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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.
Transactions on Modelling and Simulation vol 4, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X
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:
Transactions on Modelling and Simulation vol 4, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X
60 Computational Methods and Experimental Measurements
Figure 1-SCHEME OF A GENERAL CROSS SECTION
Figure 2- SCHEME OF A LONGITUDINAL ELEMENT
Transactions on Modelling and Simulation vol 4, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X
Computational Methods and Experimental Measurements 61
[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
Transactions on Modelling and Simulation vol 4, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X
62 Computational Methods and Experimental Measurements
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.
Transactions on Modelling and Simulation vol 4, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X
Computational Methods and Experimental Measurements 63
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.
Transactions on Modelling and Simulation vol 4, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X
64 Computational Methods and Experimental Measurements
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)
Transactions on Modelling and Simulation vol 4, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X
Computational Methods and Experimental Measurements 65
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.
Transactions on Modelling and Simulation vol 4, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X
66 Computational Methods and Experimental Measurements
SCALE
Figure 5 - LOCATION MAP, PARAIBA DO SUL RIVER NEAR ANTA, BRAZIL
Transactions on Modelling and Simulation vol 4, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X
Computational Methods and Experimental Measurements 67
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
Transactions on Modelling and Simulation vol 4, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X
68 Computational Methods and Experimental Measurements
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
Transactions on Modelling and Simulation vol 4, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X
236.15
Computational Methods and Experimental Measurements 69
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
Transactions on Modelling and Simulation vol 4, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X
70 Computational Methods and Experimental Measurements
SECT ION-4
234 00
^ 233.50 -z:
^ 233 00o
^ 232.50 -
LUH 719 R R
231.50-
ccD CD Lf
/
D C1 CD CD I/
/^ /
D C1 C
/
S'
D CD U
/
D C-> c
r̂
D CD I/
^
D C1 CD CD I/D C1 CD GD Ifrsi in
TIMF (SECONDS) (*10**(+02))COMPUTED OBSERVED
Figure 8 - WATER STAGE HYDROGRAPHS - SPILLWAY GATE OPENING
.__ SECTION-4
234.50 -
234 00z:
z 233.50 -0
*~ 233 00>LU[j 232.50 -
oqo c\O(
23 1 50cc
_ _ _^
D CD I/
\ \\
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^
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T̂D C1 C—D CD If— -D C•> c
—D G:J IT
fNI ID fNJ Ox) CN) CN|
COMPUTEDTIM E (SECONDS) (*
OBSERVED
Figure 9 - WATER STAGE HYDROGRAPHS - SPILLWAY GATE CLOSING
Transactions on Modelling and Simulation vol 4, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X
Computational Methods and Experimental Measurements 71
S E C T I O N - 1OZD. WW -T
/ICQ C\O\
( D Q "~7 S £\ C\\COH,, qnn nn
LUCJrv-' o o cr (-»(•>a 225.00 -
LJr r) i cr r» r>r»
onc OQ
00CC
/
/ /
I'
V
D CD C
//
/
//
/
D CD C
/
/y
/
3 CD C
/
D C3 CD CD C3 C3 C3 CD C3 CD CD CD C0 CD C3 GD G
TIME (SECONDS) l*10**(+02))10 MIN 28 MIN 30 MIN
234.50
Figure 10 - RELEASED DISCHARGES AT THE SPILLWAY
SECT I ON-4
233.50 --
233.00
232.50 -
UJw 232.
231
231
5 0 - -
00-oG)
CDG) Oo GDCD CDCDLO CD CD LO CD LOrf \r iT> \D
TIME (SECONDS) (*10**(+02))10 MIN 20 MIN
Figure 11 - WATER LEVELS RESULTS
Transactions on Modelling and Simulation vol 4, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X
72 Computational Methods and Experimental Measurements
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).
Transactions on Modelling and Simulation vol 4, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X
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