07 Rr410211 Reliabilty Engineering and Application to Power Systems

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Code No: RR410211 Set No. 1

IV B.Tech I Semester Regular Examinations, November 2007RELIABILTY ENGINEERING AND APPLICATION TO POWER

SYSTEMS(Electrical & Electronic Engineering)

Time: 3 hours Max Marks: 80Answer any FIVE Questions

All Questions carry equal marks⋆ ⋆ ⋆ ⋆ ⋆

1. (a) What is a random variable? Discuss briefly any two continuous distributionsapplicable in reliability engineering.

(b) A grain distributing centre receives its monthly allocation from one of fourmain supply depots, A,B,C and D. Is does not receive its quota partly fromone depot and parlty from another. It has been observed that the distributioncentre receives its supply equally frequently from all the four supply depots.What is the probability that in a particular month the centre will receive itsquota either from A or from C? [8+8]

2. (a) Explain the reliability evaluation of series and parallel configurations.

(b) Five elements (a,b,c,d and f) of a system are connected as shown in the figure2b, which also indicates the reliability of each element. Calculate system

reliability. [8+8]

Figure 2b

3. (a) Explain the terms

i. Reliability functin

ii. Hazard rate function and

iii. Mean-time-to failure and hence develop the relationship between them.

(b) Calculate the MTTF of the components from the data of life testing of 50

specimen samples given below: [8+8]

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Time (in hours) No. of componenets working

0 502000 452040 402080 302120 202160 102200 0

4. (a) Develop the state space diagram for two unit repairable system with the prob-

abilities assigned with hazard rate functions, and hence evaluate the limitingstate probabilities of the states, if the units have same identical capacities andtransitional rates.

(b) The following stochastic transitional matrix, P show the transitional rates indiscrete Markov chains.

i. Construct the state diagram

ii. Evaluate the limiting state probabilities1 2 3

P =12

3

0.35 0.25 0.40.1 0.5 0.4

0.15 0.25 0.6

[8+8]

5. A system has 4 generating units A,B,C and D. A,B, and C have a capacity of 40MW each while D has a capacity of 80 MW. The failure and repair rates are 0.4/yearand 9.6/year obtain the state space model with equal capcity states merged. [16]

6. (a) Explain how the loss of load expectation is computed with daily peak loadexceeding the available capacity.

(b) Consider there are two generating units of 25 MW each with a forced outagerate of 0.01 failures per day repair rate of 0.49 repaires per day. The load datais:

Daily peak load in MW: 57 52 46 41 34

No. of occurences: 42 83 107 116 47

Compute the loss of load expectation. [8+8]

7. (a) Describe with state space diagram, the Markov model of a single transmissinline under two weather environment.

(b) Two transmissin lines A and B with λN =2×10−4/day is normal weather andλw=5 × 1012 in severe weather and with repair rate, µ, of 1 per day in boththe weather supply a load. The mean duration of normal weather is 10 days

and that of severe weather is 0.1 day. Calculate the probability of failure of supply to the load. [8+8]

8. Write short notes on:

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(a) Sequential addition method

(b) Reliability evaluation in radial networks(c) Uses of Reliability

(d) Markov model and its application to reliability evation. [4+4+4+4]

⋆ ⋆ ⋆ ⋆ ⋆

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1. (a) Define the terms “mutually exclusive events” and statiscally independantevents.

(b) A factory which manufactures electric bulbs has an average output of onemillion bulbs every year. Over a 5 year period, the inspection departmentrejects 15,000 bulbs.

i. What is the probability that any of the bulbs being checked by the in-spection department will be rejected?

ii. If the inspection department detects only 95% of all substandard bulbs,what is the probability that any bulb brought by a customer will be de-fective. [7+9]

2. (a) Define the term reliability of a component and discuss which factors are to be

specified in the evaluation of component reliability.(b) a system consists of four components in parallel. This system requires that

at least three of them must function. What is the probability of the systemsuccess if the component reliability is 0.8. What is probability of the systemsuccess if five componenets are placed in parallel to perform the same function.

[8+8]

3. (a) Define the term reliability of component and discuss which factors are to bespecified in the evaluation of component reliability.

(b) The light bulbs are placed under line test. The test is terminated at t=820

hours. Eight components fail before 850 hours are elapsed. Estimate thefailure rate and MTTF for the following situations.

i. components are replaced as they fail

ii. components are not replaced when fail [8+8]

4. (a) Explain two state Markov process for calculation of steady state probabilities.

(b) Explain the state space method of system reliability evaluation. [8+8]

5. A generating station has three generators, two rated for 10 MW and the third onerated for 20 MW. The failure and reapir rates of each unit are 0.35 failures/year and

9.65 repaires per year. Obtain the state diagram and mark the various equivalenttransitional rates of equal capacity states combined. Hence evaluate the cumulativeprobability of various combined states. [16]

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6. The parameters relating to generation capacity states and load capacity states of a system are tabulated below.

Generation System

State index, i Capacity Ci Availability Departure Rates/dayMW A(Ci) λ+(Ci) λ-(Ci)

1 0 0.0000026 0.104 02 50 0.0002453 0.078 0.00113 100 0.0088474 0.052 0.00224 150 0.1415578 0.026 0.00335 200 0.8493466 0.000 0.0044

Load System

State index, j Capacity Lj Availability Departure Rates/dayMW A(Lj) λ+Lj λ-Lj

1 150 0.1 0 22 140 0.25 0 23 120 0.1 0 24 100 0.05 0 25 0 0.5 2 0

Compute

(a) The probability of failure of the system.

(b) The frequency of failure of the system. [16]

7. (a) A load is served by two independent transmission lines A and B under twoweather environment. Draw the state diagram and explain how the probabilityfailure of power supply to the load can be calculated.

(b) Discuss the various performance indices that are used for the composite systemreliability analysis. [8+8]


(a) Decomposition method

(b) Bernoulli’s trial

(c) STPM

(d) Load and energy indices [4+4+4+4]

⋆ ⋆ ⋆ ⋆ ⋆

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1. (a) With the help of suitable examples distinguish between

i. Complementary event and mutualy exclusive event.

ii. Random event and certain event.iii. Impossible event and certain event.

(b) Explain the various conditions to be satisfied by the probability density func-tion with examples. [9+7]

2. (a) The reliability of network of a system is as shown in the figure 2a. The figuresmarked indicate the reliabilities of the components. Calculate the reliabilityof the system by network reduction.

Figure 2a

(b) Explain how network reliability is evaluated for series and parallel configura-tions knowing the probabilities of the components. [8+8]

3. (a) What is the MTTR? What is the median time to repair?

(b) The time to repair a power generator is best described by the following PDF,h(t)=t2/333, 1≤ t ≤ 10 hr. Determine the probabilities that a repair will beconmpleted in 6 hr.

(c) A computer system has a constant failure rate of 0.1 per day and a constantrepair rate of 1 per day. Determine the probability that a failure and repaircycle will be less than 1 day. [4+6+6]

4. (a) Explain the various types of states in Markov chains based on their commu-nication.

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(b) Consider a Markov chain, with two states, having the one step transition

matrix of P = 0.6 0.4

0.3 0.7

(a) Construct the state space diagram

(b) Evaluate the limiting state probabilities. [8+8]

5. A generating station has three generators, two rated for 10 MW and the third onerated for 20 MW. The failure and reapir rates of each unit are 0.35 failures/year and9.65 repaires per year. Obtain the state diagram and mark the various equivalenttransitional rates of equal capacity states combined. Hence evaluate the cumulativeprobability of various combined states. [16]

6. (a) Explain the recursive relation to be used for finding the cumulative probabilitywhen the unit is remove from the system

(b) A generating station consists 4 units each having a forced outage rate of 0.15and capacity of each unit is 15 MW. Develop the cumulative capacity outageprobability table. Now if one unit is removed, devleop the modified table usingrecursive relation. [8+8]

7. (a) Explain the weighted average rate model for considering weather effects ontransmission lines for reliability analysis.

(b) Discuss the various load point reliability indices that are used for radial dis-tribution networks. [8+8]


(a) Bath-Tub curve

(b) State space diagrams

(c) Stand deviation of binomial distribution

(d) Cumulative frequency failure evaluation. [4+4+4+4]

⋆ ⋆ ⋆ ⋆ ⋆

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1. (a) Show that under certain conditions a binomial distribution can be approxi-mated by a Poissions distribution.

(b) An electric motor consists of two parts, namely stator and rotor. They aremanufactured in two different sections and then assembled together. Theprobability that a stator is defective is given as 0.06 and that a rotar is defectiveis given as 0.09. What is the probability that an assembled motor will not bedeffective? [8+8]

2. (a) Distinguish with block diagrams parallel series and mixed parallel series sys-tems and write down apropriate formulae in each case.

(b) Assume that six units can bearranged in three series and parallel configura-tions. Draw their block diagram of arrangement and estimate reliability of the

system if each has reliability of 0.85. [8+8]

3. (a) Explain the following terms:

i. Reliability

ii. MTTF

iii. MTTR

iv. MTBF

(b) Show that for exponential distribution, MTTF is the reciprocal of failure rate.

(c) A component has MTTF of 950 hours. what is its reliability for a mission

time of 100 hours. [6+5+5]4. (a) Explain two state Markov process for calculation of steady state probabilities.

(b) Explain the state space method of system reliability evaluation. [8+8]

5. A generation system consists of 2 units of 30 MW capacity each λ=0.04/year andµ=0.96/year. The daily peak loads observed are found to be as follows.

Daily peak load (MW) % time the peak has occuered50 3040 40

30 2020 10

Estimate loss load probability. [16]

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6. (a) Explain the method of calculating LOLP of a generatin system. What are theinadequacies of LOLP as an index of unreliability.

(b) A power system containes three 40MW and one 60MW capacity unit eachhaving a forced outage rate of 0.02. The annual daily peak load variationcurve is a straight line from 100% to 40% points. Estimate LOLE for a peakload of 200 MW. [8+8]

7. (a) A load is served by two independent transmission lines A and B under twoweather environment. Draw the state diagram and explain how the probabilityfailure of power supply to the load can be calculated.

(b) Discuss the various performance indices that are used for the composite systemreliability analysis. [8+8]


(a) Decomposition method

(b) Bernoulli’s trial

(c) STPM

(d) Load and energy indices [4+4+4+4]

⋆ ⋆ ⋆ ⋆ ⋆

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07 Rr410211 Reliabilty Engineering and Application to Power Systems