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A Dynamic Interval Goal Programming Approach
to the Regulation of a Lake-River System
Raimo P. Hämäläinen
Juha Mäntysaari
S ystemsAnalysis LaboratoryHelsinki University of Technology
Systems Analysis LaboratoryHelsinki University of Technology
www.sal.hut.fi
Päijänne-Kymijoki lake river system
LAKEPÄIJÄNNE
KONNIVESI
RUOTSALAINEN
RIVERKYMIJOKI
0 10 20 30 40 50km
Jyväskylä
LAKEPYHÄJÄRVILahti
Kotka
Finland
4:th largest in Finland
Control: Outflow from Päijänne
to the river Kymijoki
Inflows: forecasted
Regulation policies:
Water levels at six time points
Päijänne-Kymijoki lake river system
LakePyhäjärvi
Lake Päijänne
Inflow
xp(t)A p(t)
9 1011
2
8
7
6
4
3
12
q(t) = Control
qL1 (t)
LakesRuotsalainen and Konnivesi
1
qp(t)
q2(t)
q21 (t)
5
Gulf of Finland
q in (t)
x(t)
A(t)
q23 (t)q22 (t)
q212 (t)q211 (t)
x(t)
A(t)
Inflow
dam
lakewater flowpower plant
qL2 (t)Inflow
Need for modelling
Development of feasible regulation strategies is a dynamic control problem
– No intuitive solutions
– Planning againts long historical inflow data
– Analysis of regulation impacts
– Many interest groups multicriteria optimization in a dynamic system
Goals in terms of water levels
Users give desired water levels at:
– six different points during one year
– ideal level + acceptable interval (min, max)
78.5
78.3
78.979.02
78.9179.02
77.35
77.15
77.44 77.4477.58
77.33
77
77.5
78
78.5
79
79.5
1.1
21.1
11.2 1.3
21.3
11.4 1.5
21.5
11.6 1.7
21.7
11.8 1.9
21.9
11.1
0
1.11
21.1
1
11.1
2
1.1
NN
+m Max
Goal
Min
Constraints
Outflow from Päijänne:
Min/max flow
Fixed and hard
Max change in outflow:Soft, violation penalties
Water level in the lake Pyhäjärvi:
Fixed rule based regulation
Part of the dynamics
q q qimin max
q q qi i 1 max
x x xpi
pmin max
Criteria and penalty functions
Criterion for goal levels:
Quadratic cost for differences of goal points from regulated water levels
Penalty outside the goal interval:Quadratic difference from the limits (min or max)
Penalty for violation of change in outflow rate:
Quadratic cost outside the maximum flow limit, otherwise zero
F x xkgoal
kk
K
2
1
P max x xx x , xk
mink k k
max
k
K
2
1
Pq max q qii
N2
1
Cost function minimized =Sum of deviations from goal + penalty outside goal intervals
cx = 10
cx = 1
cx = 0.01
MinGoal
Max
xgoalk
Interval
Criteria and penalty functions
Model assumptions
• Lake dynamics
• Optimization against one to four year history
• Lower dam regulation by a given rule
• Regulator uses a rolling two goal optimization principle
• Adjustment rules
10 days
Optimal
Goal optimization
Beginning of month
Adjusted bymeasurement
Updating of inflow forecastGoal optimization
Beginning of month
Updating of inflow forecastGoal optimization
Beginning of month
Updating of inflow forecast
Beginning of month
Goal optimization
Goal
Generation of the optimal regulation strategy
Goal programming• Goal (infeasible point)
• Problem: Find a point in the feasible set closest to the goal point/set Weighted, Min Max, Lexicographic
• Aspects in regulation:
– Dynamic problem
– Goal interval (set)
Why goal programming ?
• Economic, social and environmental impacts
37 primary + 20 secondary = 57 different impacts
• For example: Power production, flood damages, number of destroyed loon nests
• Some impacts are interdependent:energy produced and the value of energy
Use of tradeoff comparison questions or criteria classification becomes difficult
ISMO spreadsheet application
Minimizes deviations from goal levels and goal intervals
Satisfies flow constraints
Simulates the regulator’s operating principles
Preference model • Set of goal levels + acceptability intervals • Optimization againts history data for a selected one to four year
period
Modifiable parameters• Flow constraints in the river• steepness of the penalty function
ISMO spreadsheet application
Use of ISMO
User
Flow constraints
Goals
Hydrological Model(1 stage=10 days)
Weather
Annual inflowprediction updated
every month(=every third stage) Dynamic
optimization overperiod of 2 goal
points
Initial strategy
Adjustment ofthe stratetegy at
every stage
Flow measurementsat every stage
Impact models
Practical regulationstrategy
Multicriteria outcomesUser
User
User Updating Initialconditions each
month
ISMO example
Inflow 1980-1984
0.00
100.00
200.00
300.00
400.00
500.00
600.00
700.00
30.12 23.5 14.10 7.3 29.7 20.12 13.5 4.10 25.2 18.7 9.12
Utopia and realistic solutionsWater level
75.0
75.5
76.0
76.5
77.0
77.5
78.0
78.5
79.0
79.5
80.0
1-Jan-80 31-Dec-80 31-Dec-81 31-Dec-82 31-Dec-83
Cu
bic
me
ters
pe
r s
ec
on
d
0
100
200
300
400
500
600
700
800
900
Me
ters Inflow
Goal points
Utopia and realistic solutionsTotal outflows
75
76
76
77
77
78
78
79
79
80
80
1-Jan-80 31-Dec-80 31-Dec-81 31-Dec-82 31-Dec-83
Cu
bic
me
ters
pe
r s
ec
on
d
150
200
250
300
350
400
450
500
550
Me
ters
Max flow
Impacts
• Nature– Spawning areas for pike fish– Water level when ice melts– number of destroyed loon nests
• Social– Recreational losses– Professional fishing: Reduction of
the water level during 10-Dec and 28-Feb
• Economic– Power production– Flood damages– Days infavourable for log floating
Comparison of impacts:
User evaluates and modifies goal levels
Aktiivinen Vertailu
Taloudelliset vaikutukset Mittari (luvut keskiarvoja/vuosi, jos ei muuta mainittu)
Vesivoimantuotanto Sähkön määrä (MWh) 1,394,842 1,394,842 Sähkön arvo (mk) 272,353,166 272,353,166 Talvella tuotettu sähkö (MWh) (talvi=sähkön hinnoituksen mukainen) 631,268 631,268 Kesällä tuotettu sähkö (MWh) 763,574 763,574 Voimalaitosten ohijuoksutusten määrä kuukausittain (MWh) 12,450 12,450
Tulvavahingot Tarkastelujakson ylin vedenkorkeus Päijänteellä (NN+m) 79.16 79.16Tarkastelujakson ylin virtaama Päijänteeltä (m^3/s) 500 500
-Maatalous Vahinkojen määrä Päijänteellä (mk) 89,691 89,691 Vahinkojen määrä Kymijoella (mk) 27,167 27,167 Tulvapeltojen pinta-ala Päijänteellä (ha) 75 75Tulvan kesto Päijänteellä (vrk) 2.5 2.5Tulvan kesto Kymijoella (vrk) 99 99
-Yhdyskunnat Vahinkojen määrä Päijänteellä (mk) 1,014,704 1,014,704 Vahinkojen määrä Kymijoella (mk) 301,624 301,624 Tulvan kesto Päijänteellä (vrk) 48.5 48.5Tulvan kesto Kymijoella (vrk) 53 53Huutorajan Päijänteellä ylittävien päivien lkm (vrk) 30.5 30.5
Historia vertailuTesti
• ISMO is implemented in MS Excel 7.0 (MS Office 95)
– Solver provides optimization routines
– 10-20 minutes for one solution
• Benefits
– Rapid development
– Easy: data input, model modification, visualisation and printing
• Users accept easily
– Excel is a commonly used office program
Spreadsheet modelling works !
• Generation of alternative regulation strategies
• Impact tables of regulation
– a key info material in decision analysis interviews and conferences
• Sensitivity tool
– individual changes for water levels and related impacts
– helps representatives to better understand the restrictions of the system
Added value
Further development
• Different information patterns
• Iterative optimization of the goal levels to produce maximum amount/value of the energy
• Now used to develop new regulation policies and their impacts
References
– Marttunen, M., Järvinen, E. A., Saukkonen, J. and Hämäläinen, R. P., “Regulation of Lake Päijänne - a Learning Process Preceding Decision-Making”, Finnish Journal of Water Economy, 6:29-37, 1999.
– Hämäläinen, R. P., Kettunen, E., Marttunen, M. and Ehtamo, H., “Evaluating a framework for multi-stakeholder decision support in water resources management”, Manuscript, 1999. (Downloadable from http://www.sal.hut.fi/Publications/pdf-files/mhamb.pdf)
– Hämäläinen, R. P. and Mäntysaari, J., “A Dynamic Interval Goal Programming Approach the Regulation of a Lake-River System”, Manuscript, 2000. (Downloadable from http://www.sal.hut.fi/Publications/pdf-files/mhama.pdf)
– Hämäläinen, R. P., “Interactive Multiple Criteria Decision Analysis in Water Resources Planning”, Home pages of the Lake Päijänne project, 1998, www.paijanne.hut.fi
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