maximization of the profit of hydropower stations used as a design tool jónas elíasson, university...
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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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