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Optimal Control of Flood Diversion in Watershed Using Nonlinear
2
Optimization
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Yan Ding and Sam S. Y. Wang
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National Center for Computational Hydroscience and Engineering, The University of Mississippi,
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University, MS 38677, U.S.A.
7 8
Abstract
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This study aims to develop a simulation-based optimization model applicable to mitigate
10
hazardous floods in storm events in a watershed which consists of a complex channel network
11
and irregular topography. A well-established model, CCHE1D, is used as the simulation model
12
to predict water stages and discharges of unsteady flood flows in a channel network, in which
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irregular (i.e. non-rectangular and non-prismatic) cross-sections are taken into account. Based on
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the variational principle, the adjoint equations are derived from the nonlinear hydrodynamic
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equations of CCHE1D, which are to establish a unique relationship between flood control
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variables and hydrodynamic variables. The internal conditions at the confluence in channel
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network for solving the adjoint equations in a watershed are obtained. An implicit numerical
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scheme (i.e. Preissman’s scheme) is implemented for discretizing and solving the adjoint
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equations with the derived internal conditions and boundary conditions. The applicability of this
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integrated optimization model is demonstrated by searching for the optimal diversion
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hydrographs for withdrawing flood waters through a single floodgate and multiple floodgates
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into detention basins. Numerical optimization results show that this integrated model is efficient
Corresponding author. Tel: (662)915-8969, Fax: (662)915-7796, Email:
[email protected]
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and robust. It is found that the single-floodgate control leads to an unfavorable speed-up in river
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flow which may create extra erosions in the channel bed; and multiple-floodgates diversion
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control diverts less flood waters, therefore can be a cost-effective control action. This simulation-
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based optimization model is capable of determining the optimal schedules of diversion discharge,
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optimal floodgate locations, minimum capacities of flood water detention basins in rivers and
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watersheds.
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Keywords: Flood Control, Adjoint Sensitivity, Optimization, Channel Network, Watershed
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2
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NOMENCLATURE A
cross-sectional area (m2)
B*
channel width on water surface (m)
g
the gravitational acceleration (m/s2)
q
volumetric rate of lateral outflow (or inflow) per unit length of the channel between two adjacent cross sections (m2/s)
Q
discharge through a cross-section (m3/s)
J
objective function
J*
augmented objective function
K
the conveyance K is defined as K= AR2/3/n (m3/s)
L
length of river reach (m)
L1 , L2
represent the continuity equation and the momentum equation, respectively
M
coefficient matrix in the adjoint equations
n
Manning’s roughness coefficient (s/m1/3)
N
coefficient matrix in the adjoint equations
P
source term in the adjoint equations
R
hydraulic radius (m)
t
time (s)
T
control period (s)
u
g−B*βQ2/A3 (m/s2)
U
A vector for the adjoint variables
V
cross section averaged velocity (m/s)
3
Wz
weighting factor which is to adjust the scale of the objective function
x
channel length coordinate (m)
x0
target location where the water stage is protective (m)
x1,x2,x3
locations of three cross-sections in a confluence
xc
location of flood diversion gate (m)
Z
water stage (m)
Zobj(x)
the maximum allowable water stage (the objective water stage) (m)
Greek Symbols
β
momentum correction factor
Dirac delta function
∆t
time increment (s)
∆x
spatial length (m)
1 , 2
eigenvalues of the adjoint equations
A , Q
the Lagrangian multipliers
Λ
2/3−4R /3B*
θ
weighting parameter of Preissmann’s scheme
Ψ
weighting parameter of Preissmann’s scheme
34 35
4
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INTRODUCTION
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Flood control to mitigate hazardous flood waters in rivers and watershed is of vital
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importance for inundation prevention, flood risk management, and water resource management.
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By operating in-stream hydraulic structures (e.g. reservoirs, dams, floodgates, spillways, etc.),
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peak flood discharges and high water stages in channels during storms can be reduced so that
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overflow and overtopping on levees, as well as the resulting inland flooding and inundation are
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eventually prevented. In case of emergency when flood waters are predicted to exceed capacities
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of river reaches, controls of flood water diversion/withdrawal through floodgates, diversion
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channels, or deliberate levee breaching can quickly mitigate flood water stages over the target
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reaches and even the entire watershed. Among them is the optimal flood control that can give the
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best cost-effective flood control schedule to minimize the risk of hazardous flood waters. It also
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enables to find the best design of capacities and locations of flood control structures, e.g.
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floodgates, floodwater detention basins (Ding and Wang 2006a, b). Meanwhile, optimal flood
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flow control methodologies can be utilized to practice best management of water resource,
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sediment transport (e.g. Nicklow and Mays 2000), and water quality (e.g. Piasecki and
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Katapodes 1997) to achieve sustainable development of economy and society that largely depend
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on limited water availability.
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The difficulty for achieving optimal flood control in rivers and watersheds is attributed to
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non-uniformities of spatially-distributed flood waters, unsteadiness and nonlinearities of its
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dynamics. As a prerequisite, a flood simulation model must be able to predict hydrodynamic
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variations of nonlinear flood flows in time and space and compute efficiently and accurately
5
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flood wave propagations of in river channels. Different from man-made open channels, river
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cross-sections, in general, are irregular, usually, non-rectangular and/or non-prismatic.
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Consequently, nonlinear optimization techniques have to be developed for establishing the
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relationship between flood control variables/actions (i.e. lateral diversion discharge schedules in
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this study) and nonlinear responses of hydrodynamic variables (i.e., water stages and discharges).
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Among them, adjoint sensitivity analysis (ASA) based on the variational principle has been
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applied to find the sensitivity of hydrodynamic variables with respect to control variables in one-
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and two- dimensional flow models (Kawahara and Kawasaki 1990, Sanders and Katopodes
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1999abc, Ding and Wang 2006a). Through solving a set of adjoint equations of hydrodynamic
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equations, the solutions of adjoint variables can readily connect the control actions with
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nonlinear and unsteady responses of flood flows, and provide an accurate measure of sensitivity
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of control performance (usually defined by an objective function). Sanders and Katapodes
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(1999a) performed an ASA of flood control by operating lateral flood water outflow in a one-
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dimensional (1-D) straight channel. They extended to a flood control by using the nonlinear two-
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dimensional (2-D) shallow water equations and Hancock’s predictor-corrector scheme (Sanders
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and Katapodes 1999b). According to these river flood control studies, ASA has shown its
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efficiency and effectiveness in solving optimal control problems which involve a large number
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of control variables, which usually are long vector variables in flow simulation model to
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determine the optimal diversion flow schedule over a several days long flood control period.
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Meanwhile, variational data assimilation (VDA) and ASA have been used for estimating
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unknown bathymetries of rivers and improving flood predictions by assimilating observed flow
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variables from measurements and satellite images (e.g. Mazauric 2003, Le Dimet et al. 2003,
6
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Mazauric et al. 2004). A similar variational approach for Lagrangian data assimilation in rivers
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applied to identification of bottom topography and initial water depth and velocity estimation
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was presented in Honnorat et al. (2007, 2009, 2010). Elhanafy et al. (2008) quantified predictive
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uncertainties in simulations of flood wave propagations in a prismatic channel of trapezoidal
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section. Gejadze et al. (2006), Gejadze and Monnier (2007), and Marin and Monnier (2009)
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proposed a ‘zoom’ model to assimilate river flood observation data into a coupled open channel
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flow model to predict respectively a flood flow in channel network by a 1-D flow model and
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local overflow/innudation in a vulnerable river reach by a 2-D model. Gejadze and Copeland
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(2006) and Gejadze et al. (2006) developed a data assimilation method for the estimation of open
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boundary conditions for the vertical 2-D Navier–Stokes equations with a free surface from fixed
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depth and velocity measurements. Lai and Monnier (2009) developed a data assimilation
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technique to simulate flood flows by using the 2-D shallow water equations and assimilating
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remote sensing observations. Hostache et al. (2010) further applied this approach to predict a
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flood inundation in a river by using remote sensing images of the river flood. Castaings et al.
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(2009) successfully applied adjoint sensitivity analysis and parameter estimation to a distributed
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hydrological model to compute flood flow in a watershed catchment. It is shown that forward
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and adjoint sensitivity analysis provide a local but extensive insight on the relation between the
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assigned model parameters and the simulated hydrological response.
100 101
This study aims to develop a simulation-based optimization model applicable to mitigate
102
hazardous floods in storm events in channel network which covers a watershed. A well-
103
established model, CCHE1D, is used as the simulation model to predict water stages and
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discharges in river reaches of the channel network, because it is computationally efficient and
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capable of predicting dynamic flood flow propagations in channel network with irregular cross-
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section shapes (i.e. non-rectangular and non-prismatic channels) (Wu and Vieira 2002, NCCHE
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2008). As an extensively verified and validated numerical model, CCHE1D has become a
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general computational software package for simulating unsteady open channel flows, sediment
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transport, morphodynamic processes, and water quality in river and watershed. The optimization
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of flood control in the paper is based on the adjoint sensitivity approach which is the way to
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establish the most accurate relationship between control actions and dynamic flood flows
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simulated by CCHE1D. The adjoint equations are derived for the nonlinear 1-D Saint-Venant
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equations which are the governing equations in CCHE1D for simulating open-channel flows
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through non-rectangular and non-prismatic cross-sections. In order to be generally applicable to
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find the adjoint solutions in channel network, the internal conditions of the adjoint equations at
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confluences (junctions) of channel network are derived on the basis of the variational analysis.
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By specifying internal conditions at confluences (junctions) for both the simulation model and
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the adjoint model, this simulation-based optimization model becomes well-integrated to be able
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to automatically find the optimal solutions (i.e. lateral flow diversion hydrographs) to best
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control flood waters occurring in all the river reaches of the channel network. Detailed numerical
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solution algorithms for solving the flow governing equations and the adjoint equations are
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developed by using an implicit time-space discretization scheme (i.e. Preissman’s scheme). For
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finding the optimal solutions for the best flood control, an efficient optimization procedure based
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on the Limited-Memory Quasi-Newton (LMQN) method (Liu and Nocedal 1989) is used.
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In this paper, effectiveness and applicability of this integrated simulation-based
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optimization tool are confirmed by searching for the optimal control variables, lateral flood
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diversion discharge hydrographs at one or multiple floodgates to withdraw flood waters from
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channels of watershed. For demonstrating the optimal flood control, a hypothetical storm event
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causing hazardous flood in the watershed is adopted; and the channel network with non-
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rectangular cross sections is sufficiently complex to represent realistic watershed topography.
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The control objective is to best prevent potential flooding waters from overflowing or
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overtopping when the capacity of target reaches is predicted to be exceeded. Cost-effectiveness
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of different control actions is analyzed by comparing the diverted flood water volume by one
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floodgate control with that by multiple floodgates control. It is found that the single-floodgate
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control leads to an unfavorable speed-up in river flow which may create extra erosions in the
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channel bed; and multiple-floodgates diversion control diverts less flood waters, therefore can be
138
a cost-effective control action. Furthermore, in order to validate the applicability of the
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optimization, a one-month optimal flood control is demonstrated by applying the CCHE1D-
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based optimization to mitigate river floods in a natural river in western Wyoming with complex
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river course and cross-section shapes. The following paragraphs will present mathematical
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formulations of the flood simulation model, the adjoint model, their internal condition, and
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boundary condition. By employing an implicit iterative solver, details on numerical
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implementation for solving both models will be addressed. Two flood control cases in a
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watershed in a hypothetical storm will be demonstrated. Conclusions on the study will be given
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finally.
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FLOOD SIMULATION MODEL
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The flood simulation model in this simulation-based optimization model is a well-
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established watershed model called CCHE1D (Wu and Vieira 2002). It contains a geographic
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information system (GIS) application software to extract channel network from a digital
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elevation model (DEM) of a watershed (Figure 1a), and 1-D open-channel flow model (diffusion
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wave model or dynamic wave model) to simulate hydrodynamic variables (water stages and
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discharges) in watershed. The dynamic wave model in CCHE1D is adopted in the optimization
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model for simulating unsteady flood flows in channel network of a watershed. The governing
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equations of this hydrodynamic model are the 1-D nonlinear Saint-Venant equations, which
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consist of a continuity equation and a momentum equation: A Q q 0, t x
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L1
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QQ Q Q 2 Z L2 ( ) ( 2 ) g g 2 0, t A x 2 A x K
(1)
(2)
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where L1 and L2 represent the continuity equation and the momentum equation, respectively; x =
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channel length coordinate; t = time; Q = Q(x,t), discharge through a cross-section varying with
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time and space; A = A(x, t), cross-sectional area; Z = Z(x,t), water stage; β = momentum
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correction factor due to nonuniformity of velocity distribution at cross sections; g = gravitational
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acceleration; q = q(x,t), volumetric rate of lateral outflow (or inflow) per unit length of the
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channel between two adjacent cross sections, in which the lateral outflow discharge is the control
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variable for flood diversion in this study; the conveyance K=AR2/3/n, where n = Manning’s
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roughness coefficient, and R = hydraulic radius. The Saint-Venant equations (1) and (2) can
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simulate open channel flows with complex cross-section shapes.
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To simulate an unsteady flood flow during a storm event, the Saint-Venant equations (1)
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and (2) are subject to initial and boundary conditions. The initial condition for the simulation is a
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base flow in a channel (i.e. the so-called “wet” channel) in which the water stages and discharges
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at the starting time are predetermined.
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As for the upstream boundary condition, a hydrograph is usually specified for giving a time
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series of flood discharges through the cross-section of an upstream inlet. At downstream outlet of
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a channel or channel network, a time series of water stages can be imposed; or a stage-discharge
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rating curve can be used for specifying water stages at downstream according to the computed
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discharge at the downstream outlet.
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As shown in Figure 1(a), surface water flow course in watershed can be represented by a
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dentritic channel network. As an internal condition in channel network, a confluence is a three-
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way junction as shown in Figure 1(b), which is a key element for solving the 1-D flow equations
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in channel network. To compute stages and discharges in a confluence, the internal conditions
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are assumed as follows:
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Z1 Z 2 Z 3 and Q3 Q1 Q2
(3)
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where Z1, Z2, and Z3 represent the stages in the cross-sections 1, 2, and 3, respectively; Q1, Q2,
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and Q3 are the discharges at the three cross-sections. The internal conditions of a confluence
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defined by (3) may create reflective waves in channel network, especially when open channel
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flow becomes transcritical or supercritical. As CCHE1D only takes into account subcritical open
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channel flows, reflective waves can be negligible. However, for the treatment of non-reflective
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boundary conditions, one may further refer to Gejadze et al. (2006), Sanders and Katopodes
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(1999c, 2000).
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OPTIMIZATION MODEL
197 198
In this simulation-based flood control model, nonlinear optimization methodologies
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based on the variational principle are adopted to build a numerical optimization algorithm for
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searching for the optimal flood diversion discharge (hydrograph) to mitigate flood water stages
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in channel network of watershed. In order to make this flood control model applicable to real
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flows in natural river and watershed with irregular channel cross-section shapes, which are
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subject to hydrograph conditions of storms at upstream, and various boundary conditions in
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downstream, an inverse model has to be built for the dynamic wave model of CCHE1D defined
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by Eqs. (1) and (2). To do so, through the definition of an objective function for control flood
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water stages in target reaches of a watershed, the so-called “continuous” differential adjoint
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model based on the differential governing equations of the CCHE1D dynamic wave model is
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developed to establish an analytical (i.e. a differential) relationship between the control action
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(lateral flood diversion) and the flood states in streams determined by stages and discharges.
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Although, without mathematical derivations on the governing equations of simulation model, a
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“discrete” adjoint model can be generated by means of automatic differentiation (e.g. Giles et al.
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2006), our preference for the continuous adjoint model is to develop an accurate sensitivity of the
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objective function with respect to the control variables, so that an efficient optimization can be
214
achieved. Moreover, a quick minimization procedure based on the Limited-Memory Quasi-
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Newton (LMQN) method (Liu and Nocedal 1989) is adopted for searching for the optimal
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solution of the lateral flood diversion hydrograph.
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Objective Function
219 220
In order to find the optimal solution of flood diversion discharge q(xc,t) at a floodgate
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located at xc, the corresponding optimization problem is to minimize the objective function J that
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is generally defined as an integration of the discrepancy between the predicted stages Z(x,t) and
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the maximum allowable water stages Zobj(x) in target reaches x0 where needs to be protected, i.e.
224
1 J LT 0,
T
0
L
0
WZ [ Z ( x, t ) Z obj ( x)]4 ( x x0 ), if Z ( x0 ) Z obj ( x0 ) if Z ( x0 ) Z
obj
(4)
( x0 )
225
where L = channel length; T = optimal control duration; Wz = weighting factor to address
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importance of target river reaches and to adjust the scale of the objective function; = Dirac
227
delta function; x0= target reaches where the water stages need to be controlled. The value of
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Zobj(x) could be constant or varying along target reaches. Notice that the objective function
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evaluates the stage discrepancies over the target reaches only where the predicted water stages
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exceed the allowable water stage Zobj(x); therefore this is a non-differential objective function
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which results in a non-smooth optimization problem (please refer to Elhanafy et al. 2008, Yu et
232
al. 2008, or Lewis and Overton 2010 for the details). Indeed, there should be more than one
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solution to minimize the function (4) if a lateral withdrawal discharge is large enough to suck
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most of flood waters out of the river channel, by which a supercritical flow may occur in
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channels. However, that is an undesirable flood control action because over-withdrawn flood
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waters need a huge detention basin to store them. Practically, one would like to search for the
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best flood diversion control by which the flood water will be withdrawn as less as possible. The
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optimal solution with this physical meaning/constraint for the flood diversion control is
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dependent on the searching direction of the minimization procedure or an appropriated initial
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guess of the control variable. The examples in the papers demonstrate that starting with a small
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non-zero guess of the lateral withdrawal discharge will guarantee to find out the physical-
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meaning-dependent optimal solutions. Detailed explanations will be presented in the following
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sections.
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Adjoint Sensitivity Analysis of Flood Flow Model
246 247
Adjoint Equations
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In general, based on the variational analysis for a nonlinear and unsteady dynamic system,
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the ASA is the best approach to precisely establish the quantitative relationship between dynamic
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control variables and adjoint variables of the complex dynamic system. As for the dynamic wave
251
model (i.e. Eqs (1) and (2)) for modeling spatio-temporal variations of flood flows, the flood
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control action in the study is to divert flood water from channels so that water stages in channels
253
are lower than a safe elevation. In order to minimize the objective function in (4) and guarantee
254
that the flow variables satisfy the governing equations (1) and (2), an augmented objective
255
function J* is introduced in terms of the Lagrangian multiplier approach, namely,
256
J* J
T 0
L 0
(A L1 Q L2 )dxdt ,
(5)
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where A and Q are two Lagrangian multipliers associated with the two nonlinear governing
258
equations, respectively. According to the stationary (necessary) condition of the optimal
14
259
solutions, the first-order gradient of the augmented objective function should be vanished, if the
260
optimal control action is the solution of the nonlinear dynamic system. Therefore, taking the
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variation of J* with respect to all flow variables (i.e. stages and discharges) and the control
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variable q(xc, t) in a time-space domain, the first-order variation of the augmented objective
263
function can be obtained as follows:
264 J *
265
T L J J J A Q q ( A )qdxdt 0 0 A Q q
A g Q Q Q Q 2 Q 2 g (1 )Q | Q | 2 3 Q Adxdt 0 0 t B * x A t A x AK 2 , T L 1 Q Q Q A 2 g Q 2 Q Qdxdt 0 0 x A x K2 A t
T
L
(6)
I 1A I 2Q dx I 3A I 4Q dt
266
where δ is the variation operator, I 1 A
267
I2
Q
Q Q A2
A g Q 2 I 3 ( * 3 ) Q B A Q I 4 A 2 Q A
(7)
268
and Λ=2/3−4R/(3B*), B* = B*(x,t), the channel width on water surface. Similar to the derivation
269
procedure given by Ding and Wang (2006a), to satisfy with the stationary condition (i.e. δJ*=0),
270
all the terms multiplied by the two variations, δA and δQ, can be set to zero, respectively,
271
including the last contour integral associated with the boundary conditions. This contour integral
272
is defined over a closed time-space rectangular domain, i.e. 0≤t≤T and 0≤x≤L. As a result, two
273
adjoint equations which describe the variations of the two adjoint variables λA and λQ in space
15
274
and time can be obtained. Through a simplification procedure, the two adjoint equations are
275
written as follows:
276
4W A g Q 2 gAV | V | Q * Z ( Z ( x, t ) Z obj ( x)) 3 ( x x0 ), if Z ( x0 ) Z obj ( x0 ) V A * B LT t x B x K2 0, if Z ( x0 ) Z obj ( x0 )
Q
277
t
A
Q 2 gA 2 V A V Q 0 x x K2
(8)
(9)
278
where the cross-section velocity V=Q/A. The adjoint equations are general formulations for the
279
inverse analysis of 1-D unsteady channel flows with arbitrary cross-sections shapes. According
280
to Le Dimet’s variational data assimilation theory (Le Dimet et al 2002), the above adjoint
281
equations are categorized into the first-order adjoint equations. Eqs. (8) and (9) are also first-
282
order partial differential equations, which can be rewritten into a compact vector form:
283
U U M NU P 0 t x
(10)
2 gAV | V | K2 , and P represents the source term related 2 gA 2 | V | K2
284
g 0 A Where U , M V B * , N 0 Q A V
285
to the objective function as shown in the right hand side of the adjoint equations. From the
286
eigenvalues of the coefficient matrix M, it is found that this adjoint model has two characteristic
287
lines with the following two real and distinct eigenvalues:
288
1, 2
1 2
V
( 1) 2 2 A V g * 4 B
(11)
289
It indicates that the adjoint system (10) is linear and hyperbolic. In the case of a flow in a
290
prismatic open channel, β=1, therefore, 1, 2 V g
A , the characteristic lines of the adjoint B*
16
291
model are the same as those of the dynamic wave model shown in Eqs. (1) and (2) (Sanders and
292
Katopodes 1999c, Chaudhry 2008). It can be found in the following content that due to the
293
transversality condition of the adjoint variables, following the same characteristic lines, the travel
294
directions of the perturbation waves of the adjoint variables are opposite to those forward time
295
directions of flood flow waves. Gejadze et al. (2006) also derived two characteristic lines for a
296
set of adjoint equations associated with a vertical two-dimensional free-surface flow. Since this
297
inverse model is strongly dependent of the governing equations (1) and (2), the adjoint equations
298
only can be solved numerically.
299 300 301
Sensitivities of Flood Diversion The sensitivities of control variables (flood diversion discharge q(xc,t)),
which is
302
identical to the gradient of the objective function J with respect to q, also can be obtained from
303
the first variation of the objective function. For simplicity, this study does not consider the actual
304
type of the control facility installed at the place of diversion; there might be a pump station or an
305
operational floodgate which is able to divert the desired amount of the lateral outflow. From the
306
first variations δJ* given in Eq. (6), the variation of the objective function J with respect to the
307
control variable q at the location xc and a discretized time tn is obtained immediately as follows:
308
J q
x xc t tn
A ( xc , tn )
(12)
309
By the formulation (12), the Lagrangian multiplier λA precisely determines the sensitivity of the
310
objective function with respect to the control variable (i.e. the flood diversion discharge).
311 312
Transversality and Boundary Conditions of Adjoint Equations
17
313
Similar to solving the hydrodynamic equations, the adjoint equations (8) and (9) must be
314
computed numerically with boundary conditions and certain conditions in time. It is well known
315
that the initial conditions for the adjoint variable at t=0 do not exist in the adjoint equations of
316
the inverse model. Only the final conditions at the end of control action, t=T, can be given,
317
which are called as the transversality conditions for inverse models. The transversality condition
318
and the boundary conditions can be obtained from the contour integral as the last term of the first
319
variation of the augmented objective function in Eq. (6).
320
I [ I1A I 2Q ]dx [ I 3A I 4Q ]dt I1A I 2Q dx I 3A I 4Q L
T
t 0
0
0
dt I1A I 2Q L
xL
0
dx I 3A I 4Q T
t T
0
x 0
dt
321
(13)
322
Considering the initial condition of river flow, by which the discharges and water stages in
323
channels are predetermined, therefore, δQ(x,0) = 0 and δA(x,0) = 0. From the specified boundary
324
condition of the discharge at an inlet of upstream, it can be derived that δQ(0,t) = 0, but δA(0,t) ≠
325
0. Similarly, At the outlet of downstream, for specified water stages or a given stage-discharge
326
rating curve, the conditions of the variations of discharge and water stage at the outlet turn to be
327
δA(L,t) = 0, but δQ(L,t) ≠ 0. Therefore, considering the definitions of the integrands in Eq. (7),
328
Eq. (13) can be extended to the following form:
329
I ( A
T
0
Q A2
L T g Q Q 2 dt ( A 2 Q )A Q Q dx ( * 3 )QA dt 0 0 A A A xL t T B x 0
Q )Q
330
(14)
331
To assure the stationary condition for the optimal solution of the dynamic system, the variations
332
of the boundary condition δI must be zero. Consequently, from the second term in Eq. (14), the
333
transversality conditions of the adjoint equations can be readily obtained, i.e.
18
334
Q ( x, T ) 0 and A ( x, T ) 0,
x [0, L]
(15)
335
Due to the conditions given at the end of control period, the adjoint equations (8) and (9) must be
336
solved backward in time. It is therefore inevitable to store the computed flow variables obtained
337
in the forward computation of a flow over all time steps before the backward computation of the
338
adjoint variables is carried out. It implies that along with the characteristic lines given in (11), the
339
perturbation waves of the adjoint variables, traveling backward in time, are opposite to the wave
340
propagation directions of flows. Furthermore, by setting the third term of Eq. (14) to zero, the
341
boundary condition of λQ at upstream (i.e. x=0) is given as follows:
342
Q (0, t ) 0,
t [0, T ]
(16)
343
This means that the Lagrangian multiplier λQ is always zero at the upstream inlet. This upstream
344
condition of the adjoint variable can be also applied to every inlet cross-section in channel
345
network. Finally, from the first term of δI in (14), the two Lagrangian multipliers at the
346
downstream outlet at x=L have the following relationship, i.e.
347
A ( L, t )
( L, t )Q( L, t ) A( L, t ) 2
Q ( L, t ),
t [0, T ]
(17)
348
By applying the transversality condition (15), the upstream condition (16), and the downstream
349
condition (17), the adjoint equations (8) and (9) become a well-defined inverse model. It should be
350
noted that the flow model and the adjoint model are suitable for modeling and optimizing
351
subcritical flows or transcritical flows, but probably not supcritical flows, since the water stages
352
at downstream are given. For treatment of non-reflective boundary conditions of the adjoint
353
variables, one may further refer to Sanders and Katopodes (1999c, 2000), Gejadze et al. (2006).
354
19
355
Internal Conditions at a Confluence in Channel Network
356
In order to solve the adjoint equations for optimal control in a watershed represented by
357
channel network, the internal conditions Eq. (3) have to be imposed at every confluence in
358
channel network which are shown in Figure 1 (b). Similar to the variational analysis of the
359
boundary conditions at the upstream and the downstream as discussed above, the variation of the
360
contour integral over boundary, i.e. δI shown in Eq. (13), can be extended at a confluence as
361
shown in Figure 1 (b). There are three terms associated with the three cross-sections at a
362
confluence added to the variation, i.e.
363
I confluence I 3A I 4Q T
0
dt I 3A I 4Q T
x x1
0
dt I 3A I 4Q T
x x2
0
x x3
dt
(18)
364
where x1, x2, and x3 represent the locations of two upstream cross-sections and one downstream
365
cross-section at the confluence. By recalling the internal conditions for discharges and stages at
366
the confluence shown in Eq. (3), and considering δZ=δA/B*, the relationship between the
367
variations of the variables at the three cross sections is readily written as follows: A1
368
369 370 371
* 1
B
A2 B
* 2
A3 B3*
and Q3 Q1 Q2
(19)
Using this relationship, therefore, Eq. (18) is rearranged as: T
0
B B I confluence I 3 | x x 2* I 3 | x x 3* I 3 | x x A1 I 4 | x x I 4 | x x Q1 I 4 | x x I 4 | x x Q2 dt *
*
1
B1
2
B1
3
1
3
2
3
(20)
372
To satisfy the stationary condition of the optimal solution, this variation on the internal boundary
373
at a confluence should be vanished. Namely, the terms before the variations δA1, δQ1 and δQ2
374
become zero, respectively. Considering the definitions of the integrands in Eq. (7), three internal
375
conditions for the adjoint equations are obtained as follows:
20
u
376
Q x x 3
uQ x x uQ x x 1
QQ QQ QQ A A A 2 2 A x x A x x A2 x x 1 2 3
377
(21)
2
(22)
B * Q 2 .The above Equations (21) and (22) are generally the internal conditions of A3
378
where u g
379
the two Lagrangian multipliers, A and Q, at a confluence. By imposing these internal
380
conditions at each confluence in channel network, this inverse model governed by the adjoint
381
equations, subject to the transversality conditions (15) and their boundary conditions (16) and
382
(17) , becomes well-posed. The adjoint equations, dependent on the flow model results, can only
383
be solved numerically.
384 385
NUMERICAL IMPLEMENTATION FOR SOLVING SIMULATION MODEL AND
386
ADJOINT EQUATIONS
387 388
The Saint-Venant equations (1) and (2) in the wave dynamic model of CCHE1D were discretized
389
by means of the implicit four-point finite difference scheme proposed by Preissmann (1960),
390
which has been widely used to simulate 1-D unsteady flows. Provided that the computational
391
domain as shown in Figure 2 is discretized in time and space, this scheme gives a set of
392
interpolation functions in time and space for evaluating values of a variable and its derivations
393
with respect to time and space, i.e. ∂ϕ/∂t and ∂ϕ/∂x, near the ith spatial step and the nth time step,
394
i.e.
395
( x, t ) [in11 (1 )in1 ] (1 )[in1 (1 )in ] ,
(23)
21
396
( x, t ) n1 1 n1 i 1 in1 i in , t t t
(24)
397
( x, t ) n1 1 n i 1 in1 i 1 in , x x x
(25)
398
where t = time increment; x = spatial length; x (ix, (i 1)x) and t (nt , (n 1)t ) ; and
399
are two weighting parameters less than or equal to one; in =the value of function at the nth
400
time step and the ith spatial step. The spatial length x is not limited to be constant; it can vary
401
with reaches. Using Preissmann’s scheme, the discretized continuity equation and the
402
momentum equation are numerically coupled. By assuming a linear relationship between the
403
incremental values of discharge and stage, Liggett and Cunge (1975) obtained a set of linear
404
algebra equations with a pentadiagonal matrix which are solved by a double sweep algorithm.
405
Wu and Vieira (2002) adopted the same approach to solve the Saint-Venant equations in
406
CCHE1D and to apply to modeling flows in channel network in watersheds and practical
407
engineering problems with various in-stream hydraulic structures such as culverts, bridge
408
crossing, measuring flume, etc.
409 410
This implicit scheme is also used for discretization of the adjoint equations (8) and (9) . The first
411
equation for the adjoint variable λA is discretized as follows:
22
t
412
n 1 A i 1
g (B )
n
* n i
n
AV | V | n 1 n 1 n n 2g (Q i 1 (1 )Q i ) (1 )(Q i 1 (1 )Q i ) 2 K i B4*WLTZ Z ( x ,t ) Z obj ( x ) 3 ( x x0 ), 0, if Z ( x0 ) Z obj ( x0 )
413
1 1 A in1 A in Vi n A in11 A in1 A in1 A in t x x 1 Q in11 Q in1 Q in1 Q in x x
A i 1
(26)
if Z ( x0 ) Z obj ( x0 )
where
Vi n Vi n11 (1 )Vi n1 (1 ) Vi n1 (1 )Vi n
414
(27)
n
* n i
AV | V | and can be calculated using the same 2 K i
415
The other terms in (26) such as ( B )
416
discretization method as shown in (27) or (23). Similarly, the second adjoint equation for λQ has
417
the following discretized form:
t
Q i 1 n
418
1 1 Q in1 Q in Ain A in11 A in1 A in1 A in t x x 1 n V i Q in11 Q in1 Q in1 Q in x x n 1 Q i 1
n
(28)
A2 | V | (Q in11 (1 )Q in1 ) (1 )(Q in1 (1 )Q in ) 0 2 g 2 K i
n i
, V
n i
A2 | V and 2 K
n
| can also be calculated using the same Preissmann’ i
419
where A
420
scheme (23) or (27). In computation of the adjoint equations, the value of is about 0.50 for
421
flood controls, and that of ψ is set to 0.55. Note that the adjoint equations are solved backward
422
in time, i.e. suppose that the adjoint variables at the (n+1) th time step are known, two
423
discretized equations are rearranged into the following algebraic equations:
23
424
ai Ai bi Q i ci Ai 1 d i Q i1 pi 0
(29)
425
ei Ai f i Q i g i Ai1 ri Q i 1 wi 0
(30)
n
n
n
n
n
n
n
n
426
where
427
ai
428
g 1 AV | V | bi * n 2g (1 )(1 ) 2 ( B ) i x K i
429
ci
430
g 1 AV | V | d i * n 2g (1 ) 2 ( B ) i x K i
1 1 n Vi t x n
1 n Vi t x
n
n1 g n1 1 n 1 n 1 pi Vi n (Q i 1 Q i ) Vi n A i * n A x x i 1 ( B ) i x t t n
431
432
AV | V | n 1 n 1 2g Q i 1 (1 )Q i 2 K i B4*WLTZ Z ( x ,t ) Z obj ( x ) 3 ( x x0 ), 0, if Z ( x0 ) Z obj ( x0 ) 1 n ei Ai x
if Z ( x0 ) Z obj ( x0 )
n
433
2 1 1 V in 2 g A | 2V | (1 )(1 ) fi t x K i
434
gi
435
1 n Ai x n
A2 | V | 1 n V i 2 g 2 (1 ) ri t x K i
24
wi 436
x
x
n i
A
n i
A
n 1 Ai
n 1 A i 1
n 1 n1 A2 | V | n Q i ( V ) i 2 g ( 1 ) 2 x t K i n n1 A2 | V | n Q i 1 ( V ) i 2 g 2 x t K i
437 438
The discretized adjoint equations, (26) and (28), are linear as the coefficients are dependent on
439
the computed flows. Note that the adjoint equations are solved backward in time, due to the
440
transversality condition. To solve these two equations, a double sweep algorithm is employed,
441
which contains two recursive loops, backward and forward computation. For the details of the
442
algorithm, one may refer to Liggett and Cunge (1975). Here, the formulations for the
443
computations are given as follows: in forward computation, the following coefficients are
444
calculated over the whole computational domain (all river reaches) in the nth time step:
445
S i 1
(ei f i S i )ci (ai bi S i ) g i (ai bi S i )ri (ei f i S i )d i
(31)
446
Ti 1
( wi f iTi )(ai bi S i ) ( pi biTi )(ei f i S i ) (ai bi S i )ri (ei f i S i )d i
(32)
447 448
In backward computation, the adjoint variables are calculated by the following formulations:
( pi biTi ) (ci A i 1 d i Q i 1 ) n
449
450
Q in Si Ain Ti
n Ai
ai bi S i
n
(33)
(34)
451
In forward computation, the coefficients in (31) and (32) are calculated recursively, with the index
452
i varying from 1 to Imax, i.e. from upstream to downstream in river reaches, where Imax is the total
453
number of cross sections. The values of S1 and T1 are given by the upstream boundary conditions
454
(16), i.e. S1= T1=0. In the backward computation, the adjoint variables λA and λQ are computed
25
455
from downstream to upstream, while the downstream condition (17) is specified to the outlet
456
cross section.
457 458
Implementation of Internal Conditions at A Confluence for Solving Adjoint Equations
459
Similar to the implementation of the internal conditions in (3) for computation of flows in a
460
confluence (Wu and Vieira 2002), the internal conditions of the two adjoint variables in (21) and
461
(22) must be specified at every confluence of channel network, when using the double sweep
462
algorithm to solve the adjoint equations. In general, both internal conditions for the flow model
463
and the adjoint model may be reflective so that non-reflective boundary conditions may need to
464
be implemented (Gejadze et al 2006). However, since this flow model (CCHE1D) is developed
465
for simulating flows in a dendritic watershed, it is supposed that an open channel flow in the
466
watershed is unidirectional, i.e. from upstream to downstream, wave reflections at confluences
467
(internal boundaries) and the outlet are negligible.
468 469
Taking into account a confluence in a dendritic channel network as shown in Figure 1(b), in the
470
forward computation by the double sweep algorithm, two upstream crosssections have the
471
relationships between the two adjoint variables λA and λQ established by (34), and the coefficients
472
Si and Ti defined at the two sections have been calculated using the formulations (31) and (32). At
473
the downstream cross-section, i.e., x=x3, of the confluence, instead of using the two formulations,
474
the coefficients Sx3 and Tx3 should be calculated accordingly by considering the internal
475
conditions (21) and (22). The derivation process for obtaining the two coefficients at the
476
downstream corsssection is briefly explained as follows: According to the relationship in (34),
477
A x1
Q x1 S x1
Tx1 S x1
(35)
26
A x2
478
Q x 2 S x2
Tx 2 S x2
(36)
479
where the subscripts x1 and x2 indicate that the variable values are given at the two upstream
480
cross sections, respectively. Substituting (35) and (36) into the internal condition (22), rearrange
481
as follows:
482
Q x1 1 A x3 1Q x3 1
(37)
483
Q x 2 2 A x3 2 Q x3 2
(38)
484
where
S xi Q 1 S xi 2 A xi
485
i
486
Q S xi 2 A x3 i Q 1 S xi 2 A xi
487
i
Txi Q 1 S xi 2 A xi
488
Substituting (37) and (38) into (21), rearrange the formulation as the same as Eq. (34), i.e.
489
Q nx3 S x3 A nx3 Tx3 the coefficients Sx3 and Tx3 can be written as
490
491
,
S x3
u x1 1 u x 2 2 u x 3 u x11 u x 2 2
Tx 3
u x1 1 u x 2 2 u x 3 u x11 u x 2 2
(39)
(40)
27
B * Q 2 defined at three cross sections of a A3
492
where ux1, ux2, and ux3 are the values of g
493
confluence, respectively. As a result, in the forward computation of the double sweep algorithm,
494
the coefficients Sx3 and Tx3 for a downstream cross section of a confluence are calculated by
495
using (39) and (40) . In the backward computation, the values of λAx3 and λQx3 are calculated by
496
using (33) and (34), and the adjoint variables at the two upstream cross sections are computed by
497
using Eqs. (35) - (38).
498 499
OPTIMIZATION ALGORITHM
500 501
In this integrated simulation-based optimization software package, the CCHE1D
502
hydrodynamic model is used to predict discharges and stages in channels, the adjoint model is to
503
compute the two Lagrangian multipliers through the adjoint equations (8) and (9), as well as the
504
gradient of the objective function, i.e. the sensitivity calculated from Eq. (12). The upstream
505
hydrographs, the cross-section geometries, and the base flow discharge are the input data for the
506
hydrodynamic model. In order to find the optimal solution of the control variable, an iterative
507
minimization procedure is generally required due to the nonlinearity of the flow control problem.
508
Moreover, since the values of the time-dependent control variable q(xc,t) at all the time steps
509
must be determined, in case of one floodgate for control, the length of the array storing q(xc,t) is
510
equal to the total number of time steps over a control period T. This requires that a minimization
511
procedure has to be able to efficiently handle a large-scale optimization problem when the
512
simulation period T is long. Zou et al. (1993) pointed out that the LMQN method is useful for
513
solving large-scale optimization problems. Ding et al. (2004) compared several minimization
514
procedures based on the LMQN and a trust region approach, and found that the procedures based
28
515
on the LMQN method have the advantages of higher convergence rate, numerical stability, and
516
computational efficiency. Amongst those algorithms, the Limited-memory Broyden, Fletcher,
517
Goldfarb, and Shanno (L-BFGS) algorithm is capable of achieving optimization in large-scale
518
problems because of the modest storage requirements using a sparse approximation to the
519
inverse Hessian matrix of the objective function (Nocedal and Wright 2006). Ding and Wang
520
(2006b) further demonstrated that the L-BFGS algorithm indeed can find out the optimal
521
solution for the flood control problem defined by (4) - a non-differential objective function at the
522
minima, even though the L-BFGS has been well known as a minimization algorithm for smooth
523
optimization problems. Yu et al. (2008), Lewis and Overtun (2010), and Skajaa (2010) proved
524
that the BFGS algorithm is robust and convergent for the non-smooth optimization problem due
525
to a locally non-differential function as shown in (4). Recently, Steward et al. (2012) have
526
compared the LBFGS with a new minimization algorithm called the limited-memory bundle
527
method (LMBM) algorithm specifically designed to address large-scale non-smooth
528
minimization problems. They have found that the LMBM yields results better than the L-BFGS,
529
despite some stability problems. The LMBM may be able to eventually overcome the
530
mathematical drawback of the L-BFGS. Investigation of the LMBM in river flood control would
531
be an interesting topic, and will be performed in the near future by the authors.
532 533
According to the physical meanings of the control variables and the limitations of their
534
values known from control devices (floodgates) and engineering experience for the capacity of
535
discharge at a spillway or a pump station for flood diversion, the identification of the values
536
subject to bound constraints about the capacities of the devices needs to be considered. There are
537
several kinds of nonlinear optimization methods available for constrained optimization, e.g.,
29
538
penalty, barrier, augmented Lagrangian methods, sequential quadratic programming (SQP), etc.
539
(Nocedal and Wright 2006). Although some optimization methods (e.g. SQP) can solve more
540
complicated constrained problems with equality and/or inequality constraints, there are still
541
limited algorithms capable of solving a large-scale optimization problem as needed in the present
542
study. L-BFGS-B is an extension of the limited memory algorithm L-BFGS, which is, at present,
543
the only limited memory quasi-Newton algorithm capable of handling both large-scale
544
optimization problems and bounds on the control variables. For further descriptions of the L-
545
BFGS and the L-BFGS-B algorithms, one may refer to Liu and Nocedal (1989) and Byrd et al.
546
(1995). In order to preserve the physical meanings of the control variables in every iteration step,
547
the L-BFGS-B procedure validated by Ding et al. (2004) is thus adopted in this study. The L-
548
BFGS-B algorithm minimizes the objective function J in Eq. (4) subject to the simple bound
549
constraints, i.e. qmin ≤ q ≤ qmax, where qmin and qmax mean, respectively, the maximum and
550
minimum lateral outflow rates in a floodgate.
551 552
In optimization process for searching for the best flood diversion rate q(t), the searching
553
process of the minimization procedure should start with a small initial value in order to avoid a
554
big diversion discharge to suddenly suck flood flows out of channels to create an unphysical and
555
unfavorable supercritical flow in river channels. Mathematically speaking, even though a large
556
withdrawal discharge which may be sufficient to dry up the channel bed could make the
557
objective function (4) equal to zero, it is not a physical solution for realistic river flood control.
558
Another advantage for searching the diversion discharge from a small value is to guarantee that
559
the optimal withdrawal hydrograph minimizes the objective function (4) to protect the target
30
560
river reaches from being overflown, and to assure that the least flood waters are diverted from
561
the river channel.
562 563
NUMERICAL EXAMPLES FOR OPTIMAL FLOOD DIVERSION CONTROL
564
As shown in Figure 3, a dendritic channel network of a watershed consisting of a main
565
stem, two branches, and two confluences is used for testing this optimal flood control model. For
566
simplicity, a compound cross-section is assumed as the channel shape for all three channels, even
567
though the CCHE1D flood model and the adjoint model are applicable to a natural river reach
568
with an irregular cross section. Three triangular hydrographs are imposed respectively at the
569
three upstream inlets of the channels. The parameters to define the triangular hydrogrphs (Qp, Qb
570
= peak and base flow discharges, respectively; T = duration of storm or period of control action;
571
Tp =time to peak discharge in a hydrograph) are listed in Table 1. The hydrograph at the inlet of
572
Channel 3 (main stem) has a slightly higher peak discharge (60 m3/s) than the other two. As
573
shown in the table, three different values of Manning’s n are specified on the three channels to
574
represent the different bed roughness. For the simulation of flood propagation, the channels are
575
divided into a total of 43 short reaches (i.e 48 cross sections) with equal spatial increment (500
576
m). The control duration for mitigating the flood water by diverting channel flow through the
577
floodgate is 50 hours, which covers the whole storm period. The time step ∆t for simulating
578
channel flows and the adjoint variables is 5 minutes. The weight factor Wz is set to 1000.
579 580
Verification of Adjoint Sensitivity
581
Based on the above adjoint sensitivity analysis (ASA), the sensitivity of the objective
582
function with respect to the control action, i.e. the flow diversion rate q(t), is determined by
31
583
Eq.(12), namely, the adjoint variable λA at the locations of floodgates. Due to nonlinear nature of
584
the adjoint equations (8) and (9), there are no directly analytical solutions for the two adjoint
585
variables. Therefore, in general, sensitivity of the function can’t be verified by comparing
586
computed values with any analytical solutions of the adjont equations. However, utilizing a
587
difference quotient as an approximation of the gradient of the function with respect to the control
588
variable, the direct sensitivity method (DSM) can be used to estimate sensitivities of the function.
589
Yeh (1986) and Yeh and Sun (1990) provided a mathematical definition of the direct sensitivity
590
and a procedure for calculating the sensitivities by using a difference scheme. A difference
591
approximation of the sensitivity of J with respect to q can be written as follows:
592
J q
x xc t tn
J ( q0 ( xc , t ) q( xc , tn )) J ( q0 ( xc , t )) q( xc , tn )
(41)
593
where q0 = an initial diversion rate, and δq(xc,tn) = a small amount of perturbed diversion rate at a
594
time tn and the location of floodgate. Sanders and Katopodes (2000) have applied the DSM to
595
verify adjoint sensitivities of a quadratic objective function for open channel flow control. They
596
produced good estimations of the sensitivities by the DSM in comparison with the results
597
obtained by the ASA. Before introducing the DSM for verification of the present adjoint
598
sensitivity, it has to be noticed that due to the non-smooth (or non-differential) nature of the
599
objective function (4), the DSM can’t be directly applied to estimate the sensitivity as Sanders
600
and Katopodes (2000) have done. But if we specify sufficient lower values for the allowable
601
water stages Zobj at target reaches x0, the calculation of the objective function (4) can rule out the
602
chance of checking the inequality in this non-smooth function, the function turns out to be
603
differential in all the range of water elevations in the target reaches. Then, the DSM can be used
604
for verification of the adjoint sensitivity under this special condition of the control water stage.
32
605
Consequently, three test cases for estimation of the sensitivities with respect to a single floodgate
606
diversion in the watershed were performed. This floodgate is located at the downstream and
607
indicated by an arrow in Figure 3. Each test case controlled only one river reach at a time with an
608
equal length of 500 m. The three target reaches are St.1, St.2, and St.6 shown in Figure 3. In the
609
direct sensitivity analysis using Eq. (41), a zero allowable stage, i.e. Zobj(x0) = 0.0 m, is specified
610
to three target reaches respectively to assure that all the variations of the water stages due to the
611
perturbations will be counted in the objective function. Having used different values for the
612
perturbed diversion rate δq at the range from 0.00001 to 10.0, we finally found convergent
613
sensitivities at δq=0.5 and 1.0. In addition, the adjoint variables are computed by using the initial
614
diversion rate q0(xc,t) equal to 0.0. Figure 4 presents the intercomparisons of the sensitivities
615
computed by the adjoint variable λA and the DSM in the three cases with different target reaches.
616
It implies that the results by the DSM are very close to those by the ASA, or in other words, the
617
adjoint sensitivities defined by the variable λA are consistent with the direct sensitivities
618
estimated by the difference quotient (41). Even though the DSM can give good estimations of the
619
sensitivities of the function, due to the tedious perturbations at every time step (showing by the
620
symbols in Figure 4) to calculate individual sensitivity value by (41), the DSM is very time-
621
consuming.
622 623
Optimal Flood Diversion with Single Floodgate
624 625
In this case, by using the ASA, only one floodgate for flood diversion control is assumed
626
to be placed at the downstream of the main stem, as shown in Figure 3. To protect all the three
627
channels from being overflowed, the maximum allowable stage Zobj is set to be 3.5 m over the
33
628
entire channel network. The optimization of the flood control problem turns out to find the
629
optimal flood diversion hydrograph q(t) by minimizing the overflow water stages over the 50-
630
hours optimization duration and the entire watershed. The searching process started with a small
631
initial value of q(t), which was 0.001 m3/s. It assures that the searching direction is from a small
632
diversion rate to a large rate in order to avoid unphysical big diversion discharge to dry up the
633
channel bed. As shown in Figure 5, the iterations for searching for the optimal lateral outflow
634
q(t) by the L-BFGS-B algorithm indicate that after less than 30 iterations, the convergent optimal
635
solution of the q(t) was found. Here, the L-BFGS-B with the bound constraint optimization (i.e.
636
q(t) < 0) prevented the lateral outflows from recharging the channel (i.e. flow from outside
637
toward the channel). The identified optimal hydrograph q(t) to schedule the flood water diversion
638
shows that the floodgate should discharge 10 m3/s flood water at the beginning of the storm; then,
639
the peak discharge of the lateral outflow reaches up to 130.0m3/s at 17.5 hour, which is a 1.5-
640
hours delay by comparing with the arrival time of the peak discharge at the upstream.
641 642
Notice that in the optimization the lateral outflow variable q(t) is a vector with the
643
number of components equal to the number of total time steps over the simulation period. Here,
644
the control vector q(t) contains 601 components at the flood duration = 50 hours, and the time
645
step = 5 min. That means that 601 values of the parameter q(t) have to be identified by the L-
646
BFGS-B optimization algorithm. For explaining the performance of the optimization procedure,
647
the optimal control model was run iteratively till the values of the control parameters was no
648
longer changed (i.e. as small as a machine precision). In the meantime, the double precision
649
arithmetic for solving the adjoint variables is adopted. Figure 6 shows the iterative histories of
650
the objective function values and the norm of its gradient (i.e. J (q) 2 ). It is found that starting
34
651
with a small guess value of the diversion rate, the objective function value decreases
652
monotonically and the control variable converges after 15 iterations using L-BFGS-B. The
653
discharge obtained after 15 iterations is considered as the optimal solution, because the gradient
654
is very close to zero and the objective function are therefore minimized. Moreover, the flood
655
control model took less than one second of CPU time for executing one iteration of the
656
optimization in a PC with 2.66GHz CPU. For this kind of optimization problem with 48 cross
657
sections in the channel network and 601 control parameters, it only took 11 seconds of CPU time
658
for the integrated simulation-based control model to find the optimal solution at the 15th L-
659
BFGS-B iteration.
660 661
In fact, optimization efficiency depends on the accuracy of the adjoint sensitivity of the
662
objective function, which is determined by the adjoint variable λA(t) at the control location as
663
shown in Eq. (12).
664
location of the floodgate as L-BFGS-B is searching for the optimal solution. At the beginning of
665
iterations, the parabolic curves of the adjoint variable reflect the temporal distribution of the
666
discrepancy between the predicted and the allowable water stages. It indicates that the lateral
667
outflow rate should increase along with the flood hydrograph. After a few L-BFGS-B iterations,
668
the peak of the adjoint variables gradually shift toward the flood peak at t=16 h. At the 10th
669
iteration, the values of λA(t) become very small and close to the machine precision, and then
670
almost reach to zero at the 20th iteration. It means that the obtained solution of the outflow
671
hydrograph in Figure 6 is indeed mathematically optimal.
Figure 7 presents the iteration process of the adjoint variable λA(t) at the
672
35
673
To look into the temporal/spatial distributions of the two adjoint variables, five
674
monitoring stations whose locations are shown in Figure 3 are chosen to visualize the variations
675
of the variables in the main stem and the two branches in the watershed. Temporal profiles of the
676
two adjoint variables at the first iteration (i.e. no lateral outflow diverted from the floodgate) are
677
plotted in Figure 8. The variations of λA(t) at the upstream of the main stem and the two branches
678
(i.e. St. 1, St.2, St. 4, and St. 5) are much larger than that at downstream (e.g. St. 3). It implies
679
that λA(t) is more sensitive at upstream than that at downstream, and therefore the diversion
680
control located at upstream should be more effective than downstream. In contrast, the changes
681
of λQ(t) at downstream are greater than those at upstream due to the boundary condition of the
682
adjoint variable at inlets, which is given by Eq. (16).
683 684
As depicted in Figure 9, the temporal variations of the controlled water stages over the
685
50-hours storm period at the five selected stations are compared with those stages without
686
diversion control. The flood control results show that the stages at almost all the stations through
687
the flood period are lower than the allowable stage (i.e. 3.5m); the flood waters in the main
688
channel and two branches over the entire storm period, therefore, are well-controlled. There is
689
only a very short period when the water stages at upstream (St. 1 and St. 4) are slightly higher
690
than the allowable stage. Meanwhile, the flood water at the downstream (St. 3) in the main
691
channel forms two small water surface waves.
692 693
Further comparisons of discharges between controlled and uncontrolled states at the five
694
monitoring stations are shown in Figure 10. Due to the flood diversion, the peak discharges at the
695
downstream of the floodgate (i.e. St. 3) is reduced significantly. However, remarkable increases
36
696
of the discharges occurred at the upstream stations, St. 2. One possible reason for the discharge
697
increasing at this station is that the control constraint of the allowable maximum stage (3.5 m)
698
over the entire watershed is so restrictive so that before the peak flood arrives, the floodgate has
699
to divert in advance a large amount of flood waters out of the channel. Consequently, the flow
700
velocities at upstream become faster than those without flood water withdrawal, and the speed-
701
up channel flows at upstream may cause unexpected erosion on river bed in the case of
702
controlling real watershed with a movable bed. To avoid this problem, one may relax the
703
constraint of the stages at upstream by constructing dikes following a longitudinal slope of the
704
river flow, by which the ability of flood prevention at upstream will be strengthened. Another
705
way is to balance the load of the lateral outflow by diverting flood waters from multiple
706
floodgates, which is discussed in the following control case.
707 708
Optimal Flood Diversion with Multiple Floodgates
709
Installation of multiple floodgates in river banks of different river reaches is a common
710
practice to sequentially divert flood waters from upstream to downstream in a storm period. As a
711
rule of thumb, flood control by operating more than floodgate in a river basin can better balance
712
the pressure of flood waters in all river reaches due to unsteadiness and non-uniformities of flood
713
flows. In this case, it is assumed that there are three floodgates equally spaced in the main stem
714
of the channel network, as shown in Figure 11, which is the same watershed as the previous case.
715
The constraint for preventing flood water overflowing is set as the same as the previous case
716
with one floodgate diversion, i.e. 3.5-m maximum allowable water elevations for the entire
717
channel network. By using L-BFGS-B, three vectors of lateral outflow hydrographs, i.e.
718
q1(t)=q(x1,t), q2(t) )=q(x2,t), and q3(t) )=q(x3,t), consisting of a totaling 1803 components (601
37
719
time steps for each outflow hydrograph) have to be identified. By this simulation-based flood
720
control model, the optimal solutions of the three diversion hydrographs at the three floodgates
721
were obtained as shown in Figure 12. Notice that the peak diversion discharge at the first
722
floodgate is about 50 m3/s, which is much less than the peak discharges value (130 m3/s) by the
723
single floodgate diversion in the previous case. It indicates that more waters need to be diverted
724
at upstream than those at downstream, and the individual hydrograph in the three gates is much
725
less than the one withdrawn by the single floodgate in the previous case. Moreover, the total
726
discharge, i.e., q1(t)+q2(t)+q3(t), withdrawn by the three gates, is even less than the one by the
727
single gate operation. It denotes that multiple floodgate control is more effective than that by a
728
single gate, and can be operated easily, because the control load is distributed along the channel.
729
By considering the total volume 13.824 million m3 of the flood waters coming from the inlet
730
over the 48-hour flood, the percentages of the diverted flood waters by the single floodgate and
731
the three floodgates are compared in Table 2. It shows that the single floodgate control operation
732
needs to divert 60.8% of the floodwaters, but the three floodgates only divert totally 47%.
733
Apparently, multiple floodgate operation is a cost-effective flood control approach if the optimal
734
control can be achieved. The results for the maximum diversion discharges and the volume of the
735
diverted flood waters can be further used as the basic data to design the minimum capacities of
736
the floodgates and the flood water detention basins.
737 738
As shown in Figure 13, controlled by the three floodgates, all the water stages at the five
739
monitoring stations, of which the locations are shown in Figure 3, didn’t exceed the allowable
740
stage (3.5m). The controlled stages in the main stem (St. 1, St. 2, and St. 3) and the two branches
741
(St. 4 and St. 5) are more stable (flatter) than these by the single floodgate control as shown in
38
742
Figure 9. As shown in Figure 14, the controlled discharges in the main stem are much smaller
743
than these without control, though the controlled discharges in the two branches (St.4 and St.5)
744
slightly increased. It means that the multiple floodgate control can avoid channel flow speeding
745
up by over-withdrawing flood waters from the main stem by a single floodgate as found in the
746
previous case, by which may cause extra river bed erosion.
747 748
To compare the optimization performance of the two control actions, Figure 15 plots the
749
values of the objective function varying in the searching processes in the single and multiple
750
floodgate control actions. It shows that the optimization algorithm (L-BFGS-B) can quickly find
751
(in about 20 iterations) the optimal flood diversion discharge hydrographs, no matter the
752
floodgate is single or multiple. However, the final value of the objective function by the three
753
floodgate control is four digits smaller than that by the single floodgate. It means that the
754
multiple floodgate control can achieve the global optimal control in the entire watershed easier
755
than the single-gate control. It also reveals that this optimization methodology and the
756
simulation-based flood control tool are capable of determining the number, locations, capacities
757
of the floodgates in rivers and watersheds.
758 759
Optimal Flood Diversion in a Natural River
760
In order to examine the capability of this simulation-based optimization for controlling
761
flood flows in a real river with a naturally complex geometry, a river reach in the East Fork
762
River in western Wyoming (Emmet et al. 1979), as shown in Figure 16, was selected. This study
763
reach is 3.3-km long, and divided into 41 cross-sections with unequal spatial length. The flow
764
control in this river is demonstrated by controlling a flow diversion during a real flood observed
39
765
in a one-month period from May 20 to June 19, 1979. The observed hydrograph at the inlet
766
shown in Figure 17 contains three flood peaks, in which the maximum peak discharge is about
767
33 m3/s. A hypothetical floodgate is installed at Section No. 5 close to the upstream (Figure 16).
768
The control target water stages or the allowable water elevations Zobj(x) vary along the reach, as
769
shown in Figure 18, which form a slope line between the minimum and maximum bank
770
elevations. This simulation-based optimization software was used for searching for the best
771
diversion schedule (hydrograph) at the hypothetical floodgate in order to control the flood water
772
stages being lower than the allowable elevations. The time step for the flow simulations is 15
773
minutes. Obtained by a previous model calibration study using observations in the river (Ding et
774
al. 2004), the bed roughness coefficients (i.e., Manning’s n) varying along the reach were used
775
for the river flow simulations. The downstream boundary condition at the outlet was given by a
776
discharge-stage rating curve.
777
Figure 19 depicts the searching process (or the history of the objective function values
778
measured by Eq. (4)) of the optimal flow diversion control using the L-BFGS-B algorithm. It
779
indicates that only through 20 L-BFGS-B iterations the value of the objective function reaches its
780
minimum. The computational CPU time for performing all the 50 searching iterations was only a
781
few minutes on a PC. The obtained optimal flood water diversion hydrograph is presented in
782
Figure 20. The optimal diversion hydrograph implies that the diversion in the first flood wave
783
plays a major role in the 30-day flood control operation. Because the last two flood waves are
784
relatively small, the diversion operation only is to cut a small amount of the peak discharge
785
(about 5 m3/s) in the two last flood flows. In Figure 21, the water elevations at three cross
786
sections with the optimal diversion control are compared with those with no control. All the
787
stages with the optimal flow control are lower than the variable target elevations Zobj(x), which
40
788
are given in Figure 18. Moreover, Figure 22 gives the comparisons of the discharges at the three
789
sections between the optimal diversion and no control. The controlled discharge at the
790
crosssection of the floodgate clearly shows that the diversion flows cut the first flood peak, and
791
end up with an almost constant value of discharge passing onto downstream. The discharges in
792
the two last floods also indicate that both two small flood flows won’t cause too much flooding
793
at the downstream; most of the river flows after the first peak can pass through this river reach.
794 795
CONCLUSIONS
796 797
In this study, nonlinear numerical optimization approach based on the variational analysis
798
for a flow simulation model is developed to determine the optimal flood diversion control
799
operation to mitigate flood water stages in channel network of a watershed. This integrated
800
simulation-based optimization model for control flood stages in watershed consists of a well-
801
established 1-D flow model called CCHE1D and its adjoint model. The adjoint equations of the
802
dynamic wave model in CCHE1D are derived based on the variational principle. The sensitivity
803
of the objective function with respect to the control variables, i.e. flood diversion discharges at
804
floodgates, is computed by the adjoint sensitivity represented by two adjoint variables. To solve
805
the two adjoint equations in watershed, the internal conditions on the confluences in channel
806
network are derived accordingly. The implicit Preissmann’s schemes are implemented for
807
solving both the flow governing equations and the adjoint equations in channel network. Both
808
the flood simulation model and the adjoint model are efficient and robust.
809
41
810
By incorporating into this 1-D flow simulation application software, this optimal control
811
model enables to solve the practical flow control problems in flood diversion control in rivers
812
and channel network of a watershed. Two examples for testing flood diversion control are
813
demonstrated for mitigating flood water stages in channel network by finding the optimal flood
814
diversion discharges at floodgates. The performance of different floodgate installations in
815
watershed is investigated by testing the effectiveness of the flood diversion control through a
816
single floodgate and multiple floodgates. It is found that the multiple floodgate control is a cost-
817
effective approach to better balance the diversion flood waters over the entire watershed, and it
818
can also avoid river flows speeding up by withdrawal of waters as found in the single floodgate
819
control. Overwithdrawing flood waters may create unexpected erosion on river bed around flood
820
diversion facility. Furthermore, a one-month optimal flood control was achieved by applying the
821
CCHE1D-based optimization to mitigate river floods in a natural river with complex river course
822
and cross-section shapes. Numerical optimization shows that the developed optimization model
823
can quickly find the optimal solutions in all the cases with various control conditions.
824
Collaborating with CCHE1D, this optimal control model becomes generally applicable for
825
practicing the optimal control of flood diversion in a river and a channel network of a watershed.
826
By evaluating the control action performance, it is demonstrated that this optimization
827
methodology and this integrated optimization software are capable of determining the best
828
schedules of the optimal diversion discharge, the optimal locations, the minimum capacities of
829
the floodgates and detention basins in river and watershed. Therefore they can assist flood water
830
managers and decision makers to seek the best planning, management and design for flood
831
diversion control in rivers and watersheds.
832
42
833
ACKNOWLEDGEMENTS
834 835
This work was a result of research sponsored by the USDA Agriculture Research Service under
836
Specific Research Agreement No. 58-6408-7-236 (monitored by the USDA-ARS National
837
Sedimentation Laboratory) and The University of Mississippi. The writers are thankful to
838
anonymous reviewers for their critical comments.
839 840
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List of Table Table 1 Hydrograph parameters, maximum stage Zobj, and Manning’s n in channel network…44 Table 2 Flood diversion volume in the flood period and its percentage in total flood water to the watershed …………………………………………………………………………………..…... 44 List of Figures Figure 1 Configuration of a confluence in a channel network in watershed (the grey color shows a sub-watershed for generating channel network from a digital elevation model of the watershed) ....................................................................................................................................................... 52 Figure 2 Computational domain ................................................................................................... 52 Figure 3 Configuration of a channel network with a floodgate .................................................... 53 Figure 4. Comparisons of sensitivities computed by adjoint sensitivity analysis (ASA) and direct sensitivity analysis (DSA) at the floodgate with respect to three targeted sections (St. 1, St. 2, and St. 6) and q0(xc,t)=0.0. ................................................................................................................... 54 Figure 5 Iterations for searching for the optimal lateral outflow discharge, i.e. q(xc,t)Δx, where Δx is the channel length where the floodgate is placed. .................................................................... 55 Figure 6 Iterations of the objective function and the norm of its gradient ................................... 56 Figure 7 Iterations of times histories of the adjoint variable λA(t) at the location of floodgate .... 57 Figure 8 Temporal profiles of the two adjoint variables at five monitoring stations at the first iteration (i.e. no lateral outflow diverted from the floodgate) ...................................................... 58 Figure 9 Comparisons of stages between controlled and uncontrolled states at five stations ...... 59 Figure 10 Comparisons of discharges between controlled and uncontrolled states at five stations ....................................................................................................................................................... 60 Figure 11 Three floodgates in channel network ........................................................................... 61
49
Figure 12 Optimal flood diversion discharges at three floodgates and comparison with the withdrawal by the single floodgate ............................................................................................... 62 Figure 13 Comparisons of stages between controlled and uncontrolled states at five stations .... 63 Figure 14 Comparisons of discharges between controlled and uncontrolled states at five stations ....................................................................................................................................................... 64 Figure 15 Comparison of objective functions between one and three floodgates control ............ 65 Figure 16. A reach of the East Fork River digitized from a DEM in the CCHE1D GUI. Note that the river flows from the south toward the north. The points represent the locations of the cross sections for the computation ......................................................................................................... 66 Figure 17. Observed hydrograph at the inlet cross in a period from May 20 to June 19, 1979.... 67 Figure 18. Allowable water elevations along the river reach ....................................................... 67 Figure 19. Iterations of the objective function by using L-BFGS-B ............................................ 68 Figure 20. The optimal diversion hydrograph at the floodgate .................................................... 68 Figure 21. Comparisons of water stages between the optimal control and no control. Note that the red lines represent the allowable water elevation Zobj at the corresponding sections. ............ 69 Figure 22. Comparisons of discharges at three cross sections between the optimal control and no control ........................................................................................................................................... 70
50
Table 1 Hydrograph parameters, maximum stage Zobj, and Manning’s n in channel network
Channel No. 1 2 3
QP (m3/s) 50.0 50.0 60.0
Qb (m3/s) 2.0 2.0 6.0
Tp (h) 16.0 16.0 16.0
T (h) 48.0 48.0 48.0
Zobj (m) 3.5 3.5 3.5
n (s/m1/3) 0.01 0.02 0.03
Table 2 Flood diversion volume in the flood period and its percentage in total flood water to the watershed
Volume (m3)
Flood Operation Total flood waters from upstream Single Floodgate
T
q(t )dt q (t )dt q (t )dt q (t )dt
Percentage of diversion (%)
13,824,000 8,398,218
60.8
2,883,711
20.9
2,475,965
17.9
1,131,258
8.2
6,490,934
47.0
0
T
0
1
T
Three Floodgate
0
2
T
0
T
0
3
q1 (t ) q2 (t ) q3 (t )dt
51
1
3
2
(a) Channel network in watershed
(b) Confluence
Figure 1 Configuration of a confluence in a channel network in watershed (the grey color shows a subwatershed for generating channel network from a digital elevation model of the watershed)
t
Δx
T
ϕ(x,t)
n+1
Δt n x i
i+1
L
Figure 2 Computational domain
52
Q Qp
Q Qp
Monitoring Station
Qb
1
t Tp
3
St.1
Zobj=3.5m
Confluence
1:2
St.4
t T
Tp
Cross-section
L1 = 4,0 00 m
Qb
T
70m
3
=
13
Q Qp
Channel No . 2
+0.0m
St.2 ,0
00
St. 5
m
1:1. 5
+2.0m
L
20m
q(t)=?
L2 = 4,500m Qb t Tp
T
St.6 St.3
Figure 3 Configuration of a channel network with a floodgate
53
0E+00
St. 1
J/q
-1E-08
-2E-08 By ASA By DSA (q = 1.0) By DSA (q = 0.5)
-3E-08
0
12
24
36
48
Hour
0E+00
St. 2
J/q
-1E-08
-2E-08 By ASA By DSA (q = 1.0) By DSA (q = 0.5)
-3E-08
0
12
24
36
48
Hour
0E+00
St. 6
J/q
-1E-08
-2E-08 By ASA By DSA (q = 1.0) By DSA (q = 0.5)
-3E-08
0
12
24
36
48
Hour
Figure 4. Comparisons of sensitivities computed by adjoint sensitivity analysis (ASA) and direct sensitivity analysis (DSA) at the floodgate with respect to three targeted sections (St. 1, St. 2, and St. 6) and q0(xc,t)=0.0.
54
100 Discharge at inlet of main stem Dischargeat two branchs
3
Discharge (m /s)
50 0
-50
Iteration= 1 Iteration= 3 Iteration= 4 Iteration= 6 Iteration= 10 Iteration= 30 Iteration= 70
-100 -150 0
12
Tp
24
Hours
36
48
Figure 5 Iterations for searching for the optimal lateral outflow discharge, i.e. q(xc,t)Δx, where Δx is the channel length where the floodgate is placed.
55
5
10
4
10
3
10
2
10
1
10
0
-2
10 Objective Function Norm of Gradient
-3
10
-4
10
-5
10
-6
10
-1
Norm of Gradient
Objective Function
10
10
-7
10
-2
10
-3
10
-8
0
20
40
60
80
10 100
Iterations of L-BFGS-B Figure 6 Iterations of the objective function and the norm of its gradient
56
1.5E-04
ITERATION= ITERATION= ITERATION= ITERATION= ITERATION= ITERATION= ITERATION=
1 3 4 5 6 10 20
A
1.0E-04
5.0E-05
0.0E+00 0
12
24
Hours
36
48
Figure 7 Iterations of times histories of the adjoint variable λA(t) at the location of floodgate
57
0.001
St. 1
0.00015
St. 1
0
Q
0.0002
A
-0.001 -0.002
0.0001
-0.003 5E-05
-0.004 0
24
Hours 36
48
-0.005
0.001
St. 2
Q
0.0002
12
0.00015
12
24
Hours 36
48
12
24
Hours 36
48
12
24
Hours 36
48
12
24
Hours 36
48
12
24
Hours 36
48
St. 2
0
A
-0.001
0.0001
-0.002 -0.003
5E-05 -0.004 0
24
Hours 36
48
-0.005
0.001
St. 3
Q
0.0002
12
0.00015
St. 3
0
A
-0.001
0.0001
-0.002 -0.003
5E-05 -0.004 0
24
Hours 36
48
-0.005
0.001
St. 4
Q
0.0002
12
0.00015
St. 4
0
A
-0.001
0.0001
-0.002 -0.003
5E-05 -0.004 0
24
Hours 36
48
-0.005
0.001
St. 5
Q
0.0002
12
0.00015
St. 5
0
A
-0.001
0.0001
-0.002 -0.003
5E-05 -0.004 0
12
(a) λA(t)
24
Hours 36
48
-0.005
(b) λQ(t)
Figure 8 Temporal profiles of the two adjoint variables at five monitoring stations at the first iteration (i.e. no lateral outflow diverted from the floodgate)
58
5
St. 1
Stage (m)
4
Allowable Stage
3 2 1
No Control Optimal Control
0
5
12
24
Hours 36
48
St. 2
Stage (m)
4 Allowable Stage
3 2 1
No Control Optimal Control
0
5
12
Hours 36
Allowable Stage
3 2 1
No Control Optimal Control
0
5
12
Stage (m)
24
Hours 36
48
St. 4
4
Allowable Stage
3 2 1
No Control Optimal Control
0
5
12
24
Hours 36
48
St. 5
4 Stage (m)
48
St. 3
4 Stage (m)
24
Allowable Stage
3 2 1
No Control Optimal Control
0
12
24
Hours 36
48
Figure 9 Comparisons of stages between controlled and uncontrolled states at five stations
59
150
Discharge (m /s)
St. 1
No Control Optimal Control
3
100
50
0
0
12
24 Hours
36
48
150
3
Discharge (m /s)
St.2
No Control Optimal Control
100
50
0
0
12
24 Hours
36
48
150
Discharge (m3/s)
St.3
No Control Optimal Control
100
50
0
0
12
24 Hours
36
48
150
3
Discharge (m /s)
St. 4
No Control Optimal Control
100
50
0
0
12
24 Hours
36
48
150
Discharge (m3/s)
St. 5
No Control Optimal Control
100
50
0
0
12
24 Hours
36
48
Figure 10 Comparisons of discharges between controlled and uncontrolled states at five stations
60
1
3 L1 = 4, 000m
q1(t)=?
Confluence
q2(t)=? L 3
=
13
,0
00
m
q3(t)=?
Channel No . 2 L2 = 4,500m
Figure 11 Three floodgates in channel network
61
Hydrograph at inlet of main stem Hydrograph at two branches
Discharge (m3/s)
50
q3 q2 q1
0 -50
Total Discharge: q1+q2+q3
-100 -150 0
Withdrawal by a single floodgate
10
Tp
20
Hours
30
40
50
Figure 12 Optimal flood diversion discharges at three floodgates and comparison with the withdrawal by the single floodgate
62
5
St. 1
Stage (m)
4
Allowable Stage
3 2 1
No Control Optimal Control
0
5
12
Hours 36
Allowable Stage
3 2 1
No Control Optimal Control
0
5
12
Stage (m)
24
Hours 36
Allowable Stage
3 2 1
No Control Optimal Control
0
5
12
24
Hours 36
48
St. 4
4
Stage (m)
48
St. 3
4
Allowable Stage
3 2 1
No Control Optimal Control
0
5
12
24
Hours 36
48
St. 5
4
Stage (m)
48
St. 2
4
Stage (m)
24
Allowable Stage
3 2 1
No Control Optimal Control
0
12
24
Hours 36
48
Figure 13 Comparisons of stages between controlled and uncontrolled states at five stations
63
150
Discharge (m /s)
St. 1
No Control Optimal Control
3
100
50
0
0
12
24 Hours
36
48
150
3
Discharge (m /s)
St.2
No Control Optimal Control
100
50
0
0
12
24 Hours
36
48
150
Discharge (m3/s)
St.3
No Control Optimal Control
100
50
0
0
12
24 Hours
36
48
150
Discharge (m3/s)
St. 4
No Control Optimal Control
100
50
0
0
12
24 Hours
36
48
150
Discharge (m3/s)
St. 5
No Control Optimal Control
100
50
0
0
12
24 Hours
36
48
Figure 14 Comparisons of discharges between controlled and uncontrolled states at five stations
64
5
10
3
Objective Function
10
10
Single Floodgate Three Floodgates
1
-1
10
-3
10
-5
10
-7
10
0
20
40
60
80
Iterations of L-BFGS-B
100
Figure 15 Comparison of objective functions between one and three floodgates control
65
Outlet
Floodgate
Inlet
Figure 16. A reach of the East Fork River digitized from a DEM in the CCHE1D GUI. Note that the river flows from the south toward the north. The points represent the locations of the cross sections for the computation
66
Figure 17. Observed hydrograph at the inlet cross in a period from May 20 to June 19, 1979
Figure 18. Allowable water elevations along the river reach
67
Figure 19. Iterations of the objective function by using L-BFGS-B
Figure 20. The optimal diversion hydrograph at the floodgate
68
Figure 21. Comparisons of water stages between the optimal control and no control. Note that the red lines represent the allowable water elevation Zobj at the corresponding sections.
69
Figure 22. Comparisons of discharges at three cross sections between the optimal control and no control
70
