A proposed method for Calculating Water Discharge and Head Losses in the Landsvirkjun Energy Management System

Abstract

In this memo a method for calculating the discharge through each turbine and the total water discharge through a power station will be presented. Head losses in turbines and waterways can be calculated simultaneously. The Búrfell power station is used an example when presenting the model used and the method in calculations.

Full text

A proposed method for Calculating Water Discharge and Head
Losses in the Landsvirkjun Energy Management System.
By Egill B. Hreinsson,
May 1986
,
.
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Below
a
method
for
calcuiating
the
discha
r
ge
through
each
turbine
and
the
·
total
water
discharge
.
through
a powe r s t
ation
will
be
presented.
Head
losses
in
turbines
and
wat
e
rways
can
be
.
calculated
simul
t
aneously.
The
Burfell
power
station
is
used
an
e x a
mp
l e
when
presenting
the
mQdel
used
and
the
method
in
calcula
t
ions.
II.
Inpu
t
da
t
a.
In
th
ese
calcula
t
io
n s
it
i s a s s umed
tha
t r e
al
po
we r
ou
t
pu
t
of
individual
g e n e
rators
is
av
ai
lable
as
measur
e me
nts
in
the
system
data
base.
Also
it
is
assumed
that
the
upper
water
level
(h
e
adwater)
is
available
as
well
as
the
lower
water
level
,
(tail
water)
.
Further
data
requirements
a'
re
the
characteristics
of
penstocks
.'
and
tunnels
to
enable
the
calcula
t
ions
of
losses.
As
an
option
the
ch
a
ract
e
ristics
of
the
approach
canal
and
tailrace
canal
are
ne
e d e
d,
if
it
is
desired
to
include
calculations
of
head
variation
and
th
e
associa
te
d
loss
e s
in
.
these
canals.
III.
Ou
tp
ut d
ata
.
In
these
calculations
th
e '
wat
e r
disch
a
rge
through
individual
turbin
e s
is
calcula
t
ed
and
the
total
discharge
th
r
ough
power
station
can
b e
calculated.
Head
los
se
s
are
also
'
calculated
and
a
br
e
akdown
is
available
betw
e
en
loss
e s
in
wa
t
erways
and
losses
in
the
turbine
itself.

IV.
Definition
of
basic
model.
.,
f
The
following
notation
serves
to
explain
the
proposed
m'
odel
and
method
used
in
the
calculation
of
water
discharge.
We
assume
for
the
sake
of
simplicity,
that
the
power
station
has
6
units
like
the
Burfellstation.
i An
index
for
the
turbine/generator
set.
Q1,Q2,.
Qi
..
Q6
Discharge
through
individual
turbines.
(in
cubic
m/sec)
P1,P2,.
Pi
..
P6
Power
'
output
of
individual
generators.
(in
MW)
QT=Q1+
...
+Q6
Total
station
discharge
PT=P1+
...
+P6
Total
station
power
output.
H1
,H2,.
Hi.
.H6
Net
head
or
difference
in
elevation
for
each
turbine
(in
meters)
.
r i
(Pi,Hi)
or
Efficency
curve
for
each
turbine.
The
efficiency
is
a
function
of
the
net
k
head
and
th
e
power
ou
t
put
of
the
turbine
itself.
The
efficiency
lies
between
a
and
1
and
can
be
stored
in
the
form
of
pr
edef
in
e d
tables.
A
constant
including
the
gravitatior.al
constant
and
the
specific
gravity
of
water.
The
following
relationship
is
·
valid
for
each
turbine
relating
power
output
and
water
·
discharge.
The
relationship
is
a
traditional
repres
e
ntation
of
hydro
e l
ectric
powe r
production
and
is
b
ase
d
on
physical
and
engineering
laws
commonly
us
ed
if'.
hydraulic
engin
ee
ring.
For
each
turbine/g
e
nerator
set
(numb
er
i)
Pi
=
ri(Pi,Hi)
k
Hi
Qi
.
(Eq
1)
where
Hi
the
net
head
for
the
turbine
Hi
= Hg -
Hdi
(Eq
2)

where
Hdi
represents
the
head
losses
for
the
'
turbine
and
Hg
· ,
. -
· \
t
i
i
""'\
..
il·
t
!
,
iI
i
r
"
;'
'I'
l:
,
\
It
I;
• ,
•
p
represents
the
gross
i.e.
the
difference
between
the
headwater
·
level
and
the
tailwater
level.
Hdi
1 2
..
=-
aQi
+
bQt
+ c(Q1+Q2+Q3)
(Eq
3)
represents
losses
in
pensto
,
cks,
partially
divided
between
total
water
discharge,
discharge
through
individual
turbines
and
discharge
through
' a
set
of
turbines
sharing
a
cOmlnon
tunnel
or
penstock.
(Assume
here
that
the
index
i
here
refers
to
turbine
nwnber
·
l,
. 2
or
3
in
the
case
of
Burfell)
.
Thus
in
depends
stations.
general,
the
.
weight
of
the
loss
a,b
and
c
upon
the
configuration
of
individual
power
For
example
when
each
turbin
'e
has
its
own
penstock
from
the
intake
to
the
tailwater,
the
loss
factors
b=c=O
in
the
above
formula.
Such
.
is
the
case
in
the
Landsv{rkjun
system
withSigalda
and
Hrauneyjafoss
hyd!electric
stations.
Alternatively,
·when
the
configUration
is
more
complex
and'
approach
tunnels
and
penstocks
are
divided
into
different
sections,
where
each
section
serves
a
set
of
turbiries,
as
is
the
case
with
the
Burfell
station,
the
factors
'
a,b
and
care
from
zero.
In
that
case
the
factor
"a"
represent
losses
in
a
penstock
dedicated
to
a
specific
turbine,
the
factor
"b"
represents
loss
e s ·
in
th
e
section
of
tunnels
shared
by
all
(6)
turbines
and
the
loss
factor
"c"
repr
e s e
nts
losses
in
the
section
of
(horizontal)
tunnel
shared
by
turbines
number
1,2
and
3.
(See
figure
I
for
the
configuration
of
the
Burfell
power
station:)
Thus
it
is
possible
with
the
above
formulation
to
model
the
head
losses
in
waterways
in
a
station
with
arbitrary
configuaration
·
of
approach
waterways
by
adding
appropriate
fa
c
tor
in
the
above
formula.

t
l
i
"
!
I
f
t
I
'\
t
;
v.
simplified
calculat
ions
.
The
above
set
of
equations
constitutes
a
set
of
nonlinear
.
of
the
following
form:
where
PI
fl(QI,Q2
....
=
P5 · =
fl(QI,Q2
•.•.
Q6,Hg)
P6 =
fl(QI,Q2
.•..
Q6,Hg)
the
set
QI,q2,
...
,Q6
variables.
The
solution
·f
or
(Eq 4)
represents
the
set
of
.w1known
QI
,Q2,
...
Q2
would
in
general
require
some
form
of
an
numerical
iterative
procedure
possibly
requiring
excessive
or
unacceptable
computation
·
time.
Using
.
predefined
tables
directly
would
be
·
unacceptible,
due
to
the
number
of
dimensions
(6).
In
principle
however
an
iterative
above,
.
could
be
implemented
in
System.
procedure,
as
m
en
tioned
the
Energy
Management
Howeyer,
in
order
to
facilitate
the
computation
of
water
discharge
(QI,Q2,
...
Q2)
in
as
simple
manner
as
possible,
the
following
alternative
set
of
equations
is
proposed,
using
an
approximation
to
avoid
any
iteration.
Qi
=
Pi/ri(Pi,Hi)
k
Hi
(Eq
5)
where
Hi,
th
e
net
(eff
ect
iv
e )
head
for
the
turbine
is
given
by:
Hi
= . Hg -
Hdi
(Eq
6)
and
the
head
losses
are:
Hdi
.1.
2.
"'2..
=
APl
+
BPt
+
C(PI+P2+P3)
(Eq
7)
The
factors
A,B,and
C
have
now
replac
e d
a,b
and
c
to
accounl

for
the
different
approximation.
Thus
in
the
calculation
of
the
head
losses
(Hdi)
the
approximation
involves
·
replacing
the
water
discharge
(Oi)
by
the
power
output
of
individual
genera
tor
s
(Pi).
Due
to
the
only
.
slightly'
.
honlinear
relationship
b
etween
Pi
and
Qi,this
introduces
· .
insignficant
errors,
as
the
following
discussion
explains
VI
Error
introduced
by
app
roxim
ation.
By
differ
entiating
the
equation
for
the
head
·
losses,
the
following
formula
·
is
obtained,
relating
percentage
error
in
to
percentage
error
in
head
losses
.
(Eq
8)
Since
the
relationship
betw
ee
n
Oi
and
Pi
from
a
linear
one
by
less
than
about
·S%,
in
the
operating
range
of
turbines,
the
error
indroduced
in
calculating
this
approximation
to
the
head
losses,
is
in
the
order
of
less
than
10%.
Furthermore,
since
the
head.
losses
themselves
are
in
the
order
of
less
than
10%
of
the
.
effective
head,
the
maximum
error
introduced
in
calculating
the
discharge
.
is
less
than
10%x10%
or
1
%.
This
error
can
be
compared
to
the
measurements
error
when
defining,
for
instanc
·
e,
the
turbine
efficiency
c':-!rve,·
and
by
.
tuning
the
factors
"A",
"B"
and
·
"C"
in
eq.
7,
.
the
error
should
be
negligible.
VII
Calculations
and
breakdown
of
losses.
Th
e
following
equations
giv
e a b r e a
kdown
of
losses,
b e
tween
losses
in
the
turbin
e
its
e
lf
(Pti)
,and
loss
es
in
waterways
(Pwi).
The
losses
are
represented
as
power
(MW)and
can
be
summed
to
obtain
the
total
losses
for
a
station.
The
maximum
ideal
.
power
output
from
a
turbine
assuming
r=100%
efficency
and
no
losses
in
waterways
equ
·
als:
Pmaxi
= k
Hg
Oi
(Eq 9)

' ..
but
the
real
powe r
output
is:
Pi
= r k
Qi
(Hg
Pi
= r k
Qi
Hg
The
total
lo
sse
s
would
the
n
equ
a
l:
Hdi)
(Eq
10)
r
kQi
Hdi
Pmaxi
-
Pi
= k
Qi
Hg
(l-r)
+ r k
Qi
,
Hdi
Thus
the
following
equation
sh
ows
the
lo
sse
s
in
turbine:
P
ti
' = k
Qi
Hg
(1
-
r)
(Eq
11)
while
th
e
lo
s
ses
in
wa
te
rways
are
given
by:
Pwi
= r k
Qi
Hdi
(Eq
1i)
The
ab
ov
e
lo
sses
percentagewise
can
b e c a
lcul
ate
d
by
the
formulas:
Pwi/Pi
and
Pti/P
i
(Eq
13)
,
VIII
An
algorithm
for
calculating
the
water
dis
c
harge
and
the
h e
ad
los
se
s.
,
Fin
a
lly,
le
t u s s ummari
ze
th
e
abov
e
discus
s
ion
in
an
,
algorithm
to
calcul
ate
both
th
e wa
te
r
disch
a
rge
and
h
ea
d
losses.
1.
Calcula
te
an
a
pproximation
to
t h e
head
l
osses
u
sin
g
equation
7.
2.
Calculate
the
wa
ter
d
isc
h
ar
ge u
sin
g
eqs.
6
and
5.
Use
pr
ede
fin
e d
2-dim
e
nsio
na
l
table
s
to
calculate
ri(Pi,Hi).
In
terp
olate,
using
li
near
in
te
rpolation,
betw
ee
n
the
valu
e s
giv
e n
in
th
e
tables.
3.
Ca l
cul
a
te
the
h
ead
l
osses
u
sing
e
q.
3.
I
.
1 I:
I . '
:
, f'
, I
: I
"
I!
II
-I
I;
I "I
" ,
ii
,

.
4.
Calculate
breakdown
of
losses
using
eqs
11
and
12.
IX
Summary
and
conclusions.
A
general
method
has
b
ee
n
presented
to
calculate
the
water
I
discharge
.
and
head
.
losses
in
a
hydroelectric
power
station.
To
avoid
the
need
for
an
iterative
·
procedure
' a
certain
aproximationwasintroduced
'
when.
calculating
the
water
discharge
involving
a
negligible
error.
2-dimensional
tables
.
should
:
by
stored
for
curve,
ri(Pi,Hi).
Predefined
the
turbine
.
1'
I, ,
I ·
,;
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i
, i
•
I

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