Computational analysis of parameters affecting economy of one gas and one steam turbine system with scheduled inspection

Dalip Singh¹, Ashok Kumar Saini^2*

Department of Mathematics, M. D. University, Rohtak, INDIA.

Department of Mathematics, BLJS College, Tosham, Bhiwani, Hariyana, INDIA.

Email: [email protected], [email protected]

Research Article

 

Abstract               Introduction: A model considering variation in demand and power production capacity has been developed for a system comprising one gas and one stream turbine. Concept of scheduled inspection at regular intervals of time for maintenance has also been introduced. Initially both the unit i.e. the gas turbine as well as the stream turbine are operating. On failure of the gas turbine, system goes to down state, whereas on failure of the stream turbine, the system may be kept in up state with only gas turbine working or put to down state according as the buyer of the power so generated is ready to pay higher amount or not. Computational work has been done for the cost benefit analysis and interesting numerical results have been obtained regarding various parameters involved which affect the economy of the system. Semi- Markov process and regenerative point technique have been used to analyze the system.

Keyword: turbine system, steep high temperature.

 

INTRODUCTION

A large number of researcher in the field of reliability have widely discussed and analysed two unit systems. Contributors for the analysis of reliability models for systems with two similar units include Tuteja et al. (1991), Rizwan et. al. (2010) and Mathew et al. (2011). Systems with two dissimilar units have also been analyzed by numerous researchers including Baohe (1997) and Taneja et. al. (2011). In most of the studies on two dissimilar units, one unit was taken as operative and other as standby. Both the dissimilar units have also been taken as operative simultaneously in some of the studies. Two units for the system discussed by them were totally dissimilar i.e. their nature was different. However, there may be practical situations where the two units are dissimilar but the nature of the work done by them is same; and failure in either of the units affects the working of the other. Such a situation was observed by the author on visiting gas turbine plants and hence a model considering variation in demand and power production capacity has been developed by Singh and Taneja (2013) wherein a power generating system comprising one gas and one steam turbine has been taken into consideration. Initially, both the gas turbine as well as the steam turbine are operative. On failure of the Gas turbine, the system goes to down state as steam turbine cannot work in this case; whereas on failure of the steam turbine, it may be kept in upstate with only gas turbine working or put to down state according as the buyer of the power so generated is ready to make higher payment or not to compensate the heavy losses. The concept of scheduled inspection is also incorporated as the same was observed while gathering information from gas turbine plant during the visit. The scheduled inspection is done at regular intervals of time for maintenance and is of three types — Minor, Path and Major Inspection. Though Singh and Taneja (2013) obtained various measures of system effectiveness, yet no computational work has been done by them for the analysis. The present paper deals with the same model but with computational work do draw interesting conclusions regarding various parameters which affect the economy of the system.

 

NOTATIONS

O_gt: Gas turbine operative

O_{gt :} Gas turbine operative after 1^st scheduled inspection/maintenance

O_gt2: Gas turbine operative after 2^nd scheduled inspection/maintenance

O_st: Steam turbine operative

O_st1: Steam turbine operative after 1^st scheduled inspection/maintenance

O_st2: Steam turbine operative after 2^nd scheduled inspection/maintenance

U_rgt: Gas turbine under repair

U_rst: Steam turbine under repair

U_Rst: Repair of the steam turbine is continuing from previous state

d_gt: Gas turbine put to down mode

 d_st: Steam turbine put to down mode

W_rgt: Gas turbine waiting for repair

W_rst: Steam turbine waiting for repair

Insp₁: First type of inspection (Minor inspection)

Insp₂: Second type of inspection (Path inspection)

Insp₃: Third type of inspection (Major inspection)

λ: Failure rate of gas turbine

α: Failure rate of steam turbine

p: Probability that there is dire demand of electricity and the customer is ready to make higher payments.

q: 1-p i.e the probability that the customer is not ready to make the payment higher than the normal rates.

g₁(t), G₁(t) : pdf and cdf of repair time of gas turbine

g₂(t), G₂(t) : pdf and cdf of repair time of steam turbine

β₁: Rate of requirement of scheduled inspection/ Maintenance

γ₁: Rate of doing minor inspection/ maintenance

γ₂: Rate of doing path inspection/maintenance

γ₃: Rate of doing major inspection/maintenance

The possible states of transition for Model will be as:

 

The possible transitions are as:

0 to 1, 0 to 2, 0 to 3, o to 5, 1 to 0, 2 to 0, 2 to 4, 2 to 1 via 4, 3 to 0, 5 to 6, 6 to 8, 6 to 9, 6 to 11, 6 to 9, 8      to 6, 8 to 10, 8 to 7 via 10, 9 to 6, 11 to 12, 12 to 14, 12 to 15, 12 to 17, 13 to 12, 14      to 12, 14 to 16, 14 to 13 via 16, 15 to 12, 17 to 0.

The epochs of entry into states 0,1,2,3,5,6,7,8,9,11,12,13,14 and 15 are regeneration points and thus 0,1,2,3,4,5,6,7,8,9,11,12,13,14 and 15 are called regenerative states. States 4, 10, and 16 are failed states. States 0,6 and12 are up states. States 2, 8, and 14 are up states in single cycle. States 5, 11, and 17 are down states due to inspection; and 1, 3, 7, 9, 13, and 15 are also down states as these are non-working states even though the steam turbine/gas turbine is operable.

TRANSITION PROBABILITIES

Mean Sojourn times:

Availability at Full Capacity in steady state

Availability in Single Cycle

Expected Down Time Excluding Failed State

Expected Time for Minor Inspection

               

Expected Time for Path Inspection

Expected Time for Major Inspection

Busy period Analysis for doing Repair

 

Expected Number of Visits of the Repairman

where

 

Cost-Benefit Analysis

Expected profit incurred to the system is the excess of revenue over cost and in steady state is given by

C₃₀ = Revenue per unit uptime with full capacity.

C_31S = Revenue per unit uptime in single cycle

C₃₂ = Loss per unit time for which the system is in down state (other than failed state)

C₃₃ = Cost per unit time for doing minor inspection/maintenance.

C₃₄ = Cost per unit time for doing path inspection/maintenance.

C₃₅ = Cost per unit time for doing major inspection/maintenance.

C₃₆ = Cost per unit time for engaging the repairman for doing repair.

C₃₇ = Cost per visit of the repairman.

COMPUTATIONAL ANALYSIS

The following particular case is considered for numerical calculations where all the distributions of times have been taken as exponential. One may, however, take that distribution which will be best fitted to the actual data. The goodness-of-fit tests given in Chapter 1 may be applied to find which one of the distributions can be fitted to the given data.

Using the estimated v­alues on the basis of gathered information and assumed values for other parameters, i.e.,

, p = 0.5;

the values of various measures of system effectiveness are obtained as:

Mean time to system failure = 277567300 hrs

Availability at full capacity (A₀) = 0.969129300

Graphical study has been made for the MTSF, availability at full capacity, availability in single cycle and the profit with respect to failure rate of steam turbine (a), revenue per unit uptime in case of working at full Capacity (C₃₀), revenue per unit uptime in single Cycle (C₃₁) for different values of probability of demand on higher payment (p) and for different values of loss during down time (C₃₂). Fig. 2 shows the behaviour of MTSF w.r.t. failure rate (a) of steam turbine for different values of probability of demand on higher payment (p). It is clear from the graph that MTSF gets decreased with the increase in the values of the failure rate (a) of steam turbine. It has higher values for lower values of probability of demand on higher payment (p).

 

Figure 2:

 

Fig. 3 reveals the behaviour of availability at full Capacity (A₀) w. r. t. failure rate (a) of steam turbine. It can be seen from the graph that availability decreases with the increase in the values of failure rate (a) of steam turbine with negligible change in availability (A₀) for different values of probability of demand on higher payment (p).

 

Figure 3:

 

Figure 4:

 

Fig. 4 depicts the behaviour of availability (AS₀) in single cycle w.r.t. failure rate (a)of steam turbine for different values of probability (p). It is clear from the graph that the availability (AS₀) increases with increase in the values of failure rate (a) of steam turbine Also, it has higher values for higher values of the probability of demand on higher payment (p). Fig. 5 shows the behaviour of the availabilities A₀, AS₀ w.r.t. failure rate (a) of steam turbine. It can be observed from the graph that the availability (A₀) decreases as failure rate (a) increaseswhereas the availability AS₀ increases with increase in the values of failure rate (a). It can also be observed that A₀ is greater than or less than AS₀according to whether failure rate (a) is lesser or greater than 0.080046053.

 Fig. 6 shows the behaviour of profit w.r.t. to failure rate

 

Figure 5:

 

(a) of steam turbine for different values of probability (p). At the initial stages when the values of probability (p) are small, the profit decreases as failure rate (a) of steam turbine increases. But after certain probability (p) level, there comes a stage where profit starts increasing with respect to failure rate (a) of steam turbine. In any case, it has higher values for higher values of probability (p).

 

 

Figure 6:

 

Fig. 7depicts the behaviour of the profit w.r.t. revenue per unit uptime when the system works at full Capacity (C₃₀) for different values of loss during down time (C₃₂). It can be interpreted that the profit increases with increase in the values of C₃₀. Following can also be concluded from the graph:

• For C₃₂ = 400000, profit is positive or zero or negative according as C₃₀> or = or < 12333.00 i.e. the price per unit of the electricity should be fixed in such a way so as to give C₃₀not less than 12333.00 to get positive profit.
• For C₃₂ = 450000, profit is positive or zero or negative according as C₃₀> or = or < 13909.03the price per unit of the electricity should be fixed in such a way so as to give C₃₀ not less than 13909.03 to get positive profit.
• For C₃₂ = 500000, profit is positive or zero or negative according as C₃₀> or = or < 15485.06 i.e. the price per unit of the electricity should be fixed in such a way so as to give C₃₀ not less than 15485.06 to get positive profit.

 

Figure 7:

 

Fig. 8 depicts the behaviour of profit w.r.t. revenue per unit uptime in single cycle (C_31S) for different values of loss during down time(C₃₂). It can be noticed that the profit increases with increase in the values of C_31S. Following observations can also be made from the graph:

• For C₃₂ = 633340, profit is positive or zero or negative according as C_31S> or = or < 1362.86 i.e. the price per unit of the electricity produced in single cycle should be fixed in such a way so as to give C_31S not less than 1362.86 to get positive profit.
• For C₃₂ = 633390, profit is positive or zero or negative according as C_31S> or = or < 7669.31 i.e. the price per unit of the electricity produced in single cycle should be fixed in such a way so as to give C_31S not less than 7669.31 to get positive profit.
• For C₃₂ = 633440, profit is positive or zero or negative according as C_31S> or = or <13975.76 i.e. the price per unit of the electricity produced in single cycle should be fixed in such a way so as to give C_31S not less than 13975.76 to get positive profit.

 

Figure 8:

 

REFERENCES

1. Su Baohe, On a two-dissimilar unit system with three modes and random check, Microelectronics Reliab., 37(1997), 1233-1238.
2. A.G. Mathew, S.M. Rizwan, M.C. Majumder, K.P. Ramachandran and G.T aneja, Reliability analysis of an identical two-unit parallel CC plant system operative with full installed capacity, International Journal of Performability Engineering, 7(2011), 179-185.
3. S.M. Rizwan, V. Khurana and G. Taneja, Reliability analysis of a hot standby industrial system, International Journal of modeling and Simulation, 30 (2010), 315-322.
4. G.Taneja, R. Narang and U.Sharma, Reliability and economic analysis of a system comprising booster and main pumps working in an oil refinery plant, Caledonian Journal of Engineering, 7 (2011),28-32.
5. R.K. Tuteja, R.T. AroraT and G. Taneja, Analysis of two-unit system with partial failures and three types of repair, Reliability Engineering and System Safety, 33(1991),199-214.
6. Singh Dalip and Gulshan Taneja, “Reliability analysis of a power generating system through gas and steam turbines with scheduled inspection,” Aryabhatta Journal of Mathamatics and Informatics, 5(2013), 373-380.

 

 

 

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