Reliability and Economic Analysis of Captive Power Plant With Reduced Capacity Текст научной статьи по специальности «Медицинские технологии»

Reliability and Economic Analysis of Captive Power Plant

With Reduced Capacity

Upasana Sharma1 and Avtar Singh2* •

department of Statistics, Punjabi University, Patiala-India, [email protected] 2*Department of Statistics, Punjabi University, Patiala-India, [email protected]

Abstract

This paper reported the performance evaluation of Captive Power plant working in the fertilizer industry with possible production capacities. The idea of reduced capacity and load sharing to use the available system optimally is analyzed. The system works on two STG's (steam turbine generators) and one gridline. Gridline can bear the load of one or both STG's on failures. At the breakdown in gridline and STG, the system work at reduced capacity. Gridline repaired on a priority basis. The semi-Markov processses and regenerative point technique are used to evaluate the reliability and economic measures such as availability, busy period of repairman, and expected no. of repairs. The graphical study shows the relationships between these measures with the failure rates of STG and gridline.

Keywords: Steam Turbine Generators, Regenerative point technique, semi-Markov process, Reduced capacity, Reliability modeling.

I. Introduction

Nowadays, Captive power plants are a reliable and beneficial energy source for power-consuming production industries. Optimizing the operations of the power-producing units in these captive power plants can boost the industry's profit. A good number of researchers have worked on various reliability models with conditions of repair and maintenance. Gupta and Goel [3] studied a two-unit cold standby system working under abnormal weather conditions. Chandrashekhar et al. [1], Goyal et al. [2], Parashar et al. [4] have analyzed two and three-unit systems. Rizwan et al. [5] worked with the reliability of the hot standby industrial system.

Singh and Taneja [7], [8] analyzed power generating systems with various types of inspections. Rajesh et. al [10], [9] studied gas turbine power plants consisting of two and three units. These attempts to the literature create a motivation for the present study, to work with the economic benefits of the captive power plant. The captive power plants are auto producers of electricity, which operates off-grid or in parallel with gridline to make consistent and quality electricity supply for industries at reasonable costs. Availability of these power generating units in any possible way (full or reduced) can make a reliable electricity supply at less cost. Keeping an eye on the above fact economic analysis of Captive Power plant working in National Fertilizer limited, Bathinda, India has done.

The present system comprises two STG's (steam turbine generators) connected in parallel with the Gridline of PSPCL (Punjab state power corporation limited). These two STGs can fulfill the electrical load for the system. On failure of any one or both of STGs, the system operates with the help of gridline. The system will work at reduced capacity when only one STG is working (one STG and gridline failed). The failure of these three units leads to complete system failure. Repair of gridline is done on a priority basis among all units, whereas the FCFS repair pattern is applied on both STGs. The reliability measure MTSF (mean time to system failure) and economic measures such as availability, a busy period of the repairman, and expected no. of repairs have

been derived using the semi-Markov processes and regenerative point techniques numerically. Also, graphical plotting was performed for these measures.

I. Assumptions for the model

• All failure time variables follow exponential distribution but repair times distributed

generally..

• Every repaired unit works as new one.

• In the given model system initially started working from state So.

II. Nomenclature & Model Description I. Notations & abbreviations

Notations Discription

A1 Constant failure rate of STG 1.

A2 Constant failure rate of STG 2.

A3 Constant failure rate of Gridline.

«1 Repair rate of STG 1.

a2 Repair rate of STG 2.

a3 Repair rate of Gridline.

G1 (t), g1 (t) c.d.f. & p.d.f of repair time of STG 1.

G2(t), g2(t) c.d.f. & p.d.f of repair time of STG 2.

G3(t), g3(t) c.d.f. & p.d.f of repair time of Gridline.

a probability of transit from S7 & Sg to S3 respectively after repair .

b probability of transit from S7 & Sg to S4 respectively after repair .

© Laplace Convolution.

© Stieltjes Convolution.

*/ * * Laplace Transformation/ Laplace Stieltjes Transformation.

Mi (t) Probability that system is working at state S, during the time interval (0 — t].

Wi (t) Probability of repairman repairing at state S, during the time interval (0 — t].

II. Symbols for States

Symbols for the states of the system:-

Si : States of the system with number i, i = 1,2,3, ...8.

Oj, Ojj, Ojjj : STG 1, STG 2, Gridline from PSPCL are in operating state.

CSjjj : Gridline (PSPCL) in cold standby state.

Fr;, Fr;j, FrJJJ : STG 1, STG 2, Gridline under repair.

FRj, FRjj, FRjjj : STG 1, STG 2, Gridline under repair from previous state.

Fwr;, Fwr;j :Failed Units STG 1, STG 2 waiting for repair.

III. State Transition Diagram

Figure 1, shows the state transitions diagram of the Captive power plant consisting of two STGs and one gridline from PSPCL. The states S0, Si, S2, S3, S4 are operating states. The states So, S1, S2, S3, S4, S5, S6, S8 are regenerative states. The states S5, Ss are reduced capacity states. The states S7, S8 are failed states. Table 1 shows the description of every state of the system.

Figure 1: State Transition Diagram

Table 1: State Discription

State notation States Discription

S0 This is the initial full capacity working state where both STGs are

working. Gridline is in a standby state.

S1 System working at full capacity where STG 1 and gridline are

working. STG 2 is in a failed state under repair.

S2 System working full capacity where STG 2 and gridline are work-

ing. STG 1 is in failed state under repair.

Sз, S4 System is operating at full capacity with gridline. Both STGs are

in a failed state.

S5 System operating at reduced capacity where only STG 2 is working.

STG 1 and gridline are in the failed state.

S6 System operating at reduced capacity where only STG 1 is working.

STG 2 and gridline are in failed states.

^ S8 These are failed states where all units are in a failed state.

IV. Transition Probabilities & Mean Sojourn Times

pij represents non-zero elements which are given below The non zero elements pij's are given as:

p01

p13

Ai + A2' A2

p24 p38

A3 + A2

A1

A1 + A3 [1 - gl (A3)],

p51 =g3 (A2 ),

p54) =b[1 - g3 (A2)],

p63) =a[1 - g3 (A1)],

p84 =b

[1 - gi (A3+A2)], [1 - g2 (A1 + A3)],

p02 p15 p26

A2

A1 + A2'

A3

A3 + A2

A3

A1 + A3 p41 =g2 (A3), p57 =[1 - g3(A2)], p62 =g3 (A1^ p67i=b[1 - g3 (A1)],

[1 - g1 (A3+A2)], [1 - g2 (A1 + A3)],

p10 =g1 (A3 + A2),

p20 =g2 (A1 + A3),

p32 =g1 (A3),

p48 =[1 - g2 (A3)]

(7) p53 =a[1 - g3(A2)],

p67 =[1 - g3 (A1)],

p83 =a,

The mean sojourn time u corresponding to regenerative state 'i' is given as:

U0

1

Ai + A2'

Ui

1

A3 + A2

[1 - gi (A3 + A2)],

U2

1

Ai + A3

[1 - gi (Ai + A3)],

1

U3 = ^ [1 - gi (A3)], 1

U6 = [1 - gi (A1)],

1

U4 = ^ [1 - gi (A3)], U8 = - gi' (0)

1

U5 = j-2 [1 - gi(A2)],

The unconditional mean time mij required by the system to transit from state'i' to any regenerative state 'j' when time is counted from the epoch of entrance into the state 'i' is mathematically stated as:

So we have mo1 + mo2 =Uo,

m41 + m48 =U4,

r b

tdQij(t) = -qij (0)

m1o + m13 + m15 =U1,

, (7) , (7) , m51 + m53 + m54 =k1,

m2o + m24 + m28 =U2,

(7) (7)

m62 + m63) + m64) =k1,

(1)

m32 + m38 =U3, m83 + m84 =U8

III. Reliability and Economic Measures for System Effectiveness I. Mean Time to System Failure (MTSF)

Assume ty (t) as a distribution function of variable time (t) lapses during the system transition from a regenerative state Si to any working or failed state where failed state act as an absorbing state. By probabilistic arguments, the following recursive relations are obtained:

<£0 (t) = Qo1(t)®01 (t) + Q02 (t)©<2 (t) <M0 = Q10 (t)©<0 (t) + Q13(t)©<3(t) + Q15 (t)©<5 (t) <2(t) = Q20 (t)©<0 (t) + Q24(t)©<4(t) + Q26 (t)©<6 (t) <3 (t) = Q32(t)©<2 (t) + Q38 (t) <4 (t) = Q41(t)©<1 (t) + Q48 (t) <5 (t) = Q51(t)©<1 (t) + Q57 (t) <6 (t) = Q62(t)©<2 (t) + Q67 (t)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

Transforming the equations(2-8) using Laplace Stieltjes Transformations to get ty** (t). Mean Time to System Failure To at steady state So is given by

To = lim1 - <0"(s)

s^0 s

Using L' Hospital's rule here, we get

where

T0 = N/D

(9) (10)

N = U0(1 - p15p51 - p26p62 - p15p26p62p57 + p13p24p32p48 + p15p26p62 - p24p13p32)

+ U1(-p62 p26 p01 + p02 p24 p41 + p01) + U2 (-p51 p02 p15 + p13 p32 p01 + p02 ) + U3 ( p13 p01 - p13p01 p26p62 + p24p02p13p41) + U4(p24p02 - p24p02p15p51 + p13p01 p24p32) + U5(p15p01 - p15p01 p26p62 + p24p02p15p41) + U6(p26p02 - p15p51 p26p02 + p13p01 p26p32)

(11)

and

D = 1 + p51 p26p15p62 + p51 p02p20p15 - p15p51 - p26p62 + p62p26p01 p10 - p13p32p24p41

- p13p32p01 p20 - p02p10p24p41 - p02p20 - p01 p10 (12)

II. Availability Analysis at Full & Reduced capacity

Let AF (t) notates the probability that system is available with full capacity to perform its intended task at a regenerative state Si at time t = 0. The availability of system at successive regenerative state Sj (j = 1,2, ..6,8) is independant from its previous transitions made. This phenomenon follows the theory of regenerative process techniques [6]. Thus following recursive relations are obtained:

AF (t) = Mo(t) + qoi(t)©AF (t) + q02 (t)©AF (t) (13)

AF (t) = Mi(t) + qio (t)©AF (t) + qi3 (t)©AF (t) + qis(t)©A5 (t) (14)

AF (t) = M2(t) + q20 (t)©AF (t) + q24 (t)©Af (t) + q26(t)©A6 (t) (15)

AF (t) = M3(t) + q32(t)©Af (t) + q38 (t)©AF (t) (16)

AF (t) = M4(t) + q41(t)©AF (t) + q48 (t)©AF (t) (17)

AF (t) = q51(t)©AF (t) + q53)(t)©AF (t) + q54)(t)©A4F (t) (18)

AF (t) = q62(t)©AF (t) + q633)(t)©AF (t) + q64)(t)©A4F (t) (19)

AF (t) = q83(t)©AF (t) + q84(t)©AF (t) (20)

Where

M0 (t) = e-(À1+À2)f, Mi (t) = e-(À1+À2)fGi"(t), M2(t) = e-(À1+À2)fG2~(t),

M3 (t) = e-(À3)fG1"(t), M4 (t) = e-(À3)fG2~(t)

Transforming the equations(13-20) using Laplace transformations to get AF* (s). we have

AF (t) = lim(sAF* (s)) (21)

The steady state availability AF of the system having full capacity is given by:

AF = lim AF (t)= N1/D1 (22)

where

N1 = ^0 [( p15 p51 - 1)( p26 p32 p63 + (1 - p62 p26 )( p38 p83 + p84 p48 ) + p26 p62 (1 + p15 p54) p41 p38 ) - p83p24p32p48 - 1) + (1 - p62p26)((p83p54)p15 - p84P53p15 - p84p13)p41 p38)

- p84 p32 p15 p26 p51 p48 (1 - p62 (1 - P^)) - (1 - p26 p32 ) p15 p54) p41 - v64) p32 p26 p41 ( p13 + p15 P54 )

- p24p32P41 (p13 + p15p53))] + ^1[p01 ((1 - p84p48 - p83p38)(1 - p26p62) - VÛp32p26(1 - p84p48)

- p83p48p32(p24 + v6lp26)) + p02((vÛp41 p26 + p24p41)(1 - p83p38) + p84p633)p41 p26p38)] + ¥2 [P02 ((1 - P83 P38)(1 - P15 P51 - P15 v54) P41) - P84 P48(1 - P51P15 ) - P84 P38 P13 P41)

+ P01 P32((P15p53) + P13)(1 - P84P48) + P15p5^P83P48)] + ^[(1 - P62P26)(P15P^P01 + P01P13

- P01P84P13P48) + (P84P63) -

P83P64))(P15P51P02P26P48 - P02P26P48) + P83P24P02P48(1 - P15P51 )

- P15P26P02P41(v54)v53) - v64)v53)) + P15P01 P48(P83p54) - P84v5^)) + P32p02p26 - P15p26P^P51P02

+ P64)P02P41P13P26 + P24P15P02P41 p£ + P24P02P41P13] + *[(1 - P83P3«)(P24P02 - P24P15P51P02

- P15v64)P51P02P26 + v64)P02P26 + P15P01 v54)) + (P13P15v53))(P01P24P32 + P01 v64)P32P26)

+ P15P84p5?P38(P01 - P02P41) + P84P63P02P26P38(1 - P15P51) + P01P^P15(P26(P62(P83P38

- 1) - v6îP32))] (23)

and

Dl = U0 ((1 - p83p38)(p24p41 pio + p41 P64p26pi0 + p20 - p51 p20pi5) + p41 pi5p38p20 (p83p54) -

p84p573)) - p84p20p48(1 - p15p51 ) + p41 p84p26p6?p38p10) + Mp41 p84p38(1 - p26p62) + p32p01 p20 (1 - p84p48) + p32p24p41 + p32p41 p64)p26 - p41 p84p38p02p20 + ) + p32p83p48(1 - p01 p10) + p41 p02p10(1 - p83p38) + p32p5?p41 p15 + p32p13p41 - p32p15p48p51 p83) + V3((1 - p26p62) (p41 p84p15p(? + p41 p84p13 - p01 p10p83 + p83 - p83p51 P15) - p41 p83pg?p15(1 - p02p02) - p41 p84p02p20(p13 + p53)p15) - p24p41 p83p02p10 - p41 PÛp83p26p02p10 + p41 p83P54p15p26p62 + p83p51 p15p02p20 + p41 p84p26p63 p02p10) + V4 (p32p24p83 (1 - p15p51 - p01 p10 ) + p84 (1 - p01 p10 - p26p62 - p15p51 ) + p32p84p15p26p51 p6? + p32p83p54) p15p01 p20 - p32p64 p26p15p51 p83 + p84p26p62(p15p51 + p01 p10) - p32p84p01 p20(p5?p15 + p13) - p84p02p20(1 - p15p51 ) + (1 - p01 p10)(p32p64)p83p26 - p32p84p26P63 ) + Mp41 p84p15p38(1 - p26p62 - p02p20) + p32p24p41 p15 + p32p01 p20p15 + p32p41 p64p26p15 - p32p15p84p48p01 p20) + k1 (p32p83p26p48(1 - p01 p10) + p41 p26p02p10(1 - p83p38) + p32p41 p5?p26p15 + p32p41 p26p13) + V8((1 - p15p51) (p32p24p48 - p38p02p20 ) + p32p64p26p48 (1 - p01 p10 ) - p41 p54) p15p38(1 - p02p20 - p26P63 ) + p38(-p26 p62 - p01 p10 - p15p51) + p26p38p62(p15p51 + p01 p10) - p41 p38p02p10(p24 + P^p26)

- p32p24p48p01 p10 - p32p64 p26p15p48p51 ) (24)

Let Af (t) notâtes the probability that system is available with reduced capacity to work at a regenerative state Si at time t = 0. The following recursive relations are obtained using the above described argument of regenrative process techniques:

Af (t) = q01 (t)©Af (t) + q02 (t)©Af (t) (25)

Af (t) = qw(t)©Af (t) + q13 (t)©Af (t) + q15 (t)©Af (t) (26)

Af (t) = q20(t)©Af (t) + q24 (t)©Af (t) + q26 (t)©Af (t) (27)

Af (t) = q32 (t)©Af (t) + q38 (t)©Af (t) (28)

Af (t) = q41 (t)©Af (t) + q48 (t)©Af (t) (29)

Af (t) = M5(t) + q51(t)©Af (t) + q573)(t)©Af (t) + q54)(t)©Af (t) (30)

Af (t) = M6(t) + q62(t)©Af (t) + q673)(t)©Af (t) + q6?(t)©Af (t) (31)

Af (t) = q83 (t)©Af (t) + q84 (t)©Af (t) (32)

where

M5 (t) = e-(À2)fG3_(t), M6 (t) = e-(A1)fG3_(t) Transforming the equations(25-32) using Laplace transformations to get Af* (s). we have

Af (t) = lim(sAf* (s)) (33)

The steady state availability Af of the system having reduced capacity is given by:

Af = lim(sAf* (s)) = N2/D1 (34)

s^ 0

where

N2 = ^5 (P24P02P15P41 - P15P01P26P62 - Pl5P01P38P83 - P84Pl5P01P48 - Pl5P01P26P32P63

, (7) , (7) (7)

+ P64 P15P41P26P02 + P15P01 - P64 P15P41P26P02P38P83 - P64 P15P01P26P32P83P48

(7) (7)

+ P84P15P41P26P02P38P63 + P84P15P02P26P32P63 P48 + P15P01P26P62P38P83 - P24P02P15P41P38P83

+ P84P15P01P26P62P48 - P24P32P83P15P01P48) + P26P02 - P26P02P38P83 + P26P32P13P01

(7) (7)

- P15P51P26P02 - P84P26P02P48 + P15P51P26P02P38P83 + P15P01P26P32P53 + P54 P15P41P26P02P38P83

(7) (7) (7)

+ P54 P15P01P26P32P83P48 + P84P51P15P26P02P48 - P84P15P41P26P02P38P53 - P84P15P01P26P32P53 P48

- P84P26P02P38P13P41 - P84P26P32P13P01P48 - ^ P15P41P26P02 ) (35) and D1 is already specified in equation 24

III. Busy Period for Repairman

Let B,(t) notates the probability that the repairman is busy on the job when the system is at a regenerative state S, at time t = 0. Using the probabilistic arguments as described above, The following recursive relations are obtained:

B0(t) = q01(t)©B1 (t) + q02 (t)©B2 (t) (36)

B1 (t) = W1(t) + q10(t)©B0 (t) + ^13 (t)©B3(t) + 915 (t)©B5(t) (37)

B2 (t) = W2(t) + q20(t)©B0 (t) + q24 (t)©B4(t) + 926 (t)©B6(t) (38)

B3(t) = q32(t)©B2 (t) + q38 (t)©B8 (t) (39)

B4(t) = q41(t)©B1 (t) + q48 (t)©B8 (t) (40)

B5(t) = W5 (t) + q51 (t)©B1 (t) + q53)(t)©B3(t) + ^^(t) (41)

B6(t) = W5 (t) + q62 (t)©B2 (t) + q63)(t)©B3(t) + ^(^(t) (42)

B8 (t) = W8(t) + q83 (t)©B3(t) + q84(t)©B4 (t) (43)

Where

W0 (t) = e-(A1+A2 )f, W1(t) = e-(A1+A2 )iG1"(t), W2 (t) = e-(A1+A2)fG2_(t),

W5(t) = e-(A2 )fG3_(t),

W6 (t) = e-(A1)fG3_(t), W8(t) = (« + b)G3_(t)

Taking Laplace transform of equations (36-43) we get B0*(s). we have

B0(t) = lim(sB5 (s)) Expected busy period of a repairman is given by

B0 = lim(B0 (t)) = N3/D1 (44)

where

N3 = U1 [(P26p674 + P24)(P02P41 - P41P83P38P02 - P83P32P48P01) + (P84P48 + P83P38) (P62P26P01 - P01) - P63)P32P26P01 (1 - P84P48) - P01 (P62P26 - 1)] + ^2[(1 - P84P48)(p5?P32P15P01

+ p13 p32 p01 + p02 + p15 p51 p02 ) - p83 p38 P02(1 - p51 P15) - p41 p15 p38 p02 ( Ps;3) P84 + p83 p^) + p15p54)(p83p32p48p01 - p41 p02) - p41 p13p84p38p02] + ^5 [(1 - p62p26)(p15p01 - p84p15p48p01

- p83p15p01 p38) + (1 - p84p48)(-p(3p32p15p26pW) + (p02p41 p15p26p6î> + p02p«p15p24) (1 - p83p38) + p15p26(p41 p63)p84p38p02 - p83p32p64)p48p01)] + M(1 - p15p51 )(-p02p84p26p48 - p83p26p38p02 + p02p26)(1 - p84p«)(pj^p32p15p26p01 p13p32p26p01 ) + p83p32p15p5^)p26p48pW] + ^8[(1 - p62p26)(p13p01 p38 + p573)p15p01 p38) + (1 - p15p51)(p02p24p48 + p02pjap26p48 + P63)P26p38p02) + (p24 + p26pja )(p13p32p48p01 + P53)P32p15p48p01) - p15p26p^p48p01 (p62 + p673)p32) + p41 p15p26p38p02(pi-Jp6Î - p^PS^ + p15P5ÎP48p01 + p41 p13p26p38p64)p02] (45) and D1 is already specified in equation 24.

IV. Expected No. of Repairs

Let Vi (t) notate no. of repairs performed by repairman in the time interval (0 to t] when the system is at regenerative state Si at time t = 0. The general formula for Vi (t) is given by

Vi (t) = E Qj)(t)®[aj + Vi (t)] (46)

j

Where Qj (t) is the probability of system transition from the regenerative state i to regenerative j and aj = 1 if the repairman starts new job at regenerative state j, otherwise aj = 0. Using Equation 46 the following recursive relations are obtained:

V0(t) = Q01 (t)®[1 + V1 (t)] + Q02(t)®[1 + V2(t)] (47)

V1 (t) = Q10 (t)©V0 (t) + Q13(t)®Vj (t) + Q15(t)®V5(t) (48)

V2 (t) = Q20 (t)®V0 (t) + Q24(t)®V4 (t) + Q26(t)®V6(t) (49)

V3 (t) = Q32 (t)®V2 (t) + Q38(t)®V8 (t) (50)

V4 (t) = Q41 (t)®V1 (t) + Q48(t)®V8 (t) (51)

V5 (t) = Q51(t)®V1 (t) + Q53)(t)®V3 (t) + Q54)(t)®V4 (t) (52)

V6 (t) = Q62(t)®V2 (t) + Q63)(t)®V3 (t) + Q64)(t)®V4 (t) (53)

V8 (t) = Q83 (t)®V3 (t) + Q84(t)®V4 (t) (54)

Taking Laplace Stieltjes Transformations of the equations (47-54) to get V0** (s). we have

V0 (t) = lim(sV0** (s)) (55)

The expected no. of repairs by repairman are given by

V0 = lim(V0(t)) = N4/D1

where

N4 = (1 - p15p51)(p38p83p26p62 - p24p32p83p48 + p84p48p26p62 - p6Ïp26p32p83p48 - p26p62 - p83p38 - p84p48 + 1) + (1 - p26p62)(-p84p38p13p41 + P54)P15p41 p38p83 - p84p15p41 p38p53)

- p54)p15p41) + p26p32p63) (p54)p15p41 + p15p51 + p84p48 - p84p48p15p51 - 1) - p^ p15p41 p26p32p5ß

(7 - p (674

and D1 is already specified in equation 24.

(7) (7)

- p64) p26p32p13p41 - p24p32p53) p15p41 - p24p32p13p41 (56)

IV. Profit Analysis

The expected total profit per unit time incurred to the system in steady state is given by

Po = Co A? + Ci AR - C2 Bo - C3V0

(57)

Where

C0 = revenue per unit up time at full capacity. Ci = revenue per unit up time at reduced capacity. C2 = cost per unit time when repairman is busy. C3 = cost per repair.

V. Particular Cases

For evaluation of above described various system performance measures and their graphical representation, the following particular cases are considered, where distribution of repair times has been taken as exponential. Let us assume that g1(t) = a1e-a1f,g2(t) = a2e-a2f,g3(t) = a3e-a3t and remaining distributions same as in general case. Therefore we have

P01 P15 P32 P51

Ai

(7) r

P83 =a H2

Ai + A2 '

: A3 A2 + A3 + ai ai

ai + A3 '

a3 a3 + A2 ' Ai

a3 + Ai

i

He =

Ai + A3 + ai ' i

Ai + a3'

P02 P20 P38

P57

(7)

Pé4

A2

Ai + A2 '

a2

Ai + A3 + a2' , A3 A3 + ai' , A2 A2 + a3 ' Ai

a3 + Ai

]

P84 =b, H3 H8

i

A3 + ai' i

ai

pio A2 + A3 + ai ' Pi3

P24 = Ai Ai + A3 + a2' P26

P4i a2 a2 + A3 ' P48

(7) A2 =a|-r-1, «3 + A2 (7) p54)

P62 _ a3 a3 + Ai' P67

Ho i Ai + A2' Hi

H4 i A3 + a2 ' H5

A2

A2 + A3 + ai ' = A3 Ai + A3 + a2 ' _ A3 A3 + a2 '

a3 + A2 Ai

]

Ai + a3 ' i

A2 + A3 + ai ' i

A2 + a3 '

a3

Estimation of Parameters

The various parameters regarding failure and repair rates involved in our studies are estimated as follows in table 2

Table 2: Faz'Zure & repair rates

Various rates

corresponding values

Failure rate of STG 1 (A1) 0.00043/hr

Failure rate of STG 2 (a2) 0.00043/hr

Failure rate of gridline (A3) 0.0067/hr

Repair rate of STG 1 (a1) 0.0065/hr

Repair rate of STG 2 (a2) 0.0063/hr

Repair rate of gridline (a3) 0.34/hr

The various costs/revenue amounts involved in our studies are assumed hypothetically. The computed values of Various reliability measures for system performance are given in table 3.

Table 3: Evaluation of various system effectiveness measures

Mean time to system failure 39140 hrs.

Availability of the system at full capacity 0.50093/hr

Availability of the system at reduced capacity 0.001044/hr

Busy period of repairman for repair time only 0.12964/hr

Expected no. of repairs 0.000350/hr

VI. Results and Discussion

Figure 2: MTSF vs )Failure rate of STG

PROFIT VS COST PER UNIT UP TIME OF THE SYSTLU(Co) FOR THE DIFFERENT VALUES OF FAILURE RATE OF STG 1 (¿¡) ¿-,=.00043, .i3=Q067. K Ct= fNR.2400, C2= INR.SiO

1Ü00

OOST Q] {¡NR.}----->

Figure 3: profit vs cost per unit up time of system

The figure 2, indicates that MTSF decreases as the falure rate of STG 1 increases and also gives lowering values for the greater values of failure rates of gridline. The graph in figure 3 interpreted that the profit increases with increasing the cost per unit up time of the system and decreases when failure rate of the STG 1 increases.

Table 4: Cut Pointfor profit w.r.t. Revenue per unit up-time of the system.

Failure rate of STG(/hr) Revenue per unit up time(C0) (Rs.) Profit (Rs.)

A1 = .00034 Ai = .0034 A1 = .034 C0 < or = or > 161 C0 < or = or > 380 C0 < or = or > 606. negative or zero or positive negative or zero or positive negative or zero or positive

VII. Conclusion

In this paper self electricity generating System is Studied. The graphical study reveals the negative relationship between failure rates of units of captive power plant and profit gained by the plant. Adding the working of system at reduced capacity results in increasing its availability and profit. The derived results enable us to find acceptable values of revenue per unit up time of the system (Table 4) corresponding to failure rates of units of system. By using this analysis and graphical representations one can procure various system effectiveness measures for similar electricity generation plant.

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