EE533 Power System Operations

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EE533 Power System Operations

• Satish J. Ranade
• Probabilistic Production Costing

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Production Modeling and Simulation

• Determine how energy resource will be utilized
• Determine production(energyOM) costs
• Determine fuel use
• Applications
• Budgeting and Planning
• Schedules
• Market position
• Risk and Reliability
• Key new idea Account for uncertainty
• Load and forecast uncertainty
• Random Outages
• Cost and Price uncertainty

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Production Modeling and Simulation
Example(Deterministic) System load 100 MW for
20 H 80 MW for 60 H 40 Mw for 20
H Generation 80 MW unit No Load 160
MBTU/H Full Load 800 MBTU/H IHR 8
/MWH Cost of fuel 1/MBTU 40 MW unit
No Load 80 MBTU/H Full Load 400
MBTU/H IHR 8 /MWH Cost of fuel
2/MBTU
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Production Modeling and Simulation

• Example(Deterministic)
• Load 40 MW Duration 20 H Unit 1 supplies 40
  MW
• Unit 2 supplies 0
• 2. Load 80 MW Duration 60 H Unit 1
  supplies 80 MW
• Unit 2 supplies 0
• 3. Load 100 MW Duration 20 H Unit 1
  supplies 80 MW
• Unit 2 supplies 20
• Unit 1 Energy 4020806080207200 MWH
• Unit 1 Fuel Cost (160 840)20(160880)8073,
  600
• Unit 2 Energy 400 MWH
• Unit 2 Fuel Cost 9600

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Production Modeling and Simulation

• Example(Deterministic)
• Load 40 MW Duration 20 H Unit 1 supplies 40
  MW
• Unit 2 supplies 0
• 2. Load 80 MW Duration 60 H Unit 1
  supplies 80 MW
• Unit 2 supplies 0
• 3. Load 100 MW Duration 20 H Unit 1
  supplies 80 MW
• Unit 2 supplies 20
• Unit 1 Energy 4020806080207200 MWH
• Unit 1 Fuel Cost (160 840)20(160880)8073,
  600
• Unit 2 Energy 400 MWH
• Unit 2 Fuel Cost 9600

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Production Modeling and Simulation

• Example Random Unit outages
• Unit 1 95 reliable Unit 2 90 reliable
• Load 40 MW Duration 20 H
• Unit 1 supplies 40 MW but only 95 of the time
• Energy 1 760 MWH
• Unit 2 needed 5 of time 1 h at 40 MW
• But unit 2 is only 90 reliable
• Unit 2 Energy 0.9140 36 MWH
• 40 MW Load not met for 0.1H

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Production Modeling and Simulation
Example Random Unit outages Unit 1 95 reliable
Unit 2 90 reliable 2. Load 80 MW Duration
60 H Unit 1 supplies 80 MW but only 95 of the
time Energy 1 4560 MWH Unit 2 needed 5 of
time 3 h at 40 MW But unit 2 is only 90
reliable Unit 2 Energy 0.9340 108 MWH 80
MW load not met for 0.3 H 40 MW not met for
2.7H
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Production Modeling and Simulation
Example Random Unit outages Unit 1 95 reliable
Unit 2 90 reliable 2. Load 100 MW Duration
20 H Unit 1 supplies 80 MW but only 95 of the
time Energy 1 1520 MWH Unit 2 needed 95 of
time 19H at 20 MW Unit 2 needed 5 of time 1 h
at 40 MW But unit 2 is only 90 reliable Unit 2
Energy 0.9140 .91920 378 MWH 100 MW load
not met for 0.1 H 60 MW not met for .9H 20 MW for
1.9H
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Production Modeling and Simulation
Example Random Unit outages Summary Fuel
Budgeting and Cost information Unit Needed(H)
Operates(H) Energy Fuel Cost MWH
MBTU 1 100 95 6840 69920 69920
2 81 72.9 522 10008 20016
Cost is 8.1 higher than when outages ignored
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Production Modeling and Simulation

• Example Random Unit outages
• Summary-- Reliability Measures
• Duration of time that all load not supplied 6
  Hours
• ( load lost gt0)
• Will later call LOLP Loss of load Probability
• Energy not supplied 238 MWH
• gt Will later call EENS Expected Energy Not
  Served

Cost is 8.1 higher than when outages ignored
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Production Modeling and Simulation
Given Probability Models for load and
generation Determine How generation will be
used Fuel Usage Energy Costs Reliability
Measures
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Production Modeling and Simulation
Approaches Operations Model Priority Loading,
ED, UC, Hydro.. Model for Load Probability
Distribution Random Process Trends
Model for Generation Availability/Forced
Outage Rate Markov model --
multistate
Analytical Methods Load Duration Curve
Stochastic Simulation Monte Carlo
Simple Operation Models Comprehensive Computation
ally Efficient Significant resources needed
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Production Modeling and Simulation
Stochastic Simulation
Analytical Methods
Complementary Distribution
Simulate realizations until convergence
Prob(Load gt x)
Load
Energy Unit 1
C2
C1
Time
Peak Load
C1 C1C2
x MW
Statistics
Load Duration Curve
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Production Modeling and Simulation
Stochastic Simulation
Analytical Methods
Complementary Distribution
Simulate realizations until convergence
Prob(Load gt x)
Load
Energy Unit 1
C2
C1
Peak Load
Time
Statistics
C1 C1C2
Practical product http//www.powergeneration.sieme
ns.com/download/pool/promod.pdf
x MW
Load Duration Curve
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Probability Random Variables
Frequency Interpretation Events Generator
Up Generator Down Availability p
Probability(Gen. Up) Time up/Observation
Time Forced outage Rate q Probability(Gen.
Up) Time down/Observation Time
1- Availability But see IEEE Standard 762
Standard Definitions for Use in Reporting
Electric Generating Unit Reliability,
Availability, and Productivity.
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Probability Random Variables
Properties P(A) 1-P(NOT A) Independent events
A and B P(AB) P(A)P(B) Example Random Unit
outages Unit 1 95 reliable Unit 2 90
reliable p10.95 p2 0.9 q10.05 q20.1 P(Both
down) q1 q2 0.005
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Probability Random Variables
Properties P(A or B or both)
P(A)P(B) Conditional Probability P(A/B) P (A
B)/P(A) Bayes Theorem Event B depends on
events A1, A2An P(Ai/B) P(B/Ai)/ ?
(P(B/Ai)P(Ai)
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Random Variables

A random variable takes on values from a
prescribed set(Sample space) according to some
probability law
Discrete event Random Variable Generator
State Capacity MW Up 0 Down 100
The RV can be continuous or discrete
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Random Variables

A random variable is a function defined on
prescribed set(Sample space) according to some
probability law
Discrete event Random Variable Generator
State Capacity MW Up 0 Down 100
The RV can be continuous or discrete
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Random Variables

Probability
Discrete Variable Continuous Variable x x1,x2,
X R Probability mass Probability
Density p1 Prob(xx1) fX(x)Prob(x X
x?x) ?pi 1
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Random Variables

Probability Distribution
Discrete Variable Continuous Variable X x1,x2,
X R Probability Density P(x) Prob(X
x) FX(x)Prob(x X) iXn, Xn x ?pi
1 i0
Complementary Distribution P(x)1-P(x)Prob(Xgtx
)
FX(x)Prob(xgt X) 1- FX(x)
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Gaussian and Exponential Distribution

• Gaussian density Exponential

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Expectation and moments

• Expectation E(X)

nth moment.
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Probabilistic Production Costing

• Given Random Load and Generation
• Determine energy from each unit
• Determine costs
• Determine Reliability metric
• Key idea
• Probability ( Load-Generation x) ?

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Probabilistic model for Load
System load 100 MW for 20 H 80 MW for 60
H 40 Mw for 20 H
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Probabilistic model for Load
System load 100 MW for 20 H 80 MW for 60
H 40 Mw for 20 H
PX(x) is also called the LOAD DURATION CURVE
and can be interpreted as time Load exceeds
abscissa value
Peak
Min
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Probabilistic model for Load
System load 100 MW for 20 H 80 MW for 60
H 40 Mw for 20 H
PX(x) is also called the LOAD DURATION CURVE
and can be interpreted as time Load exceeds
abscissa value
Wollenburg Notation
PX(x) ? Pn(x) n indicates we are
dealing with a probability
mass Note these are dimensionless numbers related
to probability TPn(x) is also used with T
representing the number of time units, e.g.,
hours in the study. This relation has units of
time.
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Probabilistic model for Load
Although we discretize load data and generation
capacities Are discrete we can also work with
continuous load and generation values and
continuous distributions If desired
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Probabilistic model for Load
Load duration curve
The area under the LDC Average load
i 8 ?Pn(x) Average Load MW i0 i 8
?TPn(x) Energy MWH i0
Lav76 MW T100H E7600MWH
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Probabilistic model for Generation Commitment
Load duration curve
If we had a single perfectly reliable unit with
capacity C120MW Energy generated by unit E1
XC1 ?TPn(x) 10020 2000MWH X0
20MW
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Probabilistic model Generation Commitment
Load duration curve
If we had a second perfectly reliable unit with
capacity C140MW Energy generated by unit E1
XC2 ?TPn(x) 100(120.820) XC1 3600 MWH
20MW
60MW
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Probabilistic Model for Generation

• Basic model
• p availability q forced outage rate
• Question What is the probability that
  generation capacity gt x MW?

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Probabilistic Model for Generation

• Outage Tables
• Classical generating system reliability models
  often used capacity outage tables
• po(Ok) Probability that Ok MW of capacity is
  forced out
• Po(Ok) Probability that capacity is forced out
• is greater than or equal to Ok

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Probabilistic Model for Generation

• Outage Tables --4 Identical 10MW units with
  availability p and FOR q

Binomial nCr
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Probabilistic Model for Generation

• Outage Tables Recursive ( Compare WW p.259)

Pon (Ok) Cumulative Outage Probability for n
units
Unit n1 has Capacity Cn and availability pn
This is Convolution We say unit n1 is
being Convolved in
For the system with n1 units Pon1 (Ok) pn
Pon (Ok) (1-pn) Pon (Ok-Cn) P( Outage
Capacity Ok with n1 units) P (Outage
Capacity Ok with n units)P(Unit n1 UP) P
(Outage Capacity Ok-Cn with n units)P(Unit n1
Down)
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Probabilistic Model for Generation

• Outage Tables For our original example with 2
  units 80 MW, p0.95 and 20 MW p0.9

• Ok Po1(Ok) Po1(Ok-20) P2o(Ok) po(Ok)
• 0 1 1 1 0.855
• 20 .05 1 .145 0.095
• 40 .05 .05 .05 0
• .05 .05 .05 0
• 0 .05 .005 0.045
• 100 0 0 0 0.005

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Probabilistic Model for Generation

• Outage Tables For our original example with 2
  units 80 MW, p0.95 and 20 MW p0.9

LDC Outage Prob
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Probabilistic Model for Generation

• Outage Tables For our original example with 2
  units 80 MW, p0.95 and 20 MW p0.9

Reliability Loss of Load Probability
P(LoadgtCapacity)
IC Installed capacity 120 MW
O Outage capacity XLoad
Sometimes written with gt
LOLP? P ( Capacity C) P(Load C)
C ? P( IC-OC) P(X C)
C ?po(OIC-C) Pn(X C) C
0.041
Convolution
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Probabilistic Production Costing
For production costing it is more convenient to
combine (convolve) Generates into the LDC
LDC Pn(x) Prob (Load X x)
Consider operation of Unit 1 Capacity C FOR q
Equivalent Load C with Prob q Load
Fictitious Load 0 with Prob 1-q
Remaining Load X-C
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Probabilistic Production Costing
For production costing it is more convenient to
combine (convolve) Generates into the LDC
LDC Pn(x) Prob (Load X x)
Remaining Load LDC Pn(x) Prob (Load
X-C x) Prob (Load X x C) Pn(x) q
Pn(x) p Pn(xC)
Pn(x) is the LDC SEEN by remaining units
accounting for the operation of Unit 1
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Probabilistic Production Costing
Discrete version with 1 MW steps Store all
curves in vector. Index k corresponds to load
level Loadk-120 So index k 0gt load 1
MW k0 gt Load -120 MW k150 gtload30 MW
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Probabilistic Production Costing
Define LDC
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Probabilistic Production Costing
Convolve Unit 1
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Probabilistic Production Costing
Convolve Unit 1
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Probabilistic Production Costing
Convolve Unit 2
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Probabilistic Production Costing
Convolve Unit 2
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Probabilistic Production Costing
ENS Energy not served will be covered by purchases