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