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


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


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

�

�

�

�

�

�

� �


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



� 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



� 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)



� 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



� 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)



� 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


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


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


Forced Outages

� Example� Load = 80 MW; duration 60 h


� 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)


Forced Outages

� Example� Load = 100 MW; duration 20 h


� 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


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


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


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

Lecture 10

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Transcript of Lecture 10