SLIDE 1 Eutrophication control in Eutrophication control in lakes lakes and and reservoirs reservoirs using using simultaneous simultaneous dynamic dynamic
- ptimization
- ptimization approaches
approaches
Maria Maria Soledad Diaz Soledad Diaz
Planta Piloto de Ingeniería Química (PLAPIQUI) Universidad Nacional del Sur – CONICET Bahía Blanca, ARGENTINA
SLIDE 2 Outline
Motivation
Objective
Biological and and biochemical biochemical determinations determinations
Global sensitivity sensitivity analysis analysis
Dynamic parameter parameter estimation estimation problem problem
Optimal control control problem problem
Simultaneous approach approach for for dynamic dynamic optimization
Discussion of
results
Conclusions
SLIDE 3 Motivation
- Eutrophication as natural process of aging of water body
Eutrophication as natural process of aging of water body
Water bodies bodies increasingly increasingly eutrophic eutrophic due due to to anthropogenic anthropogenic inputs inputs of
nutrients
- 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
SLIDE 4 Cultural Eutrophication
Main anthropogenic source source of`nutrients
: Agricultural Agricultural activities activities ( (fertilization fertilization) )
Main point point source source: : discharge discharge of
agricultural, industrial , industrial and and urban urban wastewater wastewater
- Over enrichment of nutrients (mainly P and N)
Over enrichment of nutrients (mainly P and N)
- Increase in the production levels and biomass
Increase in the production levels and biomass
- Very strong development of phytoplankton community
Very strong development of phytoplankton community
- Decrease in water depth caused by sediment accumulation
Decrease in water depth caused by sediment accumulation
SLIDE 5 Objective
Development of
ecological water water quality quality (eutrophication) (eutrophication) model model
Analysis of
the trophic trophic state state of
a water water body body through through its its composition composition and and abundance abundance of
plankton
Global sensitivity sensitivity analysis analysis and and determination determination of
sensitivity indices indices
Parameter estimation estimation based based on
available data: data:
Model validation validation
Study of
the effect effect of
nutrients concentration concentration and and environmental environmental parameters parameters
plankton population population dynamics dynamics
Determination of
bio-
restoration policies policies
SLIDE 6
Trophic classification of water bodies
Oligotrophic Oligotrophic Mesotrophic Mesotrophic Eutrophic Eutrophic Hipereutrophic Hipereutrophic
< < [
[nutrients nutrients] ] < < Productivity Productivity
> > [
[nutrients nutrients] ] > > Productivity Productivity
SLIDE 7 Trophic classification of water bodies
<1 <1 1 1-
3 3 3-
5 5 5-
10 Depth of Secchi Depth of Secchi disk disk (m) (m) >50 >50 5 5-
50 2 2-
5 1 1-
2 Superficial Superficial chlorophyll a chlorophyll a ( (µ µgl gl-
1)
) >5000 >5000 2000 2000-
5000 2000 2000 Phytoplankton Phytoplankton (cellml (cellml-
1)
) >300 >300 150 150-
300 <150 <150 Inorganic Inorganic nitrogen nitrogen ( (µ µg gl l-
1)
) >100 >100 20 20-
100 10 10-
20 1 1-
10 Inorganic Inorganic phosporus phosporus ( (µ µgl gl-
1)
)
HYPEREUTROPHIC HYPEREUTROPHIC EUTROPHIC EUTROPHIC MESOTROPHIC MESOTROPHIC OLIGOTROPHIC OLIGOTROPHIC
SLIDE 8 El Divisorio Stream Station 4 Station 3 Longitude: 61º 38´ W Latitude: 38º 25´ S 51 Provincial Route Dam Sauce Grande River Sauce Grande River Station 1 Station 2
20 m Wetland
Argentina
Paso de las Piedras Reservoir
SLIDE 9
Paso de las Piedras Reservoir
Provides Provides drinking drinking water water to to more more than than 450.000 450.000 inhabitants inhabitants from from Bah Bahí ía Blanca, Punta Alta a Blanca, Punta Alta and and to to a a petrochemical petrochemical complex complex
SLIDE 10
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
SLIDE 11 Paso de las Piedras Reservoir
Eutrophic
- Main source of nutrients:
Main source of nutrients: agricultural agricultural activities activities
High phytoplankton phytoplankton concentration concentration during during spring spring and and summer summer: : surface surface water water blooms blooms
40 80 120 160 200 240 280 320 360 0.0 0.2 0.4 0.6 0.8
O-Phosphate (mgl
Tim e (days) Eutrophication lim it O bserved data
50 100 150 200 250 300 350 0.0 5.0x10
4
1.0x10
5
1.5x10
5
2.0x10
5
2.5x10
5
Total Phytoplankton (mgl
T im e (D ays) O bserved data E utrophication lim it
SLIDE 12 Surface Surface water water blooms blooms
- 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 trophic trophic chain). chain).
In favorable favorable environmental environmental conditions conditions they they are are multiplied multiplied and and concentrated concentrated in in the the surface surface, , => => fast fast increase increase in in algal algal biomass biomass. .
SLIDE 13 Photosynthesis Photosynthesis
106 106 CO CO2
2 + 16
+ 16 NO NO3
3
+ HPO HPO4
42 2-
+ 122 H H2
2O
O + 18 + 18 H H+
+
Solar Solar radiation radiation
C C106
106H
H263
263N
N16
16P
P + 138 + 138 O O2
2
algae algae
SLIDE 14 Problems caused by water blooms
For For man man
Blockage of water-
filters
Unpleasant odor and taste and taste
Aesthetics
- Presence of potentially toxic
Presence of potentially toxic algae algae
For For ecosystem ecosystem
- Reduction of biodiversity
Reduction of biodiversity
Anoxic conditions
Shade
Blockage of
fish gills gills
SLIDE 15 Blockage of water Blockage of water-
filters
Closterium Closterium spp. spp. Staurastrum Staurastrum spp. spp.
Aulacoseira Aulacoseira spp spp.
.
SLIDE 16
Ceratium Ceratium hirundinella hirundinella Anabaena Anabaena circinalis circinalis
Unpleasant odor Unpleasant odor and taste and taste
SLIDE 17
Aesthetics Aesthetics
SLIDE 18
Toxic Cyanobacteria
SLIDE 19 Biological determinations
Qualitative analysis Qualitative analysis
Plankton net net (30 (30 µ µm) m)
to the
Observation to the
microscope of the alive and fixed microscope of the alive and fixed samples ( samples (formol formol 4%) 4%)
Determination based based on
keys
Quantitative analysis Quantitative analysis
Rutner water water sampler sampler
In situ fixation fixation with with Lugol`s Lugol`s solution solution
Phytoplankton enumeration enumeration in in inverted inverted microscope microscope by by Uterm Utermö öhl hl method method (1958) (1958)
Phytoplankton biovolume biovolume
Calculation of
mgC. .
Cyanobacteria
Diatoms
Chlorophytes
SLIDE 20 Physico Physico-
chemical determinations determinations
Nitrates
Nitrites
Ammonium
Organic Nitrogen Nitrogen
Phosphates
Organic Phosporus Phosporus
Silice
Water temperature
Solar radiation
pH
Dissolved Oxygen Oxygen
Biochemical Demand Demand
Oxygen
Depth of Secchi disk
SLIDE 21
Ecological Water Quality Model
Dynamic Dynamic mass mass balances balances for for nutrients nutrients and and phytoplankton phytoplankton Concentration Concentration gradients gradients along along water water column column height height Partial Partial Differential Differential Equations Equations System System Spatially Spatially Discretization Discretization: Horizontal : Horizontal layers layers Differential Differential Algebraic Algebraic System System (DAE) (DAE) Assumptions Assumptions
Horizontally averaged concentration Phosphorus limiting nutrient (for algae growth) Constant density Constant transversal area in lake
SLIDE 22 ) C (C Δh A k ) C (C Δh A k
r C Q
Q dt ) V d(C
j , i ij i- d N m j i ij i d i ij N k ijk i,k i ij
OUT IN OUT IN IN
ij im IN
1 1 1 1 1 − = + =
− − ∑ − + ∑ =
Mass balance for horizontal layers
dt dh h C ) C (C h Δh A k ) C (C h Δh A k
C V Q
V Q dt dC
i i ij ,j i ij i- i d N m ,j i ij i i d ij ij i i,m N k ijk i i,k ij
OUT IN OUT IN IN IN
− − − ∑ − + ∑ =
− = + = 1 1 1 1 1
⎥ ⎦ ⎤ ⎢ ⎣ ⎡ day mg
i i= horizontal layer = horizontal layer k k = Tributary inflows: El = Tributary inflows: El Divisorio Divisorio, Sauce Grande , Sauce Grande m m = Withdrawals: drinking water+complex, Sauce Grande = Withdrawals: drinking water+complex, Sauce Grande j j = Cyanobacteria, Diatoms, = Cyanobacteria, Diatoms, Chlorophyta Chlorophyta, NO , NO3
3,NH
,NH4
4, ON,
, ON, PO PO4
4,OP, BDO, DO
,OP, BDO, DO
SLIDE 23 Mass balance for horizontal layers
dt dh h C ) C (C h Δh A k
C V Q dt dC
U U Uj N k Lj Uj U U d Uj N m Uj C U V U Q Ujk U U,k Uj
IN IN IN
OUT OUT
− ∑ − + =
= ∑ = − 1 1
dt dh h C ) C (C h Δh A k r C V Q dt dC
L L Lj N m U,j Lj L L d Lj Lj L L Lj
OUT OUT
− ∑ − + + =
=1
Upper Layer Lower Layer
U i = L i =
SLIDE 24 State variables and biogeochemical processes
Boundary conditions Temperature PAR Advection and dispersion
Inflow Outflow
CBOD PO4 ON Pool OP Pool NO + NO
3 2
NH3 Dissolved oxygen Zooplankton Chlorophyta Diatoms Cyanobacteria
ATMOSPHERE WATER SEDIMENT
Nitrification Mineralization Sediment flux Death Grazing Settling Settling Settling
Settling
Mineralization D e a t h Uptake
Respiration Photosynthesis
Denitrification Death
SLIDE 25 Rate equations (rij)
i i = upper and lower layers = upper and lower layers j j = Cyanobacteria, Diatoms, = Cyanobacteria, Diatoms, Chlorophytes Chlorophytes
ij j,growth ij,growth
C * *f(N) *f(T)*f(I) k R =
ij i sedim j, sedim ij,
*C h * k R 1 =
kpj iPO C iPO C ) N ( f + = 4 4
Growth Growth
) Ij Ii (1 exp Ij I
f(I)
−
=
Respiration Respiration Death Death Settling Settling
graz , ij settling ij, ij,death ij,resp ij,growth ij
R R R R R r − − − − =
i i
j ij
T ) T T ( ) T ( f
2 2
− − =
( )
ij T r resp i resp , ij
C , k R
20 −
θ =
( )
ij T m death i death , ij
C , k R
20 −
θ =
j
Zoo graz ij ij j,graz ij,graz
C K C C k R + =
Grazing Grazing
SLIDE 26 Rate equations (rij)
i i = upper and lower layers = upper and lower layers j j = ON, OP = ON, OP
sedim ij, miner ij, ij,death ij
R r =
) C * f * k * (a R
im j 3 1 m death m, jc ij,death
∑ =
= ∑ = + ∑ =
=
3 1 m im C kmjc 3 1 m ij C * Cim 20)
* miner miner miner ij,
* * k R θ
ij i Dj sedim , j k sedim ij,
C * D ) f ( * R − = 1
Death Death Mineralization Mineralization Settling Settling
SLIDE 27 Rate equations (rij)
i i = upper and lower layers = upper and lower layers j j = PO = PO4
4
uptake ij, miner ij, ij,death ij
R R r
+
=
) C * ) f (1 * k * (a R
im po 3 1 m death m, pc ij,death
− ∑ =
=
∑ = + ∑ =
=
3 1 j im C pc km 3 1 m iOP C * im C 20)
* miner miner miner ij,
* * k R θ ) C * a * (R R
im pc 3 1 m growth , im uptake ij,
∑
= =
Death Death Mineralization Mineralization Uptake Uptake
SLIDE 28 Rate equations ( (rij) )
i i = upper and lower layers = upper and lower layers j j = NO = NO3
3
denitr ij, uptake ij, nitri ij, ij
R
R r − =
iDO nio iDO iNH4 ) Temp exp( * nitri nitri ij,nitri
C k C * C * * k R + =
− 20
θ
)) C * ) PNH ( * R * a ( R
im m growth im, nc uptake ij,
∑ − =
= 3 1
4 1 iDO C k k * iNO C * k R
no no ) Temp exp( * denitr denitr denitr ij,
+
− = 3 3 3 20
θ
Nitrification Nitrification Uptake Uptake Denitrification Denitrification
SLIDE 29 Rate equations (rij)
i i = upper and lower layers = upper and lower layers j j = NH = NH4
4
uptake ij, miner ij, ij,death ij
R R r
+
=
) C * ) f ( * k * (an R
im ON 3 1 m death m, c ij,death
− ∑ =
=
1
∑ + ∑ =
= = 3 1 m im mpc 3 1 m iON im 20)
* miner miner iner m ij,
C k C * C * * k R θ
) C * PNH * R * a ( R
im m growth im, nc uptake ij,
∑
= = 3 1 4
Death Death Mineralization Mineralization Uptake Uptake
SLIDE 30
Global sensitivity analysis
Local vs. Global Sensitivity Analysis Quantitative variance-based Global Sensitivity Analysis
Model independent Incorporate the effect of range of input variation and its pdf Allow multidimensional averaging Allow parameter grouping
SLIDE 31
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
( ) ( ) ( ) ( )
i i
x Y E V x Y V E ) Y ( V + =
( ) ( )
i
x Y V E
( ) ( )
i
x Y E V
( ) ( )
) Y ( V x Y E V S
i i =
SLIDE 32 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
( ) ( )
( )
( )
k 1 k ,..., 2 , 1 k j i 1 j i ij k 1 i i i k 1
x ,..., x f ... x , x f x f f x ,..., x f + + + + =
∑ ∑
≤ < ≤ =
∫
=
1 k i ,..., i
dx f
s l
∑ ∑
= ≤ < ≤
+ + + =
k 1 i k ,..., 2 , 1 k j i 1 ij i
V ... V V V
SLIDE 33
Sobol’ sensitivity indices Higher orders indices calculation: computationally expensive Total sensitivity index
Global sensitivity analysis
∑ ∑
= ≤ < ≤
+ + + =
k i k ,..., , k j i ij i
V V ... V V V V
1 2 1 1
1
∑ ∑
= ≤ < ≤
+ + + =
k i k ,..., , k j i ij i
S ... S S
1 2 1 1
1
( ) ( )
) Y ( V x Y V E S
i T i −
=
SLIDE 34
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
SLIDE 35 Global sensitivity analysis
) Y ( E ) ( f N
p N i i
⎯→ ⎯ ξ
∑
=1
1 ) Y ( E ) Y ( V ) ( f N
p N i i 2 1 2
1 + ⎯→ ⎯ ξ
∑
=
) Y ( E V ) , ( f ) ( f N
y p N i i ´ i i 2 1
1 + ⎯→ ⎯ ζ η ξ
∑
=
) Y ( E V ) , ( f ) ( f N
T y p N i i i ´ i 2 1
1 + ⎯→ ⎯ ζ η ξ
∑
=
) Y ( V V S
y i =
) Y ( V V S
T T
y i
− =1
Sobol’ indices Calculate
SLIDE 36 50 100 150 200 250 300 350 0.0 0.2 0.4 0.6 0.8 1.0
SCyanobacteria Time (Days) anc apc fON K1 kmn kmp kni Knit LC LD LG mD mG titam titar titamn titamp vsC vsD vsG umaxC umaxD
50 100 150 200 250 300 350 2 4 6
Sint,Cyanonacteria Time (Days)
Cyanobacteria
S ST
T -
Si
i
S Si
i
Sensitivity Indices: Si
SLIDE 37 50 100 150 200 250 300 350 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
Sint,Phosphate Time (days) anc apc fON K1 kmn kmp kni Knit LC LD mD mG titam titar titamn titamp vsC vsD vsG umaxC umaxD umaxG
50 100 150 200 250 300 350 0.0 0.2 0.4 0.6 0.8 1.0
SPhosphate Time (Days)
Phosphate
S Si
i
S ST
T -
Si
i
Sensitivity Indices: Si
SLIDE 38 50 100 150 200 250 300 350 0.0 0.2 0.4 0.6 0.8 1.0
Sj 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
50 100 150 200 250 300 350 0.0 0.2 0.4 0.6 0.8 1.0
Sj Time (Days)
Nitrate
Sensitivity Indices: Si
SLIDE 39 Dynamic Parameter Estimation Model Dynamic Parameter Estimation Model
Initial Conditions Process Model
( )
= t p u y x x f , , , , , &
( )
= t p u y x g , , , ,
x t x = ) (
Objective Function
,
Variable Bounds
xL ≤ x ≤ xU, yL ≤ y ≤ yU uL ≤ u ≤ uU, pL ≤ p ≤ pU Parameter Estimation Problem
{ }
2
σ = diag V
( ) ( )
∑ ∑ ∑
= = = −
− − = φ
NI i NV j NL k ij M ij T ij M ij
x x V x x min
1 1 1 1
2 1
SLIDE 40 Simultaneous Simultaneous Approach Approach for for Dynamic Dynamic Optimization Optimization
Nonlinear DAE optimization problem Discretization of Control and State variables Collocation on finite elements Large-Scale Nonlinear Programming Problem Interior Point Method (Biegler et al., 2002)
Process Model
( )
= t p u y x x f , , , , , &
( )
= t p u y x g , , , ,
x t x = ) (
Objective Function
,
Variable Bounds
xL ≤ x ≤ xU, yL ≤ y ≤ yU uL ≤ u ≤ uU, pL ≤ p ≤ pU Dynamic Parameter Estimation Problem
{ }
2
σ = diag V
( ) ( )
∑ ∑ ∑
= = = −
− − = φ
NI i NV j NL k ij M ij T ij M ij
x x V x x min
1 1 1 1
2 1 Initial Conditions
SLIDE 41
rSQP Algorithm
Initialization Linearization Objective 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 Convergence Optimal Solution dx = YdY + ZdZ Basis selection (Biegler et al., 2002)
SLIDE 42
Nonlinear Programming Problem
Application of Barrier method Application of Barrier method
) (x f min ) ( s.t = x c ≥ x ∑ −
= n j j
x ln x f min
1
) ( μ ) ( s.t = x c As μ 0, x*(μ) x*
Sequence of barrier problems for decreasing μ values
SLIDE 43
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 Optimal Optimal Solution Solution (Biegler et al., 2002)
SLIDE 44 Parameter Parameter Estimation Estimation Problem Problem
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)
SLIDE 45
Numerical Results
109.9 Optimal growth radiation cyano Ic (ly/d) 0.405 Max growth of diatoms kdiatom,growth (d-1) 24.52 Optimal growth radiation of diatoms Id (ly/d) 0.210 Max growth of cyanobacteria kcyanob,growth (d-1) 0.654 Max growth of chlorophytes kcyanoph,growth (d-1) 89.74 Optimal growth radiation chlorophytes Ig (ly/d) 0.343 Half-sat. conc. for oxygen lim. of nitrification Knio (mg/l) 0.015 Rate coeff. mineralization OP kOP,miner (d-1) 0.092 Rate coeff. mineralization ON kON,miner (d-1) Estimated Parameter Symbol Time horizon: 365 days – Data frequency: twice a week DAE: 20 differential equations, 60 algebraic equations, NE =40 NC=3 NLP: 10432 nonlinear equations, 52 Iterations, 4 barrier problems
(Estrada et al., 2008a,b)
SLIDE 46 0.2 0.4 0.6 0.8 1 50 100 150 200 250 300 350 400 Time (Days) Concentration (mgl -1) 0.5 1 1.5 2 50 100 150 200 250 300 350 400 Time (Days) Concentration (mgl -1)
Diatoms Phosphate
Numerical Results
SLIDE 47 0.5 1 1 .5 2 1 00 200 300 400 Tim e (days) 0.5 1 1 .5 2 2.5 1 00 200 300 400 Time (days)
Cyanobacteria Nitrate
Numerical Results
SLIDE 48
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?
SLIDE 49 Bio-restoration policies
El Divisorio Stream Station 4 Station 3 Longitude: 61º 38´ W Latitude: 38º 25´ S
51
Provincial Route
Dam
Sauce Grande River Sauce Grande River Station 1 Station 2
20 m
Wetland
Artificial wetland
Built for nitrogen and phosphorus removal (Lopez et al., 2007) Next to El Divisorio Stream Global retention: 64% (phosphate) Derivation of tributary inflows through wetland
SLIDE 50 Optimal control problem: Tributary inflows derivation
Case 1: Control variable: Tributary inflows profiles derivation to wetland for bio-remediation
st dt . ) t ( C min tf
phyto j , j 2
25 ∫ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ∑
= ) d / l ( F 5 . F
DIVISORIO WETLAND ≤
≤
DAE Eutrophication model
SLIDE 51 Case 1: Numerical results
50 100 150 200 250 300 350 0.4 0.5 0.6 0.7 0.8 0.9
240 260 280 300 320 340 360 0.54 0.56 0.58 0.60 0.62 0.64 0.66
PO4 Concentration (mg/l) Time (days)
Optimization results (PO4 reduct) No PO4 loading reduction
PO4 Conc. (mg/l) Time (days)
40 80 120 160 200 240 280 320 360 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
Cyanobacteria (mgl
Time (days) Optimization results No PO4 loading reduction
(Estrada et al., 2008d)
NE = 40, NC = 3 NLP: 10581 nonlinear equations
SLIDE 52 Optimal control problem: Inlake bio-restoration
Case 2: Control variables: Tributary inflows profiles derivation to wetland for bio-remediation and Zooplankton concentration profiles (Removal of zoo-planktivorous fish)
st dt . ) t ( C min tf
phyto j , j 2
25 ∫ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ∑
=
) d / l ( DIVISORIO WETLAND
F 5 . F ≤ ≤
) l / mg ( C .
zoo
5 01 ≤ ≤
DAE Eutrophication model
SLIDE 53 Case 2: Control and state variables profiles
50 100 150 200 250 300 350 1 2 3 4 5
Zooplankton Conc. (mg/l) Time (days)
(Estrada et al., 2008d)
50 100 150 200 250 300 350 0.0 0.4 0.8 1.2 1.6
Phyto profiles before restoration Optimal phyto profiles Diatoms Cyanobacteria Phytoplankton conc. (mg/l) Time (days)
NE = 40, NC = 3 NLP: 10741 nonlinear equations
SLIDE 54
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 optimization techniques: simultaneous approach Resolution of optimal control problem: bio-restoration policies
Conclusions
SLIDE 55 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, 2005 Biegler, L.T., A. Cervantes, A.Waechter, Advances in simultaneous strategies for dynamic process optimization. Chem. Eng. Sci. 57: 575-593, 2002 Estrada V., E. Parodi, M.S. Diaz,“Dynamic Parameter Estimation Problem For A Water Quality Model”, Chem. Eng. Transactions, 11, 247-252, 2007 Estrada 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, 2008 Estrada V., E. Parodi, M.S. Diaz, Development of eutrophication biogeochemical models: global sensitivity analysis and dynamic parameter estimation, submitted to
Estrada 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, USA Jeppesen, 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.
SLIDE 56 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 dynamic
- ptimization 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 and
their 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.
