Lecture 10
Transcript of Lecture 10
EEL 6266 Power System Operation and Control
Chapter 8
Production Cost Models
© 2002, 2004 Florida State University EEL 6266 Power System Operation and Control 2
Introduction
� Production cost models� are computational models designed to calculate information
for long-range system planning:� generation system production costs
� energy import requirements
� availability of energy for sale to other systems
� fuel consumption
� employees models of expected load patterns and simulated operation of the system’s generation � uncertainty of load forecasts
� reliability of generating units
� expected need for emergency energy and capacity supplies
� uses statistical computational methods for solving problems
© 2002, 2004 Florida State University EEL 6266 Power System Operation and Control 3
Introduction
� Stochastic production cost models� used for long range studies
� the risk of sudden, random, generating unit failures and random deviations for the mean forecasted load are treated as probability distributions
� load modeling considers the behaviors of the “expected load” patterns that cover periods of weeks, months, and/or years� the load duration cover expresses the time period that the
loading is expected to equal or exceed a given power value
� generating unit modeling includes fuel costs usually expressed over a monthly basis� generating unit scheduled maintenance outages
may involve time periods from days to years
© 2002, 2004 Florida State University EEL 6266 Power System Operation and Control 4
Introduction
� Types of production cost studies
Load ModelInterval under Consideration
Economic Dispatch Procedure
total energy or load duration
load duration or load cycles
seasons or years
months or weeks
block loading (w/o regard to incr. costs)
incremental loading
load durationor load cycles
load cycles
months, weeks, or days
weeks or days
incremental loadingwith forced outages
incremental loading with losses
Long-Range Planning
Operation Planning
Weekly Schedules
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�
�
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© 2002, 2004 Florida State University EEL 6266 Power System Operation and Control 5
Load-duration Curves
� Representation of future loads in which the impact of capacity limitations will be studied
� Building the load-duration curve� consider the expected load
pattern
� build a histogram of load for a given time period and find the load density function, p(x)
� integrate the load densityfunction to obtain the load distribution function, Pn(x)
Time (h)
Load
(M
W)
Load L (MW)
p(L)
Pro
babi
lity
that
load
= L
MW
probability densityfunction
Load L (MW)
P(L
) P
roba
bilit
ylo
ad ≥
LM
W cumulativedistribution
function
1.0
0.0( ) ( )∫∞
−=x
d1 xxpxPn
© 2002, 2004 Florida State University EEL 6266 Power System Operation and Control 6
Load-duration Curves
� Building the load duration curve� multiply the probability by the
period length to show the number of hours that load equals or exceeds a given power level, L
� common convention has theload on the vertical axis
� Block-Loading� simulates the economic dispatch
procedure with this type of load model
� generating units are ordered by cost� units are assumed to be fully loaded or
loaded up to the limitation of the load-duration curveLo
ad L
(MW
)
Hours load equalsor exceed L MW
040
500
1000
1500
8 12
© 2002, 2004 Florida State University EEL 6266 Power System Operation and Control 7
Load-duration Curves
� Block-Loading� example of the Niagara-Mohawk system
2-Mile PointMohawk #1Mohawk #2
8.18.58.7
800300200
UnitIncremental Cost ($/h)
Maximum Capacity (MW)
Rio Bravo #1Rio Bravo #2Rio Bravo #3
9.29.69.7
752520
(8) gas turbines
9.9 400
Load
L(M
W)
Hours load equals or exceed L MW0
500
1000
1500
2-Mile Point (800 MW)
Mohawk #1 (300 MW)
Mohawk #2 (200 MW)
Rio Bravo #1, #2, & #3 (120 MW)
gas turbines (280 MW out of 400 MW)
© 2002, 2004 Florida State University EEL 6266 Power System Operation and Control 8
Load-duration Curves
� Example� consider a two generating unit
system that will serve the following expected load pattern:
� construct a load-duration curvein tabular and graphic form
1008040
206020
20004800800
x-Load (MW)
Duration(h)
Energy (MWh)
Totals: 100 7600
02040
00
20
100100100
x-Load (MW)
ExactDuration (h)
T Pn(k), Hours that Load Equals
or Exceeds x
6080
100
06020
808020
100+ 0 0
100
50
0100806040200
x-Load (MW)
TP
n(x)
, Hou
rs th
at lo
ad e
qual
s or
exc
eeds
x
© 2002, 2004 Florida State University EEL 6266 Power System Operation and Control 9
Load-duration Curves
� Example� the two generating units have the following characteristics
� the fuel cost rate for each unit is a linear function of the power output
� block-load the two units onto the load-duration curve� unit #1 is used first because of its lower average cost per MWh
0800
16080080
1.01.02.0
UnitPower Output(MW)
Fuel Input
(MBtu/h)
2
1
Fuel Cost
($/MBtu)
Fuel Cost Rate($/h)
Incremental Fuel Cost ($/MWh)
40 400 2.0
160800160800
8.0
16.0
0.05
0.10
Unit Force Outage Rate
(per unit)
© 2002, 2004 Florida State University EEL 6266 Power System Operation and Control 10
Load-duration Curves
� Example� block-loaded
� unit #1 is on-line for 100 h� 80 MW output for 80 h
� 40 MW output for 20 h
� unit #2 is on-line for 20 h� 20 MW output for 20 h
� summary of results
100
50
0100806040200
x-Load (MW)
TP
n(x)
, Hou
rs th
at lo
ad e
qual
s or
exc
eeds
x
Unit 1
Unit 2
4080
20
2080
20
8006400
400
UnitLoad(MW)
Duration (h)
2
1
Energy (MWh)
Fuel Used (MBtu)
Fuel Cost ($)
Total: 7600
960064000
4800
9600
Subtotal: 7200 736006400073600
96008320078400
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Forced Outages
� Forced outage of a generator unit� the time that the unit is not available due
to a failure of some sort� represents a random event
� taken out of the total time that the unit should be available for service
� the forced outage rate is the ratio of forced outage time over the total time available� schedule outage times for maintenance are excluded in
both the total time available and the forced outage time
� Forced outage rates for all generating units must be accounted for in the expected production costs
© 2002, 2004 Florida State University EEL 6266 Power System Operation and Control 12
Forced Outages
� Example� reconsider the previous example, but now including the effects
of forced outages
� evaluate by load levels� Load = 40 MW; duration 20 h
� Unit 1: on-line for 20 h, operates for 0.95 × 20 = 19 houtput: 40 MW, energy delivered: 19 × 40 = 760 MWh
� Unit 2: on-line for 1 h, operates for 0.90 × 1 = 0.9 houtput: 40 MW, energy delivered: 0.9 × 40 = 36 MWh
� load energy = 800 MWhgeneration = 796 MWhunserved energy = 4 MWhshortage = 40 MW for 0.1 h
© 2002, 2004 Florida State University EEL 6266 Power System Operation and Control 13
Forced Outages
� Example� Load = 80 MW; duration 60 h
� Unit 1: on-line for 60 h, operates for 0.95 × 60 = 57 houtput: 80 MW, energy delivered: 57 × 80 = 4560 MWh
� Unit 2: on-line for 3 h, operates for 0.90 × 3 = 2.7 houtput: 40 MW, energy delivered: 2.7 × 40 = 108 MWh
� load energy = 4800 MWhgeneration = 4668 MWhunserved energy = 132 MWhshortage = 80 MW for 0.3 h (24 MWh) and
40 MW for 2.7 h (108 MWh)
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Forced Outages
� Example� Load = 100 MW; duration 20 h
� Unit 1: on-line for 20 h, operates for 0.95 × 20 = 19 houtput: 80 MW, energy delivered: 19 × 80 = 1520 MWh
� Unit 2: on-line for 20 h, operates as follows• Unit 1 is on-line and operating for 19 h
Unit 2: on-line for 0.90 × 19 = 17.1 houtput: 20 MW, energy delivered: 17.1 × 20 = 342 MWhshortage: 20 MW for 1.9 h
• Unit 1 is supposedly on-line, but not operating 1 hUnit 2: on-line for 0.90 × 1 = 0.9 h
output: 40 MW, energy delivered: 0.9 × 40 = 36 MWhshortage: 100 MW for 0.1 h and 60 MW for 0.9 h
� load energy = 2000 MWh; generation = 1898 MWhunserved energy = 102 MWh
• 100 MW for 0.1 h = 10 MWh; 60 MW for 0.9 h = 54 MWh;20 MW for 1.9 h = 38 MWh
© 2002, 2004 Florida State University EEL 6266 Power System Operation and Control 15
Forced Outages
� Comments� it was necessary to make an arbitrary assumption that the
second unit will be on-line for any load level that equals or exceeds the capacity of the first unit
� the enumeration of the possible states is not complete� need to separate the periods when there is excess capability,
exact matching of generation and load, and shortages
� when there is an exact matching of generation and load, it is referred to as a “zero-MW shortage”
� there are two such periods in the example� 40 MW loading, 20 h duration, unit 2 on: 0.05 × 0.9 × 20 = 0.9 h
� 80 MW loading, 60 h duration, unit 1 on: 0.95 × 0.1 × 60 = 5.7 h
� total zero-reserve expected duration: 6.6 h
© 2002, 2004 Florida State University EEL 6266 Power System Operation and Control 16
Forced Outages
� Summary of all possible states
20
Load(MW)
1
Duration(h)
EventNo.
Unit 1 Unit 2
40
1
Status Power (MW) Status Power (MW)
Combined Event
Duration (h) Consequence
2 13 04 05 16 17 08 09 110 111 012 0
6080
20100
101010101010
4040
40
8080
8080
00
00
00
00
000
0
0
0
40
40
20
17.11.90.90.1
51.35.72.70.3
17.11.90.90.1
Load satisfied
Load satisfied
0 MW shortage
40 MW shortage
Load satisfied
0 MW shortage
40 MW shortage
80 MW shortage
Load satisfied
20 MW shortage
60 MW shortage
100 MW shortage
© 2002, 2004 Florida State University EEL 6266 Power System Operation and Control 17
Forced Outages
� Summary of generation cost results
� unserved load
100
81
95.0
72.9
6840
522
UnitScheduled
Time On-line (h)
Expected Operating Time (h)
2
1
Expected Generated
Energy (MWh)
Expected Fuel Used
(MBtu)
Expected Production
Cost ($)
Total: 7362
69920
10008
69920
20016
8993679928
6.61.9
038
Unserved Demand (MW)
Duration of Shortage (h)
200
UnservedEnergy (MWh)
Total: 23812.6
12.66.0
Duration of Given Shortage or More (h)
2.80.9
1125460
40 4.11.3
0.30.1
2410100
80 0.40.1