Page 102 - Decision Making Applications in Modern Power Systems

Uncertainty analysis and risk assessment Chapter | 3 69

TTF 1 TTR 1 TTF 2 TTR 2 TTF 3

T
First cycle Second cycle k Cycles
FIGURE 3.5 Operation cycle of a repairable system.

For a system having a constant failure rate, the reliability is given by the
following equation:

2λt
RðtÞ 5 e ð3:4Þ
Unavailability or the probability of failure is another parameter that is
often used as a measure of system’s reliability. Unavailability is the steady-
state probability that the given system does not deliver the desired function-
ality. For a repairable system whose operation cycle is shown in Fig. 3.5, the
mean time to failure (TTF) (MTTF), mean time to repair (TTR) (MTTR),
the availability (A), and the unavailability or the probability of failure (U)
are given by the following equations:
k
P
TTF i
1 i51
MTTF 5 5 ð3:5Þ
λ k
k
P
TTR i
1 i51
MTTR 5 5 ð3:6Þ
μ k
μ
A 5 ð3:7Þ
μ 1 λ
λ
U 5 ð3:8Þ
μ 1 λ
where μ is the repair rate of the component, and λ is the failure rate of the
component.
An engineering system is a combination of several components that are
interconnected with one another. These components may be connected in
series, in parallel, or in standby. The components in a group are said to be in
series if the system fails due to the failure of any one of the constituent com-
ponents. The components are said to be connected in parallel if the group
fails only when all the components fail. Two components are connected in
standby if upon the failure of the first component, the second component
comes into operation. These three connections are shown in Fig. 3.6.