8/10/2019 EE220 Lecture 2

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EE

220

2

8/10/2019 EE220 Lecture 2

2/39

DA:E

8/10/2019 EE220 Lecture 2

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E

8/10/2019 EE220 Lecture 2

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8/10/2019 EE220 Lecture 2

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8/10/2019 EE220 Lecture 2

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:

A

L

H

m

m

L

H

i=1

i=1

H

L

Pick

and

suchthat

P

(

)

0

P

(

)

0

While

Do

Gi

D

Gi

D

P

P

>

M

H

L

m

M

H

M

i=1

L

M

(

)/

2

If

P

(

)

0Then

Els

e

EndW

hileG

i

D

P

=

+

>

=

=

8/10/2019 EE220 Lecture 2

7/39

:E

1

1

1

2

2

2

3

3

3

Conside

rathreegeneratorsystemwith

(

)

15

0.0

2

$/MWh

(

)

20

0.0

1

$/MWh

(

)

18

0.0

25

$/MWh

G

G

G

G

G

G

ICP

P

IC

P

P

IC

P

P

=

+

=

=

+

=

=

+

=

1

2

3

andwith

constraint

1000MW

Rewritinggenerationas

afunctionof

,

(

G

G

G

Gi

P

P

P

P

+

+

=

G1

G2

G3

),

wehave

15

20

P(

)

P

()

0.02

0.0

1

18

P(

)

0.025

=

=

=

8/10/2019 EE220 Lecture 2

8/39

:E

m

Gi

i=1

m

Gi

i=1

Pick

so

P

(

)1000

0and

P

(

)1000

0

L

L H

1

H

1

Try

20then

(20)1000

15

20

18

1000

670MW

0.02

0.01

0.025

Try

30then

(30)1000

1230MW

m

L

Gi

i m

Gi

iP P

= =

=

=

+

+

=

=

=

8/10/2019 EE220 Lecture 2

9/39

:E

Pickcon

vergencetoleran

ce

0.05$/M

Wh

Theniteratesince

0.05

(

)/2

25

H

L

M

H

Lm

=

>

=

+

=

1

1en

snce

wese

Since25

20

0.05

(25

20)/222.5

(22.5)1000

195weset

2

Gi

i

Mm

L

Gi

iP

=

=

=

=

>

=

+

=

=

=

2.5

8/10/2019 EE220 Lecture 2

10/39

:E

H *

*

Con

tinueiteratinguntil

0.05

The

solutionvalueof

,

,is23.53

$/MWh

Onc

e

isknownw

ecancalculateth

e

23.53

15

L

Gi

P

=

,min

,min

ense

Gi

Gi

Gi

Gi

>

1=400

2=250

3=150

(1

)

8.5

k

+

=

8/10/2019 EE220 Lecture 2

23/39

()

.

)(

)(

)

(

1

1

0

x

x

g

x

g

x

x

g

=

+

=

+

g

g

(Jacobian)

)(

and

,

,

(

,

,

)(

where

11

11

1

J

x

g

x

g

=

=

=

nm

m

n

n

m

n

xg

xg

x

x

x

x

gg

)(

)(

1

x

g

x

g

x

=

8/10/2019 EE220 Lecture 2

24/39

,

F

,

xL=g

1

(

1)

()

x

k

k

L

L

+

=

=

+

x

x

x

x

x

=

=

+

= i

i

gi

load

gi

i

P

P

PF

L

1

1

)

(

2

2

2

21

1

1

2

1

2

22

1

g

g

g

g

x

g

ggL

L

L

P

P

P

P

L

Hessian

Matrix

LP

P L

L

P

=

x

8/10/2019 EE220 Lecture 2

25/39

E

&

G

F

(

)

1

min(

)

min

m

T

i

gi

i

F

P

=

=

1 min

max

(supply-d

emandbalance)

(generatorcapacity

limits)

m

gi

D

i gi

gi

gi

P

P

P

P

P

=

=

8/10/2019 EE220 Lecture 2

26/39

E

&

G

:

min

gi

gi

P

P

min

gi

gi

P

P

max

gi

gi

P

P

0

giP

E

=

=

=

=

+

=

m i

g

i

gi

i

m i

gi

gi

i

D

m i

gi

m i

gi

i

P

P

P

P

P

P

PF

L

1

min

min

1

m

ax

max

1

1

min

max

)

(

)

(

)

(

)

,

,,

(

Pg

8/10/2019 EE220 Lecture 2

27/39

E

&

G

C

ma

x

maxm

in

max

if

0

,...,

2,1

0

:

gi

gi

i

i

i

i

gii

gi

P

P

dF

m

i

dPdF

PL

=

=

=

=

+

=

C1

(

)

(

)

min

max

min

ma

x

max

max

1

max

min

min

min

if

0

:

if

0

:

0

:

if

0

if

0

gi

gi

gi

gi

i

gi

gi

gi

gi

i

D

m i

gi

gi

gi

gi

gii

gi

gi

i

giig

gi

P

P

P

P

L

P

P

P

P

L

P

P

L

P

P

P

dPdF

P

P

dPdF

PL

=

=

=

=

=

=

=

=

=

=

=

+

=

=

C4

C3

C2

8/10/2019 EE220 Lecture 2

28/39

EG

AD=40

0

UnitI

ai

bi

ci

Pgim

n

Pgim

ax

1

0

.032

2.18

120.312

200MW

350MW

2

0

.038

2.407

74.074

100MW

200MW

)

200

(

)

100

(

)

380

(

)

200

(

)

400

(

074.74

407

.2

019

.0

312

.120

187

.2

016

.0

2

max2

2

min2

1

max1

1

min1

2

1

2

22

1

21

+

+

+

+

+

+

+

+

=

g

g

g

g

g

g

g

g

g

g

P

P

P

P

P

P

P

P

P

P

L

8/10/2019 EE220 Lecture 2

29/39

E

G

C

0

1

87

.2

032

.0

0

max

min

max

1

min1

1

1

=

+

+

=

g

g L

P

PL

0

)

2

00

(

0

)

100

(

0

)

380

(

0

)

200

(

0

400

0

.

.

2

max

2

2

min2

1

max

1

1

min1

2

1

2

2

2

2

=

=

+

=

=

+

=

+

=

=

=

g

g

g

g

g

g

g

g

P

P

P

P

P

P

LP

8/10/2019 EE220 Lecture 2

30/39

E

G

C

0

380

or

0

0

200

or

0

mi

1

max

1

1

min1

=

=

=

+

=

gg

PP

0

200

or

0

2

max

2

2

2

=

=

g

gP

m

i

i

,...,

2,1

all

for

0

and

0

max

i

min

=

=

=

8/10/2019 EE220 Lecture 2

31/39

E

G

0

407.2

038

.

0

0

187.2

032

.

0

21

=

=

+

=

+gg PP

C

g

g

400

0

407

.2

038

.0

0

187

.2

0

032

.0

2

1

2

1

2

1

=+

+

=

+

=

+g

g

g

g

g

g P

P

P

P

P

P

8/10/2019 EE220 Lecture 2

32/39

E

G

=

4004

07

.2

187

.2

0

1

1

1

03

8

.0

0

1

0

032

.0

21gg PP

=

$/MWhr

24.9

MW

71.

179

MW

29.

220

21gg PP

200

100

350

200

21

gg

PP

8/10/2019 EE220 Lecture 2

33/39

E

:

D

=550

=

5504

07

.2

187

.2

0

1

1

1

038.0

0

1

0

032

.0

21gg PP

2

()

=

$/MWhr

8

4

.11

MW

29.

24

8

MW

71.

30

1

21gg PP

200

100

350

200

21

gg PP

8/10/2019 EE220 Lecture 2

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E

:

2

200

max

max

2

2

2

(

200

)

0

//

lan

grangemultiplierisnotzero(bindin

g)

gP

=200

1

0

0

407

.2

038

.0

0

187

.2

0

032

.0

2

1

2

1

max

2

2

1

2

1

=

+

=

+

=

+

=

+ g

g

g

g

g

g

g

g P

P

P

P

P

P

P

P

8/10/2019 EE220 Lecture 2

35/39

E

:

=

200

5504

07

.2

187

.2

0

0

1

0

0

0

1

1

1

1

038

.0

0

0

1

0

032

.0

max21 gg PP

1 2max

2

350

200

1

3.39

3.38

g gP P

=

max

2

iscommonlyterm

edasshadowprice

8/10/2019 EE220 Lecture 2

36/39

C

8/10/2019 EE220 Lecture 2

37/39

B

.2

1

$/hr

47.

1315

074

.

74)

200

(

407

.2

)

200

(019.0

)

(

$/hr

76.

2845

312

.

120

)

350

(

187

.2

)

350

(

016

.0

)

(

2

2

2

2

1

1

=

+

+

=

=

+

+

=

gg P

FPF

=4157.894161.2

3=3.34

$/hr

23.

4161

47.

1315

78.

2845

)

(

)

(

2

2

1

1

=

+

=

+

=

g

g

T

P

F

PF

F

$/hr

89.

4157

50.

1

325

39.

2832

)

(

)

(

$/h

r

50.

1325

074

.74

)201

(

407

.2

)

201

(

019.0

)

(

$/hr

39.

2832

312

.

120

)

349

(

187

.2

)

349

(

016.0

)

(

2

2

1

1

2

2

2

2

1

1

=

+

=

+

=

=

+

+

=

=

+

+

=

g

g

T

gg

P

F

PF

F

P

FPF

8/10/2019 EE220 Lecture 2

38/39

H

2

A.

B.G

D=

D=

BE

(+5)

D(222010)

8/10/2019 EE220 Lecture 2

39/39

G

,

C

A.

B

F.

,AC.,

"EE3

58:E

"

.

EEE,

D,

2005