 # Availability Analysis of Transmission and Associated Systems

In the area of power system, various types of systems and equipment are used for transmitting electrical energy from one point to another. Two examples of such systems and equipment are transmission lines and transformers. This section presents two mathematical models to perform availability analysis of such systems [4,7,9,11].

8.6.1 Model I

This mathematical model represents a system composed of two non-identical and redundant transmission lines subject to the occurrence of common-cause failures. A common-cause failure may simply be described as any instance where multiple units fail due to a single cause [7,13,14]. In transmission lines, a common cause failure may take place due to factors such as poor weather, aircraft crash, and tornado. The system state-space diagram is shown in Figure 8.4.

The numerals in the circles and boxes denote system states. The model is subjected to the following assumptions:

• • All failure and repair rates of transmission lines are constant.
• • A repaired transmission line is as good as new.
• • All failures are statistically independent.

The following symbols are associated with the state-space diagram shown in Figure 8.4 and its associated equations:

P:(t) is the probability that the system is in state ) at time t; for ) = 0 (both transmission lines operating normally), ) = 1 (transmission line 1 failed, other operating),) = 2 (transmission line 2 failed, other operating),) = 3 (both transmission lines failed).

A,! is the transmission line 1 failure rate.

2 is the transmission line 2 failure rate. FIGURE 8.4 State-space diagram for two non-identical and redundant transmission lines.

Лс is the system common-cause failure rate. цп is the transmission line 1 repair rate. ца is the transmission line 2 repair rate.

Using the Markov method presented in Chapter 4, we write down the following equations for Figure 8.4 state-space diagram [11,13,15]: The following steady-state equations are obtained from Equations (8.44)-(8.47) by setting the derivatives with respect to time t equal to zero and using the

з

relationship У P, = 1:

3=0 where where  where P0,Pl,P2,and P2 are the steady state probabilities of the system being in states 0, 1, 2, and 3, respectively.

The system steady state availability and unavailability are given by and where

A Vss is the system steady state availability.

UAVSS is

8.6.2 Model II

This mathematical model represents a system composed of transmission lines and other equipment operating in fluctuating outdoor environments (i.e., normal and stormy). The system can malfunction under both these conditions. The system state- space diagram is shown in Figure 8.5. The numerals in circles and boxes denote system states.

The model is subjected to the following assumptions:

• • All failure, repair, and weather fluctuation rates are constant.
• • The repaired system is as good as new.
• • All failures are statistically independent.

The following symbols are associated with the state-space diagram shown in Figure 8.4 and its associated equations:

P, (r) is the probability that the system is in state / at time ?; for / = 0 (operating normally in normal weather), / = 1 (failed in normal weather), / = 2 (operating normally in stormy weather), / = 3 (failed in stormy weather).

Amv is the system constant failure rate in normal weather.

Aw is the system constant failure rate in stormy weather. цт is the system constant repair rate in normal weather. FIGURE 8.5 State-space diagram of a system operating in fluctuating outdoor environments.

цж is the system constant repair rate in stormy weather. у is the constant transition rate from normal weather to stormy weather. в is the constant transition rate from stormy weather to normal weather.

Using the Markov method presented in Chapter 4, we write down the following equations for Figure 8.4 state-space diagram [l 1,15]: The following steady-state equations are obtained from Equations (8.60)-(8-63) by setting the derivatives with respect to time t equal to zero and using the

з

relationship V' Py = 1:

j where P0,Pl,P2,and P2 are the steady state probabilities of the system being in states 0, 1, 2, and 3, respectively.

The system steady state availability and unavailability are given by and where

AVSS and UAVSS are the system steady state availability and unavailability, respectively.

# Problems

• 1. Define the following terms:
• • Forced outage
• • Forced outage rate
• • Power system reliability
• 2. Write an essay on power system reliability.
• 3. Define the following indices:
• • Average service availability index
• • Customer average interruption duration index
• • System average interruption frequency index
• 4. Describe loss of load probability (LOLP).
• 5. What are the difficulties associated with the use of LOLP?
• 6. Assume that the annual failure rate of the electricity supply is 0.9 and the mean time to electricity interruption is 4 hours. Calculate the mean number of annual down hours (i.e., service outage hours) per customer.
• 7. Assume that constant failure and repair rates of a power generator unit are
• 0.0004 failures/hour and 0.0006 repairs/hour, respectively. Calculate the steady state availability of the power generator unit.
• 8. Assume that for a power generator unit we have the following data values: •Af = 0.0002 failures/hour
• • Apm =0.0008/hour
• = 0.0006 repairs / hour
• P pm = 0.0009 / hour

Calculate the power generator unit steady state unavailability by using Equation (8.43).

• 9. Prove that the sum of Equations (8.48), (8.53), (8.55), and (8.57) is equal to unity.
• 10. Prove Equations (8.48), (8.53), (8.55), and (8.57).

# References

• 1. Billinton, R., Allan, R.N., Reliability of Electric Power Systems: An Overview, in Handbook of Reliability Engineering, edited by H. Pham, Springer-Verlag, London, 2003, pp. 511-528.
• 2. Layman, W.J., Fundamental consideration in preparing a master system plan, Electrical World, Vol. 101. 1933, pp. 778-792.
• 3. Smith, S.A., Sendee reliability measured by probabilities of outage, Electrical World, Vol. 103, 1934. pp. 371-374.
• 4. Billinton, R., Power System Reliability Evaluation, Gordon and Breach Science Publishers, New York, 1970.
• 5. Dhillon, B.S., Power System Reliability, Safety, and Management, Ann Arbor Science, Ann Arbor, Michigan, 1983.
• 6. Bilinton, R., Bibliography on the application of probability methods in power system reliability evaluation, IEEE Transactions on Power Apparatus and Systems, Vol. 91, 1972, pp. 649-660.
• 7. Dhillon, B.S., Applied Reliability and Quality: Fundamentals, Methods, and Procedures, Springer-Verlag, London, 2007.
• 8. Kueck, J.D., Kirby, B.J., Overholt, P.N., Markel, L.C., Measurement Practices for Reliability and Power Quality, Report No. ORNL/TM-2004/91. June 2004. Available from the Oak Ridge National Laboratory, Oak Ridge, Tennessee, USA.
• 9. Endrenyi, J., Reliability Modeling in Electric Power Systems, John Wiley, New York, 1978.
• 10. Kennedy, B., Power Quality Primer, McGraw Hill. New York. 2000.
• 11. Dhillon, B.S., Reliability Engineering in Systems Design and Operation, Van Nostrand Reinhold, New York, 1983.
• 12. Gangel, M.W., Ringlee, R.J., Distribution system reliability performance, IEEE Transactions on Power Apparatus and Systems, Vol. 87, 1968, pp. 1657-1665.
• 13. Billinton, R., Medicherala, T.L.P., Sachdev, M.S., Common-cause outages in multiple circuit transmission lines, IEEE Transactions on Reliability, Vol. 27, 1978, pp. 128-131.
• 14. Gangloff, W.C., Common mode failure analysis, IEEE Transactions on Power Apparatus and Systems, Vol. 94. Feb. 1975, pp. 27-30.
• 15. Dhillon, B.S., Singh, C., Engineering Reliability: New Techniques and Applications, John Wiley, New York, 1981.