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.