q - The National Center for Computational Hydroscience and

Optimal Control of Flood Water with Sediment Transport in Alluvial
Channel
Yan Ding* and Sam S. Y. Wang
National Center for Computational Hydroscience and Engineering, The University of
Mississippi, University, MS 38677, U.S.A.
ABSTRACT
Flood water in an alluvial channel (or river) conveys suspended sediments from upstream to
downstream, and entrains bed materials along with flood currents. Morphological changes on
channel/river bed trend to reduce the conveyance capability of channel, and then affect water
stages and discharges (hydrographs). It is therefore necessary to quantify the influence of
sediment transport on variations of water stages for the purpose of best flood water
management in alluvial rivers. This paper presents an integrated simulation-based
optimization model to systematically investigate optimal control of flood stages under
different conditions of channel geometries and sediment properties in alluvial channels. This
integrated system consists of a well-established one-dimensional river/channel flow model
and a non-uniform/non-equilibrium sediment transport model, as well as an adjoint sensitivity
model to find the optimal solution. The flood control is demonstrated by operating a single
floodgate or multiple gates to withdraw flood waters from alluvial channel. The optimal
withdrawal hydrographs for scheduling the floodgate operations are obtained by the
optimization model system under the constraints of objective water stages applied to a
*
Corresponding author, National Center for Computational Hydroscience and Engineering, The University of Mississippi,
University, MS 38677, U.S.A. Tel.: +1 662 915 8969; fax: +1 662 915-7796, E-mail: [email protected].
1
designated target reach. It is found that the influence of morphological changes on flood
control becomes important under certain hydrological and morphological conditions, as well
as the selection of control actions. This study shows that this simulation-based optimization
model provides an effective tool to perform best management of flood and sediment transport
in alluvial channels/ rivers, and to design the flood control structures.
Keywords: Sediment Transport, Optimization, Adjoint Sensitivity, Alluvial River Processes.
2
NOMENCLATURE
A
Abk
B*
Ctk
f
g
q
qlk
Q
Qtk
Qt*k
Qtk*
K
L
Ls
L1 , L2
n
pbk
p/
R
t
T
Utk
Wz
x
x0
xc
Z
Zobj(x)
cross-sectional area (m2)
Cross-section area contributed from sediment of size class k (m)
channel width on water surface (m)
section-averaged sediment concentration of size class k
the measuring function
the gravitational acceleration (m/s2)
volumetric rate of lateral outflow (or inflow) per unit length of the channel
between two adjacent cross sections (m2/s)
volumetric rate of lateral inflow or outflow sediment discharge per unit
length of the channel between two adjacent cross section (m2/s)
discharge through a cross-section (m3/s)
actual sediment transport rate of size class k (m3/s)
sediment transport capacity of size class k (m3/s)
potential bed material load transport capacity of size class k (m3/s)
the conveyance K is defined as K= AR2/3/n (m3/s)
length of river reach (m)
adaptation length (m)
represent the continuity equation and the momentum equation, respectively
Manning’s roughness coefficient (s/m1/3)
availability factor of sediment
porosity of the bed material
hydraulic radius (m)
time (s)
control period (s)
section averaged velocity of sediment (m/s)
weighting factor which is to adjust the scale of the objective function
channel length coordinate (m)
target location where the water stage is protective (m)
location of flood diversion gate (m)
water stage (m)
the maximum allowable water stage (the objective water stage) (m)
Greek Symbols
β
βt
δ
λA , λQ
Λ
momentum correction factor
correction coefficient
Dirac delta function
the Lagrangian multipliers
2/3−4R /3B*
3
1
1 INTRODUCTION
2
3
Flood flows in alluvial channels (rivers) convey suspended sediments from upstream to
4
downstream, and also entrain bed materials along with flood currents. During storm events,
5
due to speeding flood currents, drastic bed changes (i.e. local scour, channel degradation, and
6
aggradation) occur on alluvial river beds. Continuous sediment deposition of bed materials
7
may significantly reduce flow capacity of a channel and river and lead to chronic inundation
8
of adjacent floodplain property. Morphological changes on channels or river reaches may also
9
affect navigation, water supply, reservoir store capacity, and stability and function of in-
10
stream structures (e.g. floodgates, bridge piers, groins, etc). Therefore, in order to achieve best
11
flood control operations in alluvial channels/rivers, it is necessary to take into account the
12
influence of morphological changes induced by flood flows.
13
14
In order to best mitigate erosion and deposition impacts, an optimal sediment control
15
approaches at in-stream structures need to be developed. Nicklow and Mays [5] pointed out
16
that reservoir management release policies can be optimized by minimizing the in-stream
17
deposition height. For sediment control purpose, numerical sediment transport modeling has
18
been applied to the design of diversion works and intakes in rivers with heavy sediment
19
transport which are used in channels/rivers to withdraw water from the channel flow, and to
20
return the sediment charge to the channel with appropriate means [4]. Some previous studies
21
have been done to implement optimization theories in sediment control. Nicklow et al. [7]
22
used a genetic algorithm-based methodology to minimize bed elevation changes in rivers and
23
reservoirs of a large-scale watershed. The methodology coupled the U.S. Army Corps of
24
Engineer’s HEC-6 sediment transport simulation model with genetic algorithm procedure and
25
was subject to operational constraints on reservoir releases and storage levels. The procedure
4
26
of sediment transport model was based on the assumption that the outflow from the reservoir
27
is equal to the inflow during the same time step which neglects the storage accumulation or
28
depletion in reservoirs. In Nicklow and Mays model [5][6], though optimal release policies
29
could be achieved, a linear quadratic regulator optimization algorithm SALQR is still time
30
consuming, numerical derivative computations and the optimal policies located were often
31
local optima that depend heavily on initial user specified release policies. Some of used
32
constrains for stream bed elevation, and reservoir’s bed elevation and storage were linearized
33
in optimization module. Carriaga and Mays [1] used Differential Dynamic Programming
34
(DDP) procedure to solve the full nonlinear model by not linearizing the constraints, but
35
limiting their focus to minimizing the adverse effects (the sum of aggradation and degradation
36
depths along river reach) only in a single downstream river reach.
37
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In general, numerical optimization control is carried out in an integrated simulation-based
39
optimization system, in which simulation model is used for predicting physical variable
40
responses to variations of control actions, and an optimization model for determining optimal
41
parameters of the control actions. Due to the nature of nonlinearities and non-uniformities in
42
river flows and morphodynamic processes, predictions of flood flow and morphology require
43
a highly-accurate simulation model to compute spatiotemporal variations of flood stages and
44
bed elevation changes. The difficulties due to nonlinear system responses have to be
45
overcome to establish the relation between control actions (e.g. floodgate opening) and
46
responses of the control variables (e.g. discharge and stage).
47
48
This study aims to develop an integrated simulation-optimization system, in which a well-
49
established one-dimensional (1-D) flood flow and sediment transport model, CCHE1D [10],
50
is used as the simulation model. A nonlinear optimization model based on the adjoint
5
51
sensitivity model of the 1-D flow model has been developed for determining the nonlinear
52
dynamic responses to the flow control actions. A Limited-Memory Quasi-Newton (LMQN)
53
procedure [8] is employed to find the optimal control parameters (e.g. diversion discharge
54
hydrograph to control floodgate opening operation).
55
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By applying this optimization system, this paper presents a systematical numerical
57
investigation on the effect of bed morphological change in an alluvial channel on optimal
58
flood control. As an example, the numerical optimal flood control is demonstrated by
59
operating a single floodgate or multiple gates to withdraw flood waters from an alluvial
60
channel. The optimal withdrawal hydrographs for single or multiple floodgate operations are
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obtained by the integrated simulation-based optimization system which consists of a
62
hydrodynamic module, a sediment transport module and a bound constrained optimization
63
module. The hydrodynamic module is to solve the 1-D nonlinear Saint-Venant equations, the
64
sediment transport module computes the non-equilibrium transport equations for non-uniform
65
sediments, and the optimization module is to find the optimal solution iteratively by solving
66
the adjoint equations of the 1-D Saint-Venant equations obtained by variational approach. In
67
order to study the effectiveness and applicability of the optimal control theory, alluvial
68
channels with single and multiple floodgates are considered for different hydrological
69
conditions, sediment properties, and channel geometries. The changes in morphology due to
70
optimal control of flows are studied with various conditions of sediment sizes and channel
71
bed slopes. It is found that the influence of morphological changes on flood control becomes
72
important under certain hydrological and morphological conditions as well as the selection of
73
control actions (e.g. floodgate locations). This study shows that this simulation-based
74
optimization model is effective to perform best management of flood and sediment transport
75
in alluvial channels, rivers, and watershed to control flood flows during storms.
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2 INTEGRATED SIMULATION-BASED OPTIMIZATION MODEL
78
79
This integrated simulation-based optimization model consists of four different submodels:
80
hydrodynamic model, sediment transport and morphodynamic model, an adjoint model of the
81
hydrodynamic model, and bound constrained optimization model. In order to make this
82
optimization model applicable to most realistic flows in natural rivers, a well-established 1-D
83
river hydrodynamic and sediment transport model called CCHE1D [10] is considered here as
84
the riverine process simulation model. CCHE1D was designed to simulate steady and
85
unsteady flows and sedimentation processes in dendritic channel networks. The model
86
simulates fractional sediment transport, bed aggradation and degradation, bed material
87
composition (including hydraulic sorting and armoring), bank erosion, and the resulting
88
channel morphologic changes under unsteady flow conditions during storms. CCHE1D's
89
software design uses a watershed-based approach that provides straightforward integration
90
with existing watershed processes (rainfall-runoff and field erosion) models to produce more
91
accurate and reliable estimations of sediment loads and morphological changes in streams.
92
CCHE1D has a GIS-based graphical interface that provides support for automated spatial
93
analysis, channel network digitizing, digital mapping, and visualization of modeling results.
94
One may refer to [10] for detailed descriptions on the governing equations of CCHE1D and
95
their solution methodology for simulations of open channel flows and sediment transport in a
96
natural river and river network of a watershed. Herein, only a brief introduction about
97
mathematical formulations of one-dimensional unsteady flow model and a non-uniform/non-
98
equilibrium sediment transport model is given below.
99
100
2.1. 1-D River Hydrodynamic Model
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101
The 1-D nonlinear de Saint-Venant equations are used here for simulating open channel
102
flow and are given by
103
L1 =
∂A ∂Q
+
− q= 0
∂t ∂x
(1)
104
L2 =
QQ
∂ Q
∂ β Q2
∂Z
+g 2 = 0
( )+ ( 2 )+ g
∂t A ∂x 2 A
∂x
K
(2)
105
where, L1 and L2 represent the continuity equation (1) and the momentum equation (2),
106
respectively; x = channel length coordinate; t = time; Q=discharge through a cross-section; A=
107
cross-sectional area; Z=water stage; β=momentum correction factor; g = the gravitational
108
acceleration; q = volumetric rate of lateral outflow discharge per unit length of the channel
109
between two adjacent cross sections, which can be a control variable for flow diversion; the
110
conveyance K is defined as K= AR2/3/n, where n = Manning’s roughness coefficient, and R =
111
hydraulic radius. Eqs. (1) and (2) can simulate the open channel flow with complex cross
112
section shapes subject to initial conditions (e.g. a base flow in channels) and boundary
113
conditions at upstream (e.g. a given hydrograph for discharge inflow) and downstream (e.g. a
114
given water stage or a stage-discharge rating curve).
115
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117
118
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2.2. Sediment Transport and Morphodynamic Model
CCHE1D was developed under the concept of the non-equilibrium transport of nonuniform sediments [10]. The 1-D governing equation of sediment concentration is written as
∂ ( ACtk )
∂t
+
∂Qtk 1
+
Qtk − Qt*k =
qlk
∂x
Ls
(
)
(3)
120
where, Ctk=section-averaged sediment concentration of a sediment size class k; Qtk=actual
121
volumetric sediment transport rate of the size class k; Qt*k = sediment transport capacity of the
122
size class k; Ls = adaptation length and qlk = volumetric rate of lateral inflow or outflow
8
123
sediment discharge of the size class k per unit length of the channel between two adjacent
124
cross section. CCHE1D considers bed load and suspended load together as a total bed
125
material load and Eq. (3) is applied to bed material load and wash load. The bed material
126
transport capacity of each size class can be written in general form as
Qt*k = pbk Qtk*
127
(4)
128
where, pbk = availability factor of the k-th size class sediment and
129
load transport capacity of the size class k. In the sediment transport simulation, sediment
130
transport capacity of each size class Qtk* can be estimated by empirical sediment transport
131
formulations which were built on the equilibrium flow conditions [10]. The bed deformation
132
due to size class k is given by
(1 − p′ )
133
∂Abk
1
= Qtk − Qt*k
∂t
Ls
(
Qtk* = potential
)
bed material
(5)
134
where p/ = porosity of the bed material and Abk = cross-section area contributed from sediment
135
of size class k. Therefore, the term of the left-hand side represents bed deformation rate of
136
size class k. Substituting, Ctk=Qtk/(AβtUtk) in Eq. (3), where Utk = section averaged velocity of
137
sediment and it is approximated as section-averaged flow velocity, and βt = correction
138
coefficient that is approximated as 1, one obtains the following sediment transport equation:
∂  Qtk

∂t  βtU tk
139
 ∂Qtk 1
Qtk − Qt*k =
qlk
+
+
Ls
 ∂x
(
)
(6)
140
After the actual sediment transport rate of size class k, Qtk, is computed from Eq. (6), the
141
cross-section area changes can be computed immediately by Eq. (5). The area changes due to
142
sediment transport will be distributed on the perimeter of cross-sections.
143
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2.3. Optimization Model
9
145
For control flood water stages and bed elevation changes which are associated with
146
nonlinear and unsteady riverine processes, the nonlinear optimization approach is needed,
147
which is briefly described as follows: In general, the optimization problem for finding an
148
optimal control variable in a control action in open channel flow is to minimize an objective
149
function J that is defined as an integration of a general measure function, i.e.
J =∫
150
T
0
∫
L
0
f ( Z , Q, q, x, t )dxdt
(7)
151
where q is the lateral outflow which needs to be identified in control of flow diversion; f = the
152
measuring function; T = control period; L = length of river reach. For flood control problems,
153
Ding and Wang [3] proposed a measure function as a function of the discrepancy between the
154
predicted water stage Z(x,t) and the maximum allowable water stage Zobj(x) (the objective
155
water stage), namely,
f =
156
WZ
( Z ( x, t ) − Z obj ( x, t )) 4 δ ( x − x0 )
LT
(8)
157
where Wz=weighting factor which is to adjust the scale of the objective function; x0=target
158
locations where the water stages are protective; δ = Dirac delta function, i.e.
1
δ ( x − x0 ) = 
159
0
if Z ( x0 ) > Z obj ( x0 )
if Z ( x0 ) ≤ Z obj ( x0 )
(9)
160
Therefore, the measure function in (8) only takes account of the reach in which the predicted
161
water stage is higher than the allowable stage so as to protect the corresponding river reaches
162
from being overflowing/overtopping due to a hazardous flood. In order to minimize the
163
objective function (7) and ensure that discharge and water stage satisfies the 1-D Saint-
164
Venant equations, an augmented objective function is introduced as follows:
=
J
*
165
T
L
0
0
∫ ∫
f ( Z , Q, q, x, t )dxdt + ∫
T
0
10
∫
L
0
(λA L1 + λQ L2 )dxdt
(10)
166
where L1 and L2 represent the 1-D continuity equation (1) and the momentum equation (2),
167
respectively; λA and λQ are the Lagrangian multipliers. Note that all the flow variables are
168
influenced by bed elevation changes due to sediment transport in alluvial river flows, even
169
though the morphodynamic equations in the simulation model (i.e. CCHE1D) is not included
170
in the augmented objective functions in (10). In other words, it is assumed here that the
171
sensitivities of the flow variables with respect to bed elevation changes can be neglected. This
172
assumption is reasonable for most natural rivers with moderate sediment concentration.
173
174
By using Green’s formula, the first variation of the augmented objective function δJ* can
175
be obtained [3],
176
of A and Q can be set to zero, respectively, so as to establish the optimization conditions.
177
After some algebraic manipulations, Ding and Wang (2006) [3] obtained the following
178
adjoint equations of the de Saint-Venant equations, i.e.
179
180
181
and setting the extremum condition, all terms multiplied by the variations
 4W
obj 3
obj
∂λA Q ∂λA g ∂λQ 2 g ΛQ Q λQ  * Z ( Z − Z ) δ ( x − x0 ), if Z ( x0 ) > Z ( x0 )
+
+ *
+
=
B
LT

∂t
A ∂x B ∂x
AK 2

0, if Z ( x0 ) > Z obj ( x0 )
∂λQ
∂t
+
β Q ∂λQ
A ∂x
+A
∂λ A 2 Ag Q
−
λQ =
0
∂x
K2
(11)
(12)
where B*=channel width on water surface; g = gravitation acceleration; Λ=2/3−4R /3B*.
182
183
Therefore, the adjoint equations of the full nonlinear de Saint-Venant equations are
184
composed of the Eqs. (11) and (12). The adjoint equations are general formulations for the
185
inverse analysis of 1D unsteady channel flows with complex cross sections. From the
186
variational analysis of the augmented objective function, the sensitivity of the objective
187
function J with respect to the control variable (i.e. withdrawal discharge q(t)) is obtained as
188
suggested by Ding and Wang (2006) [3] as follows:
11
189
δ J ( xc , t )
=
T
∫0
∂f

 ∂q

x = xc

− λ A ( xc , t ) δ q( xc , t )d t


(13)
190
in which, xc = location of flood diversion gate. The gradient of the objective function with
191
respect to the lateral outflow at the control location of flood diversion at a time tn can be
192
derived immediately as follows,
δJ
δq
193
x = xc
t =tn
= −λ A ( xc , tn )
(14)
194
Obviously, through the adjoint sensitivity analysis, the sensitivity of the lateral outflow q is
195
well defined by the Lagrangian multiplier λA.
196
197
The governing equations for de Saint-Venant equations (i.e. L1 and L2), sediment transport
198
equations, and adjoint equations (Eqs. (11)-(12)) are discretized using implicit four-point
199
finite difference scheme proposed by Preissmann (1960) [9]. For the details of numerical
200
discretization of the de Saint-Venant equations and the sediment transport equation in the
201
CCHE1D model, one may refer to [10]. This same implicit scheme is also adopted for solving
202
the adjoint equations (11) and (12) for λA and λQ. Details for solving adjoint equations can be
203
found in Ding and Wang [3].
204
205
In order to minimize the objective function defined in Eq. (7) and finally identify the
206
optimal control variable, an iterative minimization procedure is required due to the
207
nonlinearity of the optimization problem. Ding et al. (2004) [2] have compared
208
comprehensively several minimization procedures, and they found that the Limited-memory
209
Broyden, Fletcher, Goldfarb, and Shanno (L-BFGS) algorithm [8] is efficient for optimization
210
in large-scale problems. The CCHE1D model serves the purpose of predicting discharge and
211
stage; the adjoint sensitivity analysis module computes the Lagrangian multipliers through the
12
212
adjoint equations (11) and (12) and uses the sensitivity formulations in Eqs. (13) – (14) to
213
calculate the gradient of the objective function.
214
215
The identification of the values with bound constraints, for example maximum diversion
216
capacity, may need to be considered. The L-BFGS-B, which is an extension of the limited
217
memory algorithm L-BFGS, is well known to successfully handle the bound constrained
218
parameter identification (Nocedal and Wright 1999 [8]). In order to preserve the physical
219
meaning of control variable in every iteration step, the limited memory algorithm for bound
220
constrained identification procedure (L-BFGS-B) is thus adopted in this study. The L-BFGS-
221
B algorithm is to minimize the objective function J in Eq. (7) subject to the following simple
222
bound constraints, i.e. qmin ≤ q ≤ qmax, where qmin and qmax mean lower and upper bounds of
223
the values of the control variable q.
224
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3
APPLICATIONS
TO
OPTIMAL
226
TRANSPORT IN ALLUVIAL CHANNEL
FLOOD
CONTROL
WITH
SEDIMENT
227
228
This simulation-optimization system has developed for a general purpose of flood control
229
in rivers with naturally complex cross-section shapes. For demonstration of the model
230
capability, it is applied to obtain the optimal solutions of flood diversion controls in a 10-km
231
long alluvial channel during a control period of 50-hour storm. The storm is assumed to have
232
a hypothetical triangular hydrograph at the upstream of the channel, in which a 100 m3/s peak
233
flood discharge occurs at 16 hour. In addition to the flood flow, a 10-m3/s base flow is
234
assumed to be flowing in the channel. In the test cases, the channel is assumed as a compound
235
channel as shown in Figure 1, with a uniform channel bed material (i.e. constant median grain
236
size, e.g. d50 = 0.127 mm). As for the boundary conditions for sediment transport at upstream,
13
237
the numerical model can take into account the clear water upstream inflows or sediment-laden
238
inflows at the inlet of the computational channel reach. In this study, we only considered the
239
open boundary conditions at downstream for both flows and sediment transport. It means that
240
the flows and sediments can go through the outlet of the computational reach with no
241
resistance from outside.
242
243
In the test cases, floodgates are to control water stages over the whole reach by operating
244
the openings of the gates, over which the stages during the flood event should be lower than
245
the allowable ones. The profile of control stages (the maximum allowable stages) Zobj(x)
246
varying along the channel is given by a slope curve as shown in the Figure 1, which means
247
the maximum water depth, 4.5m, should be limited over the entire channel. There are totally
248
21 cross-sections evenly distributed in the channel by giving a 500-m equal length between
249
two cross-sections. The minimization of the objective function defined in Eq. (7) with a
250
measuring function Eq. (8) results in the optimal lateral outflow rate satisfying allowable
251
water stages along the whole channel. The weighting factor Wz in the measure function f in Eq.
252
(8) is set to 103.
253
254
3.1 Optimal Flood Control by a Single Floodgate
255
In this test case, only one floodgate located at 7 km downstream as shown in Figure 1 is to
256
control withdrawal discharge by operating the opening of the gate so that the water stages
257
over the target reach (i.e. the whole channel in the case) are lower than the objective water
258
stages, i.e. Zobj(x) = 4.75-0.025x. The objective of the optimal flood control is to find out the
259
best solution of the floodgate discharge hydrograph so that flood stages in this 10-km alluvial
260
river reach can be minimized. By minimizing the function through the L-BFGS-B, the
261
optimal discharge has been found iteratively, and the searching process is shown in Figure 2.
14
262
Note that the identified discharge q(t) is a vector with the equal length of the total 600 time
263
steps in the simulations, by giving the time step 300 seconds. The results show that the
264
LMQN method successfully identified the control variable vector with the large number of
265
components. For explaining the optimization process, Figure 3 shows the iterative histories of
266
the objective function values and the norm of its gradient (i.e. ∇J (q) 2 ). The iterations indicate
267
that the convergent control variable has been already identified after about 15 iterations
268
through the L-BFGS-B. The obtained discharge after 15 iterations can be considered as the
269
optimal lateral outflow due to the minimized objective function and its gradient with good
270
enough accuracy. It only took 25 seconds to find the best solution of the diversion hydrograph
271
(a vector) in a desktop with an Intel 2.66GHz CPU.
272
273
Figure 4 shows the iterative history of the sensitivity of the lateral outflow (i.e. δJ/δq, or
274
negative value of λA(t) because of Eq. (14)) at the designated location of the floodgate.
275
According to the corn-shape profiles of the adjoint variable before diversion operation occurs,
276
the adjoint variable λA reflects the demand of flood water withdrawal and more flood waters
277
need to be divert at the peak of the flood. As a result, the optimal hydrograph for withdrawal
278
uniquely depends on this corn-shape profile of the sensitivity (or the adjoint variable λA).
279
Because the values of the sensitivity are eventually vanished (or very close to zero after 10
280
iterations), the obtained lateral outflow discharge in Figure 2 is indeed optimal.
281
According to the identified optimal hydrograph q(t) for flood water diversion shown in
282
Figure 2, this control diversion through the floodgate apparently is to cut down the peak flood
283
discharge in order to lower the peak flood stages in the reach. As an example, Figure 5 shows
284
the iteration history for searching for the optimal water stages at 7.0 km downstream. It
285
clearly indicates that after 15 iterations the optimal solutions is found because all the stages
286
over the 50-hour control period turn out to be lower than the objective stage (i.e. 4.75m at this
15
287
cross section), and the withdrawal action has cut the one single flood peak in the case of no
288
control into two much lower peaks. As the optimal control action is applied, all the controlled
289
flood stages over the target reach (i.e. the whole channel) indeed are assured to be lower than
290
the allowable stages.
291
292
For an alluvial channel, the bed elevations are changed with the flood flow and the
293
floodgate withdrawal operation. For the case of optimal control, Figure 6 plots profiles of
294
thalweg and thalweg changes along the channel (with a 1:40000 bed slope and a grain size
295
d50=0.127mm on bed) at the end of the flood event. The bed changes show that the flood
296
water withdrawal causes the channel degradation on the upstream reach of the floodgate due
297
to the accelerated flow by the withdrawal, as well as the aggradation on its downstream due to
298
the deceleration of downstream flow.
299
300
The optimization results in this case are also compared with those with a fixed bed (no
301
sediment transport considered). Figure 7 indicates that there is a slight difference (less than
302
1% change of withdrawal discharge) in the optimal withdrawal discharge hydrographs
303
between the two cases (i.e. with a movable and a fixed bed). Furthermore, having comparing
304
with the adjoint variable results (i.e. the values of λA, as shown in Figure 8 ) in the case of a
305
fixed bed channel, it has been found that the sediment transport in the alluvial channel didn’t
306
influence the sensitivities of the objective function with respect to withdrawal discharge. It
307
proves that the exclusion of the morphodynamic equations in the augmented object function
308
J* in Eq. (10) is appropriate in the study of flood controls as long as the morphological
309
changes are small in comparison to the flood water depth.
310
311
But more differences in the optimal water stages, as shown in Figure 9, are found at the 7
16
312
km downstream, especially at the period of flood recession. It could be the reason that the
313
withdrawal by the floodgate during the flood period caused the channel degradation (erosion),
314
and the deepened bed around the floodgate cross-section made the water elevations lower
315
during the flood recession period, because of the increased channel conveyance.
316
317
Furthermore, in the following test cases our objective is to find out the effect of optimal
318
flood control on bed morphology under different bed materials (i.e. different grain sizes) and
319
different bed slopes. Different bed materials with representative diameters d50=0.127, 0.210,
320
0.350, 0.6 and 1.25 mm and four different bed slopes (i.e. 0, 1:40000, 1:20000, and 1:13333)
321
are considered to simulate different scenarios. The single floodgate is placed in the same
322
location as shown in Figure 1. The allowable water depth along the channel for each case is
323
specified as 4.5 m, so that the allowable stages Zobj(x) are dependent on the channel bed slopes.
324
First, in order to obtain enough bed erosion/deposition, bed material with 0.127 mm diameter
325
is considered for different bed slopes. From the obtained optimal diversion discharge
326
hydrographs shown in Figure 10, it can be concluded that less water needs to be diverted as
327
the bed slope increases for the same water depth constraint along the channel. The reason
328
could be attributed to the increasing velocity and subsequent decrease in water depths with the
329
increasing bed slopes. It was further found that if the bed slope is kept increasing, and the
330
same water depths along the channel are ensured, the optimization is not required since the
331
water depths are already lower than the allowable ones (the channel flow regime may turn out
332
to be of a critical flow). Meanwhile, the bed morphology is drastically changed especially at
333
the location of the floodgate (Figure 11). The bed changes most when the maximum flow in
334
the case of the 1:40000 slope is to be diverted through the floodgate. This is obvious because
335
the higher diversion flow rate increases the upstream channel velocity and bed is eroded
336
easier. At the downstream of the gate velocity in the channel reduces and sediments are
17
337
deposited in this reach.
338
339
A second category of tests is also conducted to investigate the effect of sediment size on
340
bed erosion/deposition when optimum flood water is diverted from the channel. In this case,
341
different bed materials are considered with a fixed bed slope 1:40000. The channel thalweg
342
changes are shown in Figure 12 when the optimal discharge is diverted from the channel. It
343
can be concluded from Figure 12 that the maximum erosion/deposition takes place when the
344
sediment size is smallest, which is appropriate.
345
346
3.2 Optimal Flood Control by Three Floodgates
347
This simulation-optimization system is further applied to search for the optimal
348
withdrawal hydrographs of three floodgates operations. The setup of the channel and the
349
storm conditions are kept as the same as the previous as shown in Figure 1 (i.e.
350
slope=1:40000, d50=0.127mm, allowable water depth=4.5m). Instead of single floodgate,
351
there are three gates located at 2.0 km, 4.5 km, and 7 km of the alluvial channel; therefore
352
there are three hydrographs for water withdrawal, i.e. q1(t), q2(t), and q3(t), to be determined
353
to satisfy the flood control constraints. This nonlinear optimization has successfully found the
354
three optimal withdrawal hydrographs in less than 20 L-BFGS iterations. As shown in Figure
355
13, each discharge hydrograph, q1(t), q2(t), or q3(t), in the three gates is much less than the one
356
withdrawn by the single floodgate in the previous case. Moreover, the total discharge
357
(=q1+q2+q3) withdrawn by the three gates is also less than the one in the single gate operation
358
(denoted by the dash line in Figure 13). It indicates that multiple floodgate control can be
359
much easier (or more conservative) than the single structure control, because the control load
360
can be distributed rationally along the channel. However, the distribution feature is not even
361
in the three floodgates. In addition, in comparison of the thalweg changes along the channel
18
362
shown in Figure 14, the channel bed is less disturbed by the multiple-gate control operation
363
than that by the single floodgate.
364
365
The iterations for searching for the controlled stages at 4.5 km downstream (i.e. the
366
location of the second floodgate) are shown in Figure 15. In comparison to the control water
367
stages by the single floodgate as shown in Figure 5, the controlled stages are less undulated
368
and then more stable. Apparently, the multiple floodgate control can help stabilize the channel
369
flow and the bed morphology.
370
371
4. CONCLUSIONS
372
373
This paper presents an integrated simulation-optimization tool to systematically
374
investigate the effects of flood flows and sediment transport on the control of flood waters in
375
alluvial channels. The numerical optimal flood control is demonstrated by operating a single
376
floodgate and multiple floodgates to withdraw flood waters from channels. The optimal
377
withdrawal hydrographs for the floodgate operations are obtained by this integrated
378
simuation-optimization system which consists of a 1-D nonlinear open-channel flow module,
379
a sediment transport module, an adjoint module of the flow model, and a bound constrained
380
optimization module. In the models, (1) the flow module is to solve the well-known one-
381
dimensional nonlinear Saint-Venant equations; (2) the sediment transport module computes
382
non-equilibrium sediment transport equations for non-uniform sediment compositions; and
383
(3) the optimization module is to find the optimal solutions of withdrawal hydrographs by
384
solving the adjoint equations of the Saint-Venant equations. In order to study the effectiveness
385
and applicability of the optimal control theory, variations of sediment properties and channel
386
geometries are considered in the alluvial channels with different installation of control
19
387
structures (i.e. floodgates demonstrated in this study). The changes in morphology due to
388
optimal control of flows are studied thoroughly. It is found that the influence of
389
morphological changes on flood control varies with number of gates, hydrological and
390
morphological conditions, as well as the selection of control actions. This study shows that
391
this simulation-based optimization system is effective to perform best management of flood
392
and sediment transport in alluvial channels/rivers to control flood flows during storms, and it
393
can also facilitate engineers to achieve the best design on the locations and capacities of flood
394
control structures.
395
396
Apparently, flood control within a sediment-laden flow in a natural alluvial river (e.g. with
397
high-concentration suspended sediment) will have different characteristics from these
398
discussed in this study which is only for clear water withdrawal. Meanwhile, in some cases
399
(e.g. reservoir water release), sediment withdrawal could be operated together clear water
400
release; and then sediment withdrawal/release will become a control action.
401
sediment withdrawal in alluvial rivers with high-concentration sediment will be considered in
402
the future study.
Issues on
403
404
ACKNOWLEDGMENTS
405
This work was a result of research sponsored by the USDA Agriculture Research Service
406
under Specific Research Agreement No. 58-6408-7-236 (monitored by the USDA-ARS
407
National Sedimentation Laboratory) and The University of Mississippi. The authors thank Dr.
408
Soumendra N. Kuiry for producing test case results for this study.
409
REFERENCES
410
20
411
[1] C. C. Carriaga, and L. W. Mays: Optimal Control Approach for Sedimentation Control
412
in Alluvial Rivers, Journal of Water Resources Planning and Management, Vol. 121, No.
413
6, pp. 408-417, 1995.
414
[2] Y. Ding, Y. Jia, and S. S. Y. Wang:
Identification of the Manning’s roughness
415
coefficients in shallow water flows, Journal of Hydraulic Engineering, ASCE, Vol., No.
416
6, , pp.501-510, 2004.
417
[3] Y. Ding and S. S. Y Wang: Optimal Control of Open-Channel Flow using Adjoint
418
Sensitivity, Journal of Hydraulic Engineering, ASCE, Vol. 132, No. 11, pp. 1215-1228,
419
2006.
420
[4] D. K. Lysne, N. R. B. Olsen, H. Stole, and T. Jacobsen: Sediment control: Recent
421
Developments for Headworks, International Journal on Hydropower and Dams, Vol.2,
422
Issue 2, pp. 46-49, 1995.
423
[5] J. W. Nicklow and L. W. Mays: Optimization of Multiple Reservoir Networks for
424
Sedimentation Control, Journal of Hydraulic Engineering, 126(4), pp. 232-242, 2000.
425
[6] J. W. Nicklow and L. W. Mays: Optimal Control of Reservoir Releases to Minimize
426
Sedimentation in Rivers and reservoirs. Journal of the American Water Resources
427
Association, 37(1), pp. 197-211, 2001.
428
[7] J. W. Nicklow, O. Ozkurt, J. A. Bringer: Control of Channel Bed Morphology in Large-
429
Scale River Networks Using a Genetic Algorithm, Journal of Water resources
430
management, 17(2), pp. 113-132, 2003.
431
432
433
434
[8] J. Nocedal and S. J. Wright: Numerical optimization, P. Glynn and S. M. Robinson, eds.,
Springer-Verlag, New York, 1999.
[9] A. Preissmann: Propagation des intumescences dans les canaux et rivières, 1st Congrès
de l’Assoc. Francaise de Calcul, Grenoble, printed 1961, 433-442, 1960.
21
435
[10] W. Wu and D. A. Vieira: One-dimensional channel network model CCHE1D 3.0 -
436
Technical manual, Technical Report No. NCCHE-TR-2002-1, National Center for
437
Computational Hydroscience and Engineering, Mississippi, 2002. (Available at
438
http://www.ncche.olemiss.edu/cche1d/cche1d techmanual.pdf).
439
440
22
441
Table of Figures
442
Figure 1 A compound channel with a 1:4000 bed slope (lower: compound cross-section) .... 24
443
Figure 2 Iterations for searching the optimal withdrawal hydrograph for the flood water
444
diversion operation. .................................................................................................................. 25
445
Figure 3 Iterations of objective function and norm of objective function gradient ................. 26
446
Figure 4 Iterative history of sensitivity δJ/δq at the control point ........................................... 27
447
Figure 5. Iterations for searching for an controlled water stages at 7.0km (i.e. the location of
448
the floodgate)............................................................................................................................ 28
449
Figure 6. Profiles of thalweg and thalweg changes along the channel at the end of the flood
450
event in the optimal control case .............................................................................................. 29
451
Figure 7. Comparison of lateral discharges withdrawn from the diversion floodgate ............. 30
452
Figure 8 Comparison of the adjoint variable λA at the cross section of the floodgate at the first
453
searching iteration of L-BFGS-B ............................................................................................. 31
454
Figure 9. Comparison of water stages between the cases with and without sediment transport
455
at 7 km downstream ................................................................................................................. 32
456
Figure 10 Obtained optimal withdrawal hydrographs at the floodgate under different bed
457
slopes in which d50=0.127mm .................................................................................................. 33
458
Figure 11 The thalweg changes after the controlled flood has passed for different bed slopes
459
and constant bed material of diameter 0.127 mm .................................................................... 34
460
Figure 12 The thalweg changes after the controlled flood has passed for different bed
461
materials and constant bed slope 1:40000 ................................................................................ 35
462
Figure 13. Optimal hydrographs for the flood diversion operated by three floodgates ........... 36
463
Figure 14. Comparison of thalweg profiles after storm event by single and three floodgates. 37
464
Figure 15 Iterations for searching for the optimized water stages at 4.5 km ........................... 38
465
23
466
467
Zobj(x) = 4.75-0.025x
6
Elevation (m)
5
4
3
2
Bed slope = 1:40000; d50=0.127mm
1
0
0
2
4
x (km)
6
q(t) 8
10
1:2
468
70m
+0.0m
1:1. 5
+2.0m
20m
469
470
Figure 1 A compound channel with a 1:4000 bed slope (lower: compound cross-section)
471
24
472
473
100
Inflow Hydrograph at inlet
3
Discharge (m /s)
50
0
Iteration= 1
Iteration= 4
Iteration= 5
Iteration= 6
Iteration= 10
Iteration= 30
Iteration= 70
-50
Optimal q(t)
-100
0
12
24
36
48
Hours
474
475
Figure 2 Iterations for searching the optimal withdrawal hydrograph for the flood water
476
diversion operation.
477
25
478
479
480
10-2
4
Objective Function
Norm of Gradient
Objective Function
10
3
2
10
1
10
10
-4
10
-5
10
-6
0
10
10
-1
10
-2
10-7
0
10
20
30
40
50
Iterations of L-BFGS-B
60
10-8
70
482
483
-3
10
10-3
481
10
Norm of Gradient
105
Figure 3 Iterations of objective function and norm of objective function gradient
484
26
485
486
Iteration = 10
0.0E+00
Iteration = 6
Iteration = 5
-1.0E-03
δJ/δq
Iteration = 4
-2.0E-03
Iteration = 3
Iteration = 1
-3.0E-03
0
12
24
36
Hours
487
488
Figure 4 Iterative history of sensitivity δJ/δq at the control point
489
490
27
48
491
492
6
No Control
obj
Z
Water Stage (m)
5
= 4.57m
4
3
Optimal Control
Iteration = 1
Iteration = 5
Iteration = 15
Iteration = 30
Iteration = 70
2
1
0
24
12
36
48
Hours
493
494
Figure 5. Iterations for searching for an controlled water stages at 7.0km (i.e. the location of
495
the floodgate)
496
28
497
0.3
0.2
0.25
0.15
0.2
0.1
0.15
0.05
0
0.1
Thalweg Change (m)
Thalweg (m)
498
499
-0.05
0.05
0
-0.1
Initial thalweg
Thalweg after event
Thalweg change
-0.05
-0.15
q
-0.1
0
2
4
6
8
-0.2
10
x (km)
500
501
Figure 6. Profiles of thalweg and thalweg changes along the channel at the end of the flood
502
event in the optimal control case
503
29
504
505
0
2
1.5
-20
-40
0.5
-60
0
∆q (m3/s)
3
Discharge (m /s)
1
-0.5
-80
-1
-100
Without sediment transport
With sediment transport
Difference
-120
0
506
507
12
24
Hours
36
-1.5
-2
48
Figure 7. Comparison of lateral discharges withdrawn from the diversion floodgate
508
509
30
510
511
8E-05
No sediment transport
With sediment transport
6E-05
λA
4E-05
2E-05
0
0
12
24
36
48
512
Hours
513
Figure 8 Comparison of the adjoint variable λA at the cross section of the floodgate at the first
514
searching iteration of L-BFGS-B
515
516
517
31
518
519
5.5
Zobj=4.57m
Water Stage (m)
4.5
3.5
2.5
Without sediment transport
With sediment transport
1.5
0
12
24
36
48
520
Hours
521
Figure 9. Comparison of water stages between the cases with and without sediment transport
522
at 7 km downstream
523
524
32
525
526
100
Hydrograph at inlet
3
Discharge (m /s)
50
0
-50
Slope = 0.0
Slope = 1:40000
Slope = 1:20000
Slope = 1:13333
-100
0
12
24
36
48
Hours
527
528
Figure 10 Obtained optimal withdrawal hydrographs at the floodgate under different bed
529
slopes in which d50=0.127mm
530
531
33
532
533
0.2
Thalweg Change (m)
0.15
Slope = 1:13333
Slope = 1:20000
Slope = 1:40000
No slope
0.1
0.05
0
-0.05
-0.1
-0.15
0
534
2
6
4
8
10
Distance [km]
535
Figure 11 The thalweg changes after the controlled flood has passed for different bed slopes
536
and constant bed material of diameter 0.127 mm
537
538
34
539
540
541
Thalweg Change [m]
0.2
d50 = 0.127 mm
0.15
d50 = 0.210 mm
0.1
d50 = 0.350 mm
d50 = 0.600 mm
d50 = 1.250 mm
0.05
0
-0.05
-0.1
-0.15
0
2
4
6
8
10
Distance [km]
542
543
Figure 12 The thalweg changes after the controlled flood has passed for different bed
544
materials and constant bed slope 1:40000
545
546
547
35
548
549
550
125
Hydrograph at inlet
100
3
Discharge (m /s)
75
50
q3
q2
q1
25
0
-25
-50
Total discharge = q1 + q2 + q3
-75
-100
Discharge withdrawn by single floodgate
0
12
24
36
48
Hours
551
552
Figure 13. Optimal hydrographs for the flood diversion operated by three floodgates
553
554
36
555
556
557
0.15
Three floodgates
One floodgate
Thalweg change (m)
0.1
0.05
0
-0.05
-0.1
q2
q1
-0.15
0
2
q3(or q)
6
4
8
10
x (km)
558
559
Figure 14. Comparison of thalweg profiles after storm event by single and three floodgates
560
561
562
37
563
564
565
No Control
6
Zobj = 4.64m
Water Stage (m)
5
4
3
Optimal Control
Iteration = 1
Iteration = 5
Iteration = 15
Iteration = 30
Iteration = 70
2
1
0
12
24
36
48
Hours
566
567
Figure 15 Iterations for searching for the optimized water stages at 4.5 km
38

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