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Uncertainty analysis and risk assessment Chapter | 3 73

The intermediate optical repeaters required for reliable communication
among the end devices depend on the geographical distance between them.
Usually, repeaters are placed after every 100 150 km so as to maintain the
quality of the signal [12].

3.2.2 Monte Carlo simulation models for phasor measurement unit
and their communication networks

Reliability evaluation methods, such as Markov models [19], fault trees [20],
reliability block diagrams [21], and Petri nets [22], assume constant failure
rates and constant repair rates for obtaining the probability of failure.
Different versions of the same components may have different failure and
repair rates, and thus Monte Carlo simulation (MCS) is often used to obtain
a better estimate of the failure probability [23]. It is a probabilistic method
that uses random values of TTR and TTF for the various components of the
system. These random values are generated, and the entire cycle of real-time
events is simulated in real time. Several such cycles are iterated to obtain the
failure distribution for the various components of the system. Using this
approach, one can effortlessly model different component interconnections
and can simulate simultaneous component failures. The random values for
TTF and TTR used by the MCS are given by Eqs. (3.9) and (3.10). These
values are based on the nominal values of the component’s failure rate and
repair rate [16].

1
ðÞ
TTF 52 ln θ 1 ð3:9Þ
λ
1
ðÞ
TTR 52 ln θ 2 ð3:10Þ
μ
where θ 1 and θ 2 are random numbers between 0 and 1.
The modules M 2 , M 3 , M 4 , M 5 , M 6 , and M 7 of the PMU and the modules
C PMU , C PDC , and C 3 of the optical fiber SPCS are two component modules,
with the components connected in standby mode. The MCS model for these
modules is shown in Fig. 3.9.
The modules fail when the standby component fails before the main com-
ponent is repaired and is indicated by the checkered line (system downtime
is the duration for which the system is unavailable) in Fig. 3.9. The variables
MT and CT stand for mission time and current time, respectively. In order to
generate sufficient failure data the MT taken should be sufficiently large. All
the simulations performed in this work consider a mission time of 200 mil-
lion years. Each simulation is run for 1000 iterations (indicated by the vari-
able N) in order to obtain consistent failure data. In every iteration, initially
the values of TTF, TTR, and CT are taken to be zero. Random values of fail-
ure times (TTF 1 and TTF 2 ) and repair times (TTR 1 and TTR 2 ) are generated