comparative study of hydrodynamic and f.c.b. mascarenhas ... · comparative study of hydrodynamic...

Comparative study of hydrodynamic and

experimental models for the routing of

floods caused by hydroelectric power plant

operation

F.C.B. Mascarenhas", M.R.R. de CarvalW

o/ m'o de Jomez'ro,

, ̂ 20 (fe

.A. &

00, TZzo (fe

Brazil

ABSTRACT

The flood wave movement downstream of a hydroelectric powerplant has been recently studied mostly by the hydrodynamicapproach, resulting in computational models that try to simulatethe set of the power plant operation rules. The combined use ofphysical models can be a helpful tool in the adjustments and inthe elimination of fictitious results, mainly in those caseswhere the topography of the downstream valley is complicated. Inthis paper a comparison is made between experimentalmeasurements carried out in a physical model and results of thecounterpart hydrodynamic model, which is based on the numericalsolution of the unsteady flow equations. Important features suchas downstream morphological changes and its effects on the flowproperties are computed and analyzed. The suitable locations inthe physical model of the gauge sections turned out to be veryimportant for the correct calibration of the computationalmodel. Those locations are chosen based on the occurrence ofvery low turbulence effects on the experimental flow to assurewater level measurements with minimum oscillations that mightjeopardize the physical model behaviour. The comparativeanalysis shows the importance of using the physical model tocarry out the initial calibration of the mathematical model,despite troublesome and costly. The case study is the futurehydroelectric development system of Anta, in the Paraiba do Sulriver, Brazil, presently at preliminary design stage.

Transactions on Modelling and Simulation vol 4, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X

58 Computational Methods and Experimental Measurements

INTRODUCTION

The effect on the downstream flow due to the operation rules of

a hydroelectric power plant can be mathematically represented

through the so-called hydrodynamic computational models. Such

models simulate the unsteady flow motion by means of the flood

wave propagation and can be prepared to receive, as input data,

upstream boundary conditions related to gates and spillway dam

operations. The evaluation of these control procedures on the

downstream channel conditions, requires, among other

information, a suitable reproduction of the topography of the

area under study. The numerical model usually does not represent

exactly the topography (or area), in such case one may have

distortions on the numerical results, when compared with the

expected physical behaviour of the real situation. The use of

physical, reduced scale models has shown to be a helpful tool

for corrections of some uncertainties on the topographic

discretization process and also for the adjustment of the

mathematical model parameters. In order to achieve

representation of the real situation to be simulated by the

joint use of the two models, one has to analyze many features to

be studied in the physical model, mainly those concerned with

the location of the cross sections where flow measurements will

be carried out. For example, these gauge sections should not be

affected by local effects of the physical model itself, such as

turbulent oscillations and resistance properties of the

construction material employed in the terrain representation of

the model. The computational mathematical model is based on the

one-dimensional Saint-Venant equations, the main resistance

factor to the flow being the cross sectional conveyance, whose

values over the routing stretch are adjusted with the aid of the

measurements obtained from the physical, reduced scale, model.

The hydraulic conveyance of the cross section is considered

through a compound approach, where the section is divided into

several vertical slices, each subsection having a different

resistance from the others.


Computational Methods and Experimental Measurements 59

GOVERNING EQUATIONS

The hydrodynamic mathematical model is based on the well known

equations of the one-dimensional unsteady gradually varied flow,

also called Saint-Venant equations. These equations result from

the application of the physical laws of mass and momentum

conservation to a control volume in the channel , and the forms

presented in equations (1) and (2) are used in this work, often

called the divergent form of partial differential equations (eg.

Liggett [ 1 ] ) . The more representative variables related to a

general cross section and to a longitudinal element of the

channel, can be seen in figures 1 and 2.

continuity:

- • - '

dynam i c :

*•

In the governing equations (1) and (2) the following

definitions are applied:

x,t - space and time independent variables

z,Q - free surface elevation and discharge in the cross

sect ion

B,A - top width and wetted area of the cross section

g,q - gravity acceleration and lateral inflow per unit

length

P - correction coefficient for the non-uniform velocity

distribution in the vertical direction

v - x component of the lateral inflow velocity

Sr. - slope of the energy line

The variables p and S. are given by:



Figure 1-SCHEME OF A GENERAL CROSS SECTION

Figure 2- SCHEME OF A LONGITUDINAL ELEMENT



[J

h dzz z

V A

,_ ,5)

where

v ,h - local depth averaged velocity and depth at zz z

position

V - average velocity in the cross section A

K - cross sectional conveyance

n - resistance coefficient (Manning-Strickler )

R - hydraulic radius, relation between A and the wetted

perimeter P

It was adopted a compound conveyance of the cross section

through its division into several vertical slices, assuming

rectangular shapes for these subsections, as indicated in figure

3. Thus, the relationship between K and S^ is then distributed

over the entire section, resulting in (eg. Cunge, Holly and

Verwey [2] ) :

S J/2 = K S//2 (6)"' fm f

Considering the approximation of the vertical slices by narrow

rectangles b x h , the conveyance K is now given by:

V iK = )-J/ nL- m

b h (7)m m



NUMERICAL SOLUTION

The equations shown in the last section, continuity and dynamic,

are mathematically classified as a system of quasi-linear

partial differential equations of the hyperbolic type and in

general do not have analytical solution, except for a few

special problems. The procedure employed here is their numerical

solution using an implicit finite difference scheme with time

and space weighting factors (eg. Lyn and Goddwin [3]). In this

scheme a dependent variable f is approximated at a point M in

the discretization mesh according to figure 4. The

approximations for f and its partial derivatives with respect to

space and time are written using the values of f at the mesh

nodal points as follows:

f(x,t) = e[vf+(i-v)fH-(i-0)[vf-^^(i-v)f-] (8)

ax " AX ̂ 1+1

'ST * A. L «f V * • , 1 * • / ' \ * ^r / \ * i i 1 -I ' -* \ * ̂ /Ot At l"t"l 1 IT! 1

where

i,j - indicial notation for discrete time and space

Ax,At - spatial and temporal increments

ip,0 - weighting factors for space and time

The approximations given by the expressions (8), (9) and

(10) are applied to the Saint-Venant equations, resulting in

systems of discretized equations where the unknowns for the

unsteady problem are the variables with subscripts (i+1). The

numerical procedure starts from known initial conditions. Time

dependent boundary conditions, which can be of the type f(t),

are introduced to assure an unique solution for the transient

problem.



DATUM

Figure 3-CROSS SECTION DIVISION INTO VERTICAL SLICES

i +1i+Vi)

Ax

M

|0AtAt

j i+1

Figure 4- EVALUATION POINT M OF A VARIABLE IN THE DIS-CRETIZATION MESH.



Each set of discretized equations is solved at a given time

step by the generalized Newton-Raphson method (eg. Ralston[4]),

which allows the determination of the values of the unknown

dependent variables for every calculation point inside the

routing stretch. Using respectively the notation F and G to

represent the discretized equations of continuity (1) and

dynamic (2), one has for a generic time step "i" the following

discrete system, written as function of the unknowns at that

time step:

It is important to point out that in the solution of the

set of equations (11) by the Newton-Raphson method, the jacobian

matrices that appear w i l l have banded type structures. In fact,

any discretized equation involves no more than four unknowns at

each time step, and as the jacobian matrices elements are the

derivatives of each equation with respect to each unknown, the

following relations can be written (eg. Mascarenhas [5]):

sv\J = — (12)

= 0 for L > m and L <> m-2 (13)



where U is either F or G and x stands for any unknown

(zorQ).

TESTS IN THE PHYSICAL MODEL, CALIBRATION AND

VERIFICATION OF THE MATHEMATICAL MODEL

The physical, reduced scale, model, is related to a small

portion of the reservoir, the hydraulic structures of the power

plant (rockfill dam, spillway and power installations) and about

2 Km of the downstream channel (Paraiba do Sul river) with the

city of Anta at its neighborhoods. This model is part of the

future hydroelectric power development system of Sapucaia. The

model is three-dimensional, with fixed bed and is based on the

Froude number similitude, with no distortion on its 1:75 scale.

The local topography and bathymetry were reproduced from

surveyings on maps on a scale of 1:1000. The prototype and model

water levels had been set compatible through a model roughness

adjustments, where it was adopted a maximum difference of 20 cm

between those levels, as an acceptance criteria. The model

boundary conditions are discharges obtained from depth

measurements carried out on a triangular shaped spillway with

flow accommodation structure at the upstream l i m i t and a movable

gate at the downstream l i m i t which behaves as a sharp-crested

weir.

The first part of the studies was concerned with choosing

transverse cross sections of the physical model where water

level gauges for the level measurements were installed. The

choosing procedure was based on a careful observation of the

flow characteristics for several discharge values imposed to the

model. A representative sketch of the physical model is shown in

the figure 5.



SCALE

Figure 5 - LOCATION MAP, PARAIBA DO SUL RIVER NEAR ANTA, BRAZIL



The conveyance adjustments were made after nine model tests

under steady state conditions for the evaluation of the natural

water levels. For each test the downstream gate was positioned

in such a way to reproduce the water level in the prototype at

the last cross section adopted to the stretch (section 7). Those

adjustments were made through a trial procedure, because the

slope of the energy line has been shown to be much affected by

small water level variations due to the closeness between the

chosen sections. A number of mathematical model discharge

simulations were used to adjust on the best possible way the

numerically computed profiles to those profiles measured on the

physical model. This approximation was made in such a way that

the conveyance x water level graphical curves would have a

growing behaviour with respect to the levels, without

discontinuities. The calibration results, by means of the above

mentioned adjust ings and approximations, can be seen in the

figures 6 and 7.

The verification task was related to five unsteady tests on

the physical model. These tests were made for the case where the

downstream gate remained fixed at the position corresponding to

a discharge of 63 m /s at the prototype. The reason for that

procedure lies in the fact that the gate is unable to reproduce

automatically the rating curve of the prototype. Due to that

gate position it was observed in the physical model the

appearing of rapid flows between the sections 6 and 7,

associated to a change in the flow state, from sub-critical to

super-critical. Thus the section number 6 was adopted as the

downstream boundary condition in the verification task. Each

test was carried out starting from the steady state condition

with an initial discharge of 63 m/s, followed by an opening

procedure of the spillway gate until a position such that a

discharge of about 700 nT/s was reached. After the observation

of approximated stable conditions on the downstream water

levels, the gate was closed and returned to its same initial



position. It was measured, with the aid of capacitive

transducers coupled to graphical register instruments, the

levels at the sections 1, 4 and 6 in the physical model. The

sections 1 and 6 are respectively upstream and downstream

limits, and the intermediate section 4 can be considered

representative of the city of Anta. At this section it was made

a comparison analysis between the computed and measured water

level variations, that can be seen in the figures 8 and 9.

APPLICATIONS AND RESULTS

The mathematical model that was subjected to the described

calibration procedure was used for the water levels predictions

downstream the clam, according to three hypothetical cases of the

spillway operation, with different opening times. The starting

condition for those cases was the steady state situation with a

discharge of 63 m /s which is the minimum rate to be released

from the Anta power plant for the maintenance of sanitary

conditions at the downstream area. All the openings are carried

out until a discharge of 513 m /s is reached, which is related

to a situation of power charge rejection at the Sapucaia power

plant, that makes use of the same storage reservoir of the power

plant of Anta. The discharge hydrographs and the water level

results are shown in the figures 10 and 11, and can be used for

the determination of the operation rules of the spillway of the

Anta's dam and for the elaboration of safety plannings for the

city of Anta.

CONCLUSIONS

The use of a physical, reduced scale, model can be a helpful

tool for the calibration of some parameters of the hydrodynamic

mathematical model, in the study of hydraulic transients related

to natural flows. Despite the limitations associated to the

satisfaction of the Froude number similitude condition, the


236.15


1 2 3 4 5 6 7

000 M3/S

700 M3/S600 M3/S500 M3/S

400 M3/S

300 M3/S

200 M3/S

100 M3/5

63 M3/S

230.00OBSERVED COMPUTED

Figure 6 - WATER SURFACE PROFILES - UNSTEADY FLOW1 2

236.00

235.25 -

233.75 -

233.(

CE232.25 4

231.50--

230.75 --

230.00

34/56/7

CONVEYANCE (*10**(+03))

Figure 7 - ELEVATIONS x CONVEYANCES - ADJUSTMENT



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Figure 8 - WATER STAGE HYDROGRAPHS - SPILLWAY GATE OPENING

.__ SECTION-4

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Figure 9 - WATER STAGE HYDROGRAPHS - SPILLWAY GATE CLOSING



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TIME (SECONDS) l*10**(+02))10 MIN 28 MIN 30 MIN

234.50

Figure 10 - RELEASED DISCHARGES AT THE SPILLWAY

SECT I ON-4

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Figure 11 - WATER LEVELS RESULTS



physical model allows an acceptable representation of the

three-dimensional hydraulic behaviour of the natural phenomenon.

While the mathematical model allows the simulation of several

unsteady flow situations for large routing stretches, the

physical model, on the other hand, has been shown to be more

indicated for the determination of local properties of the

steady state flow. The comparison made in this work is situated

in a common range of application of both models. The joint use

of the physical and mathematical models certainly results in a

more accurate estimative of the conveyance variations of the

cross sections for different values of water levels.

REFERENCES

1. Liggett,J.A. Basic Equations of Unsteady Flow, chapter 2,

Unsteady Flow in Open Channels, Ed. Mahmood,K. and

Yevjevich,V., Water Resources Publications, Colorado, 1975.

2. Cunge,J.A., Hoily,P.M.Jr. and Verwey,A. Practical Aspects of

Computational River Hydraulics, Pit man Advanced Publishing

Program, London, 1980.

3. Lyn,D.A. and Goodwin,P. Stability of a General Preissmann

Scheme, Journal of Hydraulic Engineering A.S.C.E., vol 113,

No.l, pp 16-28, 1987.

4. Ralston,A. A First Course in Numerical Analysis, McGraw-Hill

Book Company, New York, 1965.

5. Mascarenhas,F.C.B. Mathematical Modeling of Waves Caused by

Breaking of Dams, Ph.D. Thesis, COPPE-UFRJ, Rio de Janeiro,

Brazil, 1990, (in Portuguese).


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