4 :

[]:,,

[]:()

[]:,/

[]:,/,,

: (!):2.

1::,,,,.(:Rippl &Sequent Peak).

2::,,

3::(Markov,Fiering ).().:

4:/.(/)

5:(.).6:

7:/.:

8::/Kuhn Tucker

9: 10: 11::

12:/

13:/

,.(2007).,. (ebook)

Loucks,D.P.,E.vanBeek,J.R.Stedinger,J.P.M.Dijkman,WaterResources Systems Planning andManagement,AnIntroduction to Methods,Models andApplications,StudiesandReports in Hydrology,UNESCOPublishing,680pages,Paris,2005(ebook)

Mays,L.W.,andY.K.Tung,Hydrosystems Engineering andManagement,McGrawHill,New York,1992.

Grigg,N.S.,WaterResources Management,McGrawHill,NewYork,1996.

https://mycourses.ntua.gr/

online mycourses (,/)

Email:cmakro@mail.ntua.gr

2012

- (DEDUCTION INDUCTION)

( , 2, 23)

Deduction

Induction

(, )

(, , )

Deduction

Induction

Deduction

Induction

Xt=k*xt-1*(1-xt-1) Xt :

X1o=0.660001 X2o=0.66

t

1t, X2t

1t-X2t

- -

:

(, )

, ,

.

( < )

(0.5)

(0.75)

(0.25)

(0.75-0.25)

1.5*(0.75-0.25)

3* (0.75-0.25)

> 3* (0.75-0.25)

-

n

Xx

n

ii

1

sX x

nxi

i

n

( )2

1

1

sx2

sx

x

xi

i

nX x

n( )

( )3 1

3

x

ii

nX x

n( )

( )4 1

4

Cn

n nsx

xx

( )

( ) /( ) ( ) ( )

3 2

2 3 2 1 2

C nn n nk

x

xx

3 4

21 2 3*

( ) * ( ) * ( ) *

( )

( )

},...,,{max.. 211 nn

iXXXTM

E T X X Xi

n

n. . min{ , ,..., } 1 1 2

1..n : n :

010

20

30

40

5-10 10-15 15-20 20-25 25-30 30-35 (m3/s)

(

%

)

0

20

40

60

80

100

5 10 15 20 25 30 (m3/s)

(

%

)

0

10

20

30

40

0 5 10 15 20 25 30 ()

(

m

3

/

s

)

0

2

4

6

8

10

12

5-10 10-15 15-20 20-25 25-30 30-35

(m3/s)

1 < 2 1 = 2 1 = 2 = 0 1 = 2

1 = 2 1 < 2 1 = 2 = 0 1 = 2

1 = 2 1 = 2 1 = - 2 1 = 2

>0

1)()()(0)()(

XXX

X

FxFFxXPxF

dxxdFXf

xFxXPF

XX

XX

)()(

)(1)(1

H x

H x

FFT 1

11

1

x

NnxF xX )(

: nx: o

0

200

400

600

800

1000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

(

m

m

)

0

200

400

600

800

1000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

(

m

m

)

Fx(800)=18/25=0.72=72%F1(800)=7/25=0.28=28%

:

Fx(1000)=25/25=1=100%F1(1000)=0/25=0=0%

:

1)( N

nxF xX

020

40

60

80

100

5 10 15 20 25 30 35

(m3/s)

(

%

)

0

10

20

30

40

0 20 40 60 80 100 (%)

(

m

3

/

s

)

0

10

20

30

40

0-5 5-10 10-15 15-20 20-25 25-30 30-35 35-40 40-45

0

20

40

60

80

100

5 10 15 20 25 30 35 40 45

010

20

30

40

0 20 40 60 80 100 (%)

(

m

3

/

s

)

16

Weibull Normal LogNormal Galton Exponential GammaPearsonIII LogPearsonIII Gumbel Max EV2-Max Gumbel Min WeibullGEV Max GEV Min Pareto GEV-Max (k spec.) GEV-Min (k spec.)

(%) - : 9

9

,

9

5

%

9

9

,

9

%

9

9

,

8

%

9

9

,

5

%

9

9

%

9

8

%

9

5

%

9

0

%

8

0

%

7

0

%

6

0

%

5

0

%

4

0

%

3

0

%

2

0

%

1

0

%

5

%

2

%

1

%

,

5

%

,

2

%

,

1

%

,

0

5

%

1.200

1.150

1.100

1.050

1.000

950

900

850

800

750

700

650

600

550

500

450

400

350

300

250

200

150

100

50

0

(Gauss)K Gumbel

010

20

30

40

-4 -3 -2 -1 0 1 2 3 4 Gauss

(

m

3

/

s

)

(%)

(%)

()

0.2% 2.3% 16% 50% 84% 97.7% 99.8%

99.8% 97.7% 84% 50% 16% 2.3% 0.2%

1.002 1.02 1.2 2 6.2 43.5 500

i z=(Xi-)/ =0, =1

(0,1)z=1, F=0,8413

i=15

z=(15-10)/5=1

z=1

=10, =5

F=84,1%

=1(1-0,8413)=6

i = 1.5

F=1-(1/1.5)=0,333

z=-0.43

(0,1) F=1-0.333 z=0.43 F=0.333 z=-0.43

F=33.3%

(Xi-10)/5=-0.43 Xi=7.8

020

40

60

80

100

-3 -2 -1 0 1 2 3

F(x) (%) -3 3

0

1

2

3

4

-3 -2 -1 0 1 2

F(x) (%)

68.3%

95.4%

99.7%

:

200 hm3 : 10 hm3

: 20 hm3 :

100 hm3 : 30 hm3

: 10 hm3 :

110 hm3 : 40 hm3

1000 .

110

94

85

65

130

...

130

100

30

. 105

90

80

80

110

...

120

110

40

1

2

3

4

5

...

1000

200

190

210

170

160

...

220

200

10

+++->=0

20+10+110+105-200=+55

20+10+94+90-190=+24

20+10+85+80-210=-15

20+10+65+80-170=+5

20+10+130+110-150=+120

............................................

20+10+110+110-220=+30

.........

: 180

: 180/1000=18%

1 101 201 301 401 501 601 701 801 9010

100

200

300

400

500

1 101 201 301 401 501 601 701 801 9010

100

200

300

400

500

1 101 201 301 401 501 601 701 801 9010

100

200

300

400

500

m (hm3) s (hm3)Q 100 30Q 100 40 200 10 30

m (hm3) s (hm3) P (%)(QA,QB)=1 Q+Q 200 70 29(QA,QB)=0 Q+Q 200 50 23(QA,QB)=-1 Q+Q 200 10 7

R n

: F=1-F1=(1-1/)

n : (1-1/)n

n (): R=1-(1-1/)n

=10 n=10

R=1-(1-1/10)10=0.65=65%

xdxexFx x

2)(

21

21)(

2)(*5.0

21)(

x

exf

H

T

T

SzxxSzxx

2/1min

2/1max

)()()()(

TS

Z1+/2 %

ST xT

N

2)(1

2T )/11()( TZTK

H

2)ln(

21

*21)(

x

YX ex

xf2

)ln(21

0

*21)(

s

Y

x

X esxF

x

x

Sxxz

ln

lnln xx xzSx lnlnln xx xzSex lnln

( )

22ln /1ln( xSS xx 2/ln

2lnln xx Sxx

Z

GUMBEL

)(

)(cxae

X exF

( )

)()()(cxaecxa

X aexf

xSa /282,1xSxc 45,0

aTc

aFcTx x ))/11ln(ln()lnln()(

xSTkxTx *)()(

))/11ln(ln(*78.045.0)( TTk

22,1 )(*1.1)(*1396.11)()( TkTkn

STxTx x

GUMBEL

)(

1)(cxae

X exF

( )

)()()(cxaecxa

X aexf

xSa /282,1xSxc 45,0

aTc

aFcTx x ))/1ln(ln()1ln(ln()(

WEIBULL

kcxX exF

)/(1)(

( )

kcxkX ec

xckxf )/(1)(*)(

)11(k

c

22

2

)11(

)21(1

k

k

kkx TcFcTx /1/1 )/1ln(*)1ln(*)(

, c, Weibull(x)

LOG PEARSON III

)(ln1)ln(*)(

)( cxX ecxxxf

dsecsx

xFx

e

csX

c )(ln1)(ln*)()(

lnx

x

x

Sxxz

ln

lnln xx xzSx lnlnln xx xzSex lnln

: 250*106 m3

: 50*106 m3

: m= 2 m3/s, s=1 m3/s

: m= 8 m3/s, s=3 m3/s

:(1) (2) 50% (3)

786 m ,

(4)

(5) .

787 m 300*106 m3 25 km2

50*106 m3

: 250*106 m3

: 50*106 m3

: 63*106 m3

: 252*106 m3

A : 300+252=552 *106 m3

A B: 50+63=113 *106 m3

:(1) =552-50=502*106 m3(2) 50% =502-250*0.5=377*106 m3(3) 786 786 m : 300*106 m3-(787-786) m* 25 km2=275*106 m3(4) (3) (5) 125*106 m3 275*106m3 377-275=102*106 m3 125-102=23 *106 m3 =275*106 m3 =113-23=90*106 m3

=1.1

: 250*106 m3

: 50*106 m3

:

21*106 m3

:

126*106 m3

A : 300+126=426 *106 m3

A B: 50+21=71 *106 m3

:(1) =426-50=376*106 m3(2) 50% =376-250*0.5=251*106 m3< 275*106 m3 =275*106 m3 =71-(275-251)*106 m3=47 *106 m3(3) 786 786 m : 300*106 m3-(787-786) m* 25 km2 =275*106m3(4) (3) (5) 125*106 m3 47 *106 m3 =275*106 m3 =0

0 =1.04

: 250*106 m3

: 50*106 m3

:

7.5*106 m3

:

0*106 m3

A : 300*106 m3 A B: 50+7.5=57.5 *106 m3

:(1) =300-50=250*106 m3(2) 50% =57.5-57.5=0*106 m3 =250-(250*0.5-57.5)*106 m3=182.5 *106 m3(3) 786 786 m : 300*106 m3-(787-786) m* 25 km2 =275*106m3(4) (3) (5) =182.5*106 m3 =0

, , .

:() (),

()

() ( )

, .

(flash floods) . . .

-

. , .

:

Q = 0.278 * C * i * A :

Q (m3/sec): C : i (mm/hr): A (km2) :

, .

: km2 : c

=100

: t (hr)

=10

1.

3.

2.

1**

b

b

taitah

1

10

1 10 100

T=2

T=5

T=10T=20T=50T=100T=200

i=a*tb-1

, t hr

,

i

m

m

/

h

r

4. :

ic (mm/hr)=*tb-1

5.

Q (m3/s)=0.278*c* ic*A

( , )

i1 i2 i3i1 i2 i3

0

100

200

300

0 1 2 3 4 5 6

(

m

3

/

s

)

0

100

200

300

0 1 2 3 4 5 6

(

m

3

/

s

)

1

10

1 10 100

T=2

T=5

T=10T=20T=50T=100T=200

=100 =10

i1 i2 i3i1 i2 i3

0

100

200

300

400

0 1 2 3 4 5 6 7 8

(

m

3

/

s

)

0

100

200

300

400

0 1 2 3 4 5 6 7 8

(

m

3

/

s

)

1

10

100

1000

10000

100000

1 10 100 1000 10000 100000 1000000 10000000

(min)

(

m

m

)

Gouadeloupe26/11/1970

38 mm

Mongolia3/7/1975401 mm

o Cherrapunji, India

1-31/7/18619300 mm

o Cherrapunji, India

8/1860-7/186126461 mm

Reunion

6-7/1/19661825 mm

05

10

15

20

1

1

0

m

i

n

r

a

i

n

f

a

l

l

(

m

m

)

21/10/1994 10:00

21/10/1994 19:30

21-22/10/1994

19:30-20:30 67.7 mm

: 17.5 mm : 29.9 mm : 82.3 mm

21/10/1994 16:00

21/10/1994 22:00

22/10/1994 04:00

110

100

1000

0 1 10 100

Rainfall duration (hr)

M

a

x

i

m

u

m

i

n

t

e

n

s

i

t

y

(

m

m

/

h

r

T=10 YEARS

T=50 YEARS

T=500 YEARS

max 10 min 17.5 mm

max 20 min 29.9 mm

max 30 min 36.3 mm

max 1 h 67.7 mm

max 2 h82.3 mm

max 3 h 93.7 mm

max 6 h 100.0 mm

max 12 h 162.1 mm

max 24 h 167.1 mm

21-22/10/1994

110

100

1000

0.1 1.0 10.0 100.0

(hr)

T = 50

T = 10

21/10/1994

20/11/1993

T = 2

13/1/1997

11/1993, 10/1994 1/1997

Weibull Gumbel Max

(%) - : Gumbel (Max)

9

9

,

9

%

9

9

,

5

%

9

8

%

9

5

%

9

0

%

8

0

%

7

0

%

6

0

%

5

0

%

4

0

%

3

0

%

2

0

%

1

0

%

5

%

2

%

1

%

,

5

%

,

2

%

,

1

%

,

0

5

%

40

38

36

34

32

30

28

26

24

22

20

18

16

14

12

10

8

6

4

2

0

Weibull Gumbel Max

(%) - : Gumbel (Max)

9

9

,

9

%

9

9

,

5

%

9

8

%

9

5

%

9

0

%

8

0

%

7

0

%

6

0

%

5

0

%

4

0

%

3

0

%

2

0

%

1

0

%

5

%

2

%

1

%

,

5

%

,

2

%

,

1

%

,

0

5

%

50

48

46

44

42

40

38

36

34

32

30

28

26

24

22

20

18

16

14

12

10

8

6

4

2

0

Weibull Gumbel Max

(%) - : Gumbel (Max)

9

9

,

9

%

9

9

,

5

%

9

8

%

9

5

%

9

0

%

8

0

%

7

0

%

6

0

%

5

0

%

4

0

%

3

0

%

2

0

%

1

0

%

5

%

2

%

1

%

,

5

%

,

2

%

,

1

%

,

0

5

%

90

85

80

75

70

65

60

55

50

45

40

35

30

25

20

15

10

5

0

Weibull Gumbel Max

(%) - : Gumbel (Max)

9

9

,

9

%

9

9

,

5

%

9

8

%

9

5

%

9

0

%

8

0

%

7

0

%

6

0

%

5

0

%

4

0

%

3

0

%

2

0

%

1

0

%

5

%

2

%

1

%

,

5

%

,

2

%

,

1

%

,

0

5

%

105

100

95

90

85

80

75

70

65

60

55

50

45

40

35

30

25

20

15

10

5

0

Weibull Gumbel Max

(%) - : Gumbel (Max)

9

9

,

9

%

9

9

,

5

%

9

8

%

9

5

%

9

0

%

8

0

%

7

0

%

6

0

%

5

0

%

4

0

%

3

0

%

2

0

%

1

0

%

5

%

2

%

1

%

,

5

%

,

2

%

,

1

%

,

0

5

%

14013513012512011511010510095908580757065605550454035302520151050

Weibull Gumbel Max

(%) - : Gumbel (Max)

9

9

,

9

%

9

9

,

5

%

9

8

%

9

5

%

9

0

%

8

0

%

7

0

%

6

0

%

5

0

%

4

0

%

3

0

%

2

0

%

1

0

%

5

%

2

%

1

%

,

5

%

,

2

%

,

1

%

,

0

5

%

15014514013513012512011511010510095908580757065605550454035302520151050

1 h 2 h 6 h

12 h 24 h 48 h

This image cannot currently be displayed.

tbaitaitbahtah

b

b

ln*)1(lnln*ln*lnln*

1

)ln(*ln*lnln*)(* dtbTcahTdtah cb

)ln(*ln*lnln*)(* 11 dtbTcaiTdtaicb

,

h (mm)

t (hr)

T ()

20

1

2

30

1

5

60

1

50

40

1

10

80

1

100

30

2

2

40

2

5

70

2

50

50

2

10

90

2

100

80

24

2

100

24

5

120

24

50

140

24

10

170

24

100

........

........

........

V10 mm

10 mm/h MY 1h

V

5 mm/h MY 2h

V

3.33 mm/h MY 3h

D (hr)

D+1 (hr)

D+2 (hr)

V=10mm* A

1

tp hrQp m3/stb hrA km2

m

3

/

s

, hr

tp

Qp

tb

tb =2.52*tp10 mm*A km2=0.5*tb hr*Qp m3/sQp m3/s =0.01 m *A*106 m2/(0.5*2.52*tp*3600 s) Qp=2.2*A/tp

tb =7 hr A =100 km2

V=10 mm*100 km2= 1 *106 m3

V=0.5*tb hr*Qp m3/s Qp m3/s = 1 *106 m3 /(0.5*7*3600 s) Qp= 79.4 m3/s

kt)(exp)f(fff c0c

f

f

0

c

,

i

(

m

m

/

h

r

)

(hr)

,

Q

(

m

3

/

s

)

(hr)

V=Q*t

V=A*(i-)*tA:

0

100

200

300

0 1 2 3 4 5 6

(

m

3

/

s

)

0

100

200

300

0 1 2 3 4 5 6

(

m

3

/

s

)

0

100

200

300

0 1 2 3 4 5 6

(

m

3

/

s

)

0

100

200

300

400

500

600

0 1 2 3 4 5 6 7 8

(

m

3

/

s

)

0

100

200

300

0 1 2 3 4 5 6

(

m

3

/

s

)

U*i3/10

Qt=Ut*i1/10+Ut-1*i2/10+Ut-2*i3/10

U*i1/10

U*i2/10

Q1=U1*i1/10+U0*i2/10Q2=U2*i1/10+U1*i2/10+U0*i3/10Q3=U3*i1/10+U2*i2/10+U1*i3/10Q4=U4*i1/10+U3*i2/10+U2*i3/10Q5=U5*i1/10+U4*i2/10+U3*i3/10Q6=U6*i1/10+U5*i2/10+U4*i3/10Q7=U6*i2/10+U5*i3/10Q8=U6*i3/10

i1 i2 i3

Q0=U0*i1/10

0

100

200

300

0 1 2 3 4 5 6

(

m

3

/

s

)

0

100

200

300

0 1 2 3 4 5 6

(

m

3

/

s

)

0

100

200

300

0 1 2 3 4 5 6

(

m

3

/

s

)

0

100

200

300

400

500

600

0 1 2 3 4 5 6 7 8

(

m

3

/

s

)

0

100

200

300

0 1 2 3 4 5 6

(

m

3

/

s

)

U*i3/10

Qt=Ut*i1/10+Ut-1*i2/10+Ut-2*i3/10

U*i1/10

U*i2/10

Q1=U1*i1/10+U0*i2/10Q2=U2*i1/10+U1*i2/10+U0*i3/10Q3=U3*i1/10+U2*i2/10+U1*i3/10Q4=U4*i1/10+U3*i2/10+U2*i3/10Q5=U5*i1/10+U4*i2/10+U3*i3/10Q6=U6*i1/10+U5*i2/10+U4*i3/10Q7=U6*i2/10+U5*i3/10Q8=U6*i3/10

i1 i2 i3

Q0=U0*i1/10

-

1

(,)

(2000/60)://(riverbasins)

()

(GameTheory):(nash equilibrium)(pareto optimal)

2 ,

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Tech. Chron. A, Greece, 1987, Vol. 7, No 3

:

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/

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Disclaimer: !

y=12.166e0.0145xR=0.7667

0

50

100

150

200

250

300

350

0 50 100 150 200 250

M

(

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26 70 97 16 5.7 0 35 71 97 8.4 1.8 0 40 72 97 0 0 0IV 60 73 97 - - -

1lb = 0,454kg1kg = 2,2 lb1 tn = 2200 lb1ft=0,305m1ft3=0,0284m3

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():1.

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W 50:=76,14lb/ft31,218tn/m3

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3. G =0,9*1060,8*106VN(50) =41*106 m3 =56W56 =1,22tn/m3 =63W63 =1,222tn/m3

1)50(ln1505049,2*4343,09,7250W

36

3

6

)50( 10*4150218,1

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mtnyrtn

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13

Morris,GregoryL.andFan,Jiahua.1998.ReservoirSedimentationHandbook,McGrawHillBookCo.,NewYork

6

: http://greek-energy.blogspot.com

: http://kpe-kastor.kas.sch.gr/energy1/alternative/hydrauliki.htm

1

CO2, SO2 /

( ) / CO2, NOx

/ CO2

1

/

: http://greek-energy.blogspot.com/2008/09/blog-post_07.html

(W)

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: http://kpe-kastor.kas.sch.gr/energy1/alternative/hydropower.htm

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: http://kpe-kastor.kas.sch.gr/energy1/alternative/hydropower.htm

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I = 1000 * 9.81 * 50 * 100 * 0.9= 44.145.000 W = 44.1 MW

E= 44.145.000 W * 3.000 hr = 132.4 GWh

-

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Mimikou, M. and Kanelopoulou, S., 1987

.

Q D

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4

-Mimikou, M. and Kaemaki, S., 1985

:

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.

: http://kpe-kastor.kas.sch.gr/energy1/alternative/hydrauliki.htm

(MW)

(km2)

Three Gorges 2011 18.300-22.500

632

Itaipu

2003 14.000 1350

Guri (Simn Bolvar)

1986 10.200 4250

Tucurui 1984 8.370 3014

4

Tucurui dam Guri (Simn Bolvar)

Itaipu Three Gorges

: , , 2011

(553,9 MW)

(925,6 MW)

(500 MW)

(879,3 MW)

(129,9 MW)

(70 MW)

: , , 2011

3.060 MW 4.000-5.000 GWh 8-10% . (437 MW), (384 MW) (375 MW) .

25

(1954-0,035) (1954- 3,8) (1955- 46,2) (1960- 300) (1966- 2805) (1969- 53) (1969- 0,46) (1974- 1020) (1981- 303) (1985-10) (1985-16) (1989-11) (1990-145) (1997-570) (1999- 3,6) (1999- 12)

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16 ( -

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4 80 Francis 320 598 21

2 75 Francis 150 237 17

1 6.2 Tube-S type 6.2 16

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(MW)

I (MW) (GWh)

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0 10 20 30 40 50 60 70 80 90 100 (%)

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- (384 MW) ( ) 440 GWh 615 GWh

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: , , 2011

0100

200

300

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: , , 2011

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: , , 2011

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, , Kuhn-Tucker. Kuhn-Tucker f, .

, :

(x, ) = f(x) + T g(x), x Rn i gi(x*) = 0 i, (x, ) f(x), (x*, *) = f(x*). x ( Lagrange).

i gi(x*) = 0 i = 1, , k df(x*)

dx + T dg(x

*)dx = 0

T

( )

, :

( Kuhn-Tucker) ,

x* * ( ) .

, (penalty functions). , , :

min (x) = f(x) + i = 1

k pi(x)

pi(x) 0 , pi(x) = 0 gi(x) 0, pi(x) > 0 gi(x) > 0. : pi(x) ( pi(x) 0

, pi(x) >> 0 ).

1:

;

Maximum

Minimum

:

(robust solution)

O

b

j

e

c

t

i

v

e

(

M

a

x

)

:

min/max f(x) = f(x1, x2, , xn)s.t. gj(x1, x2, , xn) , , = 0, j = 1, , k

ximin xi ximax , i = 1, , n ( ) :

( ) ( ) (

) :

=

( )

(x)

x1

x2

x1min

x2min

x2max

x1max

,

x1

x2

x1

x2

x1

x2

,

,

,

(f(x))

-

0

.

0

5

0

.

0

5

0

.

1

5

0

.

2

5

0

.

3

5

0

.

4

5

0

.

5

5

0

.

6

5

0

.

7

5

0.04

0.18

0.33

0.48

0.63

0.78

-0.020

0.020.040.060.080.1

0.120.140.160.180.2

0.22

-

0

.

0

5

0

.

0

5

0

.

1

5

0

.

2

5

0

.

3

5

0

.

4

5

0

.

5

5

0

.

6

5

0

.

7

5

0.04

0.08

0.13

0.18

0.23

0.28

0.33

0.38

0.43

0.48

0.53

0.58

0.63

0.68

0.73

0.78

0.83

0.2-0.22

0.18-0.2

0.16-0.18

0.14-0.16

0.12-0.14

0.1-0.12

0.08-0.1

0.06-0.08

0.04-0.06

0.02-0.04

0-0.02

-0.02-0

x1x2

f(x1, x2)

( )

f(x) Rn . , x* :

f(x*) = grad f(x*) = 0 (stationary).

, f ( ). , (f(x*) = 0) , () x*. , , .

-

5

.

0

-

3

.

5

-

2

.

0

-

0

.

5

1

.

0

2

.

5

4

.

0

-5.0

-3.0

-1.0

1.0

3.0

5.0

05

1015202530354045

50

-0.05

0.15

0.35

0.55

0.75

0

.

0

4

0

.

1

3

0

.

2

3

0

.

3

3

0

.

4

3

0

.

5

3

0

.

6

3

0

.

7

3

0

.

8

3

f(x1, x2) = x12 + x22

f(x1, x2) = 0.5(1.1x1 x2)4 + 0.5(x1 0.5)(x2 0.5)

(x1*, x2*) = (0, 0), f* = 0

(x1*, x2*) =(0.314, 0.705), f* = -0.011

(x1*, x2*) =(0.618, 0.371), f* = -0.003

f(x), k g(x) := [g1(x), ..., gk(x)]T 0. x* f = (1, ..., k) :

, , Kuhn-Tucker. Kuhn-Tucker f, .

, :

(x, ) = f(x) + T g(x), x Rn i gi(x*) = 0 i, (x, ) f(x), (x*, *) = f(x*). x ( Lagrange).

i gi(x*) = 0 i = 1, , k df(x*)

dx + T dg(x

*)dx = 0

T

( )

, :

( Kuhn-Tucker) ,

x* * ( ) .

, (penalty functions). , , :

min (x) = f(x) + i = 1

k pi(x)

pi(x) 0 , pi(x) = 0 gi(x) 0, pi(x) > 0 gi(x) > 0. : pi(x) ( pi(x) 0

, pi(x) >> 0 ).

:

. 10 hm3, , , 6 hm3. 2 hm3 1 hm3 .

, . :

: (x1), (x2) (x3).

(= , ):

: : x1 + x2 + x3 = 10 : x2 +

x3 6 :

: x1 2 : x2 1 : x3 1

, ?

, ;

; Pareto : (i)

/

(ii) /.

. - (. )

- : PARETO

.

(Random Search) , (hill

climbing) , .

, .

(.. ), .

(.. 1 hm3), .

, 113 = 1331 , .

, .

n = 10 = 0.001 hm3: 1040

: 103 104

maxf=0For j=1 to n

x2=rand[]x3=rand[]x1=10-x2-x3Calculate: f(x1,x2,x3)IF f(x1,x2,x3) > maxf

maxf= f(x1,x2,x3)maxx1=x1maxx2=x2maxx3=x3

endNext j

n:

( )

(gradient or hill-climbing methods)

,

3 ( ).

j= 1, 2 and 3 xj R ( )

; (jNBj(xj).

. NBj(xj), xj j, :

;

x1, x2, x3 R:

(- gradients)

( : df/dx=0)

dNB(x1)/dx1 =0 x1= 3 (x2=2.33, x3=8) : 13.33 6; ( 13.33

).

Hill Climbing . Q

Q Q

Q.

: .

hill climbing

R:

( Qmax=8, R=2)

Q,

dNB/dx

-

Q

;

?

Lagrange

Bj(xj), .

, . pj,

j xj, .

Pj(xj) , pj, j xj.

( : = xj, dPj(xj)/dxj, ).

:

:

() :

:

p1=3.2, p2=4.0, p3=3.9 8.77, 13.96, 18.23 155.75. x1=10.2, x2=13.6 x3=14.5 = 38.3 .

; :

Lagrange

, Kuhn-Tucker: i gi(x*) = 0 i, (x, ) f(x), (x*, *) = f(x*).

10 10 :

;

(f(x*)=0)

(. 1-2 ) Q

http://www.lindo.com/

:

( )

( df/dx=0),

, .

- (hill-climbing Lagrange).

( ) .

() .

() .

, .

.

().

.

.

. ( (Optimality Principle))

() , .

.

.

( ).

(coding)!

3 ( ).

j= 1, 2 and 3 xj.

; (j NBj(xj). Q = 10.

/ ( .

: ( ) ( )

x1=0:2, x2=3:5 x3=4:6

x1=0:2, x2=3:5 x3=4:6 ( !)

(backward dynamic programming, BDP)

: [1,4,5]

(forward dynamic programming, FDP)

FDP .

, Stage.

(state variables). 1: .

, .

. m , n (. ) mn (stage).

(curse of dimensionality).

3 xj, j :

R(x1) = (12x1 x12), R(x2) = (8x2 x22) R(x3) = (18x3 3x32). max

(TotalR(X)) () 3, () 4.

:

: , ,

: ;

C(st, xt) st, xt, t, (Dt). :

:

(.. )

() (t).

st+1 ( ).

.

ft (st+1) st+1 t

f0(s1) = 0

g(x, t) .

y x

() 23

;

;

;

Simplex

, .

: 10 20 3. 60 1 2. 10 2.

. 3

(GeneticAlgorithms)

(EvolutionaryProgramming)

(SimulatedAnnealing)

:;

()().

The Gene is by far the most sophisticated program around.

- Bill Gates, Business Week, June 27, 1994

(C.Darwin,1858) =

()

,():

:

*

.

. ( )

.

.

.

:

. ..

.

()

:;

()(fitnessevaluation )

: , (), .

.

.

:;

::

:.

:.

...:;

..:

:()

(11,6,9)101101101001

220

..100101 =[(1) 25]+[(0) 24]+[(0) 23]+[(1) 22]+[(0) 21]+[(1) 20]=37

1000=?10001=?

:

)()()Pr( j j

ii xf

xfx

1001111010110010 10111110

10010010

10011110 10011010

:

()

(Holland,1975) (Schema)=0,1

*(dontcare)

..

()

L 2L .,.

1 * * 01 1 1 01 1 0 01 0 1 01 0 0 0

:

F(x)=x2

x [1,31].

:

31,5 .

(32=?)

:

(4):

1 =01101 =13102 =11000 =24103 =01000 =8104 =10011=1910

:

F(1)=132 =169F(2)=242=576F(3)=82 =64F(4)=192=361

:1170:293

:

A1

A2 A3

A4

: .

P(A1) = 0.14

P(A2) = 0.49

P(A3) = 0.06

P(A4) = 0.31

:

:

1 =01101

2 =11000

3 =11000

4 =10011

:1=011012=110003=010004=10011

::1 2 43 4 2:

1 = 0 1 1 0 | 12 = 1 1 0 0 | 0

3 = 1 1 | 0 0 04 = 1 0 | 0 1 1

1 = 0 1 1 0 | 02 = 1 1 0 0 | 1

3 = 1 1 | 0 1 14 = 1 0 | 0 0 0

:

:

1 = 0 1 1 0 02 = 1 1 0 0 13 = 1 1 0 1 14 = 1 0 0 0 0

1 = 0 1 1 0 02 = 1 1 0 0 13 = 1 1 0 1 14 = 1 0 0 1 0

:

1 =01100=1210=>F(12)=1442 =11001=2510 =>F(25)=6253 =11011=2710 =>F(27)=7294 =10010=1810 =>F(18)=324

:1822(1170):455.5(293)

:peak

z=f(x,y)=3*(1x)^2*exp((x^2) (y+1)^2)10*(x/5 x^3 y^5)*exp(x^2y^2)1/3*exp((x+1)^2 y^2).

5 10

; :

:

P(mutation)0

.fitness().

.

()(). ().

.

:.

:

3 5 =6

()

[3,1,2]x1=3,x2=1,andx3=2.

[1,0,1] 2.

?

:

().

0.30.

30% .

312 101:302and111.

:

( 01).

,1()05,21).

112102.

312 101 301 102.

(30116.5&10219.0)

:

35.5. 30116.5/35.5= 0.4710219/35.5=0.53

[0to1],0to0.47 301.102.

.

(Genetic Programming - GP)

:

/() :

:(),,

:EPR(Giustolici andSavic,2006)

.(datafitting)

aij, (Wi), Cj,Qj,j.

Cj=iWiaij/Qj

;()

;

(SimulatedAnnealing)

(annealing) (..).,.,.,, .

: (dE),,Boltzman:

(k:)

(SimulatedAnnealing)

, .

,,,.

.,,,

Metropolis:

P(dx)=exp(dx/T)>r

:dx: T:r: [0,1]

.

.

=0,(: hillclimbing)

=

,

()

:

AntColonyOptimisation

ParticleSwarmOptimisation a

;

:,

,,.

: ,,.

:,,

:

,.

: ,.

:,.

: ,,.

:(),.

:,, (.)

:,(,),.

:,(.).

;

(Duanetal.,1992):

(effectiveness),()

(efficiency),(,).

(..,,(hillclimbing)).

. ,

.

Domi_Ma8hmatos.pdfLesson_2_Hydro-Statistics.pdfLesson_3_Reservoirs.pdfLesson_4_More_Reservoirs.pdfLesson_6_Hydroelectric_works.pdfLesson_6_Systems-Models-Simulation-Uncertainty_1.pdfLesson_6_Systems-Models-Simulation-Uncertainty_2.pdfLesson_7_Algorithms_Flowchart_Thema.pdfOptimisation_1_2013.pdfOptimisation_1_2013_PartB.pdfOptimisation_Advanced.pdf