[Abstract] [PDF] [HTML] [Linked
References]
CostBenefit analysis of two identical cold standby system subject to nonavailability of water resulting failure to produce hydroelectric power and failure due to rainfall
Ashok Kumar Saini
Associate Professor, Department of Mathematics, B. L. J. S. College, Tosham, Bhiwani, Haryana, INDIA.
Email: [email protected]
Research Article
Abstract Introduction: Hydroelectricity is the term referring to electricity generated by hydropower the production of electrical power through the use of the gravitational force of falling or flowing water. It is the most widely used form of renewable energy, accounting for 16 percent of global electricity generation – 3,427 terawatthours of electricity production in 2010 and is expected to increase about 3.1% each year for the next 25 years. Hydropower is produced in 150 countries, with the AsiaPacific region generating 32 percent of global hydropower in 2010. China is the largest hydroelectricity producer, with 721 terawatthours of production in 2010, representing around 17 percent of domestic electricity use. The cost of hydroelectricity is relatively low, making it a competitive source of renewable electricity. The average cost of electricity from a hydro plant larger than 10 megawatts is 3 to 5 U.S. cents per kilowatthour. It is also a flexible source of electricity since the amount produced by the plant can be changed up or down very quickly to adapt to changing energy demands. Water plays an important and pivotal role in producing hydroelectric power. Nonavailability of water results failure to produce hydroelectric power. Reliability is a measure of how well a system performs or meets its design requirements. It is hence the prime concern of all scientists and engineers engaged in developing such a system.. In this paper we have taken two types of failures (1) FNAW nonavailability of water resulting failure to produce Hydroelectric Power (2) FRFfailure due to Rainfall. Applying the regenerative point technique with renewal process theory the various reliability parameters MTSF, Availability, Busy period, BenefitFunction analysis have been evaluated.
Keywords: Cold Standby, FNAW nonavailability of water resulting failure to produce Hydroelectric Power, FRFfailure due to Rainfall, first come first serve, Availability, Busy period, CostBenefit analysis
INTRODUCTION
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) FNAW failure due to nonavailability of water resulting failure to produce Hydroelectric Power (2) FRFfailure due to Rainfall. When the main operative unit fails due to RainfallFRF then cold standby system becomes operative. After failure the unit undergoes repair facility of very high cost in case of FRFfailure due to Rainfall immediately. Failure due to nonavailability of water resulting failure to produce hydroelectric power may disrupt the whole life style. The repair is done on the basis of first fail first repaired.
Assumptions
 F_{1}(t) and F_{2}(t)_{ }are general failure time distributions due to nonavailability of water resulting failure to produce Hydroelectric power and Rainfall. The repair is of two types Type I, TypeII with repair time distributions as G _{1}(t) and G _{2}(t) respectively.
 The Rainfall is noninstantaneous and it cannot come simultaneously in both the units.
 Whenever the Rainfall occur within specified limit of the unit, it works as normal as before. But as soon as there occur Rainfall of higher amount the operation of the unit stops automatically.
 The repair starts immediately after detecting the Rainfall and works on the principle first fail first repaired basis.
 The repair facility does no damage to the units and after repair units are as good as new.
 The switches are perfect and instantaneous.
 All random variables are mutually independent.
 When both the units fail, we give priority to operative unit for repair.
 Repairs are perfect and failure of a unit is detected immediately and perfectly.
 The system is down when both the units are nonoperative.
Symbols for states of the System
F_{1}(t)_{ }and F_{2}(t)_{ }are the failure time distribution due to nonavailability of water resulting failure to produce Hydroelectric power and failure due to Rainfall respectively
G_{1}(t), G_{2}(t) – repair time distribution Type I, TypeII due to nonavailability of water resulting failure to produce hydroelectric power and failure due to Rainfall respectively
Superscripts: O, CS, FNAW, FRF
Operative, Cold Standby, Failure due to nonavailability of water resulting failure to produce hydroelectric power, failure due to Rainfall respectively
Subscripts: nawf, nrff, rff ur, wr, uR
Nonavailability of water resulting failure to produce hydroelectric power, No Rainfall failure, Rainfall failure, 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_{nrff}, CS_{nrff})
One unit is operative and the other unit is cold standby and there are no Rainfall failures in both the units.
1(SOFRF_{rff, ur}, O_{nrff})
The operating unit fails due to Rainfall and is under repair immediately of very costly Type I and standby unit starts operating with no Rainfall.
2(FNAW _{nrff, nawf, ur}, O_{nrff})
The operative unit fails to produce hydroelectric power due to FNAW resulting from nonavailability of water and undergoes repair of type II and the standby unit becomes operative with no Rainfall.
3(FRF_{rff, uR}, FNAW _{nrff, nawf, wr})
The first unit fails due to Rainfall and under very costly Type! repair is continued from state 1 and the other unit fails to produce hydroelectric power due to FNAW resulting from nonavailability of water and is waiting for repair of Type II.
4(FRF_{rff, uR}, FRF_{rff, wr})
The one unit fails due to Rainfall is continues under repair of very costly Type  I from state 1 and the other unit also fails due to Rainfall. is waiting for repair of very costly Type I.
5(FNAW _{nrff, nawf, uR}, FRF_{rff, wr})
The operating unit fails to produce hydroelectric power due to nonavailability of water (FNAW mode) and under repair of Type  II continues from the state 2 and the other unit fails due to Rainfall is waiting for repair of very costly Type I.
6(FNAW _{nrff,nawf,uR}, FNAW _{nrff,nawf,wr})
The operative unit fails to produce hydroelectric power due to FNAW resulting from non availability of water and under repair continues from state 2 of Type –II and the other unit is also failed to produce hydroelectric power due to FNAW resulting from nonavailability of water and is waiting for repair of TypeII and there is no Rainfall.
Figure 1: The State Transition Diagram
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 are
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) (35)
We can regard the failed state as absorbing
Taking LaplaceStiljes transform of eq. (35) and solving for
ø_{0}^{*}(s) = N_{1}(s) / D_{1}(s) (6)
where
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) and (2) it can be shown that ø_{0}^{*}(0) =1, which implies that ø_{0}^{*}(t) is a proper distribution.
MTSF = E[T] = ^{ (s)} = (D_{1}^{’}(0)  N_{1}^{’}(0)) / D_{1} (0)
s=0
= ( +p_{01} + p_{02} ) / (1  p_{01} p_{10 } p_{02} p_{20} )
where _{}
= _{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) (79)
Taking Laplace Transform of eq. (79) and solving for
= N_{2}(s) / D_{2}(s) (10)
where
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) So that (13)
The expected busy period of the server when there is FNAWfailure due to nonavailability of water resulting not to produce hydroelectric power in (0, t]
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) (1416)
Taking Laplace Transform of eq. (1416) and solving for
= N_{3}(s) / D_{3}(s) (17)
Where
N _{3}(s) = _{01}(s) _{1}(s) and
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 FNAWfailure resulting from non availability of water not to produce hydropower in (0,t] is
(t) = So that
The expected Busy period of the server when there is failure due to Rainfall when the units stops automatically 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. (1921) and solving for
= N_{4}(s) / D_{2}(s) (22)
Where
N_{4}(s) = _{02}(s) _{2}(s))
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) (2527)
Taking Laplace Transform of eq. (2527) and solving for
= N_{6}(s) / D_{3}(s) (28)
In the long run, H_{0} = (29)
COSTBENEFIT ANALYSIS
The CostBenefit analysis of the system considering mean uptime, expected busy period of the system under Rainfall when the units stops automatically, expected busy period of the server for repair of unit under nonavailability of water resulting not to produce Hydroelectric power, expected number of visits by the repairman for unit failure.
The expected total BenefitFunction incurred in (0, t] is
C (t) = Expected total revenue in (0, t]
 expected total repair cost repairing the units in (0,t ] due to FNAW failure due to nonavailability of water resulting not to produce hydroelectric power
 expected busy period of the system under Rainfall when the units automatically stop in (0,t]
 expected number of visits by the repairman for repairing of identical the 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 uptime,
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 typeII
K_{4}: cost per visit by the repairman for units repair.
CONCLUSION
After studying the system, we have analyzed graphically that when the failure rate due to nonavailability of water resulting not to produce hydroelectric power and failure rate due to Rainfall increases, the MTSF and steady state availability decreases and the CostBenefit function decreased as the failure increases.
REFERENCES
 Dhillon, B.S. and Natesen, J, Stochastic Anaysis of outdoor Power Systems in fluctuating environment, Microelectron. Reliab.. 1983; 23, 867881.
 Kan, Cheng, Reliability analysis of a system in a randomly changing environment, Acta Math. Appl. Sin. 1985, 2, pp.219228.
 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. 373378.
 Barlow, R.E. and Proschan, F., Mathematical theory of Reliability, 1965; John Wiley, New York.
 Gnedanke, B.V., Belyayar, Yu.K. and Soloyer, A.D., Mathematical Methods of Relability Theory, 1969 ; Academic Press, New York.
