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MATLAB® Modelling of Capacity Outage
Probability Table for Transmission Systems
By:By:Abhishek Chakraborty (1MJ06EE001)Abhishek Chakraborty (1MJ06EE001)
Apurva Joshi (1MJ06EE005)Apurva Joshi (1MJ06EE005)Himanshu Mishra (1MJ06EE012)Himanshu Mishra (1MJ06EE012)
Prashobh Mohan P. (1MJ06EE026)Prashobh Mohan P. (1MJ06EE026)
Dept. of Electrical and Electronics Engineering
M.V.J. College of Engineering
External Guide:Shri V.V.S.K. ChakravarthiShri V.V.S.K. ChakravarthiEngg. Officer Gr. 2
Internal Guide:Dr. B.N. SarkarDr. B.N. SarkarProfessor
IntroductionIntroduction
• Power Systems designed to provide reliable and
economic supply of electrical energy to
customers.
• Redundancy: How much & at what cost?
• Power System Planning: To resolve the dilemma
between economic and reliability constraints.
Power System PlanningPower System Planning
Power SystemPlanning
DeterministicApproach
ProbabilisticApproach
Power System Planning:Power System Planning:Deterministic ApproachDeterministic Approach
• Generating Capacity: Installed capacity equals the expected maximum demand
• Operating Capacity: Spinning capacity equals expected load demand plus a reserve equal to one or more largest units
• Network Capacity: Construct a minimum number of circuits to a load group
Power System Planning:Power System Planning:Probabilistic ApproachProbabilistic Approach
• Forced Outage Rates (FOR) of generating units is a
function of unit size and type
• The failure rate of an overhead line is a function of
length, design, location, and environment
• Planning and operating decisions are based on load
forecasting techniques
Best Approach: ProbabilisticBest Approach: Probabilistic• Probabilistic Techniques deals with cases
which are stochastic in nature
• Instills more objective assessment in decision
making process
• Why was it not used before?
– Lack of data, limitations of computing resources, lack of
realistic reliability techniques, aversion to the use of
reliability and probability techniques.
Power System Nature: StochasticPower System Nature: Stochastic
• Failure of components, plants and systems occur randomly
• Frequency, duration, and impact of failures vary from year to year
• Following data are useful for Probabilistic Approach:– System Availability– Number of incidents and hours of interruption– Excursions beyond system voltage and frequency limits
These performance measures are These performance measures are valuable because they:valuable because they:
• Identify weak areas needing reinforcement or modifications.
• Establish chronological trends in reliability performance.
• Enable previous predictions to be compared with actual operating experience.
• Monitor the response to system design changes .
Basic Approach: Adequacy Evaluation
Generation Model Load Model
Risk Model
OBJECTIVE:
Basic System Representation
Total System Generation Total System Load
Transmission lines
Hierarchical Levels
Generation Facilities
Transmission Facilities
Distribution Facilities
Hierarchical Level I(HL I)
Hierarchical Level II(HL II)
Hierarchical Level III(HL III)
Capacity Outage Probability Table
• Simple array of capacity levels and the
associated probabilities of existence.
• Considers all the possible combinations of
generating units and transmission lines.
Component States
Unit Up(0)
Unit Down(1)
Failure
Repair
Example
• Consider a plant with 3 x 5 MW units.
• Each unit has a FOR of 0.02.
• Therefore,
Individual Availability = 1 – 0.02 = 0.98.
Single Line Diagram
5 MW
5 MW
5 MW
Load
Availability = 0.98
Availability = 0.98
Availability = 0.98
Capacity Outage Probability TableSr. No. Capacity Out
of service(MW)
Capacity in service(MW)
Individual Probability
Cumulative Probability
1. 0 15 0.941190 1.000000
2. 5 10 0.057624 0.058808
3. 10 5 0.001176 0.001184
4. 15 0 0.000008 0.000008
How was the COPT formulated?
Sr. No.
Capacity Out of service(MW)
Capacity in
service(MW)
Probability Cumulative Probability
1. 0 15 0.98 x 0.98 x 0.98 = 0.941190 1.000000
2. 5 10 3C1 x 0.98 x 0.98 x 0.02 = 0.057624
0.058808
3. 10 5 3C2 x 0.98 x 0.02 x 0.02 = 0.001176
0.001184
4. 15 0 0.02 x 0.02 x 0.02 = 0.000008 0.000008
Computation of COPT when generating units are not identical
15 MW
10 MW
5 MW
Load
Availability = 0.98
Availability = 0.97
Availability = 0.96
TRUTH TABLECase No: Unit1(15mw) Unit2(10mw) Unit3(5mw)
1. 0 0 0
2. 0 0 1
3. 0 1 0
4. 0 1 1
5. 1 0 0
6. 1 0 1
7. 1 1 0
8. 1 1 1
Unit available = 0 Unit unavailable = 1
Final COPTSL no Capacity in
Service, MWCapacity out of Service, MW
Individual Probability
Cumulative Probability
1. 30 0 0.912576 1.000000
2. 25 5 0.038024 0.087424
3. 20 10 0.028224 0.049400
4. 15 15 0.019800 0.021176
5. 10 20 0.000776 0.001376
6. 5 25 0.000576 0.000600
7. 0 30 0.000024 0.000024
Transmission line Network
Plant No. of Units Capacity
(MW)
Unavailabilit
y
1 4 20 0.01
2 2 30 0.05
Total 6 140
Generation Data:
Line Availability Unavailability
1 0.99636033 0.00363967
2 0.99545455 0.00454545
3 0.99658703 0.00341297
Transmission Line Statistics:
NETWORK CONFIGURATIONS
COPT for Transmission linesState Lines out Probability
1 0 0.98844633
2 1 0.00361076
3 2 0.00451345
4 3 0.00339509
5 1,2 0.00001649
6 1,3 0.00001237
7 2,3 0.00001546
8 1,2,3 0.00000006
Combined COPT(Generation and Transmission)
Total System Load
Transmission lines
Total System Generation
Data Required
Total Generation = 6 x 40 MW
Each having Availability of 0.98
Individual line carrying capacity of 160 MW
Generation COPTSTATE NO. OF
GENERATORS ON OUTAGE
CAPACITY AVAILABLE
PROBABLITY
1 0 240 0.88584238
2 1 200 0.10847049
3 2 160 0.00553421
4 3 120 0.00015059
5 4 80 0.00000230
6 5 40 0.00000002
7 6 0 0.00000000
State Probability of Transmission Lines
STATE NO. OF LNES ON OUTAGE
CAPACITY AVAILABLE
MW
PROBABLITY
1 0 320 0.999144
2 1 160 0.999574
3 2 0 0.000428
Combined COPTSTATE CONDITION CAPACITY
AVAILABLE MW
PROBABILITY
1 0G 0L 240 0.88508444
2 0G 1L 160 0.00075778
3 0G 2L 0 0.00000016
4 1G 0L 200 0.10837768
5 1G 1L 160 0.00009279
6 1G 2L 0 0.00000002
7 2G 0L 160 0.00552947
8 2G 1L 160 0.00000473
9 2G 2L 0 0.00000000
10 3G 0L 120 0.00015046
11 3G 1L 120 0.00000013
12 3G 2L 0 0.00000000
13 4G 0L 80 0.00000230
STATE CONDITION CAPACITY AVAILABLE MW
PROBABLITY
14 4G 1L 80 0.00000000
15 4G 2L 0 0.00000000
16 5G 0L 40 0.00000002
17 5G 1L 40 0.00000000
18 5G 2L 0 0.00000000
19 6G 0L 0 0.00000000
20 6G 1L 0 0.00000000
21 6G 2L 0 0.00000000
Single Line Diagram of the Roy Billinton Test System (RBTS)
Generating Unit Locations
Unit No. Bus Rating Type
1 1 40 Thermal
2 1 40 Thermal
3 1 10 Thermal
4 1 20 Thermal
5 2 5 Hydro
6 2 5 Hydro
7 2 40 Hydro
8 2 20 Hydro
9 2 20 Hydro
10 2 20 Hydro
11 2 20 Hydro
System Reliability Data
Unit Size Type No. of Units Forced Outage Rate
5 Hydro 2 0.010
10 Thermal 1 0.020
20 Hydro 4 0.015
20 Thermal 1 0.025
40 Hydro 1 0.020
40 Thermal 2 0.030
System Line Data
Line
Buses Impedance (p.u.)CurrentRating(p.u.)From To R X B/2
1,6 1 3 0.0342 0.180 0.0106 0.85
2,7 2 4 0.1140 0.600 0.0352 0.71
3 1 2 0.0912 0.480 0.0282 0.71
4 3 4 0.0228 0.120 0.0071 0.71
5 3 5 0.0228 0.120 0.0071 0.71
8 4 5 0.0228 0.120 0.0071 0.71
9 5 6 0.0228 0.120 0.0071 0.71
System Bus Data
BusLoad (p.u.)
ScheduledGeneration
(p.u.)Qmax Qmin Vint Vmax Vmin
P Q
1 0.00 0.00 1.0 0.50 -0.40 1.05 1.05 0.97
2 0.20 0.00 1.2 0.75 -0.40 1.05 1.05 0.97
3 0.85 0.00 0.0 0.00 0.00 1.00 1.05 0.97
4 0.40 0.00 0.0 0.00 0.00 1.00 1.05 0.97
5 0.20 0.00 0.0 0.00 0.00 1.00 1.05 0.97
6 0.20 0.00 0.0 0.00 0.00 1.00 1.05 0.97
The Algorithm – “Reliability.m”
• Step 1: Start
• Step 2: Input Line Data, Bus Data, Availability
Details for all components
• Step 3: Call “GeneratorCOPT.m”
The Algorithm – “GeneratorCOPT.m”
• Step 1: Retrieve No. of Units, Power Rating of each unit,
Availability of each unit
• Step 2: Form Truth Table to accommodate all states
• Step 3: Initialize matrix such that
– Column 1: Capacity in service
– Column 2: Capacity Out of service
– Column 3: State Probability
– Column 4: Cumulative Probability
The Algorithm – “GeneratorCOPT.m”
• Step 4: Initialize counter i = 0
• Step 5: Check all elements of ith row
– If element is 0, add generator rating to column 1
and multiply availability to column 3
– If element is 1, add generator rating to column 2
and multiply unavailability to column 3
The Algorithm – “GeneratorCOPT.m”
• Step 6: Add all similar capacities
• Step 7: Sort the table in descending order of
available capacities. Truncate the table.
• Step 8: Return table to main program
Reliability
Comparing Results
Input Data
Input Data
Formation of Truth Table
0 – Unit Available1 – Unit Unavailable
Unit 1 Unit 2 Unit 3
Initialization of COPT matrix
Capacity Out of ServiceState ProbabilityCapacity In Service Cumulative Probability
Forming the Crude COPT
PR: Power Rating of the unitA: Availability of unit
Counter ‘i’
Unit 1: ‘Available’3 MW
Working of the loop
Working of the loop
Capacity In Service State Probability
Counter ‘i’
Unit 2: ‘Available’3 MW
Working of the loop
Working of the loop
Capacity In Service State Probability
Counter ‘i’
Unit 3: ‘Available’5 MW
Working of the loop
Working of the loop
Capacity In Service State Probability
Counter ‘i’
Unit 1: ‘Available’3 MW
Unit 2: ‘Available’3 MW
Unit 3: ‘Unavailable’5 MW
Working of the loop
Working of the loop
Capacity Out of Service State Probability
Counter ‘i’
Working of the loop
Counter ‘i’
Counter ‘i’
Counter ‘i’
Counter ‘i’
Counter ‘i’
Counter ‘i’
Counter ‘i’
Working of the loop
Adding the similar capacities
Working of the loop
Working of the loop
Working of the loop
Working of the loop
Working of the loop
Working of the loop
Sorting the table
Truncating the table
Function PathFinder()PathFinder()
• 1 is the Start Node• 4 is the End Node
1 2
3 4
Function
PathFinder()
Finds all the possible
paths between the:
Start Node
and the
End Node
of a given Network
Step I: Adjacency Matrix
Adjacency Matrix:1 22 33 4
1 00 1 1 122 1 00 0 133 1 0 00 14 1 1 1 00
1 2
3 4
ith Node is Connected to jth Node; Adjacency (i,j) Adjacency (i,j) = 1
ith Node is Not Connected to jth Node; Adjacency (i,j)Adjacency (i,j) = 0
MATLAB® Code
Step II: Truth Table
• Start Node and End Node must be present
11 22 33 441 11 1 11 1 0 11 0 1 11 0 0 1
1 2
3 4
MATLAB® Code
Step III: Permutations for Each Case
• Case 1: 2: Present3: PresentBy Permutation:11 22 33 4411 33 22 44
1 2
3 4
11 22 33 44Refer Adjacency Matrix:
Adjacency (1,2)=1
Adjacency (2,3)=0
Adjacency (3,4)=1
11 33 22 44Refer Adjacency Matrix:
Adjacency (1,3)=1
Adjacency (3,2)=0
Adjacency (2,4)=1
• Case II:2: Present3: AbsentBy Permutation:11 22 44
1 2
3 4
1 2
3 4
• Case III:2: Absent3: PresentBy Permutation:11 33 44
11 22 44Refer Adjacency Matrix:
Adjacency (1,2)=1
Adjacency (2,4)=1
11 33 44Refer Adjacency Matrix:
Adjacency (1,3)=1
Adjacency (3,4)=1
• Case IV:2: Absent3: Absent
By Permutation:11 44Refer Adjacency Matrix:
Adjacency (1,4)=1
1 2
3 4
Possible Paths are:1 2 41 3 41 4
1 2
3 4
MATLAB® Code
Adds the name of the Node
Permutations
MATLAB® Code
Refers Adjacency Matrix for existence of path
Function CompositeCOPT()CompositeCOPT()
• Function CompositeCOPT() CompositeCOPT() finds Capacity Outage Probability Table for Hierarchical Level II System
• Calls function PathFinder()PathFinder() to find all the suitable transmission paths connecting to a particular load bus
• Calls function NewtonRaphsonLF()NewtonRaphsonLF() to perform AC load flow studies, to check line flow limit violations
Brief Description
• 11 Generator Bus
• 55 Load Bus
• 11224455:
Transmission Path in
Service.
1 2
43
5
• COPT of Bus 1 with Bus 2 are combined, considering transmission line availability in between
• Violation of line capabilities are checked
• This is then reduced to a Single Node, having a Single COPT
• This is repeated till the load bus is reached
1 2
43
5
1 2
4
5
COMBINECOMBINE
1122
4
5
AA
1122
4
5
AA
A4
5
BBCOMBINECOMBINE
A4
5
BB11224455
COPTCOPTCOMBINECOMBINE
MATLAB® Code
Formation of combined COPT
Loading COPT of Node 2
Loading COPT of Node 1Calling function PathFinder
MATLAB® Code
Bubble Sort
Combining Probabilities of Like States
MATLAB® Code
OUTPUT: LOAD FLOW
OUTPUT: LOAD FLOW
OUTPUT: COMPOSITE SYSTEM COPTFOR LOAD BUS 3:
STATE PROBABILITY FOR THE TRANSMISSION PATH: 1 3___________________________________________________________________CAPACITY AVAILABLE CAPACITY UNAVAILABLE PROBABILITY (MW) (MW) -------------------------------------------------------------------110 0 0.88104935100 10 0.0179806090 20 0.0225910180 30 0.0004610470 40 0.0544979060 50 0.0011122050 60 0.0013973840 70 0.0000285230 80 0.0008427520 90 0.0000172010 100 0.000021610 110 0.00000044
CONCLUSION
• It was desired to develop the Capacity Outage Probability Table for a composite Generation and Transmission System.
• The following studies have been made in that connection:– COPT for a power system has been formed taking
into consideration the generation and transmission facilities.
CONCLUSION
– The COPT so developed has been verified for several cases as available in the published literature
– MATLAB® Software has been used to execute the programme
– Using the designed programme, the COPT for any general system can be formulated
– However, the limitation for the programme is that the system should contain not more than 20 transmission lines.
FUNCTIONS APPROVED BYM/s. MathWorks
• COPT for Hierarchical Level I Studies:http://www.mathworks.com/matlabcentral/
fileexchange/27599-generator-capacity-outage-probability-table
• Function “PathFinder”:http://www.mathworks.com/matlabcentral/
fileexchange/27438-find-all-the-possiblepaths-between-a-start-and-an-end-node-of-a-graph
REFERENCE: TEXTBOOKS
• Billinton R., Allan R.N., ‘Reliability Evaluation of Power Systems’, Second Edition, Plenum Press, New York, 1996.
• Billinton R., Allan R.N., ‘Reliability Evaluation of Engineering Systems’, Second Edition, Plenum Press, New York, 1992.
• Billinton R., ‘Power System Reliability Evaluation’, Gordon and Breach Science Publishers, New York, 1970.
REFERENCE: PAPERS
• Billinton R., Karki B., Karki R., Gokaraju R., ‘Operating Reserve
Assessment of Wind Integrated Power Systems’, Power System Research
Group, University of Saskatchewan, Canada.
• R. Billinton, S. Kumar, N. Chowdhury, K. Chu, K. Debnath, L. Goel, E.
Khan, P. Kos, G. Nourbakhsh, J. Oteng-Adjei, ‘A Reliability Test System
For Educational Purposes – Basic Data’, IEEE Transactions On Power
Systems, Vol. 4, No. 3, August 1989
• R. Billinton, S. Kumar, N. Chowdhury, K. Chu, K. Debnath, L. Goel, E.
Khan, P. Kos, G. Nourbakhsh, J. Oteng-Adjei, ‘A Reliability Test System
For Educational Purposes – Basic Results’, IEEE Transactions On Power
Systems, Vol. 5, No. 1, February 1990
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