Question

A system consists of two independent, identical generators of 50MW each. Each generator has a failure rate of 0.02 per day and mean repair time of 8 hours. These generators are supplying a load which varies as follows: 1 to 6 hours: the load is constant at 25MW 7 to 10 hours: the load is constant at 50MW 11 to 24 hours: the load is constant at 85MW Assume that at time 0 both generators are up. a. Find the probability of loss of load (i.e. load not being fully supplied) at hour 8 .

Answer 1

To find the probability of loss of load at hour 8, we need to consider the failure and repair times of the generators along with the load requirements. The probability of loss of load at hour 8 is 0.986407.

Step-by-step explanation:

To find the probability of loss of load at hour 8, we need to consider the failure and repair times of the generators along with the load requirements.

Step 1:

Calculate the probability that at least one generator fails within the time period of 0 to 8 hours.

Probability of failure for each generator = 0.02

Probability of success for each generator = 1 - probability of failure = 1 - 0.02 = 0.98

Probability that at least one generator fails = 1 - (probability of both generators working) = 1 - (0.98 * 0.98) = 0.0396

Step 2:

Calculate the probability that the load is not fully supplied during the time period of 0 to 8 hours.

The load requirement during this period is 25MW.

Probability of the load not being fully supplied = probability that at least one generator fails * probability that the total power supplied by the working generator is less than 25MW

Assuming the generators are independent, we can multiply their probabilities together.

Probability that the total power supplied by a working generator is less than 25MW = probability of the power being less than 25MW for one generator = probability of failure for one generator * probability of the repair time being less than 8 hours = 0.02 * (8/24) = 0.00666

Probability of the load not being fully supplied = 0.0396 * 0.00666 = 0.000263

Step 3:

Calculate the probability of loss of load at hour 8.

The load requirement at hour 8 is 50MW.

Probability of the load being less than 50MW = probability of the load not being fully supplied + probability that the total power supplied by one working generator is between 25MW and 50MW

Probability that the total power supplied by one working generator is between 25MW and 50MW = probability of the power being between 25MW and 50MW for one generator = probability of the failure for one generator * probability of the repair time being between 8 hours and 16 hours = 0.02 * (16/24) = 0.01333

Probability of the load being less than 50MW = 0.000263 + 0.01333 = 0.013593

Probability of loss of load at hour 8 = 1 - probability of the load being less than 50MW = 1 - 0.013593 = 0.986407