Optimal Control of Flow and Sediment in River and Watershed - PowerPoint Presentation

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 Ding 1 , Moustafa Elgohry 2 , Mustafa Altinakar 4 , and Sam S. Y. Wang 3 Ph.D . Dr. Eng., Research Associate Professor, UM-NCCHE Graduate Student, UM-NCCHE Ph.D., Research Professor and Director, UM-NCCHE Ph.D., P.E., F. ASCE, Frederick A. P. Barnard Distinguished Professor Emeritus&, Director Emeritus, UM-NCCHE

Outline IntroductionMathematical models for Flow and Sediment Transport in Alluvial Channel: CCHE1DOptimal Control of Flood Flow and Morphological ChangeOptimization Procedure Based on Adjoint Sensitivity Applications to Flood Diversion Control Scenarios with Morphological ChangesApplications to Sediment Control in Rivers and WatershedsConcluding Remarks

Flood and Channel Degradation/ Aggredation Flooded Street, Mississippi River Flood of 1927River Bank Erosion Levee Failure, 1993 flood. Missouri. Cedar Rapids, Iowa, June 14, 2008

Flooding and Flood ControlFlood Gate, West Atchafalaya Basin, Charenton Floodgate, LA The Bonnet Carr é Spillway, the southern-most floodway in the Mississippi River and Tributaries system, has historically been the first floodway in the Lower Mississippi River Valley opened during floods. The USACE’s hydraulic engineers rely on discharge and gauge readings at Red River Landing, about 200 miles above New Orleans, to determine when to open the spillway. The discharge takes two days to reach the city from the landing. As flows increase, bays are opened at Bonnet Carr é to divert them. The spillway (highlighted in green)

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

Flow and Sediment Transport Control Problems Perform an optimally-scheduled water delivery for irrigation to meet the demand of water resources in irrigation canals  Optimal Water Resource Management (only flow control) 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 Flow Control (probably with sediment transport) To release reservoir sediments to river reaches downstream for managing sediment transport and morphological changes  Best Sediment Release Management

Difficulties in Optimal Control of Flow and Sediments Temporally/spatially non-uniform flow and morphodynamics Requires a forecasting model which can accurately predict complex water flows and morphodynamic processes in space and time in rivers and watershedsNonlinearity of flow and sediment control Nonlinear process control, Nonlinear optimization Difficulties to establish the relationship between control actions and responses of the hydrodynamic and morphodynamic variables Requirement of Efficient Simulation 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 an 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  Optimal 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 Elevation Model (DEM) Rainfall-Runoff Simulation Upland Soil Erosion (AGNPS or SWAT) Channel Network Flow and Sediment Routing (CCHE1D) Channel Network and Sub-basin Definition (TOPAZ) 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 Model Non-equilibrium transport of non-uniform sediments A = cross-section area; C tk = section-averaged sediment concentration of size class k ; Q tk = actual sediment transport rate; Q t*k = sediment transport capacity; L s = adaptation length and Q lk = lateral inflow or outflow sediment discharge per unit channel length; U t =section averaged velocity of sediment Non-uniform Total-Load Transport Non-equilibrium SedTran Model Coupled SedTran 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 FlowControl lateral flow at a certain location x0: Real-time flow diversion rate q(x0 , t) at a spillwayControl 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 controlControl bed friction (roughness n ):

Z obj Objective Function for Flood ControlTo 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; Z obj (x) =maximum allowable water stage in river bank (levee) (or objective water stage); x 0 = target location where the water stage is protective;  = Dirac delta function

Sensitivity Analysis- Establishing A Relationship between Control Actions and System VariablesCompute 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.

Variational Analysis - to Obtain Adjoint Equations Extended Objective Function where  A and  Q are the Lagrangian multipliers Fig. 1: Solution domain Necessary Condition on the conditions that

Variation of Extended Objective Function where Top width of channel

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 )

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 where 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: 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 (i.e. backward in time)

Variations of J with Respect to Control Variables – Formulations of Sensitivities 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 Nonuniform /nonequilibrium Sediment Transport Model: CCHE1D 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 If Governing equation for the nonequilibrium transport of nonuniform sediment is (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 (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 A b is satisfied with the sediment continuity equation Local minimum theory : Necessary Condition : If q is the true value, then J S (q)=0 ; Sufficient Condition : If the Hessian matrix  2 J S (q) is positive definite , then q l is a local minimizer of f S . (8) 21

Optimization Model can be taken equal to The objective function for control of morphological changes can be written as     and measuring function as, (9) (10) 22 where i.e. the sediment transport capacity. 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, (11) where  3 is the Lagrangian multiplier for the sediment transport equation Necessary Condition on the condition that (12) (13) 23

Adjoint Equation for Sediment Control 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 * , δ J * must be equal zero which means all terms multiplied by δ Q t m ust be set to zero which leads to the following equation, which is the adjoint equation for the Lagrangian multiplier λ S (14) ( 15) (16) (17) (18) 24 Figure 1. Solution domain

Boundary Conditions and Transversality Condition Boundary Conditions of Adjoint Equation 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, Figure 1. Solution domain (19) (20) 25

Sediment Transport Control Actions and Sensitivity Lateral Sediment Discharge Upstream Sediment Discharge In this study, f S is not a function in q l , thus the sensitivity is based on the values of λ S . 26 Downstream Sediment Discharge Lateral Outflow q l Q t (0,t) Q t ( L,t )

Numerical Techniques 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 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 Four Major Modules Flow Solver Sediment Transport Sensitivity Solver Minimization Process Operational Flow Chart

Application Optimal Flood Control in Alluvial Rivers and Watersheds(with sediment transport but no sediment control)

Optimal Control of Flood Diversion Rate A Hypothetic Single Channel Storm Event: A Triangular Hydrograph Divert clear water q(t)=? Cross Section 100m 3 /s 48 hours 16 hours

Optimal Lateral Outflow and Objective Function (Case 1) Iterations of optimal lateral outflow Objective function and Norm of gradient of the function Optimal Outflow q

Comparison of Water Stages in Space and Time (Case 1) Water Stage without Control Optimal Control of Lateral Outflow Allowable Stage Z 0 =3.5

Thalweg Change after Storm

Water Stage and Lateral Discharge 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

Effects of bed slope Comparison of water stages between controlled and uncontrolled states for different bed slopes Slope = 0.0 Slope = Slope = Slope =

Lateral Discharges for Different Bed Slopes Slope = 0.0 Slope = Slope = Slope =

Optimal Withdrawal Hydrographs under Different Slopes bed material of diameter: 0.127 mm

Performance Fig. 12: Iterations for objective function and norm of objective function gradient for different bed slopes: (a) 0.0; (b) ; (c) and (d) (a) (b) (c) (d)

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

Bed Changes: Sediment Size Effect Fig. 14: The channel bed after the controlled flood has passed for different bed materials and constant bed slope

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: Z 0 =3.5m q 1 q 2 q 3

Optimal Discharges for Multiple Floodgates Z 0 =3.5m q 1 q 2 q 3

Comparison of Thalweg Changes after Storm

L 3 = 1 3 , 0 0 0 m L 2 = 4 , 5 0 0 m L 1 = 4 , 0 0 0 m 1 2 3 C h a n n e l N o . Optimal Control of Multiple Lateral Outflows in a Channel Network Z 0 =3.5m q 3 (t)=? Compound Channel Section q 2 (t)=? q 1 (t)=?

Optimal Lateral Outflow Rates and Objective Function Optimal lateral outflow rates at three diversions Comparison of objective functionOne Diversion Three Diversions

Comparisons of Stages L 3 = 1 3 , 0 0 0 m L 2 = 4 , 5 0 0 m L 1 = 4 , 0 0 0 m 1 2 3 C h a n n e l N o .

Comparison of Thalweg along Main Channel D i s t a n c e [ k m ] T h a l w e g [ m ] 0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 3 . 5 4 . 0 4 . 5 I n i t i a l T h a l w e g N o C o n t r o l C a s e 1 ( G a t e 1 a t 1 . 5 k m d s ) C a s e 2 ( G a t e 1 a t 2 . 5 k m d s ) C a s e 3 ( G a t e 1 a t 3 . 5 k m d s ) C a s e 4 ( G a t e 1 a t 4 . 5 k m d s ) C a s e 3 G a t e 2 G a t e 3 C a s e 1 C a s e 2 J u n c t i o n J u n c t i o n 4 . 5 k m 1 . 5 k m 2 . 5 k m C a s e 4 3 . 5 k m D i s t a n c e [ k m ] T h a l w e g [ m ] 0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 4 . 5 5 5 . 5 1 . 9 2 . 1 2 . 3 2 . 5 2 . 7 2 . 9 3 . 1 3 . 3 I n i t i a l T h a l w e g N o C o n t r o l C a s e 1 ( G a t e 1 a t 1 . 5 k m d s ) C a s e 2 ( G a t e 1 a t 2 . 5 k m d s ) C a s e 3 ( G a t e 1 a t 3 . 5 k m d s ) C a s e 4 ( G a t e 1 a t 4 . 5 k m d s ) C a s e 3 C a s e 1 C a s e 2 4 . 5 k m 1 . 5 k m 2 . 5 k m c a s e 4 3 . 5 k m

Application Optimal Control of Sediment Transport and Morphological Changes in Alluvial Rivers and Watersheds(No Flow Control)

Hypothetical Case (1): Dam Removal 3 km S = 0.3 % Q = 50 m 3 /s Q s = 10 kg/s 1:2 1:2 10 m 20 m L=7 km S 0 =0.5 % q s = ? Excess Deposition Problem Downstream Control Objective: To minimize morphological change downstream The inflow and sediment discharge were assumed 50 m 3 /s and 10 kg/s respectively. Sediment Properties: Uniform sediment of d = 20 mm Simulation time = 1 week Bed load adaptation length = 125 m, suspended load adaptation coefficient = 0.1, and mixing-layer thickness = 0.05 m.

Hypothetical Case (1): Model Results51

Hypothetical Case (2): Reservoir Sediment Release L=7 km S 0 =0.5 % Given Q = Q(t) 1:2 1:2 10 m 20 m Q s = ? 52 Control Objective: To minimize morphological change downstream Simulation time = 1 year Sediment Properties: Uniform sediment of d = 20 mm Bed 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): Conotrol Reservoir Sediment ReleaseThis case has been tested under three different scenarios: Regular operating conditions: The dam release discharge was assumed to be 10 m3/s.Stage operating conditions: The case has been again tested under stage dam release flow discharge Storm operating conditions: the case has been again tested under storm dam release flow discharge Figure (2) Stage reservoir water release Figure (3) Stage reservoir water release Scenario 2 Scenario 3

Hypothetical Case (2) – Scenario (1): Model Results54 Clear water release rate at upstream Q(t) = 10 m3/s

Hypothetical Case (2) – Scenario (2): Model Results55 Flow Release Condition Optimal Sediment Release Solution Morphological Changes after one year

Hypothetical Case (2) – Scenario (3): Model Results56 Upstream flood flow (given) Morphological changes after storm

Case (3): Sandy River Reach and Marmot Dam Removal Source: Stillwater Science, 1999

Why Are Some Dams Being Removed? 58Financial issue : Operating and maintenance costs outweigh the benefits - including hydropower, flood control, irrigation, or recreation, Functional issue: where the dam no longer serves any useful purpose, Ecological issues: restoring flows for fish and wildlife, reinstating the natural sediment and nutrient flow, Safety issues: eliminating safety risks, Recreational issues: restoring opportunities for recreation. 

Dam Removal Impacts 59The impacts of removal have been addressed by different studies. Generally they can be divided into main categories. (1) Short-Term Ecological Impacts of Dam Removal Sediment Release, Increased Sediment Concentration and Contaminated Sediment, and(2) Long-Term Impacts of Dam Removal (Flow change regimes, temperature, sediment transport and water quality) Objective of control in this case : Minimize the morphological changes (erosion and deposition) at downstream by diverting extra sediments from the reservoir (dredging?)

Case Study - Overview Sandy River longitudinal profile (Source: PGE photogrametry, 1999)

Case Study - Overview Reservoir deposition profile (Source: PGE photogrametry, 1999) Reservoir sediment size composition (Stillwater Science, 1999)

Case Study - Overview Sandy River longitudinal profile (Source: PGE photogrametry, 1999)

Bed Material Properties and Model Parameters ParameterValue Roughness 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 & Sutherland 1983 ) 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 0.2m Porosity 0.25 Simulation time step, Δt 0.5, 1, 3 and 6 minutes Cross 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 0.125 2 0.18393 0.125 0.25 3 0.36785 0.25 0.5 4 0.73570 0.5 1.0 5 1.47140 1.0 2.0 6 2.94281 2.0 4.0 7 5.88562 4.0 8.0 8 11.77124 8.0 16.0 9 23.54247 16.0 32.0 10 47.08494 32.0 64.0 11 94.16989 64.0 128.0 12 188.33980 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

Simulation Results: Different Roughness Coefficient

Simulation Results: Different Adaptation Length

Simulation Results: Bed load adaptation length (1/3) Parameter Value Roughness 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 minute Sediment Size Class 12 Simulation parameters and associated values

Simulation Results: Bed Evolution

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. Engineering difficulty: how to divert the sediments based on the optimal schedule?

Optimal sediment diversion after Dam Removal Engineering difficulty: how to divert the sediments based on the optimal schedule?

Application of Developed Model: Scenario (3) Results

Application of the Simulation-based Optimization Model To Channel Network – Problem Setup Compound Channel Section L 3 = 1 3 , 0 0 0 m L 2 = 4 , 5 0 0 m L 1 = 4 , 0 0 0 m 1 2 3 C h a n n e l N o . q s (t )=? Confluence Parameters for a 2-day Storm Sediment Properties: Uniform sediment of d = 20 mm

Application of Developed Model To Channel Network – Internal Condition at a Confluence Internal Boundary Condition 2 1 3

Optimal Results for Controlling Morphological Changes in Channel Network Results after 70 iterations Optimal Solution of Sediment Release Comparison of Thalweg Changes along the main channel

Minimization Process for Channel Network: History of Objective Function

Conclusions 79An 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.

Conclusions The Adjoint Sensitivity Analysis provides the optimal withdrawal of water from non-alluvial as well as alluvial channels.The morphological changes associated with the flood diversion can be quantified with help of the model.The adaptive control framework is general and available for practicing a variety of flow control actions in open channel, e.g., flood diversion, dam gate 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.This simulation-based model can practice optimal control with multiple constraints (stages and bed changes) and multiple objectives (flood control and morphological control)

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.