Eutrophication control in lakes and reservoirs using...
Transcript of 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
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
Paso de las Piedras Reservoir
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
Rate equations (rij)
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
Rate equations (rij)
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
MineralizationMineralization
UptakeUptake
Rate equations ((rij))
ii = upper and lower layers= upper and lower layers
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
Rate equations (rij)
ii = upper and lower layers= upper and lower layers
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
MineralizationMineralization
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
Global sensitivity analysis
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 =
Global sensitivity analysis
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
Global sensitivity analysis
∑ ∑= ≤<≤
+++=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)
Global sensitivity analysis
Global sensitivity analysis
)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
Sensitivity Indices: Si
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
Sensitivity Indices: Si
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.
References
López, N., Alioto, Schefer, Belleggia, Siniscalchi, E.Parodi. Diseño de un humedal artificial para remoción de nutrientes de un afluente al Embalse Paso de las Piedras (Argentina). AIDIS, Uruguay, 10-15, 2007. Parodi, E.R., V. Estrada, N. Trobbiani, G. Argañaraz Bonini, Análisis del estado trófico del Embalse Paso de las Piedras (Buenos Aires, Argentina). Ecología en tiempos de Cambio. 178, 2004.Raghunathan, A., M.S. Diaz, L.T. Biegler, An MPEC formulation for dynamicoptimization of distillation operations, Comp. Chem. Eng., 28, 2037, 2004.Rodriguez, M., M.S. Diaz, Dynamic modelling and optimisation of cryogenic systems, Applied Thermal Engineering, 27, 1182-1190, 2007.Sobol´, I. M., Global sensitivity indices for nonlinear mathematical models andtheir Monte Carlo estimates. Math. Comput. Simulation 55, 271-280, 2001.Søndergaard, M., Jeppesen, E. Anthropogenic impacts on lake and stream ecosystems, and approaches to restoration. J. Applied Ecology, 44, 1089–1094, 2007.