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Adaptive Control of Flood Diversion in an Open

Channel and Channel Network

National Center for Computational Hydroscience

and Engineering The University of Mississippi

Presented by

- Yan Ding

National Center for Computational Hydroscience

and Engineering The 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

Flood Damage

Flooded Street, Mississippi River Flood of 1927

From L.S.U. Library at the URL

http//www.lib.lsu.edu/mmarti3/smith/pages/mainst

reet.htm

An Example of Flood Diversion The Bonnet

Carre Spillway

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.

From http//www.mvn.usace.army.mil/pao/bcarre/bca

rre.htm

Scheduled Water Delivery and Pollutant Disposal

- Optimal Water Delivery

To give an optimal water delivery through

irrigation canals to irrigation areas

- 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

- 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

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

Integrated Watershed Channel Network Modeling

with CCHE1D

de Saint Venant Equations - Dynamic Wave

where Q discharge Zwater stage

ACross-sectional Area qLateral

outflow ?correction factor

Rhydraulic radius n Mannings roughness

Initial Conditions and Boundary Conditions

I.C. (Base Flow)

B.C.s

Upstream

(Hydrograph)

Downstream

or

(Stage-discharge rating curve)

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 Tcontrol duration L channel length

ttime xdistance along channel Zpredicted

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

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

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.

Variational Analysis - To Obtain Adjoint Equations

Extended Objective Function

where ?A and ?Q are the Lagrangian multipliers

Necessary Condition

on the conditions that

Fig. Solution domain

Variation of Extended Objective Function

where

Top width of channel

Variations of J with Respect to Control Variables

Formulations of Sensitivities

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

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

Considering the contour integral in ?J, This

term ?I needs to be zero.

Transversality (Final) Conditions

Backward Computation

Upstream B.C.

Downstream B.C.

Fig. Solution domain

Internal Boundary Conditions for Channel

Network

I.B.C.s of Flow Model

I.B.C.s of Adjoint Equations

Fig. Confluence

Numerical Techniques

1-D Time-Space Discretization (Preissmann, 1961)

where ? and ? are two weighting parameters in

time and space, respectively ?ttime increment

?xspatial length

Solver of the resulting linear algebraic

equations (Pentadiagonal Matrix)

Double Sweep Algorithm based on the Gauss

Elimination

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 - qk1 qk ?kdk
- ?k the step length of line search
- sufficient decrease and curvature

conditions - dk the search direction (descent direction)
- Bk n?n symmetric positive definite matrix
- For the Steepest Descent Method Bk I
- Newtons Method Bk? 2J(nk)
- Quasi-Newton Method
- Bk an approximation of the Hessian ?

2J(nk)

L-BFGS (One of LMQN method) (1)

- Difficulties of Newtons method in large-scale

optimization problem obtain the inverse Hessian

matrix, because - the Hessian is fully dense, or,
- the Hessian cannot be computed.
- BFGS (Broyden, Fletcher, Goldfarb, and Shanno,

1970) - Constructs the inverse Hessian

approximation ,

However, all of the vector pairs (sk, yk) have to

be stored.

L-BFGS (2)

- Updating process of the Hessian by using the

information from the last m Q-N iterations

Where m is the number of Q-N updates supplied by

the user, Generally (Nocedal Wright, 1999),

3 ? m ? 7 and

Only m vector pairs (si, yi),i1, m, need to be

stored

Flow chart of Finding optimal control variable

by using LMQN procedure

- 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

Objective Function

Optimal Control of Flood Diversion Rate ( Case

1) - A Hypothetic Single Channel

Lateral Outflow

Z03.5m

Cross-section

A Triangular Hydrograph

This value is used for solving adjoint equations

Optimal Lateral Outflow and Objective Function

(Case 1)

Optimal Outflow q

Objective function and Norm of gradient of the

function

Iterations of optimal lateral outflow

Comparison of Water Stages in Space and Time

(Case 1)

Allowable Stage Z03.5

No Control

Optimal Control of Lateral Outflow

Comparison of Discharge in Time and Space (Case 1)

No Control

Optimal Control of Lateral Outflow

Sensitivity ?J/?q(x,t)

Fast searching

Sensitivity of q in time and space at the 1st

iteration

Iterative history of sensitivity at the control

point

Optimal Control of Lateral Outflow (Case 2)

Under the limitation of the maximum lateral

outflow rate

Application of the quasi-Newton method with bound

constraints (L-BFGS-B)

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

Optimal Lateral Outflow with Constraint

Comparison of optimal lateral outflow rates

between Case 1 and Case 2

Iterations of optimal lateral outflow

Controlled Stage and Discharge in the Channel

(Case 2)

Allowable stage Z03.5m

Stage in time and space

Discharge in time and space

Optimal Control of Lateral Outflows Multiple

Lateral Outflows (Case 3)

Condition of control

Suppose that there are three flood gates (or

spillways) in upstream, middle reach, and

downstream.

Optimal Lateral Outflow Rates in Three Diversions

(Case 3)

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)

Allowable stage Z03.5m

Stage in time and space

Discharge in time and space

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

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

Optimal Control of Lateral Outflow (Case 4)

Allowable stage changing along with the channel

Condition of Allowable Stage

Suppose that the tolerable stage is changing

along with the channel as a function of channel

length, e.g, Z0(x) 5.5-0.2x, x(km).

Mississippi river levee, from http//images.usace.

army.mil/main.html

Optimal Lateral Outflow and Stage in the Channel

(Case 4)

Allowable stage Z0(x)5.5-0.0002x

Stage in time and space

Iterations of Lateral Outflow Rate

Control of Flood Diversion in A Channel Network

Optimal Control of One Lateral Outflow in a

Channel Network (Case 5)

Z03.5m

q(t)?

Compound Channel Section

Confluence

Optimal Lateral Outflow and Objective Function

(Case 5 Channel Network)

Comparisons of Stages (Case 5)

Comparisons of Discharges (Case 5)

Discharge increased !!

Discharge increased !!

Optimal Control of Multiple Lateral Outflows in a

Channel Network (Case 6)

1

3

q1(t)?

m

0

0

0

,

4

1

L

q2(t)?

Z03.5m

L

3

1

3

,

0

0

0

m

2

.

o

l

N

e

n

n

h

a

C

m

q3(t)?

0

5

0

4

,

L

2

Compound Channel Section

Optimal Lateral Outflow Rates and Objective

Function (Case 6)

One Diversion

Three Diversions

Optimal lateral outflow rates at three diversions

Comparison of objective function

Comparisons of Stages (Case 6)

Comparisons of Discharges (Case 6)

Flood Diversion Control in River Flow (Real

Storms)

Allowable Elevations along the River and Rating

Curve at Outlet

Zobj (x)

Z-Q

Optimal Control of One Flood Gate in River Flow

Comparison of Stages

Optimal diversion hydrograph

Storm Hydrograph

Comparisons of Water Stages

Comparisons of Discharges

Iterations of Objective Functions

Optimal Control of Two Floodgates in River Flow

Comparisons of Water Stages (Two Floodgates)

Comparisons of Discharges (Two Floodgates)

Comparison of Objective Functions

Data Flows for Optimal Control Based on the

CCHE1D Flow Model

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.

Optimal Water Delivery

- Objective To deliver a desired water to a target

area quickly without any large water stage

fluctuations Scheduled Water Delivery - Objective Function To evaluate the discrepancy

between predicted and objective water stages, a

objective function is defined as

where Tcontrol duration L channel length

ttime xdistance along channel Zpredicted

water stage Zobj(x) objective water stage in

cross-section ?Ztolerable difference of

objective water stage.

Optimal Control of Upstream Discharge

Z0-3.5m, ?Z0.1m

Q(0,t)?

Sensitivity of Q(0,t)

Constraint of Q Q(0,t) 100.0m3/s

Optimal Water Release at Upstream

Initial Estimation

Optimal Discharge

Controlled Stages and Discharges

Objective stage 3.5m, Tolerable difference

0.1m

Stage in time and space

Discharge in time and space

Objective Function J

- A general form of objective function for optimal

control is introduced as follows

where X physical variables, e.g.

stage, discharge, and/or velocity in open

channel q control variables (or

control forces), e.g., discharge at upstream or

downstream, lateral outflow, or

location of control structure t0 the

starting time of control, tf the

final time of control x1, x2 target

area for control r used-defined

measured function, e.g., discrepancy of stage

between predicted and target values

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

Optimal Control of Open Channel Flow A

Brief Review

Note CGM Conjugate Gradient Method

ASM Adjoint Sensitivity Method LBFGS

Limited memory Broyden, Fletcher, Goldfarb, and

Shanno Algorithm

CCHE1D Sediment Transport Model

- 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