# Measurement of Reliability Parameters for a Power

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IJRET: International Journal of Research in Engineering and Technology   eISSN: 2319-1163 | pISSN: 2321-7308    _______________________________________________________________________________________ Volume: 03 Issue: 09 | Sep-2014, Available @ http://www.ijret.org 101 MEASUREMENT OF RELIABILITY PARAMETERS FOR A POWER PLANT BY USING ALGEBRA OF LOGICS Reena Garg 1   1  Assistant Professor Mathematics, YMCA University of Science & Technology, Faridabad  –   121006 Abstract  In this paper, the author has considered a power plant to evaluate its reliability parameters by using algebra of logics. The system configuration has been shown in fig-1. In this power plant, there are two identical generators 1 G  and 2 G  in standby redundancy. Only one generator is sufficient to fulfill the power requirements of whole system. The author has used four change over devices CO 1  through CO 4  to feed power to four machine 321 ,,  M  M  M   and 4  M  .It is assumed that the cables used, to connect all equipments, are hundred percent reliable and we need to operate two machines ,, 21  M  M  or 3  M  and  4  M    simultaneously. These machines can be operated by switching devices 1 SD  through 4 SD . Keywords:   power plant, identical generators, standby redundancy, changes over devices. --------------------------------------------------------------------***------------------------------------------------------------------ 1. INTRODUCTION In the beginning, the whole power plant is operable and there is no repair facility to repair a failed unit/component of power plant. The author has obtained the reliability of the system as a whole and reliability function for the system in two cases, i.e., when failure rates follow weibull or exponential time distribution. An important reliability parameter, mean time to failure (M.T.T.F.), has also been evaluated in both the cases. In the end of paper, the author has been given a numerical computation and its graphical illustration to highlight the important results for their practical importance. The results obtained in this study, can be applied for the power plant of similar configuration. Fig-1 : System Configuration  IJRET: International Journal of Research in Engineering and Technology   eISSN: 2319-1163 | pISSN: 2321-7308    _______________________________________________________________________________________ Volume: 03 Issue: 09 | Sep-2014, Available @ http://www.ijret.org 102 2. ASSUMPTIONS The following assumptions have been associated with this model: (1)   Initially, all the components are good. (2)   There is no repair facility available. (3)   Failure times of all components of the power plant are arbitrary. (4)   Failures are statistically-independent. (5)   Reliability of each component of power plant is known in advance. (6)   Transition of power from one component to other is all reliable. (7)   Every component of the system remains either in good or in bad state. (8)   Only one generator is sufficient to fullfil the power requirement of whole power plant. (9)   Mostly, we need two machines M 1  and M 2  or M 3  and M 4  to operate simultaneously. 3. NOTATIONS The following notations have been used throughout this  paper: 21 ,  x x  : States of generators 21 , GG  respectively; 6543 ,,,  x x x x  : States of change over devices CO 1  through CO 4 ; 10987 ,,,  x x x x  : States of switching devices 1 SD  through 4 SD ; i  x   :  Negation of state i  x , i  ; i  x  : =1, in good state; =0 in bad state;  /  : Conjunction/Disjunction; i  R  : Reliability of the component corresponding to state i  x ; i S   : i R i   ,1 ; S   R  : Reliability of the power plant as a whole;     t  Rt  R SW SE  /  : Reliability functions when failures follow Weibull/exponential time distribution. 4. FORMULATION OF MATHEMATICAL MODEL By making use of Boolean function technique, the conditions of capability of the successful operation of the  power plant in terms of logical matrix are expressed as:    106432 96432 85432 75432 106431 96431 85431 75431 1021  ,  x x x x x  x x x x x  x x x x x  x x x x x  x x x x x  x x x x x  x x x x x  x x x x x  x x x F    …(1)   5. SOLUTION OF THE MODEL Using algebra of logics, we may write equation (1) as:         ,,,, , 10521431021  x x x x f   x x x x x F      …(2)  Where    1062962852752106196185175110521  ,,,  x x x x x x x x x x x x x x x x x x x x x x x x x x x x f     …(3)  Substituting   7511    x x x B     …(4)     8512    x x x B     …(5)     9613    x x x B     …(6)     10614    x x x B     …(7)     7525    x x x B     …(8)     8526    x x x B     …(9)     9627    x x x B     …(10)     10628    x x x B     …(11)    IJRET: International Journal of Research in Engineering and Technology   eISSN: 2319-1163 | pISSN: 2321-7308    _______________________________________________________________________________________ Volume: 03 Issue: 09 | Sep-2014, Available @ http://www.ijret.org 103 in equation (3), we have    8765432110521  ,,,  B B B B B B B B x x x x  f     …(12)  Using orthogonalisation algorithm, equation (12) may be written as        87654321 7654321 654321 54321 4321 32121110521  ,,,  B B B B B B B B  B B B B B B B  B B B B B B  B B B B B  B B B B  B B B B B B x x x x f     …(13)   Now we have  7515111    x x x x x x B     85175151121    x x x x x x x x x B B       8751    x x x x     …(14)  Similarly, we obtain  987651 9651 321    x x x x x x  x x x x  B B B   …(15)    10987651 109651 4321    x x x x x x x  x x x x x  B B B B …(16)     752154321    x x x x B B B B B    (17)  109876521 876521 87521 654321    x x x x x x x x  x x x x x x  x x x x x  B B B B B B   …(18)    9876521 96521 7654321    x x x x x x x  x x x x x  B B B B B B B   …(19)  And   109876521 1096521 87654321    x x x x x x x x  x x x x x x  B B B B B B B B   …(20)  Therefore, equation (13) becomes, subjected to equations (14) through (20):     109876521 1096521 9876521 96521 109876521 876521 87521 7521 10987651 109651 987651 9651 8751 75110521  ,,,  x x x x x x x x  x x x x x x  x x x x x x x  x x x x x  x x x x x x x x  x x x x x x  x x x x x  x x x x  x x x x x x x  x x x x x  x x x x x x  x x x x  x x x x  x x x x x x x  f     …(21)    IJRET: International Journal of Research in Engineering and Technology   eISSN: 2319-1163 | pISSN: 2321-7308    _______________________________________________________________________________________ Volume: 03 Issue: 09 | Sep-2014, Available @ http://www.ijret.org 104 Using equation (21), equation (2) gives     10987654321 109654321 987654321 9654321 10987654321 87654321 8754321 754321 1098765431 10965431 98765431 965431 875431 75431 1021  ,,  x x x x x x x x x x  x x x x x x x x  x x x x x x x x x  x x x x x x x  x x x x x x x x x x  x x x x x x x x  x x x x x x x  x x x x x x  x x x x x x x x x  x x x x x x x  x x x x x x x x  x x x x x x  x x x x x x  x x x x x  x x x F    …(23)  Since, R.H.S. of equation (23) is disjunction of pair-wise disjoint conjunctions, therefore, the reliability of considered power plant will be:    1, 1021    x x x F  P  R r S     1061957986515961785175143  R R RS S S  R R R R RS  R R RS  R R R R R R R R     85217685271752110865197  R R R RS S  R R RS S  R R RS  R R R R RS S      965287110865219796251  R R R RS S S  R R R R R RS S  R R RS S       1065298711062951  R R R RS S S S  R R RS S S    , Where, .10,2,1,1    i RS  ii  or,  852106175296185175143  R R R R R R R R R R R R R R R R R R R R R S      108651109651986511062962  R R R R R R R R R R R R R R R R R R R R R     108652187652110987651  R R R R R R R R R R R R R R R R R R R     987652986521976521109876521  R R R R R R R R R R R R R R R R R R R R R R R R R R     108765210865211076521109621  R R R R R R R R R R R R R R R R R R R R R R R     10987652110986521097652  R R R R R R R R R R R R R R R R R R R R     10651752198765196518751  R R R R R R R R R R R R R R R R R R R R R R  

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