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Adaptive Control of Flood Diversion in an Open Channel and Channel Network
Yan Ding
National Center for Computational Hydroscience and EngineeringThe University of Mississippi
Presented by
National Center for Computational Hydroscience and EngineeringThe University of Mississippi
April 29, 2005
Outline
• Introduction
• Nonlinear Models for Forecasting Flood Events
• Adjoint Sensitivity Analysis and Boundary Conditions for Adjoint Equations
• Optimization Procedures
• Applications to a Variety of Flood Diversion Control Scenarios
• Conclusions and Future Research Topics
Flooded Street, Mississippi River Flood of 1927From: L.S.U. Library at the URL: http://www.lib.lsu.edu/~mmarti3/smith/pages/mainstreet.htm
Flood Damage
Floodways and flow distribution during major floods in the Lower Mississippi River Valley
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 Diversion – The Bonnet Carre’ Spillway
From: http://www.mvn.usace.army.mil/pao/bcarre/bcarre.htm
Scheduled Water Delivery and Pollutant Disposal
•Flow-Optimized Discharges (Scheduled Disposal)Discharging pollutants to waters only during high river flows may mimic the pattern of natural diffuse pollutant loads in waters (such as nutrients or suspended solids exports from the catchment).– Scheduled disposal
•Optimal Pollutant Discharge
To find a optimal discharge to meet regulation for water quality protection, e.g., a tolerable amount of pollutant into water body
•Optimal Water Delivery To give an optimal water delivery through irrigation canals to irrigation areas
Applicability of Flow Control Problems
• Prevent levee of river from breaching or overflowing during flood season by using the most secure or efficient approach, e.g., operating dam discharge, diverting flood, etc.
Optimal Flood Control Adaptive Control• Perform an optimally-scheduled water delivery for irrigation to
meet the demand of water resource in irrigation canal Optimal Water Resource Management• To give the best pollutant disposal by controlling pollutant
discharge to obey the policy of water quality protection Best Environmental Management
Goal: Real Time Adaptive Control of Open Channel Flow
Difficulties in Control of Open Channel Flow
• Temporally/spatially non-uniform open channel flow Requires that a forecasting model can predict accurately complex water
flows in space and time in single channel and channel network
• Nonlinearity of flow control Nonlinear process control, Nonlinear optimization Difficulties to establish the relationship between control actions and
responses of the hydrodynamic variables
• Requirement of fast flow solver and optimization 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 making decision of flood mitigation, it is crucial for decision makers to 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
General Analysis Frameworks of Optimal Theories
Objective Function
Optimal Theory
Minimization Procedure
e.g.,Weight Least-Square Method
1. Sensitivity Analysis 2. Maximum likelihood 3. Extended Kalman Filter
1. CG Method 2. Newton Method 3. LMQN (LBFGS, LBFGS-B, etc) 4. Trust Region Method 5. Linear Programming
Integrated Watershed & Channel Network Modeling with CCHE1D
Digital ElevationModel (DEM)
Rainfall-Runoff SimulationUpland Soil Erosion(AGNPS or SWAT)
Channel Network Flow and Sediment Routing
(CCHE1D)
Channel Network andSub-basin Definition
(TOPAZ)
de Saint Venant Equations- Dynamic Wave
01
qx
Q
t
AL
02 2
2
2
fgSx
Zg
A
Q
xA
Q
t
L
3/42
2 ||
RA
QQnS f
A=Cross-sectional Area; q=Lateral outflow;=correction factor; R=hydraulic radius n = Manning’s roughness
where Q = discharge; Z=water stage;
Initial Conditions and Boundary Conditions
],0[),0,(
],0[),0,(
0
0
LxxAA
LxxQQ
t
t
I.C. (Base Flow)
B.C.s
],0[),,0(0
TttQQx
],0[),,( TttLZZLx
Upstream
Downstream
)(ZQLx
or (Stage-discharge rating curve)
(Hydrograph)
or open downstream boundary
Control Actions - Available Control Variables in Open Channel Flow
• Control lateral flow at a certain location x0: Real-time flow diversion rate q(x0, t) at a spillway
• Control lateral flow at the optimal location x: Real-time levee breaching rate q(x, t) at the optimal location
• Control upstream discharge Q(0, t): real-time reservoir release
• Control downstream stage Z(L, t): real-time gate operation
• Control downstream discharge Q(L, t): real-time pump rate control
• Control bed friction (roughness n):
An Objective Function for Flood Control
To evaluate the discrepancy between predicted and maximum allowable 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
)()(),()](),([
00
0004
xZxZif
xZxZifxxxZtxZLT
Wr
obj
objobjZ
dxdttxqQZrJT L
0 0
),,,,(
Mathematical Framework for Optimal Control
• The optimazition is to find the control variable q satisfying a dynamic system such that
where Q and Z are satisfied with the continuity equation and momentum equation, respectively (i.e., de Saint Venant Equations)
• Local minimum theory : Necessary Condition: If n* is the true value, then J(n*)=0; Sufficient Condition: If the Hessian matrix 2J(n*) is
positive definite, then n* is a strict local minimizer of f
)),,,(min()( qZQJqf
Sensitivity Analysis- Establishing A Relationship between Control Actions and System Variables
• Compute the gradient of objective function, q(X, q), i.e., sensitivity of control variable through
1. Influence Coefficient Method (Yeh, 1986): Parameter perturbation trial-and-error; lower accuracy
2. Sensitivity Equation Method (Ding, Jia, & Wang, 2004)
Directly compute the sensitivity ∂X/∂q by solving the sensitivity equations Drawback: different control variables have different forms in the equations, no
general measures for system perturbations; The number of sensitivity equations = the number of control variables.
Merit: Forward computation, no worry about the storage of codes
3. Adjoint Sensitivity Method (Ding and Wang, 2003)
Solve the governing equations and their associated adjoint equations sequentially. Merit: general measures for sensitivity, limited number of the adjoint equations
(=number of the governing equations) regardless of the number of control variables. Drawback: Backward computation, has to save the time histories of physical variables
before the computation of the adjoint equations.
x
t
A B
CD
O L
T
Variational Analysis- To 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
2{
ZQLZQL
Variation of Extended Objective Function
0
*
])([])[(
)(
||2
)||21
(
)||)1(2
(
)(
23
2
2
0 0
0 0 2
0 0 22
0 0 3
3/2
3
2
2*
0 0
dtQA
QA
A
QdxQ
AA
A
Q
qdxdt
ndxdtnK
QgQ
QdxdtK
Qg
xA
Q
tAx
AdxdtnK
QQRg
xA
Q
tA
Q
xB
g
t
dxdtqq
rQ
Q
rδA
A
Z
Z
r
QAQQ
QA
T L
A
T L Q
T L QQQA
T L QQQQA
T LJ
where
*;3
42;
*
*3/2
BB
RB
n
ARK Top width of channel
Variations of J with Respect to Control Variables – Formulations of Sensitivities
dttQQ
rtOQJ
T
x
A ),0()()),((0
0
ndxdtnK
QgQ
n
rnJ
T L Q
0 0 2
||2)(
dxdttxqq
rtxqJ
T L
A ),()),((0 0
dttLAA
Q
B
g
A
rtLAJ
Lx
T
Q ),()()),((0 3
2
*
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 the Full Nonlinear Saint Venant Equations
Q
r
A
Q
A
r
AK
QQg
xB
g
xA
Q
tQQAA
2*
||)1(2
Q
r
K
QgA
xA
xA
Q
t QAQQ
2
2
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 Conditions
x
t
A B
CD
O L
T
Fig. Solution domain
Considering the contour integral in J*, This term I needs to be zero.
0
])()[(])[(23
2
*2
DACDBCAB
QAQQ
QA dtQA
QA
A
Q
B
gdxQ
AA
A
QI
],0[,0),(
],0[,0),(
LxTx
LxTx
A
Q
Transversality (Final) Conditions
],0[,0),0( TttQ
],0[),,(),(2
TttLQ
AtL AQ
Upstream B.C.
Downstream B.C.
Backward Computation
Internal Boundary Conditions – for Channel Network
213
321
QQQ
ZZZ
I.B.C.s of Adjoint Equations
32
22
12
A
Q
A
Q
A
Q QA
QA
QA
I.B.C.s of Flow Model
Fig. Confluence
3
3
2*
2
3
2*
1
3
2*
xx
Q
xx
Q
xx
Q A
QBg
A
QBg
A
QBg
Numerical Techniques
])1()[1(])1([),( 111
1ni
ni
ni
nitx
ttt
tx ni
ni
ni
ni
1
11
1 )1(),(
xxx
tx ni
ni
ni
ni
111
1 )1(),(
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 Optimization
• CG Method (Fletcher-Reeves method) (Fletcher 1987) The convergence direction of minimization is considered as
the gradient of objective function.• Trust Region Method (e.g Sakawa-Shindo method) considering the first order derivative of performance function
only, stable in most of practical problems (Ding et al 2004)• Limited-Memory Quasi-Newton Method (LMQN) Newton-like method, applicable for large-scale computation,
considering the second order derivative of objective function (the approximate Hessian matrix) (Ding & Wang 2005)
• Others
Minimization Procedures
• Limited-Memory Quasi-Newton Method (LMQN) Newton-like method, applicable for large-scale computation
(with a large number of control parameters), considering the second order derivative of objective function (the approximate Hessian matrix)
Algorithms:
BFGS (named after its inventors, Broyden, Fletcher, Goldfarb, and Shanno)
L-BFGS (unconstrained optimization)
L-BFGS-B (bound constrained optimization)
Limited-Memory Quasi-Newton Method (LMQN) (Basic Concept 1)
Given the iteration of a line search method for parameter q
qk+1 = qk + kdk
k = the step length of line search sufficient decrease and curvature conditions
dk = the search direction (descent direction)
Bk = nn symmetric positive definite matrix
For the Steepest Descent Method: Bk = I
Newton’s Method: Bk= 2J(nk) Quasi-Newton Method:
Bk= an approximation of the Hessian 2J(nk)
)(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-B
• The purpose of the L-BFGS-B method is to minimize the objective function J(q) , i.e.,
min J(q),subject to the following simple bound constraint,
qmin q qmax,where the vectors qmin and qmax mean lower and upper
bounds on the control variables.
• L-BFGS-B is an extension of the limited memory algorithm (L-BFGS) (Liu & Nocedal, 1989) for bound constrained optimization (Byrd et al, 1995) .
Flooding and Flood Control
Levee Failure, 1993 flood. Missouri. Flood Gate, West Atchafalaya Basin, Charenton Floodgate, Louisiana
Control of Flood Diversion in A Single Channel – 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 ( Case 1) - 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 (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 (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 (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 ∂J/∂q(x,t)
Hours
A
0 10 20 30 40 500
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) –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 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 (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 – 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 (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 (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 Objective Functions
Case qmax Number of floodgate
1 N/A 1
2 50.0m3/s 1
3 N/A 3
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
Case Diversion Volume
(m3)
Percentage of Diversion
(%)
1 3,952,231 41.3
2 3,743,379 39.1
3 3,180,661 33.2
Control 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 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 (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)
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)
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 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 Function (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)
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)
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)
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 Curve 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 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 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 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 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)
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)
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 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 CCHE1D 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
Conclusions
• The Adjoint Sensitivity Analysis provides the nonlinear flow control with comprehensive and accurate measures of sensitivities on control actions.
• The control model is capable of solving a large-scale flow control problem efficiently.
• The integrated flow model (the CCHE1D) and the adjoint equations are suitable for computing channel network with complex geometries; By taking the advantages of the flow model in dealing with channel network, this control model can be applied readily to realistic flow control problems in natural streams and channel network.
• The adaptive control framework is general and available for practicing a variety of flow control actions in open channel, e.g., flood diversion, damgate operation, and water delivery.
• The control model also can assist engineers to plan the best locations and capacities of floodgates from hydrodynamic point of view.
Research Topics In the Future
• Find a real case to apply the model to flood control problem or water delivery problem;
• Flood control with water security management;• Develop further modules for other process controls,
e.g. water disposal control, water quality control, sediment transport and morphological process control;
• Flow controls with uncertainties under natural conditions
• Others
Acknowledgements
This work was a result of research sponsored by the USDA Agriculture Research Service under Specific Research Agreement No. 58-6408-2-0062 (monitored by the USDA-ARS National Sedimentation Laboratory) and The University of Mississippi.
Special appreciation is expressed to Dr. Sam S. Y. Wang, Dr. Mustafa Altinakar, Dr. Weiming Wu, and Dr. Dalmo Vieira for their comments and cooperation.
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