eutrophication control in lakes and reservoirs using...

Eutrophication control in Eutrophication control in lakeslakes andand reservoirsreservoirs

usingusing simultaneoussimultaneous dynamicdynamicoptimizationoptimization approachesapproaches

MariaMaria Soledad DiazSoledad Diaz

Planta Piloto de Ingeniería Química (PLAPIQUI) Universidad Nacional del Sur – CONICET

Bahía Blanca, ARGENTINA

Outline

MotivationMotivation

ObjectiveObjective

BiologicalBiological andand biochemicalbiochemical determinationsdeterminations

Global Global sensitivitysensitivity analysisanalysis

DynamicDynamic parameterparameter estimationestimation problemproblem

OptimalOptimal control control problemproblem

SimultaneousSimultaneous approachapproach forfor dynamicdynamic optimizationoptimization

DiscussionDiscussion ofof resultsresults

ConclusionsConclusions

Motivation

Eutrophication as natural process of aging of water bodyEutrophication as natural process of aging of water body

WaterWater bodiesbodies increasinglyincreasingly eutrophiceutrophic duedue toto anthropogenicanthropogenicinputsinputs ofof nutrientsnutrients

Application of restoration strategies requires systematic study,Application of restoration strategies requires systematic study,modeling and optimization of eutrophication processes modeling and optimization of eutrophication processes

Cultural Eutrophication

Main anthropogenic Main anthropogenic sourcesource of`nutrientsof`nutrients: : AgriculturalAgricultural activitiesactivities((fertilizationfertilization) )

MainMain pointpoint sourcesource: : dischargedischarge ofof agriculturalagricultural, industrial , industrial andandurbanurban wastewaterwastewater

Over enrichment of nutrients (mainly P and N)Over enrichment of nutrients (mainly P and N)

Increase in the production levels and biomassIncrease in the production levels and biomass

Very strong development of phytoplankton communityVery strong development of phytoplankton community

Decrease in water depth caused by sediment accumulationDecrease in water depth caused by sediment accumulation

Objective

DevelopmentDevelopment ofof ecologicalecological waterwater qualityquality (eutrophication) (eutrophication) modelmodel

AnalysisAnalysis ofof thethe trophictrophic statestate ofof a a waterwater body body throughthrough itsits compositioncomposition andandabundanceabundance ofof planktonplankton

Global Global sensitivitysensitivity analysisanalysis andand determinationdetermination ofof sensitivitysensitivity indicesindices

ParameterParameter estimationestimation basedbased onon availableavailable data: data:

ModelModel validationvalidation

StudyStudy ofof thethe effecteffect ofof nutrientsnutrients concentrationconcentration andand environmentalenvironmental parametersparametersonon plankton plankton populationpopulation dynamicsdynamics

DeterminationDetermination ofof optimaloptimal biobio--restorationrestoration policiespolicies

Trophic classification of water bodies

OligotrophicOligotrophic

MesotrophicMesotrophic

EutrophicEutrophic

HipereutrophicHipereutrophic

< < [[nutrientsnutrients]]< < ProductivityProductivity

> > [[nutrientsnutrients]]> > ProductivityProductivity

Trophic classification of water bodies

<1<111--3333--5555--1010Depth of Secchi Depth of Secchi

diskdisk(m)(m)

>50>5055--505022--5511--22Superficial Superficial

chlorophyll achlorophyll a((µµglgl--11))

>5000>500020002000--5000500020002000PhytoplanktonPhytoplankton

(cellml(cellml--11))

>300>300150150--300300<150<150Inorganic Inorganic nitrogennitrogen

((µµggll--11))

>100>1002020--1001001010--202011--1010Inorganic Inorganic phosporusphosporus

((µµglgl--11))

HYPEREUTROPHICHYPEREUTROPHICEUTROPHICEUTROPHICMESOTROPHICMESOTROPHICOLIGOTROPHICOLIGOTROPHIC

El DivisorioStream

Station 4

Station 3

Longitude: 61º 38´ WLatitude: 38º 25´ S

51Provincial Route

Dam

Sauce GrandeRiver

Sauce GrandeRiver

Station 1

Station 2

20 m

Wetland

Arg

entin

a

Paso de las Piedras Reservoir


ProvidesProvides drinkingdrinking waterwater toto more more thanthan 450.000 450.000 inhabitantsinhabitantsfromfrom BahBahíía Blanca, Punta Alta a Blanca, Punta Alta andand toto a a petrochemicalpetrochemical complexcomplex

Lake Characteristics

Area of drainage basin Perimeter of coastline Surface Mean depth

1620 km2 60 km 36 km2 8.2 m

Maximum depth Maximum volume Retention time

28 m 328 Hm3 4 years


EutrophicEutrophic

Main source of nutrients: Main source of nutrients: agriculturalagricultural activitiesactivities

HighHigh phytoplanktonphytoplankton concentrationconcentrationduringduring springspring andand summersummer: : surfacesurface waterwater bloomsblooms

0 40 80 120 160 200 240 280 320 3600.0

0.2

0.4

0.6

0.8

O-P

hosp

hate

(mgl

-1)

T im e (days)

E u troph ica tion lim it O bserved da ta

0 50 100 150 200 250 300 3500 .0

5 .0x10 4

1 .0x10 5

1 .5x10 5

2 .0x10 5

2 .5x10 5

Tota

l Phy

topl

ankt

on (m

gl-1)

T im e (D a ys)

O b se rve d d a ta E u trop h ica tio n lim it

SurfaceSurface waterwater bloomsblooms

Natural phenomena caused by Natural phenomena caused by phytoplankton.phytoplankton.

Phytoplankton: microscopic floating Phytoplankton: microscopic floating algae (first link of the algae (first link of the trophictrophicchain).chain).

InIn favorable favorable environmentalenvironmentalconditionsconditions theythey are are multipliedmultiplied andandconcentratedconcentrated in in thethe surfacesurface, , => => fastfast increaseincrease in in algalalgal biomassbiomass..

PhotosynthesisPhotosynthesis

106 106 COCO22 + 16 + 16 NONO33-- + + HPOHPO44

22-- + 122 + 122 HH22OO + 18 + 18 HH++

Solar Solar radiationradiationCC106106HH263263NN1616P P + 138 + 138 OO22

algaealgae

Problems caused by water blooms

ForFor manman

Blockage of waterBlockage of water--filtersfilters

Unpleasant odorUnpleasant odor and tasteand taste

AestheticsAesthetics

Presence of potentially toxic Presence of potentially toxic algaealgae

ForFor ecosystemecosystem

Reduction of biodiversityReduction of biodiversity

Anoxic conditions Anoxic conditions

ShadeShade

BlockageBlockage ofof fishfish gillsgills

Blockage of waterBlockage of water--filtersfilters

ClosteriumClosterium spp.spp.

StaurastrumStaurastrum spp.spp.

AulacoseiraAulacoseira sppspp..

CeratiumCeratium hirundinellahirundinella AnabaenaAnabaena circinaliscircinalis

Unpleasant odorUnpleasant odor and tasteand taste

AestheticsAesthetics

Toxic Cyanobacteria

Biological determinationsQualitative analysisQualitative analysis

Plankton Plankton netnet (30 (30 µµm)m)Observation to the optical Observation to the optical microscope of the alive and fixed microscope of the alive and fixed samples (samples (formolformol 4%)4%)DeterminationDetermination basedbased onon keyskeys

Quantitative analysisQuantitative analysis

RutnerRutner waterwater samplersamplerIn situIn situ fixationfixation withwith Lugol`sLugol`s solutionsolutionPhytoplanktonPhytoplankton enumerationenumeration in in invertedinverted microscopemicroscope by by UtermUtermööhlhlmethodmethod (1958)(1958)PhytoplanktonPhytoplankton biovolumebiovolumeCalculationCalculation ofof mgCmgC..

CyanobacteriaCyanobacteria

DiatomsDiatoms

ChlorophytesChlorophytes

PhysicoPhysico--chemicalchemical determinationsdeterminations

NitratesNitrates

NitritesNitrites

AmmoniumAmmonium

OrganicOrganic NitrogenNitrogen

PhosphatesPhosphates

OrganicOrganic PhosporusPhosporus

SiliceSilice

Water temperatureWater temperature

Solar radiationSolar radiation

pHpH

DissolvedDissolved OxygenOxygen

BiochemicalBiochemical DemandDemandofof OxygenOxygen

Depth of Secchi diskDepth of Secchi disk

Ecological Water Quality Model

DynamicDynamic massmass balances balances forfor nutrientsnutrients andandphytoplanktonphytoplankton

ConcentrationConcentration gradientsgradients alongalong waterwater columncolumnheightheight

PartialPartial DifferentialDifferential EquationsEquations SystemSystem

SpatiallySpatially DiscretizationDiscretization: Horizontal : Horizontal layerslayers

DifferentialDifferential AlgebraicAlgebraic SystemSystem (DAE)(DAE)

AssumptionsAssumptions

Horizontally averagedconcentration

Phosphorus limitingnutrient (for algaegrowth)

Constant density

Constant transversal area in lake

)C(CΔh

Ak)C(CΔh

Ak-VrCQ-CQ dt

)Vd(Cj,iij

i-

dN

mjiij

i

diij

N

kijki,k

iij OUTINOUTININ

ijim

IN1

111

1−

=+

=−−∑ −+∑=

Mass balance for horizontal layers

dtdh

hC

)C(ChΔhAk)C(C

hΔhAk-rC

VQ

-CV

Q

dtdC i

i

ij,jiij

i-i

dN

m,jiij

ii

dijij

ii,mN

kijk

i

i,kij OUTIN

OUTININ

IN−−−∑ −+∑= −

=+

=1

111

1

⎥⎦

⎤⎢⎣

⎡daymg

ii= horizontal layer= horizontal layer

kk = Tributary inflows: El = Tributary inflows: El DivisorioDivisorio, Sauce Grande, Sauce Grande

mm = Withdrawals: drinking water+complex, Sauce Grande= Withdrawals: drinking water+complex, Sauce Grande

jj = Cyanobacteria, Diatoms, = Cyanobacteria, Diatoms, ChlorophytaChlorophyta, NO, NO33,NH,NH44, ON,, ON,

POPO44,OP, BDO, DO,OP, BDO, DO

Mass balance for horizontal layers

dtdh

hC

)C(ChΔhAk-rC

VQ

dt

dC UU

UjN

kLjUj

UU

dUj

N

mUjC

UVUQ

UjkU

U,kUj ININ

IN OUT OUT −∑ −+==

∑=

−1 1

dtdh

hC

)C(ChΔhAkrC

VQ

dtdC L

L

LjN

mU,jLj

LL

dLjLj

LLLj OUT OUT

−∑ −++==1

Upper Layer

Lower LayerUi =

Li =

State variables and biogeochemical processes

Boundaryconditions Temperature PARAdvection and

dispersion

Inflow Outflow

CBOD PO4

ON PoolOP Pool

NO + NO3 2NH3

Dissolved oxygenZooplankton

Chlorophyta

Diatoms

Cyanobacteria

ATMOSPHERE

WATER

SEDIMENT

Nitrification

Mineralization

Sed

imen

t flu

x

Death

Grazing

Set

tling

Set

tling

Set

tling

Settl

ing

MineralizationD

eath

Upt

ake

Respiration

Photosynthesis

Denitrification

Death

Rate equations (rij)

ii = upper and lower layers= upper and lower layers

jj = Cyanobacteria, Diatoms, = Cyanobacteria, Diatoms,

ChlorophytesChlorophytes

ijj,growthij,growth C**f(N)*f(T)*f(I) k R =

iji

sedimj,sedimij, *Ch

* kR 1=

kpjiPOCiPOC )N(f

+=

44

GrowthGrowth

)IjIi(1 exp

IjI f(I) −=

RespirationRespiration

DeathDeath

SettlingSettling

graz,ijsettlingij,ij,deathij,respij,growthij RRRR Rr −−−−=

i

i

opt

optjij

T

)TT()T(f 2

2−−=

( )ij

Trrespiresp,ij C,kR 20−θ=

( )ij

Tmdeathideath,ij C,kR 20−θ=

jZoograzij

ijj,grazij,graz C

KCC

k R+

=GrazingGrazing


i i = upper and lower layers= upper and lower layers

jj = ON, OP= ON, OP

sedimij,minerij,ij,deathij -R-R Rr =

)C*f*k*(a R imj3

1mdeathm,jcij,death ∑=

=

∑=

+

∑== 3

1mimCkmjc

3

1mijC*Cim

20)-exp(Temp*minerminerminerij, ** kR θ

iji

Djsedim,jksedimij, C*

D)f(*

R−

=1

DeathDeath

MineralizationMineralization

SettlingSettling


i i = upper and lower layers= upper and lower layers

jj = PO= PO44

uptakeij,minerij,ij,deathij -RR Rr +=

)C*)f(1*k*(a R impo3

1mdeathm,pcij,death −∑=

=

∑=

+

∑== 3

1jimCpckm

3

1miOPC*imC

20)-exp(Temp*minerminerminerij, ** kR θ

)C*a*(RR impc3

1mgrowth,imuptakeij, ∑

==

DeathDeath


UptakeUptake

Rate equations ((rij))


j j = NO= NO33

denitrij,uptakeij,nitriij,ij R-RR r −=

iDOnioiDOiNH4)Tempexp(*nitrinitriij,nitri Ck

C*C** k R

+= − 20θ

))C*)PNH(*R*a( R imm

growthim,ncuptakeij, ∑ −==

3

141

iDOCkk*iNOC*kR

nono)Tempexp(*denitrdenitrdenitrij, +

−=3

3320θ

NitrificationNitrification

UptakeUptake

DenitrificationDenitrification



jj = NH= NH44

uptakeij,minerij,ij,deathij -RR Rr +=

)C*)f(*k*(an R imON3

1mdeathm,cij,death −∑=

=1

∑+

∑=

=

=3

1mimmpc

3

1miONim

20)-exp(Temp*minerminerinermij,Ck

C*C** kR θ

)C*PNH*R*a(R imm

growthim,ncuptakeij, ∑=

=3

14

DeathDeath


UptakeUptake

Global sensitivity analysis

Local vs. Global Sensitivity Analysis

Quantitative variance-based Global Sensitivity AnalysisModel independentIncorporate the effect of range of input variation and its pdfAllow multidimensional averagingAllow parameter grouping


Given model output Y=f(x), x vector of input factors

Output variance

mean output variance that remains if xi fixed (known)expected reduction in output variance if xi fixed

Sensitivity index input xi

( )( ) ( )( )ii xYEVxYVE)Y(V +=

( )( )ixYVE

( )( )ixYEV

( )( ))Y(VxYEV

S ii =


Decompose model output Y=f(x), as the sum of terms of increasing dimensionality

If input parameters are mutually independent ( )unique decomposition of f such that the summands are orthogonal.

Vi, Vij, V1,2,…,k : Variance of fi, fij, f1,2,…,k

( ) ( ) ( ) ( )k1k,...,2,1kji1

jiijk

1iii0k1 x,...,xf...x,xfxffx,...,xf ++++= ∑∑

≤<≤=

∫ =1

0ki,...,i 0dxf sl

∑ ∑= ≤<≤

+++=k

1ik,...,2,1

kji1iji V...VVV

Sobol’ sensitivity indices

Higher orders indices calculation: computationally expensive

Total sensitivity index


∑ ∑= ≤<≤

+++=k

i

k,...,,

kji

ijiV

V...

VV

VV

1

21

11

∑ ∑= ≤<≤

+++=k

ik,...,,

kjiiji S...SS

121

11

( )( ))Y(V

xYVES iT

i−=

Calculation of Sobol’ sensitivity indices (Monte Carlo)

Generate two independent random sets ξ and ξ´, let ξ = (η, ζ) ; ξ´ = (η´, ζ´)

Evaluate

f (η, ζ)f (η, ζ´) f (η´, ζ)



)Y(E)(fN

pN

ii ⎯→⎯ξ∑

=1

1

)Y(E)Y(V)(fN

pN

ii

2

1

21+⎯→⎯ξ∑

=

)Y(EV),(f)(fN y

pN

ii

´ii

2

1

1+⎯→⎯ζηξ∑

=

)Y(EV),(f)(fN

Ty

pN

iii

´i

2

1

1+⎯→⎯ζηξ∑

=

)Y(VV

S yi =

)Y(V

VS

TT yi

−=1

Sobol’ indices

Calculate

50 100 150 200 250 300 3500.0

0.2

0.4

0.6

0.8

1.0

S Cya

noba

cter

ia

Time (Days)

anc apc fON K1 kmn kmp kni Knit LC LD LG mD mG titam titar titamn titamp vsC vsD vsG umaxC umaxD

50 100 150 200 250 300 3500

2

4

6

S int,C

yano

nact

eria

Time (Days)

Cyanobacteria

SSTT -- SSii

SSii

Sensitivity Indices: Si

50 100 150 200 250 300 3500.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

S int,P

hosp

hate

Time (days)

anc apc fON K1 kmn kmp kni Knit LC LD mD mG titam titar titamn titamp vsC vsD vsG umaxC umaxD umaxG

50 100 150 200 250 300 3500.0

0.2

0.4

0.6

0.8

1.0

S Phos

phat

e

Time (Days)

PhosphateSSii

SSTT -- SSii


50 100 150 200 250 300 3500.0

0.2

0.4

0.6

0.8

1.0

S j

Time (Days)

anc apc fON K1 kmn kmp kni Knit LC LD LG mD mG titam titar titamn titamp vsC vsD vsG umaxC umaxD umaxG

Diatoms

50 100 150 200 250 300 3500.0

0.2

0.4

0.6

0.8

1.0

S j

Time (Days)

Nitrate


Dynamic Parameter Estimation ModelDynamic Parameter Estimation Model

Initial Conditions

Process Model

( ) 0=tpuyxxf ,,,,,&

( ) 0=tpuyxg ,,,,

00 xtx =)(

Objective Function

,

Variable BoundsxL ≤ x ≤ xU, yL ≤ y ≤ yU

uL ≤ u ≤ uU, pL ≤ p ≤ pU

Parameter EstimationProblem

{ }2σ= diagV

( ) ( )∑ ∑ ∑= = =

− −−=φNI

i

NV

j

NL

kij

Mij

Tij

Mij xxVxxmin

1 1 1

121

SimultaneousSimultaneous ApproachApproach forfor DynamicDynamic OptimizationOptimization

Nonlinear DAE optimization problem

Discretization of Control and State variables Collocation on finite elements

Large-Scale Nonlinear Programming ProblemInterior Point Method (Biegler et al., 2002)

Process Model

( ) 0=tpuyxxf ,,,,,&

( ) 0=tpuyxg ,,,,

00 xtx =)(

Objective Function

,

Variable BoundsxL ≤ x ≤ xU, yL ≤ y ≤ yU

uL ≤ u ≤ uU, pL ≤ p ≤ pU

Dynamic Parameter EstimationProblem

{ }2σ= diagV

( ) ( )∑ ∑ ∑= = =

− −−=φNI

i

NV

j

NL

kij

Mij

Tij

Mij xxVxxmin

1 1 1

121

Initial Conditions

rSQP Algorithm

Initialization

LinearizationObjective function, constraints

and gradients at xk

Step for dependent variables (dY)(Linear system)

Step for independent variables (dZ)QP in null space (inequality constraints)

Line search or Trust region

Check ConvergenceOptimal Solution

dx = YdY + ZdZ

Basis selection

(Biegler et al., 2002)

Nonlinear Programming Problem

Application of Barrier methodApplication of Barrier method

)(xfmin

0)(s.t =xc

0 ≥x

∑−=

n

jjxlnxfmin

1)( μ

0)(s.t =xc

As μ 0, x*(μ) x*

Sequence of barrier problems for decreasing μ values

Barrier Method Algorithm: Primal-Dual Approach

Initialization

Step for dependent variables (dY)(Linear system)

Step for independent variables (dZ)Unconstrained QP in null space

Line search or Trust region

Check Barrier Problem Convergence

Check NLP Convergence

dx = YdY + ZdZ

Step for dual variables dv

Update μ, ε

Update B

B, μ,ε

No

Update B

OptimalOptimalSolutionSolution

(Biegler et al., 2002)

ParameterParameter EstimationEstimation ProblemProblem

Input data

Descriptive data for lake

Temperature, solar radiation, lake depth time profiles

Inflows and outflows time profiles

Nutrients concentration profiles in inflows

Initial conditions

State variables profiles for upper and lower layer (measured)

Numerical Results

109.9Optimal growth radiation cyanoIc (ly/d)0.405Max growth of diatomskdiatom,growth (d-1)24.52Optimal growth radiation of diatomsId (ly/d)

0.210Max growth of cyanobacteriakcyanob,growth (d-1)

0.654Max growth of chlorophyteskcyanoph,growth (d-1)89.74Optimal growth radiation chlorophytesIg (ly/d)

0.343Half-sat. conc. for oxygen lim. of nitrification

Knio (mg/l)0.015Rate coeff. mineralization OPkOP,miner (d-1)

0.092Rate coeff. mineralization ONkON,miner (d-1)

EstimatedParameterSymbol

Time horizon: 365 days – Data frequency: twice a weekDAE: 20 differential equations, 60 algebraic equations, NE =40 NC=3NLP: 10432 nonlinear equations, 52 Iterations, 4 barrier problems

(Estrada et al., 2008a,b)

0

0.2

0.4

0.6

0.8

1

0 50 100 150 200 250 300 350 400

Time (Days)

Con

cent

ratio

n (m

gl-1

)

0

0.5

1

1.5

2

0 50 100 150 200 250 300 350 400

Time (Days)

Con

cent

ratio

n (m

gl-1

)

Diatoms

Phosphate

Numerical Results

0

0.5

1

1.5

2

0 100 200 300 400

T im e (days)

00.5

1

1.52

2.5

0 100 200 300 400

Time (days)

Cyanobacteria

Nitrate

Numerical Results

Bio-restoration policies

Excessive nutrients that promote algal growth were identified as the most important problem in 44% of all U.S. lakes surveyed in 1998 (U.S. EPA 2000)

Nutrient management:– How much do nutrients have to be reduced to eliminate algal blooms?– How long will it take for lake water quality to improve once controls

are in place?– How successful will restoration be, based on water quality management

goals?– Are proposed lake management goals realistic and cost effective?

Bio-restoration policies

El DivisorioStream

Station 4

Station 3

Longitude: 61º 38´ WLatitude: 38º 25´ S

51Provincial Route

Dam

Sauce GrandeRiver

Sauce GrandeRiver

Station 1

Station 2

20 m

Wetland

Artificial wetlandBuilt for nitrogen and phosphorus removal (Lopez et al., 2007)

Next to El DivisorioStream

Global retention: 64% (phosphate)

Derivation of tributary inflows through wetland

Optimal control problem: Tributary inflows derivation

Case 1: Control variable: Tributary inflows profiles derivation to wetland for bio-remediation

st

dt.)t(Cmin tf

phytoj,j

2

0 250∫ ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−∑

=

)d/l(F5.0F0 DIVISORIOWETLAND ≤≤

DAE Eutrophication model

Case 1: Numerical results

0 50 100 150 200 250 300 350

0.4

0.5

0.6

0.7

0.8

0.9

240 260 280 300 320 340 3600.54

0.56

0.58

0.60

0.62

0.64

0.66

PO

4 Con

cent

ratio

n (m

g/l)

Time (days)

Optimization results (PO4 reduct) No PO4 loading reduction

PO

4 C

onc.

(mg/

l)

Time (days)

0 40 80 120 160 200 240 280 320 3600.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Cya

noba

cter

ia (m

gl-1)

Time (days)

Optimization results No PO4 loading reduction

(Estrada et al., 2008d)

NE = 40, NC = 3 NLP: 10581 nonlinear equations

Optimal control problem: Inlake bio-restoration

Case 2: Control variables: Tributary inflows profiles derivationto wetland for bio-remediation and Zooplankton concentration profiles (Removal of zoo-planktivorous fish)

st

dt.)t(Cmin tf

phytoj,j

2

0 250∫ ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−∑

=

)d/l(DIVISORIOWETLAND F5.0F0 ≤≤

)l/mg(C. zoo 5010 ≤≤

DAE Eutrophication model

Case 2: Control and state variables profiles

0 50 100 150 200 250 300 350

0

1

2

3

4

5

Zoop

lank

ton

Con

c. (m

g/l)

Time (days)

(Estrada et al., 2008d)

0 50 100 150 200 250 300 3500.0

0.4

0.8

1.2

1.6

Phyto profiles before restoration Optimal phyto profiles

Diatoms

Cyanobacteria

Phy

topl

ankt

on c

onc.

(mg/

l)Time (days)

NE = 40, NC = 3 NLP: 10741 nonlinear equations

Biological and physico chemical determinations at two depth level in Paso de las Piedras Reservoir. Current data collection at eight levels.

Development of rigorous eutrophication model

Global sensitivity analysis: ranking of input parameters

Formulation of parameter estimation problem subject to DAE system

Parameter estimation problem solved with advanced dynamic optimizationtechniques: simultaneous approach

Resolution of optimal control problem: bio-restoration policies

Conclusions

References

Arhonditsis, G. B. and Brett, M. T., Eutrophication model for Lake Washington (USA) Part. I. Model description and sensitivity análisis. Ecol. Model. 187, 140-178, 2005Biegler, L.T., A. Cervantes, A.Waechter, Advances in simultaneous strategies for dynamic process optimization. Chem. Eng. Sci. 57: 575-593, 2002Estrada V., E. Parodi, M.S. Diaz,“Dynamic Parameter Estimation Problem For A Water Quality Model”, Chem. Eng. Transactions, 11, 247-252, 2007Estrada V., E. Parodi, M.S. Diaz, Developing a Lake Eutrophication Model And Determining Biogeochemical Parameters: A Large Scale Parameter Estimation Problem, Comp. Aided Chem. Eng., 23, 1113-1118, 2008Estrada V., E. Parodi, M.S. Diaz, Development of eutrophication biogeochemical models: global sensitivity analysis and dynamic parameter estimation, submitted to J. Appl. Ecology, 2008Estrada V., E. Parodi, M.S. Diaz, A simultaneous dynamic optimization approach for addressing the control problem of algae growth in water reservoirs through biogeochemical models, FOCAPO, June 2008, Boston, USAJeppesen, E., Søndergaard, M., Jensen, Havens, Anneville, Hampton, Hilt, Kangur, Köhler, Lammens, Lauridsen, Portielje, Schelske, Straile, Tatrai, Willén, Winder. Lake responses to reduced nutrient loading: an analysis of contemporary long-term data from 35 case studies.Freshwater Biology, 50, 1747–1771, 2005.

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