© 2007 Science Publications Reliability Parameters of a Power Generating System with Shared Load 1

This study presents a model, based on power generating system with shared load. The whole generating system consists of three subsystems viz: subsystem A , subsystem B and subsystem C. The subsystem A consists of one generating unit and one inbuilt transformer. The subsystem B also contains the same units and is connected in parallel to subsystem A. The output of this power system goes through the subsystem C that consists of one outer transformer and which may be further distributed as desired. The system has three types of states, viz: normal, degraded and failed. All types of failure rates and repair rates of inbuilt transformers are exponential while other repair rates are distributed quite generally. Supplementary Variable Technique has been employed to obtain various state probabilities and then the reliability parameters have been evaluated for the whole generating system.


INRODUCTION
Electric energy demand has been rapidly increasing all over the world. This is attributed to greater industrialization and large-scale use of electric energy for agricultural purpose. The demand is likely to increase exponentially for many more decades to come. There are no signs of saturation in the foreseeable future. Electric supply authorities are likely to pay more attention to improve the utilization of generating equipment. The reliability of electric supply in India is very low. The public is likely to become more and more conscious of its rights to get uninterrupted supply at proper voltage. This would force the electric supply undertakings to analyze the system and take corrective measures to improve reliability.
Keeping these points in view, the author has considered a mathematical model by which the reliability of the generating system can be improved. The whole generating system consists of three subsystems viz, subsystem A, subsystem B and subsystem C. The subsystem A consists of one generating unit and one inbuilt transformer. The subsystem B, arranged in parallel with subsystem A, is a redundant system and also consists of one generating unit and one inbuilt transformer. The output of these two subsystems goes through the subsystem C that consists of one outer transformer and the electric supply may be further distributed from this subsystem C as desired. The power required at subsystem C is shared by two subsystems, A and B which will increase the availability of power in comparison to that which is, instead, produced by a single subsystem A. Supplementary Variable technique has been employed to evaluate various reliability parameters of the generating system. The mathematical model of the whole system is shown in the state transition diagram.

ASSUMPTIONS
At time t = 0, the system is in operable state. The subsystem B works as the redundant unit. All the failure rates and repair rates of unit A2 and unit B2 (inbuilt transformers) are exponential while other repair rates are distributed quite generally. The failure rate of all units is distinct. After the complete breakdown of the system, the repair rate is assumed as same. The system is in degraded state after the failure of either unit of subsystem A or subsystem B, or completely subsystem A or subsystem B. All the units recover their functioning perfectly after repair.
Formulation of the model: The probabilities given above are mutually exclusive and provide the complete markovian characteristic of the process. Therefore, using continuity arguments and elementary probability considerations, one get the following difference differential equations governing the stochastic behaviour of the complex system, which is discrete in space and continuous in time:

OPERATIONAL AVAILABILITY AND NON AVAILABILITY
The Laplace Transform of the probabilities that the system is in operable and down state at time t, are given as follows: provided that the limit on the RHS exists, the time independent up and down state probabilities are obtained as follows:  g (s) g (s) g (s) g (s) g (s) g (s) g (s) (s )

OPERATIONAL AVAILABILITY AND NON AVAILABILITY
The Laplace Transform of the probabilities that the system is in operable and down state at time t, are given as follows:  Table 1 and Fig. 1. Reliability analysis: Setting a = 0.001, b = 0.002, 1 = 0.011, 2 = 0.015, = 0.05 in the Eq. 72 one may obtain the variations in reliability of the system with time as shown in Table 2 and Fig. 2.

MTTF analysis:
Setting the values 2 = 0.02, 1 = 0.01, b = 0.002 and taking different values of a in the Eq. 73 one may obtain the variations of MTTF of the system against the failure rate of unit A 1 , ( a ) as shown in Table 3 and   obtain the variations of MTTF of the system against the failure rate of unit A 1 ( a ) as shown in Table 5 and Fig. 5. Fig. 7: STATE TRANSITION DIAGRAM Setting the values 1 = 0.01, a = 0.001, b = 0.002 and taking different values of in the Eq. 73 one may obtain the variations of MTTF of the system against the failure rate of unit B 2 ( 2 ) as shown in Table 6 and Fig. 6.

CONCLUSION
The concept of redundancy of the generating unit can be applied to the areas where the electricity requirements are increasing at an alarming rate. The findings in Table 1 and 2 depicts that the system is available and reliable for a longer time period. The failures are considered to occur purely by chance for a component which is operating within its useful life period. So, under these conditions the calculations shown in Table 3-6 shows apparently that the failure rate is the reciprocal of the mean time to system failure (MTTF).

ACKNOWLEDGMENT
Authors are grateful to Rear Admiral Dr. Ajay Sharma (Director General) and Prof. G.S. Sandhu (Director), of Krishna Institute of Engineering & Technology, Ghaziabad (India) for their constant encouragement, and providing necessary facilities (including financial support) to carry out the present work.

NOTATIONS
i, : Failure and constant repair rate of unit A 2 or B 2 (i = 1, 2). j , 1 (x): Failure and repair rate of unit A 2 or B 1 (j = a, b). : Failure rate of subsystem C. (z): Repair rate when the system is in failed state.

(y):
Repair rate of both units of subsystem A or subsystem B. P 1 (t): Probability that the system is in operable state at time t. P i (x, t) : Probability that the system is in degraded state at time t and elapsed repair time lies between x and x + , where 9i = 2, 3) P j (y, t) : Probability that the system is in degraded state at time t and elapsed repair time lies between y and y + , where (j = 6, 12) P k (z, t) : Probability that the system is in failed state at time t and elapsed repair time lies between z and z + , where (k = 4, 7, 8, 9, 10,l 13, 14, 15, 16, 17) P m (t): Probability that the system is in degraded state at time t where (m = 5, 11) 0 , Otherwise stated.