILURE Ø 0 (t) = Q 01 (t)[s] Ø 1 (t) + Q 02 (t)[s] Ø 2 (t) Ø 1 (t) = Q 10 (t)[s] Ø 0 (t) + Q 13 (t) + Q 14 (t) Ø 2 (t) = Q 20 (t)[s] Ø 0 (t) + Q 25 (t) + Q 26 (t) (3-5) We can regard the failed state as absorbing Taking Laplace-Stiljes transform of eq. (3-5) and solving for ø 0 (s) = N 1 (s) / D 1 (s) (6) N 1 (s) = Q 01 [ Q 13 (s) + Q 14 (s) ] + Q 02 [ Q 25 (s) + Q 26 (s) ] D 1 (s) = 1 - Q 01 Q 10 - Q 02 Q 20 Making use of relations (1) & (2) it can be shown that ø 0 (0) =1, which implies that ø 0 (t) is a proper distribution. MTSF = E[T] = (s) s=0 = (D 1 (0) - N 1 (0)) / D 1 (0) = ( +p 01 + p 02 ) / (1 - p 01 p 10 - p 02 p 20 ) = 1 + 2, 1= 0 + 3 + 4 + + AVAILABILITY ANALYSIS Let M i (t) be the probability of the system having started from state i is up at time t without making any other regenerative state. By probabilistic arguments, we have The value of M 0 (t), M 1 (t), M 2 (t) can be found easily. The point wise availability A i (t) have the following recursive relations A 0 (t) = M 0 (t) + q 01 (t)[c]a 1 (t) + q 02 (t)[c]a 2 (t) A 1 (t) = M 1 (t) + q 10 (t)[c]a 0 (t) + q 11 (3) (t)[c]a 1 (t)+ q 11 (4) (t)[c]a 1 (t), A 2 (t) = M 2 (t) + q 20 (t)[c]a 0 (t) + [q 22 (5) (t)[c]+ q 22 (6) (t)] [c]a 2 (t) Taking Laplace Transform of eq. (7-9) and solving for = N 2 (s) / D 2 (s) (10) N 2 (s) = 0(s)(1-11 (3) (s) - 11 (4) (s)) (1-22 (5) (s)- 22 (6) (s)) + (s) 1(s) [ 1-22 (5) (s)- 22 (6) (s)] + 02(s) 2(s)(1-11 (3) (s) - 11 (4) (s)) D 2 (s) = (1-11 (3) (s)- 11 (4) (s)) { 1-22 (5) (s) - 22 (6) (s) )[1-( 01(s) 10 (s))(1-11 (3) (s)- 11 (4) (s))] The steady state availability A 0 = = = Using L Hospitals rule, we get A 0 = = (11) The expected up time of the system in (0,t] is (t) = So that (12) The expected down time of the system in (0,t] is (t) = t- (t) o that (13) The expected busy period of the server when there is FLRCE- failure due to leak in the Russian cryogenic engine on the third stage of the vehicle in (0, t] 30
R 0 (t) = q 01 (t)[c]r 1 (t) + q 02 (t)[c]r 2 (t) R 1 (t) = S 1 (t) + q 01 (t)[c]r 1 (t) + [q 11 (3) (t) + q 11 (4) (t)[c]r 1 (t) R 2 (t) =q 20 (t)[c]r 0 (t)+ [q 22 (6) (t)+ q 22 (5) (t)][c]r 2 (t) (14-16) Taking Laplace Transform of eq. (14-16) and solving for N 3 (s) = 01(s) 1 (s) and = N 3 (s) / D 3 (s) (17) D 3 (s)= (1-11 (3) (s)- 11 (4) (s)) 01 (s) is already defined. In the long run, R 0 = (18) The expected period of the system under FLRCE- fails due leak in the Russian cryogenic engine on the third stage of the vehicle in (0,t] is (t) = So that The expected Busy period of the server when there is Failure caused by an anomaly on the Fuel Booster Turbo Pump (FBTP) of the third stage in (0,t] B 0 (t) = q 01 (t)[c]b 1 (t) + q 02 (t)[c]b 2 (t) B 1 (t) = q 01 (t)[c]b 1 (t) + [q 11 (3) (t)+ q 11 (4) (t)] [c]b 1 (t), B 2 (t) =T 2 (t) + q 02 (t)[c] B 2 (t) + [q 22 (5) (t)+ q 22 (6) (t)] [c]b 2 (t) T 2 (t) = e - λ 1 t G 2 (t) (19-21) Taking Laplace Transform of eq. (19-21) and solving for Where N 4 (s) = 02(s) 2 (s)) = N 4 (s) / D 2 (s) (22) And D 2 (s) is already defined. In steady state, B 0 = (23) The expected busy period of the server for repair in (0,t] is (t) = So that (24) The expected number of visits by the repairman for repairing the identical units in (0,t] H 0 (t) = Q 01 (t)[s][1+ H 1 (t)] + Q 02 (t)[s][1+ H 2 (t)] H 1 (t) = Q 10 (t)[s]h 0 (t)] + [Q 11 (3) (t)+ Q 11 (4) (t)] [s]h 1 (t), H 2 (t) =Q 20 (t)[s]h 0 (t)+[q 22 (5) (t)+q 22 (6) (t)][c]h 2 (t) (25-27) Taking Laplace Transform of eq. (25-27) and solving for = N 6 (s) / D 3 (s) (28) In the long run, H 0 = (29) COST-BENEFIT ANALYSIS The Cost-Benefit analysis of the system considering mean up-time, expected busy period of the system when there is failure due to leak in the Russian cryogenic engine on the third stage of the vehicle when the units stops automatically, expected busy period of the server for repair of unit failure caused by an anomaly on the Fuel Booster Turbo Pump (FBTP) of the third stage, expected number of visits by the repairman for unit failure. The expected total Benefit-Function incurred in (0,t] is C (t) = Expected total revenue in (0,t] - expected total repair cost when there is failure due to leak in the Russian cryogenic engine on the third stage of the vehicle when the units automatically stop in (0,t] - expected total repairing cost of the units when there is failure caused by an anomaly on the Fuel Booster Turbo Pump (FBTP) of the third stage in (0,t ] - expected number of visits by the repairman for repairing of identical units in (0,t] The expected total cost per unit time in steady state is C = = = K 1 A 0 - K 2 R 0 - K 3 B 0 - K 4 H 0 Where K 1 - revenue per unit up-time, K 2 - cost per unit time for which the system is under repair of type- I K 3 - cost per unit time for which the system is under repair of type-ii K 4 - cost per visit by the repairman for units repair. CONCLUSION After studying the system, we have analyzed graphically that when The failure due to leak in the Russian cryogenic engine on the third stage of the vehicle rate, failure caused by an anomaly on the Fuel Booster Turbo Pump (FBTP) of the third stage rate increases, the MTSF and steady state availability decreases and the Cost-Benefit function decreased as the failure increases and ultimately risk of failure of satellite increases. REFERENCES [1] Dhillon, B.S. and Natesen, J, Stochastic Anaysis of outdoor Power Systems in fluctuating 31
environment, Microelectron. Reliab..1983; 23, 867-881. [2] Kan, Cheng, Reliability analysis of a system in a randomly changing environment, Acta Math. Appl. Sin. 1985, 2, pp.219-228. [3] Cao, Jinhua, Stochatic Behaviour of a Man Machine System operating under changing environment subject to a Markov Process with two states, Microelectron. Reliab.,1989; 28, pp. 373-378. [4] Barlow, R.E. and Proschan, F., Mathematical theory of Reliability, 1965; John Wiley, New York. [5] Gnedanke, B.V., Belyayar, Yu.K. and Soloyer, A.D., Mathematical Methods of Relability Theory, 1969 ; Academic Press, New York. 32