Optimal Control of Flow and Sediment in River and Watershed

1. Optimal Control of Flow and Sediment in River and Watershed • Yan Ding1, MoustafaElgohry2, Mustafa Altinakar4,and Sam S. Y. Wang3 • 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 National Center for Computational Hydroscience and Engineering (NCCHE) The University of Mississippi Presented in 35th IAHR World Congress, September 8-13,2013, Chengdu, China

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

3. Flooding and Flood Control The spillway (highlighted in green) The Bonnet Carré 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 Carré to divert them. Flood Gate, West Atchafalaya Basin, Charenton Floodgate, LA

4. Sediment Control Xiao Land Di Reservoir, Yellow River, China Yellow River Reservoir Sediment Release at 9:00am, Clear Water Release at 10:00am, 6/19/2010

5. 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

6. 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 watersheds • Nonlinearity 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.

7. 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

8. Integrated Watershed & Channel Network Modeling with CCHE1D Rainfall-Runoff Simulation Upland Soil Erosion (AGNPS or SWAT) Channel Network and Sub-basin Definition (TOPAZ) Digital Elevation Model (DEM) Channel Network Flow and Sediment Routing (CCHE1D) Principal Features Dynamic Wave Model for Flood Wave Prediction • 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 where Q = discharge; Z=water stage; A=Cross-sectional Area; q=Lateral outflow; =correction factor; R=hydraulic radius n = Manning’s roughness • Boundary Conditions • Initial Conditions (Base Flows) • Internal Flow Conditions for Channel Network

9. CCHE1D Sediment Transport Model Principal Features Non-equilibrium transport of non-uniform sediments • 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 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 andQlk= lateral inflow or outflow sediment discharge per unit channel length; Ut=section averaged velocity of sediment

10. 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 Zobj

11. 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.

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

13. 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)

14. 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)

15. Variations of J with Respect to Control Variables – Formulations of Sensitivities Q(0,t) Q(L,t) Lateral Outflow q(x,t) 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.

16. 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. Sediment Control Model • Optimization Model • Adjoint sensitivity model • Sediment Transport Simulation Model • CCHE1D

17. A Nonuniform/nonequilibrium Sediment Transport Model: CCHE1D • Governing equation for the nonequilibrium transport of nonuniform sediment is (1) • In which the depth-average concentration and the sediment transport rate can be expressed as • and Eq. (1) becomes, (2) (3) • The bed deformation is determined with (4) (5) If

18. 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) where f is a measuring function and can be defined as, • (7) • 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 qlis a local minimizer of fS. • (8)

19. Optimization Model Consider the equation of morphological change: The objective function for control of morphological changes can be written as and measuring function as, • (9) • (10) where can be taken equal to i.e. the sediment transport capacity. 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.

20. Adjoint Equation for Sediment Control Taking the first variation of the augmented objective function, i.e. • (14) (15) • (16) Figure 1. Solution domain 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 • (17) • For minimizing J* , δJ* must be equal zero which means all terms multiplied by δQtmust be set to zero which leads to the following equation, • (18) which is the adjoint equation for the Lagrangian multiplier λS

21. Sediment Transport Control Actions and Sensitivity Qt(0,t) • Lateral Sediment Discharge Qt(L,t) Lateral Outflow ql • Upstream Sediment Discharge • Downstream Sediment Discharge • In this study,fS is not a function in ql, thus the sensitivity is based on the values of λS.

22. 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)

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

24. Optimal Control of Flood Diversion Rate A Hypothetic Single Channel 100m3/s Divert clear water q(t)=? 48 hours 16 hours Storm Event: A Triangular Hydrograph Cross Section

25. 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

26. Thalweg Change after Storm

27. Water Stage and Lateral Discharge The lateral discharge comparison for with and without sediment transport consideration at bed slope The water stage comparison for with and without sediment transport consideration at bed slope

28. Z0=3.5m q1 q2 q3 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.

29. Optimal Discharges for Multiple Floodgates Z0=3.5m q1 q2 q3

30. Comparison of Thalweg Changes after Storm

31. Optimal Control of Multiple Lateral Outflows in a Channel Network 1 3 q1(t)=? m 0 0 0 , 4 = 1 L q2(t)=? Z0=3.5m L = 3 1 3 , 0 0 0 m 2 . o N l e n n a h C q3(t)=? m 0 0 5 , 4 = L 2 Compound Channel Section

32. Optimal Lateral Outflow Rates and Objective Function One Diversion Three Diversions Optimal lateral outflow rates at three diversions Comparison of objective function

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

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

35. Hypothetical Case (2): Reservoir Sediment Release Excess Erosion Problem Downstream 20 m 1:2 1:2 Given Q = Q(t) 10 m S0=0.5 % Qs= ? L=7 km 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.

36. Hypothetical Case (2): ConotrolReservoir Sediment Release This 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 Scenario 3 Scenario 2 Figure (3) Stage reservoir water release Figure (2) Stage reservoir water release

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

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

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

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

41. Dam Removal Impacts The 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?)

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

43. Bed Material Properties and Model Parameters

44. Sediment size classes used in the simulations

45. Upstream Discharge Hydrograph Simulation Period 10/19/2007 – 09/30/2008

46. Simulation Results: Bed load adaptation length (1/3) Simulation parameters and associated values

47. Simulation Results: Bed Evolution

48. 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?

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

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