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OCR for page 45
A Flood Control of Dam Reservoir
by Conjugate Gradient and Finite Element Methods
M. Kawahara and T. Kawasaki
Chuo University
Tokyo, Japan
Abstract
A flood control of dam reservoir
by the combined method of conjugate
gradient and finite element methods
is presented. For the numerical
integration procedure, the two step
explicit scheme originally presented
by the authors' group was effectively
used. Using the numerical computation
based on the imaginary river basin
and estimated hydrograph, it is seen
that the water elevation can be
controlled by the discharge of the
dam gate to obtain almost the flat
water surface. It is detected out
that the flow rate of the dam gate
should discharge in advance before
the peak value of the flood arrives
to control the wave propagation
caused by the sudden close of the dam
gate. This paper presents the
strategy how to operate the dam gate
knowing the flood configuration
beforehand.
1. Introduction
It is necessary to construct a
large dam to protect the human
property from a flood caused by heavy
rain fall on the mountains. Normally
a reservoir is set up by the dam. The
flood propagates through the
reservoir from upstream to
downstream. To protect the downstream
area, the gate equiped to the dam
will be closed. The bore would be
generated sometimes in case that the
capacity of the reservoir is not
satisfactory enough if the dam gate
would have been shut suddenly. There
45
are several practical examples that
the human property at the upstream
area was destroyed by the reflected
bore which seems to be caused by the
dam gate operation. Thus, it is
strictly necessary and practically
important that the dam gate should be
controlled to secure the operation
that the water elevation of the
reservoir would be as small as
possible. Is it possible to obtain
almost the flat water elevation of
the reservoir during the flood by way
of controlling the dam gate? The
answer in this paper is affirmative.
The flood propagation through the
reservoir can be expressed by the
shallow water equation. One
dimensional linear equation with the
hydrograph as the upstream boundary
condition is used. This is because
the present paper aims mainly at
presenting the numerical controlled
method. It is simple to extend the
present method to the two dimensional
case. The optimal control system can
be established introducing the water
elevation of reservoir as the state
function and the discharge of the dam
gate as the control function. The
quadratic functional of the water
elevation and the control discharge
is chosen as the performance
function. Because the hydrograph of
the flood is given at the upstream of
the reservoir as the time function
over the interval to be analyzed, the
problem is resulted to the so called
quadratic tracking control problem.
Conventionally, the shallow water
equation is descritized by the finite
difference method or method of
characteristics- 5 ~ . The
OCR for page 46
discretization of the shallow water
equation is carried out by the finite
element methodist ~ 7 ~ in this paper.
For the computation of the optimal
control, the conjugate gradient
method is effectively
employed'~'-'=°'. The Lagrangian
function is introduced to express the
constraints of the state equation. To
solve the time dependent equation for
the Lagrange function, the backward
integration should be introduced. To
do this, a two step scheme has been
originated similar to the forward two
step scheme which was presented by
the authors' group in the previous
papered c 7 ~ To determine the
amplitude of the direction vector,
the line search method is used. To
show the adaptability of the control
method presented in this paper,
several numerical examples are
carried out. Comparing with the
results obtained by the dynamic
programming technique, it is found
that the present method is more
suitable for the practical
computation because the computer core
storage in the present method is much
smaller than that in the dynamic
programming. This paper detected out
a possibility that the control of the
dam gate can be performed in the
manner that almost the flat water
elevation of the reservoir at the
flood can be obtained on the
condition that the hydrograph of the
flood flowing to the reservoir is
known in advance.
The flood propagation behavior in
a dam reservoir can be expressed by
the linearized shallow water
equation. Consider one dimensional
channel with X coordinate and time t.
Denoting mean discharge and water
elevation as q and {, one dimensional
equations of motion and continuity
can be written in the following forms:
gh ~d; = 0 (1)
dq = 0 (2)
By t
at
t
at
where g, h are gravity acceleration
46
and water depth respectively.
The flood is given as the upstream
boundary conditions of mean discharge:
q = ^q on S1 (3)
where superscripted ^ represents a
function given on the boundary. The
flood control in a dam reservior is
assumed to be carried out by the
discharge decided by the optimal
control of the operation of the water
gate equiped on the dam. Thus, it is
expressed that
q = q on SO (4)
where superscripted - denotes a
function determined by the optimal
control analysis. The initial
condition are given as:
~ = r0
q = qO
at t=to
The governing
equations (1) and (2)
descritized by the
finite element method
about a piece of one
dimensional element
shown in Figure 1 can
be described as
follows.
3 6 1 q 1+
~6 3 1~ (= 1
The usual
leads to the
as:
(5)
X
—7—
a b
|
Figure 1
1 1
2 2
1 1
2 2
superposition procedure
finite element equation
[M]{X(t)}+[H]{X(t)}
+[A]{F(t)}+[B]{U(t)}={O} (8)
where
{X(t)} = ~ ``t't (9)
in which q(t) and ((t) mean discharge
and water elevation at all nodal
points of the flow domain to be
analyzed. The boundary condition (3)
is transformed to the term [A]{F(t)},
OCR for page 47
{F(t)} = ~ 4(t)t (10) following equations.
where q(t) denotes the discharge of
flood at the upstream point. The
control term [BJ{U(t)} is derived
from equation (4), and
{P(t)}=- ~6 H
=([M]-1[H])T{P(t)}+[K]{X(t)} (16)
{P(t~)}={O}
{U(t)} = ~ clot)| (11) The gradient
where q(t) represents control
discharge applied at the point
corresponding to the dam. The initial
condition can be described as follows:
{X(to)} = ~ ~o~t't. (12)
3. Optimal Control Theory
The optimal control theory employed
in this paper is the quadratic
control theory. The problem can be
converted to determine an optimal
function {U(t)} that minimizes the
performance function:
; tat{ ((t)} T ~ S ~ { ~ ( t)}
+{U(t)}T[R]{U(t)})dt (13)
under the state equation:
{X(t)}=-[M]-1tH]{X(t)}
-[M]-1[A]{F(t)}
-tM3-1[BJ{U(t)} (14)
with the initial condition {X(to)},
where [S],tR] are weighting matrices
and to' to are starting and final
times respectively.
To obtain the optimum control, the
conjugate gradient method has been
used. To apply the conjugate gradient
method, the Hamiltoniam is introduced
as:
H=-1({ {(t)}TtS]{ {(t)}
+{U(t)}TtR]{U(t)})
+{P(t)}T(-[M]-1tH]{X(t)}
-[M]-~tA3{F(t)}
-~M]-1[BJ{U(t)}) (15)
where {P(t)} denotes Lagrange
multiplier. Euler equation and
transversality condition lead to the
47
(17)
of the performance
function {J=(t)} is given as:
{J~(t)}=--~u~
=tR]tU(t)}+(~M]-1tBj)T{P(t)} (18)
The gradient of the performance
function is used to determine that
the convergence is obtained. If the
gradient comes to almost zero, the
optimal control {U(t)} can be
obtained.
To obtain the optimal control
solution, differential equation (14)
with (12) and equation (16) with (17)
must be solved. Moreover, equation
(16) must be solved from to to to
because the initial condition (17) is
given at the final time to. To solve
these equations, the time marching
numerical integration scheme is
introduced. The total time interval
to be analyzed is divided into short
time interval At by a plenty of time
points n. For equation (14), the
forward two step explicit method can
be applied as follows'' [73
For the first step:
{X(t)~+1'2~=[M]-l[M]{X(t)~]
- 2tM]-~H]{X(t)~) (19)
and for the second step:
{X(t)~+1~=[M]-l[M]{X(t)~)
-
- ~t~M]-1[H]{X(t )~+1~2} ( 20)
starting from the initial condition
equation (12). For equation (16), the
backward two step explicit method is
used:
For the first step:
{P(t )~+1~2}= ( [M]-1 [M] )T{P(t )~]
+ 2(([M]-~[H])T{P(t)~]
+[K3{X(t)~}) (21)
OCR for page 48
OCR for page 51
OCR for page 52
OCR for page 53
OCR for page 54
OCR for page 55
OCR for page 56
Representative terms from entire chapter:
flood control
and for the second step:
{P(t)~+l}=( [M]-1tM] ~T[ptt)~)
+ Att ( [M]-l[H] )TIptt)~+l,2)
+[K]{X(t)~+1'2~) (22)
starting from the initial condition
equation (17~. In equations
(19~-(22), the lumped coefficient
matrix [M] is introduced to obtain
the full explicit scheme. The mixed
coefficient matrix [M] is also used
as:
[M]=etM]+(1-e)[M] (23)
where e is referred to as the lumping
parameter.
4. Comoutational Algorithm
The conjugate gradient method is
successfully applied for the
computational algorithm. To express
the procedure of equations (19) and
(20) with equation (12), the
abbreviated form is introduced as:
{Xi)={X(t:U'(t))) (24)
where subscripted i means the
function is the value in the ith
iteration cycle and Ui(t) means the
optimal control function assumed at
the ith iteration. Thus, equation
(24) represents to solve equation
(14) with (12) by the procedure of
equations (19) and (20) assumming the
control function as Ui(t). Similarly,
the abbreviated form:
{Pi)={P(t:Ui(t))] (25)
is introduced to express the
procedure to solve equations (16)
with (17) by equations (21) and (22)
assumming the control function as
Ui(t). Using those notations, the
computational algorithm can be
described as follows.
~ ~—~ ~ ~ ~ ~ ~ ~ ~ ~ ~
Figure 2
Em,
48
1) Assume initial control function
Uo(t), t C[to,t~]
2) Solve {Xo(t)~={X(t:Uo(t))}
3) Solve {Po(t)~={P(t:Uo(t)~}
4) Compute {So(t)~=-{J=ott)}
=~~tR]{Uo(t)~+(tM]-1[B])T[Po~t)~)
5) Determine amplitude ~' by
minimizing J[Uitt)+ niS'(t)~.
6) Compute {Ui+~(t)}
=tUi(t)~+ n'tSi~t)}
7) Solve {Xi+~(t)~={X(t:Ui+~(t)))
8) Solve {P'+~(t)~={P(t:Ui+~(t)~)
9) Compute {J=i'=(t)]
=tR]tUi+~(t)~+(tM]-~[B])T[Pi+~(t)}
10) If J=i+~(t)
U(l)=Uco)+ a S. If J(U(o))>J(U(l)),
the direction is right. And the
amplitude ~ doubles the step size,
then J(U<2~) is computed by
U(2,=U(=,+2 ~ S. If J(U`~.)>J(U<2~)'
the amplitude ~ doubles the step size
again and determine Us 3 ) , then
continue the same procedure. If
J(U`~
(a
it'
lOm
L ~ ~
S1 1~ 0.26667m x 150 = 40m ~ SC
S1: (A INFLOW POINT
(A OBSERVATION POINT
Sc: (3) CONTROL POINT
1 ~ (3)
20m )1( 20m
.....
. . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
............................
. . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
............................
. . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
_ j
=,o. Do
Figure 4 Reservoir model-1
( 1 ) I NFLOW PC I NT (40M)
l
2- 00 4. OG S. GO 8. 00 10. 00 12. 00
T I ME (SEC)
Figure 5 Hydrograph of inflow(model-1)
(3) C0NTRCL PCINT (ON)
: O P T 1 M A L O U T F L O 14
--: NORMAL OUTFLOW
O _ _ _
O
~ A
~ ° /
J o
Lo
~ o
o
~0.00 2-00 4.00 6.00 8.00 10.00 12.00
T I ME (SEC)
Figure 6 Control discharge
at the control point(model-1)
available for the more complicated
control problem.
2) Test example No.2
The control results computed by the
conjugate gradient method is compared
with the ones by the dynamic
programming which was originally used
for the structural control by one of
the authorship ~ . For the computation,
the reservoir shown in Figure 8 is
employed. Total lengh L is 3km and
water depth is 6Om. Total numbers of
nodal points and elements are 31 and
30 respectively. Total time lOmin. is
- - - : OPT I MAL OUTFLON
-----: NORMAL OUTFLOW
~ ~ ~ I NFECH PO I NT (40M)
2.oo 4.oo 6.oo 8.oo 1o.oo 12.oo
TIME (SEC)
(2) CBSERVAT ION PO I NT (2OM)
_ o
~ o
_ .
z
.,,
-
~ o
o
L'J
as
o
o _ , . . .
3 I. 30 2. Do 4.00 6.00 8.oo
TIME [SEC)
. . . .
lo. Do i2. Do
( ~ ) c a N T ROL P ~ I N T ~ O M ~
4. Do 6. Do 8. Do lo. Do t2. Do
T I HE (SEC)
Figure 7 Water elevations at each point
(model-1)
divided into short time interval
l.Osec. For the lumping parameter
e=0.9 is used.
The computation has been carried
out specifying the discharge at the
inflow point as a function shown in
Figure 9. The resulted control
obtained at the control point is
represented in Figure 10. The dotted
line shows the outflow at the control
point without control. The solid line
is the optimal control discharge
computed. The computed water
elevations at inflow, observation and
control points are illustrated in
Figure 11. The water elevation at
every point is computed smaller than
that without control, which is
expressed by the dotted line. The
same problem has been computed by the
method of the dynamic programming.
The resulted control and water
elevations are represented in Figures
12 and 13. All results are completely
coincident with the results obtained
by the conjugate gradient method.
In case of the method of dynamic
programming, 2.5 times as long
intervals as that of the conjugate
gradient method can be employed.
However, all of the intermediate
values of the computation must be
stored in this type of tracking
problems. Thus, a large amount of
core storage capacity must be
required. Considering this, the
method of the dynamic programming is
not suitable for the large scale
problem.
60m
S1 1~ 0.1 km x 30 = 3km~ SC
S1: (A FLOOD INFLOW POINT
A OBSERVATION POINT
SC: (3) FLOOD CONTROL POINT
(1) (2) (3)
~ ~1.5km )< 1.5km—~ IGATE
n ...................... '
., ~
Figure 8 Reservoir model-2
(1) FLOOD INFLOW POINT (SKY)
,~,,
to
O _ 1 1 1 1 1 1 1 1
to. Do 2. Do 4. Do 6. Do 8. Go 10. Do
T I ME (M I N)
Figure 9 Hydrograph of inflow(model-2)
51
FLOOD CONTROL POINT (OKM)
:OPTIMAL OUTFLOW
-----: NORMAL OUTFLON
—
I ~
\
~ \ \
/ \ \
~ \ \
/ ~ \
\
/ \ ~
\
o ~
No. Go 2. Do 4. Go 6. Do
T I ME (M I N)
l
8. Go 1 o. Do
Figure 10 Control discharge by conjugate
gradient method(model-2-)
oOPTIMAL OUTFLOW
-----I NORMAL OUTFLOW
(1)FLOOD INFLOW POINT (SKY)
_ ~
CC
Lo o
c] .
C .
0. 00
CC .
o
C .
3 ,0. 00 2- 00
2.00 4.00 6.00 8- 00 10- 00
T I ME (M I N)
(2)0BSERVATION POINT (105KM)
~ _
/
1 1 1
2. Go 4. Do 6. Do
TIME (MIN)
1 1 1 1 1
8. Do 10. Do
3)FLOOO CONTROL POINT (OKN)
l
l
4. Do 6. Do 8.00 10.00
TIME (MIN)
Figure 11 Water elevations at each
reference point(model-2-)
.
(3) FLCOD CONTROL PC I NT (OKM) 7. Flood Control
:OPTIMAL OUTFLOW
-----: NORMAL OUTFLON
.,
/ ~
/ \
~~ '
t \ \
./ \ \
/ ~\ ~~`
l
4. Do 6. Do 8. Do 10. Go
TIME (MIN)
Figure 12 Control discharge by the
dynamic programming(model-2-)
o OPTIMAL OUTFLOW
-----: NORMAL OUTFLOW
( 1 ) FLOOD I NFLOW PO I NT (3KN)
o
~ o / ~
, '
JO
LIJ
JO l l l
3 -10. 00 2- 00 4.00 6- 00 8.00 10.00
T I ME (M I N)
(2) OBSERVAT I CN PO I NT ( 1 D SKY)
i i 1 1 1 1 1 1 1 1
2. Do 4. Do 6. Do 8. Go 10. Do
TIME (MIN)
(3) FLOOD CONTROL PO I NT (OKM)
l
2- 00 4- 00 6- 00
TIME (MIN)
Figure 13 Water elevations at each
reference point(model-2-)
52
The flood control problem by a dam
based on the imaginal river basin and
estimated hydrograph using the
observed data is carried out as a
practical example to show the
practicability of the present method.
The river basin used is shown in
Figure 14 by water depth and width.
The total length of the model is 3Okm
long. The maximum width of the
reservoir is 135Om and the upstream
river is 5Om wide. On the most
upstream side of the model the flood
inflow discharge is specified and it
is referred to as the flood inflow
discharge point. The flood control
point is set on the most downstream
side where the dam is assumed to be
equipped. The water elevation
computed is expressed on the
referrence point at the center of the
model which is called as the
observation point.
Total numbers of nodal points and
elements are 61 and 60 respectively.
Total time interval used is 48 hours,
which is divided into short time
intervals 13.824sec. The steady state
computation has been carried out to
get the steady state water elevation
~ (s(t)~. In the practical
computation, it is more suitable to
modify the perfo~lL`ance function as:
2 ~ o (I (t)) (to)) T
S](~(t)~-~`s(t)~)
+{U(t))T[R]tU(t)~)dt (28)
where the weighting matrix [S] is set
unit matrix and the weighting
coefficient for fR] matrix is R=1.0 x
1O-7. The lumping parameter is chosen
as e=O.9.
Specifying the flow discharge at
the inflow point as the time
dependent function:
q = $(t) on So
where q(t) is shown in Figure 15, the
optimum control discharge at the
control point can be computed as
q = q(t) on SO
where q(t) is expressed in Figure 16.
(1)
FLOOD
I NPLOW
POINT
50m .
~ !
T
WIDTH
S.
0. 00 3. 00
(1) FLOOD INFLOW POINT (30KM)
o
o
to
o
_ C~
Z °
o
_ o
3 ° _
Lo o
me 0
~4 0
to
o
_
to
0
.
no. JO
(2)
OBSERVATION
POINT
1. 15 km -1' - 15km
5m, .
~ ' r
(3)
PLOOD
CONTROL
POINT
v 1
35m
~ ,~~, 1
Sc
l
b.00 9.00 12.00 15.00 18.00 21.00 24.00 27.00 30.00 (km)
Figure 14 Reservoir model-3
A 2300m3/s
~ 1~l -
12.00 24.00 36.00 48.00
TIME(HOUR)
Figure 15 Hydrograph of inflow
(model-3)
In this figure, the outflow discharge
at the control point without control
is also expressed by the dotted line.
An outlook of the uncontrolled
discharge is almost conincident with
that of the inflow discharge shown in
Figure 15 until the maximum peak
value arrives. The duration to arrive
the peak value of the outlow
corresponds to that of inflow.
53
However, the uncertain oscillation of
the discharge can be found after the
peak value of the flood, which is
caused by the reflection of the wave
by the dam. By the optimum control of
the discharge, the maximum peak value
can be reduced. Moreover, the
oscillation of the discharge after
the peak value can be eliminated. It
is important to note that the control
of discharge must start before the
peak value of the flood will arrive
to the dam.
The computed water elevations at
the inflow, observation and control
points are illustrated in Figure 17.
The dotted lines show the computed
water elevations without control. At
the inflow point, which represents
the upstream area, oscillation of the
water elevation can be found in case
of the flow without control. It is
detected out that there is a
possibility that the oscillation of
the water elevation can cause severe
damages to the human properties
around upstream area. For the water
elevation controlled at the dam, the
upstream oscillation has completely
been eliminated. The variation of the
water elevation controlled is smaller
than that of uncontrolled. The water
(3) FLOOD CONTROL PO INT (OKM]
: O P T I M A L ~ U T. F. L O
-: NORMAL OUTFLOW
ll
ll
1 1
!
it,
°o.oo 12.00 24.00 36.00 48.00
TIME(HOUR]
Figure 16 Control discharge
at the control point
(model-3-)
elevations computed at every
referrence point show almost flat
water elevation excluding the
duration after the peak value of the
flood.
In the practical problem, it is
usually seen that the maximum ability
of the flow rate of the dam gate is
limited. In Figures 18 and 19,
control example of which maximum flow
rate is limited as lOOOm3/s is
illustrated. In Figures 20 and 21,
control example of which maximum flow
rate is limited as 1500m3/s is
illustrated. Figures 18 and 20 show
the control discharge at the control
point. Figures 19 and 21 are the
computed water elevations at each
reference point compared with the
water elevations without control. In
these examples, it is also detected
out that the water elevations can be
controlled by the discharge of the
dam gate. It is also seen that the
secondary oscillation of the water
elevation at the upstream has been
eliminated.
8. Conclusion
This paper has presented the
optimum control method for the wave
propagation caused by the flood
: OPT I MAL OUTFLOW
-o NORMAL OUTFLOW
( 1 ) FLOOD I NFLOW PO I NT [30KM)
CC
Led
o
3 o _
_G
1\
1 1
h I
~1~
Ill
ll
1
12.00 24.00
T I ME (HOUR)
1 1
36.00 48.00
(2] OBSERVAT I ON PO I NT ( 1 SKY]
rim
//
1 1 1 1 1 1
on 12. Do 24. Do 36. Do 48. Do
T I ME (HOUR)
(3] FLOOD CONTROL PO INT (OHM]
-
~_
~ o
Z o
~ _
1_
Lad o
~ o
~ o
LLJ
~ o
~ o
,O. OO 12. OO 24. OO SS. OO
T I ME (HOUR)
l
48- 00
Figure 17 Water elevations
at each reference point
(model-3-)
54
(3) FLOOD CONTROL PO I NT (OKM)
:OPTIMAL OUTFLOW
-: NORMAL OUTFLOW
:OPTIMAL OUTFLOW
---: NORMAL OUTFLOW
1D ~ NFLON PO ~ NT (30KM)
2.00 24.00 36.00 48.00 '°~°°
TIME(HOUR)
Figure 18 Control discharge
at the control point
(model-3-)
[outflow < lOOOm3/s ~
through the reservoir set up by a
dam. It is detected out that the
control of the dam can be effectively
performed by the conjugate gradient
method combined with the finite
element method. Comparing with the
dynamic programming, the computer
core storage of this method can be
extraordinary reduced. For the
forward and backward numerical
integrations in time, the two step
scheme can be effectively introduced.
For the determination of the
magnitude of the gradient vector, the
line search method is shown to be one
of the most efficient method of the
analysis.
Using the numerical computation
based on the practical basin and
estimated hydrograph, it has been
cleared that the water elevation can
be controlled by the discharge of the
dam to reduce the peak value and to
eliminate the secondary wave
propagation toward the upstream of
the dam reservoir. To control the
wave propagation generated by the
reflection of the sudden close of dam
gate, it is necessary to discharge
through the dam gate in advance
before the peak value of the flood
arrives. The strategy how to open and
shut the dam gate can be determined
by the present method knowing the
flood configuration beforehand.
12.00 24.00 36.00 48.00
T I ME (HOUR)
(2) OBSERVAT I ON PO I NT ( 1 5KM)
_ ,
=
— o
Z o .
cc
I1J o
J o .
o
to
3 o
10. 00
,/~N
jY-/'i'\\,/\~
.
1 1 1 1 1 1 1 1
12.00 24.00 36.00 48.00
T I ME (HOUR)
(3) FLOOD CONTROL PO INT (OKM)
,0. 00 12.00 24.00 36.00 48.00
TIME(HOUR)
Figure 19 Water elevations
at each reference point
(model-3-42>)
55
t3) FLOOD CONTROL POINT (OKM)
: ~ P T I M A L a U TF L ON
-: NORMAL OUTFLOW
to
to
o ,,
O 1 1
_ ~ _ it
(in 1 1
~ 1
Z o _ I ~
1~ o 1 1
— O ~ 1
3 o ~ 1
D ~ ~ !~ C o
.
°o.oo 12.00 24.00 36.00 48.00
TIMEtHOUR)
Figure 20 Control discharge
at the control point
(model-3-)
"outflow < 1500m3/s]
References
t1] D.A.Hughes and H.C.Murrell:"Non-linear runoff
routing-A comparison of solution methods",
Jour.Hydro.,Vol.85,pp339-347,1986
[2] B.Hunt :"A symptotic solution for dam break on
sloping channel", Proc.ASCE, Vol.lO9,No.HY12,
ppl698-1706,1983
[3] V.M.Ponce and A.J.Tsivoglou :"Modeling gradual
dam breaches", Proc.ASCE,Vol.107,No.HY7,pp829-
838,1981
[4] A.O.Akan and B.C.Yen : "Diffusion-wave flood
routing in channel networks",Proc.ASCE,Vol.107,
No.HY6,pp719-732,1981
[5] V.M.Ponce : "Linear reservoirs and numerical
diffusion",Proc.ASCE,Vol.106,No.HY5,pp691-699,
1980
[6] M.Kawahara and T.Umetsu:"Pinite element method
for moving boundary problems in river flown,
Int.J.Num.Meth.Eluid, Vol.6,pp365-386, 1986
t7] N.Kawahara, H.Hirano, K.Tsubota and K.Inagaki:
"Selective lumping finite element method for
shallow water flow",Int.J.Num.Meth.Pluid,Vol.2,
PP89-112~1982
[81 H.Kanoh : "Theory and Computational Methods in
Optimization", Corona Publishing Co., 1987
~9] A.E.Bryson and Y.C.Ho:''APplied optimal control"
Hemisphere Publishing Corporation, 1975
[10] R.P.Stengel : "Stochastic optimal control",
John Wiley & Sons, 1986
t11] M.Kawahara and T.kawasaki:"A flood control of
dam reservoir by conjugate gradient method
and finite element method", Proc.7th Int.Conf
FEN in Plow Problems,pp629-634,1989
[12] M.Kawahara and K.Pukazawa:"Optimal control of
structures subjected to earthquake loads
using dynamic programming", J. Structural
Engineering,Vol.34A,J.S.C.E.1988
: OPT I MAL OUTFLOW
NORMAL OUTFLOW
(1) FLOOD INFLOW POINT (SOKM)
l~--~v'
'1
1)
=10. 00 12.00 24.00 36.00 48.00
T I ME (HOUR)
(2) OBSERVATION POINT (15KM)
-
Z o A-' ~
~ ~ 1
> ~
Lid o
J o _
~ o
TIC to
3 o _
10.00 12.00
1 1 1 1 1
24.00 36.00 48.00
T I ME (HOUR)
(3) FLOOD CONTROL POINT (OKM)
-
~ o
Z o
o _
_
o
J o _
~ o
I1J
cc to
3 o _
to. 00 12.00 24.00 36.00 48. 00
_^ !! {~=__ _
T I ME (HOUR)
Figure 21 Water elevations
at each reference point
(model-3-)
56
