There is a confusion in defining repair rate. From Richard Brown's book, Power Distribution Reliability, it is said that repair rate is the inverse of repair duration, which is defined as the mean time needed to repair the system SINCE it fails (page 164). But in the Markov process example (page 204), he used repair rate as the rate of transition from switched state (1b to 0 and 2b to 0) to normal state, not from the fail state to normal state (1a to 0 and 2a to 0). I attached the screenshoot of his book.
Yes, I find the presentation confusing too. One problem being that the text mentions sigma2, but the figure never even shows sigma2, only sigma1.
Another problem is that from state 0, a failure of LS2 would require SW1 to be opened, but not SW2 to be closed, so this is in fact not similar to what happens with a failure in LS1. Just as you cannot restore power to LS1 until the problem in LS1 has been resolved, you equally cannot restore power to LS2 until the problem in LS2 has been resolved. You cannot resolve the LS2 problem by closing SW2.
So, closing SW2 should never be required, if there is a problem in LS2. The only reason to ever close SW2 would be if there was a problem in LS1, as they already described in the first example, or at SS1 itself.
I think that to transition from a failure in LS1 back to state 0 requires two actions: open SW2 and close SW1.
To transition from a failure in LS2 back to state 0 requires only one action: close SW1.
Now to your point.
To transition from any failure state back to a normal state, you do need to go from 1b to 0 or 2b to 0. This is because the Markov process has to factor in the time it takes to repair the line segment. They are trying to show the total disruption time, for the failed line segment, which includes the time it takes to first throw the switch or switches, the time it takes to repair the line segment, and the time it takes to throw the switch or switches again.
Albert is in right. Here you have a switching process whith transition. It menas that it is three state Markov model for sections. Maby the transition rate sigma1 is the same for both switches and the line is supplied from both side ??
Srete
Hi Johanno,
Frankly, in the material you have exposed, there is no statement about the repair rate. Basically, you are right, one should take into account the sum of the two times for passage from nb to na which (for n=1) equals to
THEAVERAGETIME1 = 1/\sigma1 + 1/\mu1,
so, the rate equals, according to the author's definition the inverse value, i.e.
THEREPAIRRATE1= \sigma1 \cdot \mu1 / ( \sigma1 + \mu1).
It is very hard to guess, why the sigmas are omitted. Suggestion - ask prof. Brown:)
Best regards.
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