Module 4-2 Methods of Quantitative Reliability Analysis

Transcription

1 Module 4-2 Methods of Quantitative Reliability Analysis Chanan Singh Texas A&M University

2 METHODS OF QUANTITATIVE RELIABILITY ANALYSIS ANALYTICAL METHODS - STATE SPACE USING MARKOV PROCESSES - NETWORK REDUCTION - MIN CUT SETS MONTE CARLO SIMULATION - NONSEQUENTAL - RANDOM SAMPLING - TIME SEQUENTIAL CONCEPT OF RELIABILITY COHERENCE

3 EXAMPLE SYSTEM Generator Transmission Load B1 B2 B B3 B4

4 EXAMPLE SYSTEM Generators: Each generator either has full capacity of 5 MW or MW when failed. Failure rate of each generator is.1/day and mean-repair-time is 12 hours Transmission Lines: The failure rate of each transmission line is assumed to be 1 f/y during the normal weather and 1 f/y during the adverse weather. The mean down time is 8 hours. Capacity of each line is 1 MW. Weather: The weather fluctuates between normal and adverse state with mean duration of normal state 2 hours and that of adverse state 6 hours. Breakers: Breakers are assumed perfectly reliable except that the pair B1&B2 or B3&B4 may not open on fault on the transmission line with probability.1. Load: Load fluctuates between two states, 14 MW and 5 MW with mean duration in each state of 8hr and 16hr respectively.

5 EXAMPLE SYSTEM Generator Transmission Load B1 B2 B B3 B4 FOR THE DESCRIBED SYSTEM, HOW CAN YOU CALCULATE THE FOLLOWING BASIC RELIABILITY INDICES? 1. Loss of load probability 2. Frequency of loss of load 3. Mean duration of loss of load

6 1. System State Description & Equivalents The first task is to obtain probabilities for the generators, transmission lines and loads, which are independent parts of the system Generators µ / year λ.1/ day 36.5/ year

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8 Merging Identical Capacity States Equivalent transition rates:

9 Transition rate matrix is: R G 3λ µ 3λ ( µ + 2λ) 2µ 2λ (2µ + λ) 3µ λ 3µ If we substitute values for µ and λ obtained in the beginning into above matrix, transition rate matrix for the generator system is: R G

10 1. 2. Transmission Lines During the normal weather During the adverse weather λ 1 / year λʹ 1 / year µ / year

11 If all the breakers are perfectly reliable, for the two-transmission-line system, there will be 4 states.

12 If breakers may not open on command:

13 Merging of states: Equivalent transition rates: P5µ + P6µ λ12 µ P5 + P6 P 1(.9λ +.9λ) λ λ P1 P λ + P λ λ λ 2 3 Copyright 34by Chanan Singh For educational purpose P2 + only P3 P 1(.1λ +.1λ ) µ 21. 2λ P1 P2µ + P3µ µ 32 µ P + P 2 P4 ( µ + µ ) µ 43 2µ P 4 3

14 1.2.1 Weather Transition rate from normal weather to adverse weather is Transition rate from adverse weather to normal weather is: 1 N / year S / year 876

15 Transition rate matrix of transmission system is: R T - (2λ + µ µ S N) 1.8λ - ( µ + λ + 2µ S N) λ - (2µ + S N).2λ - ( µ + N) S N N N - (2λ' + S) ' 1.8λ µ - ( µ + λ ' + S) λ' 2µ - (2µ + S) µ N.2λ' - ( µ + S)

16 1.3 Load λ / year λ / year Transition rate matrix: R L

17 2 Steady State Probabilities, Frequency and Mean Duration of Loss of Load Generation System In order to get the steady probability of each state, we can write: Using the RG we obtained to solve above equation, we get the steady state probability of each state. If generators are independent probabilities can be calculated by product rule also. Probabilities calculated in either way are the same. P P Copyright by Chanan d Singh For educational purpose λ only + u µ λ + µ λ µ

18 2. 2. Transmission System We have the following equations:

19 Using the RT we obtained to solve previous equation, we get the steady state probability of each state: P P P P P P P P

20 We can also reduce the eight-state transmission transition diagram to a three-state diagram with respect to the capacities of the states:

21 For the reduced model, the following results apply: λ λ P 1.8λ + P ' 1.8λ ' ' T ' ' P1 + P5 P λ + P λ ' ' ' T ' ' P2 + P µ µ P µ + P µ ' ' T µ ' ' P2 + P6 ' ' P 2µ + P 2µ T ' ' ' ' P3 + P7 + P4 + P8 195 λ P.2λ + P '.2λ ' ' T ' ' P1 + P µ P µ + P µ ' ' T ' ' ' ' P3 + P7 + P4 + P8 29

22 2. 3. Load The following equations apply: Using the RL we obtained to solve above equations we get the steady state probability in each state: P1 P2 L L

23 2. 4. Solution for the System Steady state probability, frequency and mean time of loss of load could be found using the following table:

24 System State Generation, transmission, load system state Probability of system state Transition to the states with loss of load Loss of load ,3,4 ( λ, λ λ ) 12 T 13T, Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes ( λ 13T ) No ,15 No ( λ 21,λ ) L 23T Yes ,18 No ( λ 21,λ ) L 13T ,18 ( λ,λ ) Yes ,21,22, No ( λ ) 21 L, λ13 T, λ34g ,21,23 No ( λ, λ λ ) Yes Yes Yes Yes L 23T 12G 21 L 23T, 34G No No

25 We can calculate the probability of states having no load loss. Those probabilities are obtained for the generators, transmission lines and loads as independent. From previous Table, we can get the steady state probability of the loss of load as follows. P1- ( ) P

26 The frequency of loss of load is: Values needed for F that are calculated previously:

27 The mean time of loss of load is:

28 Cut Set Method A cut set is a set of components or conditions that cause system failure. A min cut set is a cut set that does not contain any cut set as a subset. In this presentation a cut set implies a min cut set. The term component will be used to indicate both a physical component as well as a condition. Components in a given cut set are in parallel, as they all need to fail to cause system failure. Cut sets are in series as any cut set can cause system failure.

29 Frequency & Duration Equations For Cut Sets First Order Cut Set: One component involved r r λ λ csk i csk i where

30 Frequency & Duration Equations For Cut Sets Second Order Cut Set k:two components involved λ csk λiλ j ( ri + rj ) 1 + λ r i i + λ r j j r csk r i rr i + j r j where

31 Frequency & Duration Equations For Cut Sets Second Order Cut Set with Components subject to Normal and Adverse Weather.

32 Combining n Cut Sets λ λ + λ +! + T cs1 cs 2 λ csn r T ( λ r + λ r +! + cs1 cs1 cs 2 cs 2 λ csn r csn )/ λ T

33 APPLICATION OF CUT SET METHOD TO EXAMPLE SYSTEM Cut set 1: One line failure and breaker stuck. Cut set 2: One generator failure and load changes from 5 to 14

34 APPLICATION OF CUT SET METHOD TO EXAMPLE SYSTEM Cut set 3:One line failure (breaker not stuck) and load changes from 5 to 14. Cut set 4: Two lines fail(breaker not stuck) For each line Applying the equation for second order cut set exposed to fluctuating environment,

35 APPLICATION OF CUT SET METHOD TO EXAMPLE SYSTEM Cut set 4: Two lines fail(breaker not stuck) For the system For each line Applying the equation for second order cut set exposed to fluctuating environment,