Optimal Control of Flood Diversion in Watershed Optimal Control of Flood Diversion in Watershed
Using Nonlinear OptimizationUsing Nonlinear Optimization
National Center for Computational Hydroscience and EngineeringThe University of Mississippi
Yan Ding, Ph.D.1 and Sam S. Y. Wang, Ph.D. P.E.2
1Research Assistant Professor, 2 F.A.P. Barnard Distinguished Professor and Director
National Center for Computational Hydroscience and Engingeerin, The University of Mississippi, Oxford, MS 38677
Presented at Conference of 50 Years of Soil and Water Research In a Changing Agricultural Environment, Oxford, MS, Sept 4, 2008
OutlineOutline
IntroductionIntroduction
Nonlinear Models for Forecasting Flood EventsNonlinear Models for Forecasting Flood Events
Nonlinear Optimization Scheme for Finding the Nonlinear Optimization Scheme for Finding the Optimal Flood Diversion Hydrograph to Mitigate Optimal Flood Diversion Hydrograph to Mitigate Hazardous Storm WatersHazardous Storm Waters
Applications to a Variety of Flood Diversion Control Applications to a Variety of Flood Diversion Control ScenariosScenarios
Conclusions and Future Research Topics Conclusions and Future Research Topics
The spillway (highlighted in green) stretches from the Mississippi River, at right, northward to Lake Ponchartrain, on the left of the photo.
An Example of Flood DiversionAn Example of Flood Diversion – The Bonnet Carre’ Spillway – The Bonnet Carre’ Spillway
Flooded Street, Mississippi River Flood of 1927
The Bonnet Carré Spillway, the southern-most floodway in the Mississippi River and Tributaries system, has historically been the first floodway in the Lower Mississippi River Valley opened during floods. The USACE’s hydraulic engineers rely on discharge and gauge readings at Red River Landing, about 200 miles above New Orleans, to determine when to open the spillway. The discharge takes two days to reach the city from the landing. As flows increase, bays are opened at Bonnet Carré to divert them.
Difficulties in Optimal Control of Open Channel Flow Difficulties in Optimal Control of Open Channel Flow
Temporally/spatially non-uniform open channel flowTemporally/spatially non-uniform open channel flow Requires that a forecasting model can predict accurately complex water Requires that a forecasting model can predict accurately complex water
flows in space and time in single channel and channel networkflows in space and time in single channel and channel network
Nonlinearity of flow controlNonlinearity of flow control Nonlinear process control, Nonlinear optimizationNonlinear process control, Nonlinear optimization Difficulties to establish the relationship between control actions and Difficulties to establish the relationship between control actions and
responses of the hydrodynamic variablesresponses of the hydrodynamic variables
Requirement of fast flow solver and optimization Requirement of fast flow solver and optimization In case of fast propagation of flood wave, a very short time is available for In case of fast propagation of flood wave, a very short time is available for
predicting the flood flow at downstream. Due to the limited time for predicting the flood flow at downstream. Due to the limited time for making decision of flood mitigation, it is crucial for decision makers to making decision of flood mitigation, it is crucial for decision makers to have a very efficient forecasting model and a control model.have a very efficient forecasting model and a control model.
Objectives
Theoretically, • Through adjoint sensitivity analysis, make nonlinear optimization
capable of flow control in complex channel shape and channel network in watershed
Real-Time Nonlinear Adaptive Control Applicable to unsteady river flows
• Establish a general numerical model for controlling hazardous floods so as to make it applicable to a variety of control scenarios
Flexible Control System; and a general tool for real-time flow control
For Engineering Applications,• Integrate the control model with the CCHE1D flow model,
• Apply to practical problems
Integrated Watershed & Channel Network Modeling with Integrated Watershed & Channel Network Modeling with CCHE1DCCHE1D
Digital ElevationModel (DEM)
Rainfall-Runoff Simulation
Upland Soil Erosion(AGNPS or SWAT)
Channel Network Flow and Sediment Routing
(CCHE1D)
Channel Network andSub-basin Definition
(TOPAZ)
q(t)
=?
01
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02 2
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fgSx
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Dynamic Wave ModelDynamic Wave Modelfor Flood Wave Predictionfor Flood Wave Prediction
A=Cross-sectional Area; q=Lateral outflow;=correction factor; R=hydraulic radius n = Manning’s roughness
where Q = discharge; Z=water stage;
• Boundary Conditions
• Initial Conditions (Base Flows)
• Internal Flow Conditions for Channel Network
A Typical Hydrograph by USGS
Control Actions Control Actions - Available Control Variables in Open Channel Flow- Available Control Variables in Open Channel Flow
Control lateral flow at a certain location Control lateral flow at a certain location xx00: Real-time flow : Real-time flow diversion rate diversion rate q(xq(x00, t), t) at a spillwayat a spillway
Control lateral flow at the optimal location Control lateral flow at the optimal location xx: Real-time levee : Real-time levee breaching rate breaching rate q(x, t)q(x, t) at the optimal locationat the optimal location
Control upstream discharge Control upstream discharge Q(0, t)Q(0, t):: real-time reservoir release real-time reservoir release
Control downstream stage Control downstream stage Z(L, t)Z(L, t):: real-time gate operation real-time gate operation
Control downstream discharge Control downstream discharge Q(L, t)Q(L, t):: real-time pump rate real-time pump rate controlcontrol
Control bed friction (roughness Control bed friction (roughness nn): ):
+2.0m
+0.0m
20m
70m
Zobj
An Objective Function for Flood ControlAn Objective Function for Flood Control
To evaluate the discrepancy between predicted and maximum To evaluate the discrepancy between predicted and maximum allowable stages, a weighted form is defined asallowable stages, a weighted form is defined as
where T=control duration; L = channel length; t=time; x=distance along channel; Z=predicted water stage; Zobj(x) =maximum allowable water stage in river bank (levee) (or objective water stage); x0= target location where the water stage is protective; = Dirac delta function
)()(,0
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00
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Mathematical Framework for Optimal ControlMathematical Framework for Optimal Control
The optimazition is to find the control variable The optimazition is to find the control variable qq satisfying satisfying a dynamic system such thata dynamic system such that
where where QQ and and ZZ are satisfied with the continuity equation are satisfied with the continuity equation and momentum equation, respectively (i.e., de Saint and momentum equation, respectively (i.e., de Saint Venant Equations)Venant Equations)
Local minimum theory :Local minimum theory : Necessary ConditionNecessary Condition: If : If nn** is the true value, then is the true value, then J(nJ(n**)=0)=0;; Sufficient ConditionSufficient Condition: If the Hessian matrix : If the Hessian matrix 22J(nJ(n**) is ) is
positive definitepositive definite, then , then nn** is a strict local minimizer of is a strict local minimizer of ff
)),,,(min()( qZQJqf
Sensitivity AnalysisSensitivity Analysis- - Establishing A Relationship between Control Actions and System VariablesEstablishing A Relationship between Control Actions and System Variables
Compute the gradient of objective function, Compute the gradient of objective function, qq((XX, q),, q), i.e., sensitivity of i.e., sensitivity of control variable throughcontrol variable through
1. 1. Influence Coefficient MethodInfluence Coefficient Method (Yeh, 1986):(Yeh, 1986): Parameter perturbation trial-and-error; lower accuracyParameter perturbation trial-and-error; lower accuracy
2. 2. Sensitivity Equation Method Sensitivity Equation Method (Ding, Jia, & Wang, 2004)(Ding, Jia, & Wang, 2004)
Directly compute the sensitivity Directly compute the sensitivity ∂X/∂q∂X/∂q by solving the sensitivity equations by solving the sensitivity equations Drawback: different control variables have different forms in the equations, no Drawback: different control variables have different forms in the equations, no
general measures for system perturbations; The number of sensitivity equations = the general measures for system perturbations; The number of sensitivity equations = the number of control variables.number of control variables.
Merit: Forward computation, no worry about the storage of codesMerit: Forward computation, no worry about the storage of codes
3. 3. Adjoint Sensitivity Method Adjoint Sensitivity Method (Ding and Wang, 2003)(Ding and Wang, 2003)
Solve the governing equations and their associated adjoint equations sequentially. Solve the governing equations and their associated adjoint equations sequentially. Merit: general measures for sensitivity, limited number of the adjoint equations Merit: general measures for sensitivity, limited number of the adjoint equations
(=number of the governing equations) regardless of the number of control variables.(=number of the governing equations) regardless of the number of control variables. Drawback: Backward computation, has to save the time histories of physical variables Drawback: Backward computation, has to save the time histories of physical variables
before the computation of the adjoint equations.before the computation of the adjoint equations.
x
t
A B
CD
O L
T
Variational AnalysisVariational Analysis- - To Obtain Adjoint EquationsTo Obtain Adjoint Equations
Extended Objective Function
dxdtLLJJT L
QA 0 0 21
* )(
where A and Q are the Lagrangian multipliers
Fig. Solution domain
Necessary Condition
0* JJ on the conditions that
0),(0),(
1
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Variation of Extended Objective FunctionVariation of Extended Objective Function
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where
*;3
42;
*
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BB
RB
n
ARK Top width of channel
Variations of Variations of JJ with Respect to Control Variables with Respect to Control Variables – Formulations of Sensitivities– Formulations of Sensitivities
dttQQ
rtOQJ
T
x
A ),0()()),((0
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Lateral Outflow
Upstream Discharge
Downstream Section Area or Stage
Bed Friction
Remarks: Control actions for open channel flows may rely on one control variable or a rational combination of these variables. Therefore, a variety of control scenarios principally can be integrated into a general control model of open channel flow.
General Formulations of Adjoint Equations for General Formulations of Adjoint Equations for the Full Nonlinear Saint Venant Equationsthe Full Nonlinear Saint Venant Equations
Q
r
A
Q
A
r
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QQg
xB
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xA
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t QAQQ
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According to the extremum condition, all terms multiplied by A and Q can be set to zero, respectively, so as to obtain the equations of the two Lagrangian multipliers, i.e, adjoint equations (Ding & Wang 2003)
Transversality Conditions and Boundary Transversality Conditions and Boundary ConditionsConditions
x
t
A B
CD
O L
T
Fig. Solution domain
Considering the contour integral in J*, This term I needs to be zero.
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Transversality (Final) Conditions
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Downstream B.C.
Backward Computation
Internal Boundary Conditions Internal Boundary Conditions – – for Channel Networkfor Channel Network
213
321
QQQ
ZZZ
I.B.C.s of Adjoint Equations
32
22
12
A
Q
A
Q
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QA
QA
I.B.C.s of Flow Model
Fig. Confluence
3
3
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Numerical TechniquesNumerical Techniques
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1-D Time-Space Discretization (Preissmann, 1961)
Solver of the resulting linear algebraic equations (Pentadiagonal Matrix)
Double Sweep Algorithm based on the Gauss Elimination
where and are two weighting parameters in time and space, respectively;t=time increment; x=spatial length
Minimization Procedures for Nonlinear Minimization Procedures for Nonlinear OptimizationOptimization
CG MethodCG Method ( (Fletcher-Reeves methodFletcher-Reeves method) () (Fletcher 1987Fletcher 1987)) The convergence direction of minimization is considered as The convergence direction of minimization is considered as
the gradient of objective functionthe gradient of objective function.. Trust Region MethodTrust Region Method (e.g (e.g Sakawa-Shindo methodSakawa-Shindo method)) considering the considering the first order derivativefirst order derivative of performance function of performance function
only, stable in most of practical problems (only, stable in most of practical problems (Ding et al 2004Ding et al 2004)) Limited-Memory Quasi-Newton MethodLimited-Memory Quasi-Newton Method (LMQN) (LMQN) Newton-like method, applicable for large-scale computation, Newton-like method, applicable for large-scale computation,
considering the considering the second order derivativesecond order derivative of objective function of objective function (the approximate Hessian matrix) (Ding & Wang 2005)(the approximate Hessian matrix) (Ding & Wang 2005)
OthersOthers
Minimization ProceduresMinimization Procedures
Limited-Memory Quasi-Newton Method (LMQN)Limited-Memory Quasi-Newton Method (LMQN) Newton-like method, applicable for large-scale computation Newton-like method, applicable for large-scale computation
(with a large number of control parameters), considering the (with a large number of control parameters), considering the second order derivativesecond order derivative of objective function (the approximate of objective function (the approximate Hessian matrix)Hessian matrix)
AlgorithmsAlgorithms::
BFGS (named after its inventors, BFGS (named after its inventors, BBroyden, royden, FFletcher, letcher, GGoldfarb, and oldfarb, and SShanno)hanno)
L-BFGSL-BFGS (unconstrained optimization) (unconstrained optimization)
L-BFGS-BL-BFGS-B (bound constrained optimization) (bound constrained optimization)
Limited-Memory Quasi-Newton Method (LMQN)Limited-Memory Quasi-Newton Method (LMQN) (Basic Concept 1) (Basic Concept 1)
Given the iteration of a line search method for parameter Given the iteration of a line search method for parameter qq
qqk+1k+1 = = qqkk + + kkddkk
kk = the step length of line search = the step length of line search
sufficient decrease and curvature conditions sufficient decrease and curvature conditions
ddkk = the search direction (descent direction) = the search direction (descent direction)
BBkk = = nnnn symmetric positive definite matrixsymmetric positive definite matrix
For For the Steepest Descent Methodthe Steepest Descent Method: : BBkk = I = I
Newton’s MethodNewton’s Method: : BBkk== 22J(nJ(nkk))
Quasi-Newton MethodQuasi-Newton Method: :
BBkk= an approximation of the Hessian = an approximation of the Hessian 22J(nJ(nkk))
)(1kkkk nJBd
.
qi
qj
.q* d1
Contour of J
Flow chart of Finding optimal control variable by using LMQN procedure
Set the initial q
k=0
Solve the initial state vector X0 Flow Model (CCHE1D)
Calculation of objective function J0, gradient g0, and search
direction d0
Calculation of )( 11 kk qJg
||gk+1||max{1,||qk+1||}
Calculation of Jk+1
Stop
Yes
No
Calculate kkkk dqq 1 Line Search
Solver of Adjoint Equations
Calculation of 111 kkk gHd
Update Hessian matrix by the recursive iteration
nlk
lk
lk
q
qqMax
)( 1 Yes
Yes
Solve the state vector Xk+1
L-BFGS
JkJ 1
Flow Model (CCHE1D)
Solver of Adjoint Equations
Three Major Modules• Flow Solver• Sensitivity Solver• Minimization Process
L-BFGS-BL-BFGS-B
The purpose of the L-BFGS-B method is to The purpose of the L-BFGS-B method is to minimize the objective function minimize the objective function J(q)J(q) , i.e., , i.e.,
min J(q),min J(q),subject to the following simple bound constraint,subject to the following simple bound constraint,
qqminmin q q q qmaxmax,,
where the vectors where the vectors qqminmin and and qqmaxmax mean lower and upper mean lower and upper bounds on the control variables.bounds on the control variables.
L-BFGS-B is an extension of the limited memory L-BFGS-B is an extension of the limited memory algorithm (L-BFGS) (algorithm (L-BFGS) (Liu & Nocedal, 1989) Liu & Nocedal, 1989) for bound for bound constrained optimization (constrained optimization (Byrd et al, 1995)Byrd et al, 1995) . .
Flooding and Flood ControlFlooding and Flood Control
Levee Failure, 1993 flood. Missouri. Flood Gate, West Atchafalaya Basin, Charenton Floodgate, Louisiana
Control of Flood Diversion in A Single Channel Control of Flood Diversion in A Single Channel – A Simplified Problem– A Simplified Problem
q(xc,t) = ?
xc
No Control
Zobj(x0,t)
Under Control
Z(x0,t) A Tolerable Stage
t
Objective Function
dxdttxqQZfLT
JT L
0 0
),,,,(1
obj
objobjZ
ZZif
ZZifxxxZtxZWf
,0
),()](),([ 04
Optimal Control of Flood Diversion Rate Optimal Control of Flood Diversion Rate ( Case 1) -( Case 1) - A Hypothetic Single Channel A Hypothetic Single Channel
Time
Dis
char
ge
TpTd
Qp
Qb
+2.0m
+0.0m
20m
70m
1:2
1:1.
5
A Triangular HydrographCross-section
Parameter L x t n QP Qb Tp Td Z0 Wz
Unit (km) (km) (min) s/m1/3 (m3/s) (m3/s) (hour) (hour) m
Value 10.0 0.5 5.0 1.0(0.55*) 0.5 0.03 100.0 10.0 16.0 48.0 3.5 103
* This value is used for solving adjoint equations
Lateral Outflow
Z0=3.5m
Optimal Lateral Outflow and Objective Function Optimal Lateral Outflow and Objective Function (Case 1)(Case 1)
Hours
Dis
char
ge(m
3 /s)
0 10 20 30 40 50-100
-50
0
50
100
Iteration= 1Iteration= 3Iteration= 4Iteration= 5Iteration= 6Iteration= 10Iteration= 30Iteration= 70
Hydrograph at inlet
Iterations of L-BFGS-BO
bjec
tive
Fun
ctio
n
Nor
mof
Gra
dien
t
0 10 20 30 40 50 60 7010-3
10-2
10-1
100
101
102
103
104
105
10-8
10-7
10-6
10-5
10-4
10-3
10-2
Objective FunctionNorm of Gradient
Iterations of optimal lateral outflowObjective function and Norm of gradient of the function
Optimal Outflow q
Comparison of Water Stages in Space and Time Comparison of Water Stages in Space and Time (Case 1)(Case 1)
Km
01
23
45
67
89
10Hours
012
2436
48
Wat
erS
tage
(m)
0
1
2
3
4
5
No Control Optimal Control of Lateral Outflow
Km
01
23
45
67
89
10Hours
012
2436
48
Wat
erS
tage
(m)
0
1
2
3
4
5
Lateral Outflow
Allowable Stage Z0=3.5
Comparison of Discharge in Time and Space Comparison of Discharge in Time and Space (Case 1)(Case 1)
Km
0 1 2 3 4 5 6 7 8 9 10 Hours0
1224
3648
Dis
cha
rge
(m3/s
)
20
40
60
80
100
Lateral Outflow
Km
0 1 2 3 4 5 6 7 8 9 10 Hours0
1224
3648
Dis
char
ge(m
3 /s)
20
40
60
80
100
No Control Optimal Control of Lateral Outflow
Sensitivity Sensitivity ∂J/∂q(x,t)∂J/∂q(x,t)
Hours
A0 10 20 30 40 50
0
5E-06
1E-05
1.5E-05
2E-05
2.5E-05ITERATION= 1ITERATION= 3ITERATION= 4ITERATION= 5ITERATION= 6ITERATION= 10ITERATION= 30
Km
01
23
45
67
89
10 Hours
012
2436
48
A0
2E-05
4E-05
Lateral Discharge
Sensitivity of q in time and space at the 1st iteration
Iterative history of sensitivity at the control point
Fast searching
Optimal Control of Lateral Outflow (Case 2) Optimal Control of Lateral Outflow (Case 2) –Under the limitation of the maximum lateral outflow rate–Under the limitation of the maximum lateral outflow rate
Suppose that the maximum lateral outflow rate is specified due to the limited capacity of flood gate or pump station, e.g. q 50.0 m3/s
Bound Constraints:
Application of the quasi-Newton method with bound constraints (L-BFGS-B)
Lateral Outflow q≤q0
Z0=-3.5m
Optimal Lateral Outflow with ConstraintOptimal Lateral Outflow with Constraint
Hours
Dis
char
ge(m
3 /s)
0 10 20 30 40 50-100
-50
0
50
100
Iteration= 1Iteration= 3Iteration= 4Iteration= 5Iteration= 6Iteration= 10Iteration= 30Iteration= 70
Hydrograph at inlet
Hours
Dis
char
ge(m
3 /s)
0 10 20 30 40 50-100
-50
0
50
100
Case 1Case 2
Hydrograph at inlet
Iterations of optimal lateral outflow Comparison of optimal lateral outflow rates between Case 1 and Case 2
Controlled Stage and Discharge in the Channel Controlled Stage and Discharge in the Channel (Case 2)(Case 2)
Km
01
23
45
67
89
10Hours
012
2436
48 Wat
erS
tage
(m)
0
1
2
3
4
5
Lateral Outflow
Km
0 1 2 3 4 5 6 7 8 9 10 Hours0
1224
3648
Dis
char
ge(m
3 /s)
20
40
60
80
100
Lateral Outflow
Stage in time and space Discharge in time and space
Allowable stage Z0=3.5m
Optimal Control of Lateral Outflows Optimal Control of Lateral Outflows – Multiple Lateral Outflows (Case 3)– Multiple Lateral Outflows (Case 3)
Suppose that there are three flood gates (or spillways) in upstream, middle reach, and downstream.
Condition of control:
Z0=3.5m
q1 q2q3
Optimal Lateral Outflow Rates in Three Diversions Optimal Lateral Outflow Rates in Three Diversions (Case 3)(Case 3)
Hours
Dis
char
ge(m
3 /s)
0 10 20 30 40 50-100
-50
0
50
100
q1
q2
q3
Hydrograph at inlet
HoursD
isch
arge
(m3 /s
)0 10 20 30 40 50
-100
-50
0
50
100
Lateral Outflow
Hydrograph at inlet
Optimal lateral outflow rates of three floodgates (Case 4)
Optimal lateral outflow of only one gate (=q1) (Case 1)
Controlled Stage and Discharge by Three Diversions Controlled Stage and Discharge by Three Diversions (Case 3)(Case 3)
Km
01
23
45
67
89
10Hours
012
2436
48
Wat
erS
tage
(m)
0
1
2
3
4
5
q 1
q 2
q 3 Km
01
23
45
67
89
10 Hours0
1224
3648
Dis
char
ge(m
3 /s)
20
40
60
80
100
q 3Lateral Outflo
w: q 1
q 2
Stage in time and space Discharge in time and space
Allowable stage Z0=3.5m
Comparisons of Diversion Percentages and Comparisons of Diversion Percentages and Objective Functions Objective Functions
Case Case qqmaxmax Number of Number of floodgatefloodgate
11 N/AN/A 11
22 50.050.0mm33/s/s 11
33 N/AN/A 33
Iterations of L-BFGS-B
Obj
ectiv
eF
unct
ion
0 10 20 30 40 50 60 7010-7
10-5
10-3
10-1
101
103
Case 1Case 2Case 43
CaseCase DiversionDiversion Volume Volume
(m(m33))
Percentage ofPercentage of Diversion Diversion
(%)(%)
11 3,952,2313,952,231 41.341.3
22 3,743,3793,743,379 39.139.1
33 3,180,6613,180,661 33.233.2
Control of Flood Diversion in A Channel NetworkControl of Flood Diversion in A Channel Network
L3 = 13,000m
L2 = 4,500m
L1
=4,
000m
1
2
3
Channel No.
Optimal Control of One Lateral Outflow in a Channel Optimal Control of One Lateral Outflow in a Channel Network (Case 5) Network (Case 5)
Channel No.
QP (m3/s)
Qb (m3/s)
Tp (hour)
Td (hour)
Z0 (m)
1 50.0 2.0 16.0 48.0 3.5 2 50.0 2.0 16.0 48.0 3.5 3 60.0 6.0 16.0 48.0 3.5
+2.0m
+0.0m
20m
70m
1:2
1:1.
5
Z0=3.5m
q(t)=?
Compound Channel Section
Time
Dis
char
ge
TpTd
Qp
Qb
Time
Dis
char
ge
TpTd
Qp
Qb
Time
Dis
char
ge
TpTd
Qp
Qb
Confluence
Optimal Lateral Outflow and Objective Function Optimal Lateral Outflow and Objective Function (Case 5: Channel Network)(Case 5: Channel Network)
Hours
Dis
char
ge(m
3 /s)
0 10 20 30 40 50-150
-100
-50
0
50
Iteration= 1Iteration= 3Iteration= 4Iteration= 6Iteration= 10Iteration= 30Iteration= 70
Hydrograph at inlet of main stem
Tp
Hydrograph at two branchs
Iterations of L-BFGS-B
Obj
ectiv
eF
unct
ion
Nor
mof
Gra
dien
t
0 10 20 30 40 50 60 7010-3
10-2
10-1
100
101
102
103
104
105
10-8
10-7
10-6
10-5
10-4
10-3
10-2
Objective FunctionNorm of Gradient
L3 =13,000m
L2 = 4,500m
L 1=
4,00
0m
1
2
3
Channel No.
Hours
Wat
erS
tage
(m)
0 10 20 30 40 500
1
2
3
4
5
No ControlOptimal Control
Allowable Stage
Comparisons of Stages (Case 5)Comparisons of Stages (Case 5)
Hours
Wat
erS
tage
(m)
0 10 20 30 40 500
1
2
3
4
5
No ControlOptimal Control
Allowable Stage
Hours
Wat
erS
tage
(m)
0 10 20 30 40 500
1
2
3
4
5
No ControlOptimal Control
Allowable Stage
Hours
Wat
erS
tage
(m)
0 10 20 30 40 500
1
2
3
4
5
No ControlOptimal Control
Allowable Stage
L3 =13,000m
L2 = 4,500m
L 1=
4,00
0m
1
2
3
Channel No.
Comparisons of Discharges (Case 5)Comparisons of Discharges (Case 5)
Hours
Dis
char
ge(m
3 /s)
0 10 20 30 40 500
50
100
150No ControlOptimal Control
Hours
Dis
char
ge(m
3 /s)
0 10 20 30 40 500
50
100
150No ControlOptimal Control
Hours
Dis
char
ge(m
3 /s)
0 10 20 30 40 500
50
100
150No ControlOptimal Control
Discharge increased !!
Hours
Dis
char
ge(m
3 /s)
0 10 20 30 40 500
50
100
150
No ControlOptimal Control
Discharge increased !!
L3 = 13,000m
L2 = 4,500m
L1
=4,
000m
1
2
3
Channel No.
Optimal Control of Multiple Lateral Outflows in a Optimal Control of Multiple Lateral Outflows in a Channel Network (Case 6) Channel Network (Case 6)
Channel No.
QP (m3/s)
Qb (m3/s)
Tp (hour)
Td (hour)
Z0 (m)
1 50.0 2.0 16.0 48.0 3.5 2 50.0 2.0 16.0 48.0 3.5 3 60.0 6.0 16.0 48.0 3.5
+2.0m
+0.0m
20m
70m
1:2
1:1.
5
Z0=3.5m
q3(t)=?
Compound Channel Section
Time
Dis
char
ge
TpTd
Qp
Qb
Time
Dis
char
ge
TpTd
Qp
Qb
Time
Dis
char
ge
TpTd
Qp
Qb
q2(t)=?
q1(t)=?
Optimal Lateral Outflow Rates and Objective FunctionOptimal Lateral Outflow Rates and Objective Function (Case 6) (Case 6)
Hours
Dis
char
ge(m
3 /s)
0 10 20 30 40 50-150
-100
-50
0
50
q1
q2
q3
Hydrograph at inlet of main stem
Tp
Hydrograph at two branchs
Iterations of L-BFGS-B
Obj
ectiv
eF
unct
ion
0 20 40 60 80 100
10-6
10-4
10-2
100
102
Case 5Case 6
Optimal lateral outflow rates at three diversions
Comparison of objective function
One Diversion
Three Diversions
Comparisons of Stages (Case 6)Comparisons of Stages (Case 6)
Hours
Wat
erS
tage
(m)
0 10 20 30 40 500
1
2
3
4
5
No ControlOptimal Control
Allowable Stage
Hours
Wat
erS
tage
(m)
0 10 20 30 40 500
1
2
3
4
5
No ControlOptimal Control
Allowable Stage
Hours
Wat
erS
tage
(m)
0 10 20 30 40 500
1
2
3
4
5
No ControlOptimal Control
Allowable Stage
L3 =13,000m
L2 = 4,500m
L 1=
4,00
0m
1
2
3
Channel No.
Comparisons of Discharges (Case 6)Comparisons of Discharges (Case 6)
L3 =13,000m
L2 = 4,500m
L 1=
4,00
0m
1
2
3
Channel No.
Hours
Dis
char
ge(m
3 /s)
0 10 20 30 40 500
50
100
150No ControlOptimal Control
Hours
Dis
char
ge(m
3 /s)
0 10 20 30 40 500
50
100
150No ControlOptimal Control
Hours
Dis
char
ge(m
3 /s)
0 10 20 30 40 500
50
100
150No ControlOptimal Control
Flood Diversion Control in River Flow (Real Storms)Flood Diversion Control in River Flow (Real Storms)
Days
Dis
cha
rge
(m3/s
)
0 5 10 15 20 25 300
5
10
15
20
25
30
35
40
Allowable Elevations along the River and Rating Allowable Elevations along the River and Rating Curve at OutletCurve at Outlet
Allowable Elevations at Cross Sections
4
5
6
7
8
9
10
0 500 1000 1500 2000 2500 3000 3500
X (m)
Ele
vati
on
(m
)
Maximum Bank Elevation (m)
Minimum Elevation (m)
Allowable Elevation (m)
Rating Curve at Outlet
4
4.5
5
5.5
6
6.5
7
7.5
0 5 10 15 20 25 30 35 40 45 50
Discharge (m3/s)
Wa
ter
Ele
vat
ion
(m
)
Rating Curve by Regression
Measured Data
Zobj (x) Z-Q
Optimal Control of One Flood Gate in River FlowOptimal Control of One Flood Gate in River Flow
Days
Dis
cha
rge
(m3/s
)
0 5 10 15 20 25 300
5
10
15
20
25
30
35
40
Days
Dis
cha
rge
(m3/s
)
0 5 10 15 20 25 30-30
-25
-20
-15
-10
-5
0
Optimal diversion hydrograph
Storm Hydrograph
Comparison of Stages
Comparisons of Water StagesComparisons of Water Stages
Days
Wa
ter
Sta
ge
(m)
0 5 10 15 20 25 305.4
5.6
5.8
6
6.2
6.4
6.6
6.8
Stage without controlControlled stage
Allowable stage
Days
Wa
ter
Sta
ge
(m)
0 5 10 15 20 25 306.4
6.6
6.8
7
7.2
7.4
7.6
7.8
8
8.2
Stage without controlControlled stage
Allowable stage
Days
Wa
ter
Sta
ge
(m)
0 5 10 15 20 25 307.4
7.6
7.8
8
8.2
8.4
8.6
8.8
9
Stage without controlControlled stage
Allowable stage
Comparisons of DischargesComparisons of Discharges
Days
Dis
cha
rge
(m3/s
)
0 5 10 15 20 25 300
5
10
15
20
25
30
35
40
Discharge without controlControlled discharge
Days
Dis
cha
rge
(m3/s
)
0 5 10 15 20 25 300
5
10
15
20
25
30
35
40
Discharge without controlControlled discharge
Days
Dis
cha
rge
(m3/s
)
0 5 10 15 20 25 300
5
10
15
20
25
30
35
40
Discharge without controlControlled discharge
Optimal Control of Two Floodgates in River FlowOptimal Control of Two Floodgates in River Flow
Days
Dis
cha
rge
(m3/s
)
0 5 10 15 20 25 30-30
-25
-20
-15
-10
-5
0
Days
Dis
cha
rge
(m3/s
)
0 5 10 15 20 25 30-30
-25
-20
-15
-10
-5
0
Days
Dis
cha
rge
(m3/s
)
0 5 10 15 20 25 300
5
10
15
20
25
30
35
40
Comparisons of Water Stages (Two Floodgates)Comparisons of Water Stages (Two Floodgates)
Days
Wa
ter
Sta
ge
(m)
0 5 10 15 20 25 305.4
5.6
5.8
6
6.2
6.4
6.6
6.8
Stage without controlControlled stage
Allowable stage
Days
Wa
ter
Sta
ge
(m)
0 5 10 15 20 25 306.4
6.6
6.8
7
7.2
7.4
7.6
7.8
8
8.2
Stage without controlControlled stage
Allowable stage
Days
Wa
ter
Sta
ge
(m)
0 5 10 15 20 25 307.4
7.6
7.8
8
8.2
8.4
8.6
8.8
9
Stage without controlControlled stage
Allowable stage
Comparisons of Discharges (Two Floodgates)Comparisons of Discharges (Two Floodgates)
Days
Dis
cha
rge
(m3/s
)
0 5 10 15 20 25 300
5
10
15
20
25
30
35
40
Discharge without controlControlled discharge
Days
Dis
cha
rge
(m3/s
)
0 5 10 15 20 25 300
5
10
15
20
25
30
35
40
Discharge without controlControlled discharge
Days
Dis
cha
rge
(m3/s
)
0 5 10 15 20 25 300
5
10
15
20
25
30
35
40
Discharge without controlControlled discharge
Comparison of Objective FunctionsComparison of Objective Functions
Iterations of L-BFGS-B
Ob
ject
ive
Fu
nct
ion
0 10 20 30 40 5010-7
10-5
10-3
10-1
101
103
Control of One FloodgateControl of Two Floodgates
Data Flows for Optimal Control Based on the Data Flows for Optimal Control Based on the CCHE1D Flow ModelCCHE1D Flow Model
Model of Optimal Flow Control Based on
the CCHE1D
Input data for the CCHE1D, e.g., *.bc, *.bf
Objective data: Filename: case.obs
Initial control variable data Filename: case.cnt
Control data of L-BFGS-B: Filename: case.lbf
Output data from the CCHE1D
Results of control variables: Filename: case.par iterate.dat
Results of objective Function: Filename: case.per
History output at every nodal point: case_long.plt
Input Data Output Data
ConclusionsConclusions
The Adjoint Sensitivity Analysis provides the nonlinear flow control with The Adjoint Sensitivity Analysis provides the nonlinear flow control with comprehensive and accurate measures of sensitivities on control actions.comprehensive and accurate measures of sensitivities on control actions.
The control model is capable of solving a large-scale flow control problem The control model is capable of solving a large-scale flow control problem efficiently.efficiently.
The integrated flow model (the CCHE1D) and the adjoint equations are suitable The integrated flow model (the CCHE1D) and the adjoint equations are suitable for computing channel network with complex geometries; By taking the for computing channel network with complex geometries; By taking the advantages of the flow model in dealing with channel network, this control advantages of the flow model in dealing with channel network, this control model can be applied readily to realistic flow control problems in natural model can be applied readily to realistic flow control problems in natural streams and channel network. streams and channel network.
The adaptive control framework is general and available for practicing a variety The adaptive control framework is general and available for practicing a variety of flow control actions in open channel, e.g., flood diversion, damgate of flow control actions in open channel, e.g., flood diversion, damgate operation, and water delivery.operation, and water delivery.
The control model also can assist engineers to plan the best locations and The control model also can assist engineers to plan the best locations and capacities of floodgates from hydrodynamic point of view.capacities of floodgates from hydrodynamic point of view.
Research Topics In the FutureResearch Topics In the Future
Control of Flood and Bed ChangesControl of Flood and Bed Changes Find a real case to apply the model to flood control Find a real case to apply the model to flood control
problem or water delivery problem;problem or water delivery problem; Flood control with water security management;Flood control with water security management; Develop further modules for other process controls, Develop further modules for other process controls,
e.g. water disposal control, water quality control, e.g. water disposal control, water quality control, sediment transport and morphological process control;sediment transport and morphological process control;
Flow controls with uncertainties under natural Flow controls with uncertainties under natural conditionsconditions
OthersOthers
AcknowledgementsAcknowledgements
This work was a result of research sponsored by the This work was a result of research sponsored by the USDA Agriculture Research Service under Specific USDA Agriculture Research Service under Specific Research Agreement No. 58-6408-2-0062 (monitored Research Agreement No. 58-6408-2-0062 (monitored by the USDA-ARS National Sedimentation by the USDA-ARS National Sedimentation Laboratory) and The University of Mississippi. Laboratory) and The University of Mississippi.
Special appreciation is expressed to Dr. Sam S. Y. Special appreciation is expressed to Dr. Sam S. Y. Wang, Dr. Mustafa Altinakar, Dr. Weiming Wu, and Wang, Dr. Mustafa Altinakar, Dr. Weiming Wu, and Dr. Dalmo Vieira for their comments and Dr. Dalmo Vieira for their comments and cooperation.cooperation.
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