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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

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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)},

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{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)

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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)J(U<2,) OCR for page 45
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`~ OCR for page 45
(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

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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-) .

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(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.

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(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

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(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

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(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

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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