Maximization of the profit of Maximization of the profit of hydropower stations used as a hydropower stations used as a

design tooldesign tool

Jónas Elíasson, University of IcelandJónas Elíasson, University of Iceland

Research grants: National Power Company of IcelandResearch grants: National Power Company of IcelandIcelandic Research CouncilIcelandic Research Council

Optimization of hydropower Optimization of hydropower stations as a design toolstations as a design tool

1.Introduction, state of the art

2.Different objectives

3.Global optimization

4.Global optimization results, theoretical remarks

5.Case studiesFljotsdalur

1.State of the art1.State of the art

Local optimization common knowledge but:Difficult calculationsSearch for optimium time consuming

Global optimization Difficult in spreadsheets or matlabEasy in HYDRA using Genetic Algorithms

HYDRA software produced by Univ. of Icel in cooperation with NPCI and Tech. Univ. of Vienna

1.Objective Function1.Objective Function

Calculate income T(x1,..xn) from energy sales

Calculate the project investment cost C(x1,..xn) according to the design parameters x1,..xn

Add operation and maintenance cost e.g. C(x1,..xn) to form G(x1,..xn)

Calculate the net present value of the project, called the profit : NPV(x1,..xn)

NPV(x1,..xn) = T(x1,..xn) - G(x1,..xn)

1. Basic equations1. Basic equationsNPV = Net present value

Cr

)r1(1vC

r

)r1(1EkGTNPV

NN

e

0xdx

NPVNPVd

n

1ii

i

0x

NPV

i

Ex. 1: Local optimizationEx. 1: Local optimization

Headrace tunnel diameter

11e

i dx

dCa

dx

dEk0

x

NPV

r

1aCEk

r

1rvCEk

r

rvCE

r

kNPV ee

e

By use of Chezy’s formula of flow resistance and putting construction

cost equal to k3 times area we get

7/2

2e

3v

3

ek

C

Qe5,2

ak

kTA

Tk = duration time of max. load Ce = Chezy´s coef = water unit weight e = efficiency = Area coef

Using HYDRA givesUsing HYDRA gives

Global optimisation of layout and design parameters Dam height, tunnel diameter, position of

structures, etc. Similarity in calculations between projects Easy to create and compare different project

arrangements Simple to upgrade calculations when conditions

change, such as New unit or market prices Extended inflow series …...

2: Different objectives2: Different objectives

Choice of o.f. rises several question: What about resource utilization policyIs max profit the objective of the owner ?Is a more competitive price wanted ? Ex. 2

Environmental considerations ? Ex. 3

Ex. 2: Long term marginal costEx. 2: Long term marginal cost

Partial utilization may spoil the resource Large schemes more economical than small L. t. m. c. used in Iceland and Norway dC/dE = J, does this comply with HYDRA Use J instead of sales price to trans. net = ke

)1(J)1(dE

dCk0

r

r)(11))1(CEk(

dE

de

N

e

Ex. 3: Environmental policyEx. 3: Environmental policy

Total conservation => No development: Partial conservation => restrictions

on storage size (elevation H)on roads and earth workson layout of development scheme

Resulting effect in HYDRA:higher unit prices and e.g. xi < Vmax(H)total ban on some development schemes

3. Global optimisation3. Global optimisation

Select parameters and points to optimise, ex. 4 Run genetic algorithm on the objective

function Get list of best solutions Get a detailed description of any listed

solution Specific data Cost summary Bill of quantities...

3. Genetic Algorithm3. Genetic Algorithm

Based on Charles Darwin’s ‘Survival of the fittest’ Each arrangement is considered an individual (variables

encoded as genes in a chromosome) with a fitness equal to the profit it produces

Weak individuals tend to die before reproducing, while the stronger ones live longer and bear many offsprings

These offsprings often inherit the qualities that enabled their parents to survive

As time passes the individuals in each generation become fitter, i.e. the objective function comes closer to its optimum value.

0010110101

3. Program Objects 3. Program Objects how HYDRA is constructed

Pin PoutObjectQin, Hin, ... Qout, Hout, ...

Stand alone objects ‘plugged into’ the Hydra Able to recieve, modify, and return hydraulic

parameters such as head and discharge Return scheduled construction cost, plus

energy and power capacity

Hydra Model

External DatabaseTopographical Data

Geological Data

Hydrological Data

Design ParametersLayout

Existing Power System

Unit-Prices

Additional Cost

Plant Capacity

NPV Profit

Market Characteristics

Capacity Addition

Market Prices

Construction Cost

Project Investment

O & M Cost

CAD

Revenue Charges

Reserve Plant Cost

Structures

?

Steps in calculation

Not existing connection

Genetic Algorithm

Maximum Profit CAD

Input Data

Structure object

External application

Ex. 4: A small power stationEx. 4: A small power station Create and arrange stand alone structure objects together

Tailrace tunnel

Dam

Powerhouse

P1

P2

P3

Pr. shaft

P1 P3P2 P2

Dam Pr. shaft Powerhouse Tailrace tunnel

Ex. 4: Math and HYDRA Ex. 4: Math and HYDRA

P 50 50 50 50 50 20 20

G 100 100 100 100 100 200 200

0.001 0.005 0.01 0.025 0.05 0.025 0.05

D 4,0 4.0 4.0 4.0 3.9 4.0 4.0 3.9

H1 543,0 543 543 543 543 543 543 543

H2 48,2 44 49 49 48 42 50 44

H3 44,9 39 46 46 45 37 46 39

NPV 28594 28580 28594 28594 28590 28569 28593 28576

dNPV - -14 0 0 -4 -25 -1 -18

Math

sol

Selection of parametersSelection of parametersNo of generations and mutation probability

Development of Best Solution

27.000

27.200

27.400

27.600

27.800

28.000

28.200

28.400

28.600

28.800

29.000

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96

Generation

NP

V [

Mk

r]

50/0,01

50/0,025

50/0,05

50/0,005

50/0,001

10/0,01

4. Theoretical considerations4. Theoretical considerationsEx. 5: Average and break even power costEx. 5: Average and break even power cost

Is the average cost of power optimized ? Lowest price non-profit utility can offer ? ko = ke when profit H = 0

HκG

Gkk

T

Gk

k/T

G

E

Gk e0e

ea

aa0

5. Fljótsdalsvirkjun5. Fljótsdalsvirkjun Comparision of HYDRA results to actual project planning

study: Eliasson, Jensson and Ludvigsson; Optimal design of hydropower plants; Hydropower’97, Trondheim, Norway, July 1997

Same project with a detailed optimisation of al possible diversions into Eyjabakkar storage reservoir: Eliasson, J., Ludvigsson, G., Doujak, E., Ólsen, A. and Matthias, H. – B.; A proposal to exploit optimally the hydropower of Fljotsdalur Iceland, Waterpower 99 Conference, Las Vegas, Nevada, USA, July 7 – 9, 19991, (Published by ASCE on a CD-ROM ISBN 0-7844-0440-2)

Eyjabakkar todayEyjabakkar today

Eyjabakkar storageEyjabakkar storage

Fljótsdalsvirkjun - object mapFljótsdalsvirkjun - object map

FVHV

Eyjabakkarstorage

Grjótárdiversion

Hölknárdiversion

Laugarár diversion

Intake

Headracetunnel I

Headracetunnel II

Headracetunnel III

Headracetunnel IV

Penstock

Powerstation F

Tailracetunnel

Accesstunnel

Canal

Headracetunnel

Flow

}VV2 diversion

GSV

YSV

KV

GHV

Tail-racecanal

Powerstation K}Diversions

Dams

Tunnels

Penstock

Volume and area of Eyjabakkar storage

0,0

200,0

400,0

600,0

800,0

1000,0

1200,0

645 650 655 660 665 670 675 680

Eleveation (m.a.s.l.)

Vol

ume

[Gl]

0,0

10,0

20,0

30,0

40,0

50,0

60,0

Are

a [k

m2]

Volume

Area

Fljótsdalsvirkjun - dataFljótsdalsvirkjun - data

0

20

40

60

80

100

120

140

160

180

200

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43

Year

Gl/y

ear

Hafskurð

Hafkvísl

GrjotSaud

Laugará

Fellsá

Keldá

YSaudá

Annual flow 1950-93

Fljótsdalsvirkjun - dataFljótsdalsvirkjun - data

165 Hafursárskurður 2-week flow series in selected years

0

0,5

1

1,5

2

2,5

3

3,5

1 6 11 16 21 26

506263646574

7

50

62

6465 63

m3/s

time interval no

Fljótsdalsvirkjun - dataFljótsdalsvirkjun - data

Power station F 1997Power station F 1997Various diversions - optimal designVarious diversions - optimal design

Mikr = million icelandic kr GWh = Gigawatthours/annum kr/k = kronur/kwhEl = Elevation meters above sea level A.C. = Access tunnel diameter

Scheme Profit Hd Qd Pow Energ Inv. B.ev Stor. A.c.

M.IKR m m3/s MW GWha M.IKR kr/k El.. mVV2 13,821 577 28.4 144 1,025 17,949 1.30 666,0 4.3VV2+KV 16,542 588 33.1 171 1,212 21,040 1.29 667,5 4.8VV2+KV+GSV 17,846 583 35.5 181 1,284 22,015 1.28 667,5 4.8VV2+KV+GSV+YSV 20,154 596 40.0 205 1,466 25,320 1.29 670,0 5VV2+KV+GSV+YSV+GHV 20,636 582 42.0 214 1,527 26,702 1.30 670,5 5VV2+KV+GSV+YSV+GHV+FV 21,250 584 44.0 225 1,604 28,412 1.32 671,5 5.1VV2+KV+GSV+YSV+GHV+FV+HV 22,056 586 46.1 236 1,678 29,900 1.33 672,5 5.2

Power station K 1997 Power station K 1997 Various diversions - optimal designVarious diversions - optimal design

Scheme Profit Hd Qd Pow Ener-gy

Inv. Breakeven

Storage. Tun-nel

Mio IKR* m m3/s MW GWh/a

MioIKR

kr/kWh

El.m.asl.

m

No diversion 12.826 573 27,0 135 966 17.095 1,32 664,5 4,2KV 14.889 569 30,4 152 1.083 18.711 1,29 664,5 4,3KV+GSV 15.746 587 32,0 164 1.170 20.509 1,31 664,5 4,8KV+GSV+YSV 16.932 583 34,5 176 1.252 21.867 1,30 664,5 4,8KV+GSV+YSV+GHV 17.048 582 35,5 181 1.288 22.832 1,32 664,5 4,8KV+GSV+YSV+GHV+FV 17.564 580 36,5 185 1.324 23.439 1,32 664,5 4,8KV+GSV+YSV+GHV+FV+HV 18.072 575 37,8 190 1.356 23.920 1,31 664,5 4,8

Mikr = million icelandic kr GWh = Gigawatthours/annum kr/k = kronur/kwhEl = Elevation meters above sea level A.C. = Access tunnel diameter

Description Bestun Fast SkýrslaAðkomugöng Lengd m 1188 1188 1000Eyjabakkalón Flatarmál km2 48 48 43 Hámarks lónhæð m.a.s.l. 665 665 664,5 Lágmarks lónhæð m.a.s.l. 648 648 648 Nothæft lónrými Gl 574 574 500 Krónuhæð m.a.s.l. 668,7 668,7 668,5 Hámarks stífluhæð m 23,7 23,7 25 Lengd stíflu m 3813 3813 4100 Lengd yfirfalls m 234 234 360Aðrennslisgöng Heildarlengd m 30530 30530 31231 Meðal þvermál m 4,1 5 5Fallgöng Lengd m 451 451 450 Þvermál m 2,9 2,9 2,9Stöðvarhús Fjöldi hverfla 2 2 2 Hæð hverfla m.a.s.l. 40 40 41 Þrýstihæð við hönnunarrennsli m.a.s.l. 502,3 579,4 570,3 Meðal þrýstihæð m.a.s.l. 568,9 598,8 575,9Frárennslisgöng Lengd m 1285 1285 1200 Þvermál m 4,1 6,4 6,4

Meðalrennsli m3/s 32,3 32,3 31,3Hönnunarrennsli m3/s 43,3 43,3 42Afl MW 190 220 210Orka GWh/a 1.392 1465 1.400

Main dimensions !!!!

Economy of best schemes 1999Economy of best schemes 1999

Po En Prof Inv Price 0.pr. Eyjab.Scheme

MW GWh/a Mkr Mkr k/kwh kr/kWh m.y.s.

Fljótsdalsvirkjun I 210 1220 14.159 20.793 1,9 1,2 664,5HÁV+KV+GSV+YSV+GHV+FV+HV2 198 1387 17.174 22.076 1,9 1,1 671,5

HÁV+KV+GSV+YSV+FV2 176 1232 10.568 19.024 1,6 1,05 669HÁV+KV+GSV+YSV+GHV+FV2 182 1274 5.138 19.923 1,3 1,05 669

VV2 + KV +GSV +YSV1 206 1445 16.850 23.573 1,9 1,15 665,71 Fljótsdalsvirkjun II 2 Kiðufellsvirkjun

Optimized 210 MW 1999 Optimized 210 MW 1999

Po En Prof Inv Price 0.p. Eyjab.Scheme

MW GWh/a Mkr Mkr k/kwh kr/kWh m.y.s.

Fljótsdalsvirkjun I 210 1220 14.159 20.793 1,9 1,2 664,5HÁV+KV+GSV+YSV2 210 1224 14.151 20.262 1,9 1, 1 669,5VV2 + KV +GSV +YSV1 206 1445 16.850 23.573 1,9 1,15 665,71 Fljótsdalsvirkjun II 2 Kiðufellsvirkjun

ConclusionConclusion

Genetic algorithm can maximize the ‘badly behaving’ objective function easily

Enviromental consideration can be included and different objectives used

HYDRA gives us main dimensions of all structures and water levels

It is possible to optimize on tender prices Case studies have shown HYDRA effective

Thank you for listeningThank you for listening

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