Optimal Control of Flow and Sediment in River and Watershed

35th IAHR World Congress, September 8-13,2013, Chengdu, China

Optimal Control of Flow and Sediment in River and Watershed

National Center for Computational Hydroscience and Engineering (NCCHE)The University of Mississippi

Presented in 35th IAHR World Congress, September 8-13,2013, Chengdu, China

Yan Ding1, Moustafa Elgohry2, Mustafa Altinakar4, and Sam S. Y. Wang3

1. Ph.D. Dr. Eng., Research Associate Professor, UM-NCCHE2. Graduate Student, UM-NCCHE3. Ph.D., Research Professor and Director, UM-NCCHE4. Ph.D., P.E., F. ASCE, Frederick A. P. Barnard Distinguished Professor Emeritus&, Director Emeritus, UM-

NCCHE


Outline

• Introduction

• Mathematical models for Flow and Sediment Transport in Alluvial Channel:

CCHE1D

• Optimal Control of Flood Flow and Morphological Change

• Optimization Procedure Based on Adjoint Sensitivity

• Applications to Flood Diversion Control Scenarios with Morphological

Changes

• Applications to Sediment Control in Rivers and Watersheds

• Concluding Remarks


Flooding and Flood Control

Levee Failure, 1993 flood. Missouri. Flood Gate, West Atchafalaya Basin, Charenton Floodgate, Louisiana


Sediment Control

Reservoir Sediment Release at 9:00am, Clear Water Release at 10:00am, 6/19/2010

Xiao Land Di Reservoir, Yellow River, China Yellow River


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 DischargeTo 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 overflowing or breaching 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 simulation-based optimization 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

• Sediment Control: Minimize morphological changes due to flood control actions

Optimal Control with multiple constraints and objectives

For Engineering Applications,• Integrate the control model with the CCHE1D flow model, • Apply to practical problems


Integrated Watershed & Channel Network Modeling with CCHE1D

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)

01

qxQ

tAL

02 2

2

2

fgSxZg

AQ

xAQ

tL

3/42

2 ||RA

QQnS f

Dynamic Wave Model for 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

Hydrodynamic Modeling in Channel Network Non-uniform Total-Load Transport Non-equilibrium Transport Model Coupled Sediment Transport Equations Solution Bank Erosion and Mass Failure Several Methods for Determination of Sediment-Related

Parameters

Principal Features


CCHE1D Sediment Transport ModelNon-equilibrium transport of non-uniform sediments

*

1tk tktk lkt k

t tk s

Q Q Q Q qt U x L

/tk tk t tkC Q A U

*' 11 bk

tk t ks

Ap Q Qt L

*

1tk tktk lkt k

s

AC Q Q Q qt x L

**

bk tkt kQ p Q

A= cross-section area; Ctk= section-averaged sediment concentration of size class k; Qtk= actual sediment transport rate; Qt*k= sediment transport capacity; Ls= adaptation length and Qlk = lateral inflow or outflow sediment discharge per unit channel length; Ut=section averaged velocity of sediment

Non-uniform Total-Load Transport Non-equilibrium Transport Model Coupled Sediment Transport Equations

Solution (Direct Solution Technique) Bank Erosion and Mass Failure Several Methods for Determination of

Sediment-Related Parameters

Principal Features


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):


+2.0m

+0.0m

20m

70m

Zobj

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

40 0 0

0 0

[ ( , ) ( )] ( ), ( ) ( )

0, ( ) ( )

obj objZ

obj

W Z x t Z x x x if Z x Z xr LT

if Z x Z x

0 0( , , , , )

T LFJ r Z Q q x t dxdt


Sensitivity Analysis- Establishing A Relationship between Control Actions and System Variables

• Compute the gradient of objective function with respect to control variable

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

*1 20 0

( )T L

F F A QJ J L L dxdt where A and Q are the Lagrangian multipliers

Fig. 1: Solution domain

Necessary Condition* 0F FJ J

on the conditions that

1

2

( , ) 0( , ) 0{L Q Z

L Q Z


Variation of Extended Objective Function

0 0

2/32

* 2 3 30 0

2 20 0

20 0

0 0

* ( )

2 (1 ) | |( )

2 | |1( )

2 | |

( )

T L

T L Q Q Q QA

T L Q Q QA

T L Q

T L

A

r Z r rδA Q q dxdtF Z A Q qg R Q Qg Q Q Adxdt

t B x A t A x nKg QQ Qdxdt

x A t A x KgQ Q

ndxdtnK

qdxd

J

2

2 3 2[( ) ] [ ( ) ]

0

Q QA Q A Q

t

QQ QA Q dx A Q dtA A A A

where

*;3

42; *

*3/2

BB

RBn

ARK Top width of channel


Adjoint Equations for the Full Nonlinear Saint Venant Equations

* 2

2 (1 ) | |Q QA A g AV Vg r Q rVt x B x K A A Q

2

2

2Q QAQ

gA V rA Vt x x K Q

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)


Adjoint Equations: Linear, Hyperbolic, and of First-Order

The obtained adjoint equations are first-order partial differential equations, which can be rewritten into a compact vector form

0

PNUUMU

xt

Q

A

U

VABgV

*M

2

2

2

||20

||20

KVgA

KVAVg

Nwhere

P represents the source term related to the objective function

This adjoint model has two characteristic lines with the following two real and distinct eigenvalues:

*2

2

2,1 4)1(

21

BAgVV

*2,1 BAgV

In the case of a flow in a prismatic open channel, β=1, therefore

Wave celerity is the same as the open channel flow. But propagation direction in time is opposite to the flow.


Variations of J with Respect to Control Variables – Formulations of Sensitivities

00

( (0, )) ( ) (0, )T

Ax

rJ Q t Q t dtQ

ndxdtnK

QgQnrnJ

T L Q

0 0 2

||2)(

dttLAAQ

Bg

ArtLAJ

Lx

T

Q ),()()),((0 3

2

*

dxdttxqqrtxqJ

T L

A ),()),((0 0

Lateral Outflow

Upstream Discharge

Downstream Section Area or Stage

Bed Roughness

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.

Q(0,t)

Q(L,t)

q(x,t)


Optimal Control of Sediment Transport and Morphological Changes

The developed model is coupling an adjoint sensitivity model with a sediment transport simulation model (CCHE1D) to mitigate morphological changes.

Different optimization algorithms have been used to estimate the value of the diverted or imposed sediment along river reach (control actions) to minimize the morphological changes under different practices and applications.

Optimization Model

Adjoint sensitivity model

Sediment Transport Simulation Model

CCHE1D

Sediment Control Model


A Sediment Transport Model, CCHE1D Overview

1tk tktk t k k

s

AC Q Q Q qt x L

/ ( )tk tk t tkC Q AU

In which the depth-average concentration and the sediment transport rate can be expressed as

and Eq. (1) becomes,

The bed deformation is determined with

' 11 bktk t k

s

Ap Q Q

t L

* 0bktk t k

AQ Qt

If

Governing equation for the nonequilibrium transport of nonuniform sediment is

3 *1( ) ( ) 0t t

t t t ls

Q QL Q Q Q qt U x L

(1)

(2)

(3)

(4)

(5)


Objective Function for Sediment Control and Minimization of Morphological Changes (Strong Control Condition)

To evaluate the bed area change, a weighted form is defined as

0 0

( , , , )T L

S b lJ f A q x t dxdt

2

0b

SW Af x xLT t

(6)

(7)

where f is a measuring function and can be defined as,

The optimization is to find the control variable q satisfying a dynamic system such that

where Abis satisfied with the sediment continuity equation

Local minimum theory : Necessary Condition: If q is the true value, then JS(q)=0; Sufficient Condition: If the Hessian matrix 2JS(q) is positive definite,

then ql is a local minimizer of fS.

( ) min( ( , )),S b lf q J A q (8)

23


Optimization Model

objtQ can be taken equal to *tQ

The objective function for control of morphological changes can be written as

and measuring function as,

0 0

( , , , )T L

S S t lJ f Q q x t dxdt

2

02 2

1 , ,(1 ')

objS t t

s

Wf Q x t Q x t x xLT p L

(9)

(10)

24

where i.e. the sediment transport capacity.

' 11 bt t

s

Ap Q Qt L

Consider the equation of morphological change:

It means that for minimizing morphological change in a cross section, it is needed to make sediment transport rate in the section close to the sediment transport capacity.


Variational Analysis for Optimal Control of Morphological Changes

An augmented objective function,

*3 3

0 0 0 0 0 0

( , , , )T L T L T L

S S S t l SJ J L dxdt f Q q x t dxdt L dxdt (11)

where 3 is the Lagrangian multiplier for the sediment transport equation

Necessary Condition

0* JJ

on the condition that

(12)

(13)

25

3 *1( ) ( ) 0t t

t t t ls

Q QL Q Q Q qt U x L


Adjoint Equation for Sediment Control

*3

0 0

T L

S S SJ f L dxdt

S SS t l

t l

f fdf Q qQ q

31 1 0t t

t t ls

Q QL Q Q qU t x L

*

0 0 0 0 0 0

T L T L T LS S S S S S t

S t l t S l S tt l s

f f QJ Q q dxdt Q dxdt q dxdt dx Q dtQ q U t x L U

Taking the first variation of the augmented objective function, i.e.

By using Green’s theorem and the variation operator δ in time-space domain shown in Fig. (1), the first variation of the augmented function can be obtained

For minimizing J* , δJ* must be equal zero which means all terms multiplied by δQt must be set to zero which leads to the following equation,

which is the adjoint equation for the Lagrangian multiplier λS

(14)

(15)

(16)

(17)

(18) 02 2

2 1 ,(1 ')

objS S St t

s s

W Q x t Q x x xU t x L LT p L

26

Figure 1. Solution domain


Boundary Conditions and Transversality Condition

Boundary Conditions of Adjoint Equation

, 0S L t 0,t T

, 0S x T Lx ,0

which result in the following boundary conditions for the adjoint equation,

The contour integral in Eq. (17) needs to be zero so as to satisfy the minimum condition of the performance function J*, namely,

0 0 0 00 0

( ) 0L T L T

S t S t S tS t S t S t

AB BC CD DA t x L t T x

Q Q Qdx Q dt dx Q dt dx Q dtU U U

Figure 1. Solution domain

(19)

(20)

27


Application of Developed Model To Channel Network – Internal Condition at a Confluence

Internal Boundary Condition

21

3 1 20 0 0

T T T

S t S t S tx x xQ dt Q dt Q dt

3 1 2t t tQ Q Q

1 2 3

0

0T

S t S t S tx x xQ Q Q dt

3

3 1 2t t tx x x

Q Q Q

1 2 3 1 3 2

0

0T

S t S t S t S tx x x x x xQ Q Q Q dt

1 3 2 3

0

0T

S S S Sx x x xdt 1 2 3

S S Sx x x


Sediment Transport Control Actions and Sensitivity

00

( (0, )) ( ) (0, )T

SS t S t

t x

fJ Q t Q t dtQ

Lateral Sediment Discharge

Upstream Sediment Discharge

0 0( ( , )) ( , )

T LS

S l S ll

fJ q x t q x t dxdtq

In this study, fS is not a function in ql, thus the sensitivity is based on the values of λS.29

0( ( , )) ( ) ( , )

TS

S t S tt x L

fJ Q L t Q L t dtQ

Downstream Sediment Discharge

Lateral Outflow ql

Qt(0,t)

Qt(L,t)


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)


Finding optimal control variable using LMQN procedure

Set the initial q, k=0, , j

Compute flow variables and update bathymetry (X0)

Flow and Sediment Transport Model

(CCHE1D)

Calculation of objective function Jk, gradient J q , and search

direction dk

Calculation of )( 11 kk qJg

1

1or k

k j

J q

J

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

Yes

No

Compute flow variables and update bathymetry (Xk+1)

L-BFGS

Solver of Adjoint Equations

Flow and Sediment Transport Model

(CCHE1D)

q is convergent

Four Major Modules

• Flow Solver• Sediment Transport• Sensitivity Solver• Minimization Process

Operational Flow Chart


Application

Optimal Flood Control in Alluvial Rivers and Watersheds


Optimal Control of Flood Diversion Rate A Hypothetic Single Channel

Time

Dis

char

ge

Tp Td

Qp

Qb

+2.0m

+0.0m

20m

70m

1:2

1:1.

5

Storm Event: A Triangular Hydrograph

x (km)

Elev

atio

n(m

)

0 2 4 6 8 100

1

2

3

4

5

6 Zobj(x) = 4.75-0.025x

Bed slope = 1:40000; d50=0.127mm

q(t)q(x)=

Cross Section

100m3/s

48 hours16 hours


Hours

Dis

char

ge(m

3 /s)

0 12 24 36 48

-100

-50

0

50

100

Iteration= 1Iteration= 4Iteration= 5Iteration= 6Iteration= 10Iteration= 30Iteration= 70

Inflow Hydrograph at inlet

Optimal q(t)

Optimal Lateral Outflow and Objective Function (Case 1)

Iterations of L-BFGS-BO

bjec

tive

Func

tion

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

erSt

age

(m)

012345

Water Stage without Control Optimal Control of Lateral Outflow

Km

01

23

45

67

89

10Hours

012

2436

48

Wat

erSt

age

(m)

012345

Lateral Outflow

Allowable Stage Z0=3.5


Thalweg Changes

x (km)

Thalweg(m)

ThalwegChange(m)

0 2 4 6 8 10-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

Initial thalwegThalweg after eventThalweg change

q


Water Stage and Lateral Discharge

Hours

Wat

erSt

age

(m)

0 12 24 36 481.5

2.5

3.5

4.5

5.5

Without sediment transportWith sediment transport

Zobj=4.57m

HoursD

isch

arge

(m3 /s

)0 12 24 36 48

-80

-60

-40

-20

0

Without sediment transportWith sediment transport

The water stage comparison for with and without sediment transport consideration at bed slope

The lateral discharge comparison for with and without sediment transport consideration at bed slope 32.5 10

32.5 10


Effects of bed slope

Hours

Wate

rStag

e(m)

0 12 24 36 481

2

3

4

5

6

Iteration = 1Iteration = 5Iteration = 15Iteration = 30Iteration = 70

No Control

Optimal Control

Hours

Wate

rStag

e(m)

0 12 24 36 481

2

3

4

5

6


No Control

Optimal Control

Hours

Wate

rStag

e(m)

0 12 24 36 481

2

3

4

5

6


No Control

Optimal Control

Hours

Wate

rStag

e(m)

0 12 24 36 481

2

3

4

5

6


No Control

Optimal Control

Comparison of water stages between controlled and uncontrolled states for different bed slopes

Slope = 0.0Slope = 32.5 10

Slope = 35.0 10 Slope = 37.5 10


Lateral Discharges for Different Bed Slopes

Hours

Disc

harg

e(m

3 /s)

0 12 24 36 48

-50

0

50

100


Hydrograph at inlet

Hours

Dis

char

ge(m

3 /s)

0 12 24 36 48

-100

-50

0

50

100


Hydrograph at inlet

Hours

Disc

harg

e(m

3 /s)

0 12 24 36 48

-100

-50

0

50

100


Hydrograph at inlet

Hours

Dis

char

ge(m

3 /s)

0 12 24 36 48

-100

-50

0

50

100


Hydrograph at inlet

Slope = 0.0Slope = 32.5 10

Slope = 35.0 10 Slope = 37.5 10


Optimal Withdrawal Hydrographs under Different Slopes

Hours

Dis

char

ge(m

3 /s)

0 12 24 36 48

-100

-50

0

50

100

Slope = 0.0Slope = 1:40000Slope = 1:20000Slope = 1:13333

Hydrograph at inlet

bed material of diameter: 0.127 mm


Bed Changes: Slope Effect

Fig. 13: The channel bed after the controlled flood has passed for different bed slopes and constant bed material of diameter 0.127 mm

Distance [km]

Tha

lweg

Cha

nge

(m)

0 2 4 6 8 10-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

No slopeSlope = 1:40000Slope = 1:20000Slope = 1:13333


Bed Changes: Sediment Size Effect

Fig. 14: The channel bed after the controlled flood has passed for different bed materials and constant bed slope 32.5 10

Distance [km]

Thal

wegC

hang

e[m

]

0 2 4 6 8 10-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

d50 = 0.210 mm

d50 = 0.127 mm

d50 = 0.600 mmd50 = 0.350 mm

d50 = 1.250 mm


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 Discharges for Multiple Floodgates

Hours

Discharge(m

3 /s)

0 12 24 36 48-100

-75

-50

-25

0

25

50

75

100

125

q2

q3

q1

Discharge withdrawn by single floodgate

Total discharge = q1 + q2 + q3

Hydrograph at inlet


Comparison of Thalweg Change

x (km)

Thalwegchange(m)

0 2 4 6 8 10-0.15

-0.1

-0.05

0

0.05

0.1

0.15

Three floodgatesOne floodgate

q2q1 q3(or q)


L3 = 13,000m

L2 = 4,500m

L 1=

4,00

0m

1

2

3

Channel No.

Optimal Control of Multiple Lateral Outflows in a Channel Network

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

Tp Td

Qp

Qb

Time

Dis

char

ge

Tp Td

Qp

Qb

Time

Dis

char

ge

Tp Td

Qp

Qb

q2(t)=?

q1(t)=?


Optimal Lateral Outflow Rates and Objective Function

Iterations of L-BFGS-B

Obj

ectiv

eFu

nctio

n0 20 40 60 80 100

10-6

10-4

10-2

100

102Case 5Case 6

Optimal lateral outflow rates at three diversions

Comparison of objective function

One Diversion

Three Diversions

Hours

Dis

char

ge(m

3 /s)

0 12 24 36 48-50

-40

-30

-20

-10

0

10

20

30

40

50

60

q1q2q3

Hydrograph of inlfow at main stem

Hydrograph at two branches


Comparisons of Stages

L3 =13,000m

L2 = 4,500m

L 1=

4,00

0m

1

2

3

Channel No.

Hours

Wat

erSt

age

(m)

0 12 24 36 480

1

2

3

4

5

6

Optimal ControlNo Control

Allowable Stage

Hours

Wat

erSt

age

(m)

0 12 24 36 480

1

2

3

4

5

6


Allowable Stage

Hours

Wat

erSt

age

(m)

0 12 24 36 480

1

2

3

4

5

6


Allowable Stage


Comparison of Thalweg along Main Channel

Distance [k m]

Th

alw

eg[m

]

0 1 2 3 4 5 6 7 8 9 10 11 12 130.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

Initial ThalwegNo ControlCase 1 (Gate 1 at 1.5 k m ds)Case 2 (Gate 1 at 2.5 k m ds)Case 3 (Gate 1 at 3.5 k m ds)Case 4 (Gate 1 at 4.5 k m ds)

Case 3

Gate 2

Gate 3

C ase 1C ase 2

Jun

cti o

n

Jun c

tion

4.5 k m

1.5 k m

2.5 k m

Case 4

3.5 k m

Distance [km]

Tha

lwe g

[m]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.51.9

2.1

2.3

2.5

2.7

2.9

3.1

3.3

Initial ThalwegNo ControlCase 1 (Gate 1 at 1.5 km ds)Case 2 (Gate 1 at 2.5 km ds)Case 3 (Gate 1 at 3.5 km ds)Case 4 (Gate 1 at 4.5 km ds)

Case 3

Case 1

Case 2

4.5 km

1.5 km

2.5 km

case 43.5 km


Application

Optimal Control of Morphological Changes in Alluvial Rivers and Watersheds


Hypothetical Case (1): Dam Removal

3 km

S = 0.3 %

Q = 50 m3/sQs= 10 kg/s 1:2 1:2

10 m

20 m

L=7 km

S0=0.5 %

qs = ?

Excess Deposition

Problem Downstream

Control Objective: To minimize morphological change downstream The inflow and sediment discharge were assumed 50 m3/s and 10 kg/s respectively.Sediment Properties: Uniform sediment of d = 20 mmSimulation time = 1 weekBed load adaptation length = 125 m, suspended load adaptation coefficient = 0.1, and mixing-layer thickness = 0.05 m.


Hypothetical Case (1): Model Results

54


Hypothetical Case (2): Reservoir Sediment Release

L=7 km

S0=0.5 %

Q = Q(t)

1:2 1:2

10 m

20 m

Qs= ?

55

Control Objective: To minimize morphological change downstream

Simulation time = 1 yearSediment Properties: Uniform sediment of d = 20 mmBed load adaptation length = 125 m, suspended load adaptation coefficient = 0.1, and mixing-layer thickness = 0.05 m.

Excess Erosion Problem

Downstream


Hypothetical Case (2): Reservoir Sediment Release

This case has been tested under three different scenarios:

1. Regular operating conditions: The dam release discharge was assumed to be 10 m3/s.

2. Stage operating conditions: The case has been again tested under stage dam release flow discharge

3. Storm operating conditions: the case has been again tested under storm dam release flow discharge

Time (days)

Upstreamflowdischarge(m3/s)

0 50 100 150 200 250 300 350 4000

5

10

15

20

25

30

35

40

45

50

55

60

Figure (2) Stage reservoir water release Figure (3) Stage reservoir water releaseTime (hour)

Upstreamflowdischarge(m3/s)

0 5 10 15 20 25 30 35 40 450

5

10

15

20

25

30

35

40

45

50

55

60


Hypothetical Case (2) – Scenario (1): Model Results

57

Clear water release rate at upstream Q(t) = 10 m3/s



58

Time (days)

Upstreamsedimentdischarge(kg/s)

0 50 100 150 200 250 300 350 4000

5

10

15

20

Distance downstream (km)

Thalwegchange(m)

0 1 2 3 4 5 6 7-10

-8

-6

-4

-2

0

2

No control (clear water)With control



59

Time (hours)

Upstream

sedimentdischarge(kg/s)

0 10 20 30 400

5

10

15

20

25

30

35

No limit on reservior release capacityReservior release capacity = 15 kg/s

Distance downstream (km)

Thalwegchange(m)

0 1 2 3 4 5 6 7-0.4

-0.2

0

0.2

No control (clear water)With control


Case (3): Sandy River Reach and Marmot Dam Removal

Source: Stillwater Science, 1999


Case Study - Overview

Sandy River longitudinal profile (Source: PGE photogrametry, 1999)



Sandy River longitudinal profile (Source: PGE photogrametry, 1999)



Sandy River longitudinal profile (Stillwater Science, 1999)

0.01 0.1 1 10 100 10000.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

90.0

100.0

Unit 1 - Average

Unit 1 - Upper bound

Unit 1 - Lower bound


Flow and Bed Material Properties

The simulation tested formulas and parameters value

Parameter ValueRoughness Coefficient (Manning’s n)

0.03-0.06 upstream and 0.04-0.06 downstream

Sediment transport equation Wu-Wang-Jia’s formula (Wu et al. 2000), SEDTRA module (Garbrecht et al. 1995), Modified Ackers-White formula (Proffit and Sutherland 1983), and Engelund and Hansen’s formula (Engelund and Hansen 1967)

Bed load adaptation length 250, 350, 500 and 1000m

Suspended load adaptation coefficient

0.25, 0.5 and 1.0

Mixing-layer thickness 0.05, 0.1 and 0.2mPorosity

0.25Simulation time step, Δt

0.5, 1, 3 and 6 minutesCross sections spacing, Δx

Varying (12m-325m)


Sediment size classes used in the simulations

Number of size

Representative size (mm)

Lower limit (mm)

Upper limit (mm)

1 0.09196 0.0625 0.1252 0.18393 0.125 0.253 0.36785 0.25 0.54 0.73570 0.5 1.05 1.47140 1.0 2.06 2.94281 2.0 4.07 5.88562 4.0 8.08 11.77124 8.0 16.09 23.54247 16.0 32.0

10 47.08494 32.0 64.011 94.16989 64.0 128.012 188.33980 128.0 256.0


Upstream Discharge Hydrograph

Simulation Period 10/19/2007 – 09/30/2008


Downstream Water Depth Hydrograph

Simulation Period 10/19/2007 – 09/30/2008


Simulation Results: Different Sediment transport formulas

Distance from Marmot Dam (km)

Average

Bed

ElevationChange(m)

-2.5 -2 -1.5 -1 -0.5 0-12

-10

-8

-6

-4

-2

0

2

ObservationsSEDTRA moduleWu et al. s formulaModified Ackers-White formulaEngelund and Hansen formula

DamLocation

(a)


Average

Bed

ElevationChange(m)

0 2 4 6 8 10 12 14 16 18-2

0

2

4

6

8

ObservationsSEDTRA moduleWu et al. s formulaModified Ackers-White formulaEngelund and Hansen formula

Dam

Location

(b)


Simulation Results: Different Roughness Coefficient

Distance From Marmot Dam (km)

Average

Bed

ElevationChange(m)

-2.5 -2 -1.5 -1 -0.5 0-12

-10

-8

-6

-4

-2

0

2

Observations after 1 yearn = 0.06-0.03 upstream & 0.04 downstreamn = 0.04 upstream & 0.04 downstreamn = 0.04 upstream & 0.06 downstream

Dam

Location

(a)


Average

Bed

ElevationChange(m)

0 2 4 6 8 10 12 14 16 18-4

-2

0

2

4

6

8

10

Observations after 1 yearn = 0.06-0.03 upstream & 0.04 downstreamn = 0.04 upstream & 0.04 downstreamn = 0.04 upstream & 0.06 downstreamD

amLocation

(b)


Simulation Results: Different Adaptation Length


Average

Bed

ElevationChange(m)

-2.5 -2 -1.5 -1 -0.5 0-12

-10

-8

-6

-4

-2

0

2

ObservationsL = 250 mL = 350L = 500 mL = 1000 m

Dam

Location

(a)


Average

Bed

ElevationChange(m)

0 2 4 6 8 10 12 14 16 18-2

0

2

4

6

8

ObservationsL = 250 mL = 350L = 500 mL = 1000 mD

amLocation

(b)


Simulation Results: Bed load adaptation length (1/3)

Parameter ValueRoughness Coefficient (Manning’s n)

0.04 upstream and 0.06 downstream

Sediment transport equation

Wu-Wang-Jia’s formula (Wu et al. 2000)

Bed load adaptation length 350 m

Suspended load adaptation coefficient 0.5

Mixing-layer thickness 0.05 m

Simulation time step 0.5 minuteSediment Size Class 12

Simulation parameters and associated values


Simulation Results: Bed Evolution


Average

Bed

ElevationChange(m)

-2.5 -2 -1.5 -1 -0.5 0-12

-10

-8

-6

-4

-2

0

2

Observations after 1 year15 Days3 months6 months9 months1 year

Dam

Location

(a)


Average

Bed

ElevationChange(m)

0 2 4 6 8 10 12 14 16 18-2

0

2

4

6

8

Observations after 1 year15 Days3 months6 months9 months1 year

Dam

Location

(b)


Application of Developed Model after Dam Removal

Calculate the required diverted sediment after Marmot dam removal at the location of the dam to mitigate the excess deposition downstream. Simulating period is one year immediately after dam removal.

Date

SedimentDischarge(m3/s)

12-04-2007 03-03-2008 06-01-2008 08-30-20080

0.1

0.2

0.3

0.4Natural sediment flushDiverted sediment under optimal control


Application of Developed Model: Results

Distance Downstream from Marmot Dam (km)

AverageBed

ElevationChange(m)

0 2 4 6 8 10 12 14 16 18-1

0

1

2

3

4

5

Without ControlWith Control


Application of the Simulation-based Optimization Model To Channel Network – Problem Setup

+2.0m

+0.0m

20m

70m

1:2

1:1.

5

Compound Channel Section

L3 = 13,000m

L2 = 4,500m

L 1=

4,00

0m

1

2

3

Channel No.

qs(t)=?

Time

Dis

char

ge

Tp Td

Qp

Qb

Time

Dis

char

ge

Tp Td

Qp

Qb

Time

Dis

char

ge

Tp Td

Qp

Qb

Confluence

Channel No.

QP (m3/s)

Qb (m3/s)

Tp (hour)

Td (hour)

1 50.0 2.0 16.0 48.0 2 50.0 2.0 16.0 48.0 3 60.0 6.0 16.0 48.0

Parameters for a 2-day Storm

Sediment Properties: Uniform sediment of d = 20 mm


Optimal Results for Controlling Morphological Changes in Channel Network

Results after 70 iterations

Time [hours]

Upstreamsedimentdischarge[kg/s]

0 4 8 12 16 20 24 28 32 36 40 44 480

0.01

0.02

0.03

0.04

0.05

0.06

Distance [km]

Thalwegchange[m]

0 2 4 6 8 10 12-0.02

-0.015

-0.01

-0.005

0

0.005

Without ControlWith ControlOptimal Solution of Sediment Release

Comparison of Thalweg Changes along the main channel


Minimization Process for Channel Network: History of Objective Function

Iterations

Objectivefunction

0 10 20 30 40 50 60 70100

101

102

103

104


Conclusions

82

An optimal procedure to minimize bed changes in open-channels was developed. It is based on adjoint sensitivity analysis for a one-dimensional sediment transport model, CCHE1D. The optimization module includes a numerical solver for the adjoint equation and an optimization procedure.

The model has been validated and applied to different sediment problems in alluvial rivers under different scenarios. The model has the flexibility to control the rate of bed deformation cross-sectional area under different control variables i.e. side inflow/outflow, upstream or downstream sediment discharge conditions.

Different optimization algorithms has been tested and a Limited Memory Quasi-Newton (L-BFGS-B) algorithm was the fastest convergent one.

The model has been applied to sedimentation problems and the results demonstrated that the model is able to mitigate the morphological changes effectively. The developed approach for real world cases such as optimal sediment diversion after dam removal has been elaborated.


Acknowledgements

This work was a result of research sponsored by the USDA Agriculture Research Service under Specific Research Agreement No. 58-6408-7-236 (monitored by the USDA-ARS National Sedimentation Laboratory) and The University of Mississippi.

Documents

Optimal Control of Flow and Sediment in River and Watershed