Raabe J. Hydro power - the design, use, and function of hydromechanical, hydraulic, and electrical еquipment
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Illserting this into the right h213d side of
(4.5-
15)
and then redllcing
10
31'
yields thc live
volun~c of
n
pumpcd sturage plant to covcr a certain peak load
I:,
1'
=
{P,,,
(1
-
2)
A,,,
!I
T/[(l
+
2)
il,,,]
1
'I2
.
4.5.2.4.
Econon~ical aspects
of
the
layout
of
a
yllrnpcd storage plant
The econon~ical background of
a
pumped storage plant is based on the fact that peak
load
can bc sold at
a
higher electricity ratc and hcnce a higher earning than the generation
of
the ~urnped storage needed for this peak load would cost at the low electricity rate of
base load.
During the period
t,
the flow
Qp
is pumped into an upper reservoir. Durinz the period
I,
the flow
QT
passes the turbine from the head water to the tailwater reservoir.
In
the following
,
target release, rainfall, seepage and evaporation are neglected. Hence
f
P
the volume pumped into the upper basin during the working cycle Vp
=
5
Qpdt has to
1
I'
0
be equal to the \~olume VT
=
QTdt
passing through the turbines during this cycle.
0
The instantaneous output or input power with respect to head
H,
discharge
Q
snd
q
whilst turbining (T) and whilst pumping
(P)
respectively, is
Pr
=
e
9
VTNT
QT,
PP
=
e
g
HP
Qplllp.
(4.5- 17)
With kc, and
li,.,.
as the electricity rates during turbining
(T)
and pumping
(P)
respec-
tively, the net balance per
jvorking cycle is obtained as
1
.,-
IP
S
=
J
pTkeTdt
-
J
pPkepdt.
(4.5- 18)
0 0
This surplus should be above a limit, which has to cover the
OMR
cost of the plant and
its annuity due to its initial cost. Including in the
efficiencies
also those of the penstock,
the motor-generator and the grid (as the domestic consumer
often pays the lion's share),
the surplus now reads
(AV=
1/,
=
V,,)
where the efficiencies with the prime are supposed to include all the losses within the
corresponding operating mode. From this it is seen, that the economy of
a
pumped
storage plant rises with
3)
Rising storage volu~nc
A
b!
b)
Rising efficiency due to turbining
17;.
c) Rising head clue to turbining
H,.
Usually the head is given with the topography in the
ncighbourhocd of the plant.
As
a rough rule it can be said that the product of herid and
volume in simple storage plants does not vary as much in nature as the head itself does.
d)
Rising ratio of electricity rate due to turbining to that due to pumping, ranging always
above
"onew.
The positive values rvitllin the bracket of (4.5- 19) lead to the essential criterion for the
economical feasibility of
a
pumped storage scheme:
ke,/keT
<
t1;q;.
Example:
q;
=
0,91,
1;
=
0,865. This gives kc,/kc,
<
0,787.
.vGlllcn,ann
treats the economic details of machines used in pumped storage systems
(4.201,
Ponicelli
IJ.?ll
their planning.
lisu.llly whcn making, as in thc above, profit the guide line of planning. the capitalist econorny
to the power
of
given conditions. However, in many pumped storage plants these conditions
;~rc
no
J)i!igcr co~~\itlcili?
.:I
pics~~tit. f:or
C,(;I~I?~>IC
tlic m;~.jorily
irf
p~~nli>cd stulagc plants in West
(~L,I.I~I;II~~. o!~ce
,)ti:
oi
111c i-ti~iiccrs
or
tliis
i~lc:~, :lrc clcgr;~clccl to
;III
ctliiBrgclicy s[;ilus bo
~ISCC!
ill
!Ii:
d:i~,.c
of
;I,X.~~IC:~~:II
t';~ilol.c
01'
OII~
of
llic
1)ig
~it~clc;~r [jo\ve-r ~I;!IIIS.
!\I
Ic;I,>~
in cfcvc.lo1~ctl colitl~
l.i(-;
rhc bcncfi~s
;i1\\1
possibility
vr
pi~rr~ping
by
~ncaris
of
chc:~]?
;lnd
cxct.ssi\.c b;:sc In;lil
S,CL'III
to (lis;il)~ci~r
marc
atid
marc.
This 11i.1y hc
;I
coliscqucnco of tlic rising
applii.;~tion
c:f
clcctricity to ~iisht lic;~ting, to opcratc chcmica! pl;~nts :iround tllc clock ctc.
Those
pl:ints
in
Lvhich punipcd storoyzc sets (binary or ternary ones) have beeti installed as supple.
mentary c(evices. ,~pply thcir sets ncarly cxclt~r;ivcly to turbining.
Countries like thc
ljSSK
\\.llich prob;ibly for ic!cological rc:!sons don't have diffcrcnt electricity rntcs
bccwec!i basc
and
peak lo;~tl strpply, ncvcrthclcss now install pumpcd storrlgc plants, probably as
a
quick substitute for
dcficicnt thcrmal.powcr plant.
'
4.6.
'The
problems
of
optimizing
the
cross
section
of
water
ways
4.6.
I.
General
remarks
Wdtcr
niay
mcnns here the structural mcrnber of
a
tiiversion or channel power plant,
rcql~irecl
to con?,ey ihe water from the upper reservoir or the intake structure to the power
house and from the
latter
to
the tail water of the river.
Strictly speaking tbe surgc tank rnust also be included into this consideration. Its main designs are:
a)
tiit
do~tblc ch:!niDcr design (Fig. 3.1.2),
5)
thc diffcrcntial surge tank wit11 tlirottlc (Fig. 3.1.2), c) the
surge tii~tk
\\it11
ii~ro:tlc (Fig. 3.1.3). the lattcr now mainly applied in r\ustri;i in the form ofThoma's
rcturn
!low
ihrottii:
[J.27!,
;11ic!
the Joli~ison design (Fig.
3.1.2),
which brings together the advantages
and
dciiccs of tile diffcrcntiai a~ld throttle clesigns, namely cutti~ig and damping
thc
arnplitildcs of
lcvel oscill;:tions In thc
\l.ir_re
tank.
In
remote arcas according to
Rrekke
(3.171 air chambers ex-
ca::;tted
in
the rock
arc
used to an incrcnsing degree.
The
d~mcnsiol~lng
sf
the
crabs
sectional arc3 of such
R
surge rclicf dcvicc clocs
1101
follow that of the
com~non \later
\\a?
but re5ults from cot~sideration of dynamic stability in tlic utmost critical cases
of
1c~u1:i:ion. St21 ring
\i
1111
tllc c;li
I/
\\fork of
Tlrottlrr.
\\
ho
mudc
the first s1:lternent of the minimum
cross sectional
area
of
a simple sllrge tank [I
.50].
much research
\\
ork
;inti
thcoretical considerations
have
becn published
ir?
this respect. from which the papers of
Serbrr
[4.23].
Tnglr.c~.l<c,r
(4.241,
Brekke
[3.17].
and tho book of
E.
.Closorlj.i
ahout high head plants
[1.54]
rnny be
mentioned.
For calculating
surge tiink fiiotion, see Cap.
11.3.8.
However, in the following. consideration
exclusively
concentrntes on the dimensioning
of
the cross sectional area of the channel, penstocks (or pressure shafts), tunnel or tailwater
gallery at stationary rated flow.
To
approxinlate real conditions,
a
differencr: is made bct\vcen full running closed ducts
such
as
pens:ocks, psessure shafts or tunnels
and
partly
filled,
unpressurized ducts such
3s c11atlnt.l~ or tail \\later galleries.
The basic pl inciplc nov, ;ipplicc! to ih~ optimum
dimensioning
of cross sectional are21 runs
as
follows:
Irnqine ,i g~\en rated
fl~nl
through the batcr
v:lp
For
3
ver) slnall available cross sectional area
the first cbst of thc nr,ltcr
\\.iIv
rchulting from rnatcrinls used, construction and ground acquisition
(if
needcd at
all).
is srn~ll.
As
then the velocity and the loss linked to it
are
high, the capitalized cost
due tc
it
durinp :he deprecintioi~ period is also high. On the other hand the initial cost iucreases witb
cross sectional area and the loss dependent cost dccreascs with incrcnscd cross sectional area.
Hence
the resulting cost must have
rt
minimum, which yields the optimum size
of
the water
way,
aJ
cilculated bclow.
d
4.6.2. The optimization
of
tllc channel sectiorl
4.6.2.1.
Thc rectangular channcl section as
a
model and
the
resulting loss
The real unpressurized tunnel
o:
open channel considered now may be trapezoidal,
horse-shoe-shaped or rectangular with real
wetted cross section A,, and a real wetted
U,,
14.251.
Hence the real hydraulic diameter is: d,,,
=
4
Are/Ure. The real
is
now
replaced by a rectangular channel with the same wetted cross sectio1131
area
A
=
A,, and hence the same mean flow velocity
c
=
Q/A for the given ratcd flow
Q
and with the same hydraulic diameter d,
=
dhr,, as the real channel.
he
imaginary rectangular channel now servinz as a model has
a
width
b
and a wetted
depth
d.
Hence the hydraulic diameter d, of the rectangular channel is d,
=
4
bd/(2
d
+
b).
Inserting
h
=
-4/d
gives d,
=
4
A/{2
d
+
Aid).
Making this equal the hydraulic diameter of
the real channel d,,.
=
4
Ar,/Ur,, and with
A
=
A,,, the following relation is obtained for
the wetted circumference Ure of the real channel as a function of the rectangular channel's
depth d: Ur,
=
2d
+
Aid.
Usually
A
=
Are is fixed for a given rated flow
Q
via continuity
A
=
Qlc, using
a
certain mean velocity c, which limits sedimentation or erosion. Once the
optimum depth of the rectangular model channel has been found as
do,
the above gives
the optimum of the real
cllannel's wetted periphery (perimeter)
Finding
do,
will be the following task. According to the above the hydraulic diameter of
the rectangular channe! becomes,
if the width is reduced to d by
O
=
A/d
in which
d
is the depth and
A
thc cross section of the channel.
4.6.2.2.
The
cash value of energy loss
Jn general the drop in level
11,.
of a channel resl~lts from
h,
=
1.
Lc2/(2
g
d,),
(4.6-
3)
in which
c
is the mean velocity in the channel,
L
its length, and
1.
its resistance coefficient.
The latter depznds, according to Bazin, on the wall roughness
k,
and the hydra~tlic
diameter
d,
as follows:
in
which the wall roughness,
k,,
has the following values as a function of the wall material
after
Kutter-
[3.18];
0,06 for planed wooden planks; 0.16 for unplaned wocden planks or
hewn stones (asher masonry); 0,47 for rock masonry wails; 1,3 for pure earthen mascjnry
of
regular structure; 1,75 for boulders.
For
a
hydro power plant the head loss from
(4.6-3)
must be treated like a head loss of
a machine, having efficiency
q.
Then the power loss results from the given rated flow
Q
as
P
=
10-3~QghL,rl
kW.
The
cash value of energy loss during the depreciation period of
z
years at the instant when
Operation starts becomes
in
\\
tiich
I;,,
is thc clcctrici
ty
ratc,
r
-=
I
ILI
"
the
reciprocal
of
the capitill recovt:ry factor
a
'
(see
(2.2
-
5)
and
(4.2-
3)).
cp
the lu;id factor
;IS
the
shnrc of n!lnunl hocrs
u~ider
rated
load,
y
thc dcnsity, a~iil
11
the overall efiicicncy of plant and grid.
inserting
\I,,
from
(4.6-
3)
with the hydraulic dianlctcr from (4.6-2) illto (4.6.-
5)
gives the
capitaliircd energy loss
4.6.2.3.
The
initial cost
of
the
ch;~nncl
The initial cost of a channel is assumed to consist of the follov~ing terms:
a)
K,, the cost
due to the wall materiai, excavation and construction,
b)
K,,
the cost for ground acquisi-
tion and
indernriities.
K, is assumed to be proportional to thc volume of the chiinnel material. Hence with
respect to
b
=
Ald:
K,
=
k;
(2
d
+
A14
Lt,
where
t
is the mean
wall
thic!<ness
of
the
channel and
k;
the specific cost
of
the
~;\11
material per unit volume including the
construction of an
excavaiion for the channel. The thickness
t
of channel wall is trtken
as
a
mean value. Assuming the channel wall acts as
a
gravity dam, the stability condition
bet ween the dam-weigh
t-induced- and the water-pressure-indilced tilting moment yields
t
-
d(~i~~)"',
Q,
being the dam's density. Hence
The cost
Kg
for the ground is assumed to be proportional to the channel's ground surface.
Thus
K,=
k2Lb
=
kzLA/d. (4.6-
8)
Strictly spellking
k,
is
split into
a
groiind rate and an indemnity rate.
1
4.6.2.4.
Thc
resulting cost
and
the
optimum
depth
J,,
1
The resilltinp cost of
a
channel from (4.6-6) thro~rgh (4.6-8) is
I
K,,,=C,+
C,/d+C,d+C,d2,
where
Co
=
k
A~L(Q/@~)~'~,
I
-
C,
=
kl
LA
+
(8,7618)
I.
Q
cp
rl
c
Lk, Q3/nZ,
(4.6-
12)
The condition
c7K,,,jiY
=
0,
gives the following third degree cquation for
the
optirnurn
depth
do,
=
d:
I
f
4
Introducing
d
=
x
-
e/3,
this reduces to
x3
+
3px
+
2q
=
0,
where
I
2
Obviously equation (4.6- 14) has only one real root. Hence the original case of
Cllrdano's
solution is given
[1.26].
With the discriminant
A
=qz
+p3,
160
tilt:
solution of (4.6- 14) reads
112
113
.dop=(-q+d
)
+(-q-~~1~)~/~-e/3.
Because of the topography, this proposcd optimization may be limited by site conditions,
e.g. a given limitation of the chailnel width
b.
4.6.3.
Tire
opti~llization
of
the
diameter
of
a
pressurized duct
4.6.3.1.
Tlie duct
with
cylindrical cross section as a model
For closed pressurized ducts, a pipe with a cylindrical cross section may serve as a model.
~ollo\i~ing the directions in Cap. 4.6.2.1, the model is assumed to have the same cross
sectional area A as the real duct with
A,,
and the same hydraulic diameter
d,
=
d
as the
real
dilct with d,,,. Hence the optimized wetted periphery of the real duct
may
be
&tained by the relation (4.6- 1).
4.6.3.2.
The
loss, its cash value, initial cost
Consider a developed turbulent pipe flow in a straight pipe of length
L
and diameter
d.
M'ith the mean velocity
c
depending on the given rated discharge
(2
and the pipe's cross
section A, the pressure
drop along this pipe reads (4.261 (with
I.
as the loss coefficient
mainly depending on the Reynolds number and the roughness, but occasionally also on
the fluid
[4.27; 4.28; 4.301)
following the considerations which lead to (4.6-5), the cash value of energy loss during
the depreciation period in the instant the plant starts operating reads now per unit
Iengtll
with
the meanings of symbols corresponding to those due to (4.6-5), except that
d
now
is
the
pipe diameter.
The first cost is proportional to the mass of pipe per unit length namely,
m
=
n
g,ds,
s
being
the wall thickness, assumed as small compared to
d.
s
results as follows from the
admissible tensile stress
a,,
in the longitudinal pipe section
s
=
ptil(2
a,,).
A
certain
minimum pipe wall thickness
s
is advisable [4.31]. By experiment,
Surber
[4.32] deter-
mined
s
for a lining of a pressure shaft.
Matt
[4.33]
considered prestressed walls
of
concrete. In the above,
p
is the maximum internal pressure inclusive water hammer.
With a specific cost factor
k,
that includes also transportation and erection, the pipe ccst
is k,~~71d~~/(2a~,),
Q,
being the density of the pipe wall material. The excavation of a
Pressure shaft or a tunnel but also that of
a
penstock within
a
ditch along a slope is
expressed by the
volulnc of a cylinder larger in diameter than the pipe
@(n/4)
d2
and a
Specific cost factor k3. Thus the initial cost of the pipe's unit length
4.6.3.3.
The
resulting cost and
optimum
diameter
From (4.6-19) and (4.6-20), the resulting cost of pipe
is
K,,,,=
~,d~
+
~~d-',
I:,
=
(S,70:'2)
(4,
r)'
)I
rp
2
k,,
Q3
1,.
(4.6
-
23)
The n~ininium cost rcsults from
ith',,,,/c1ll
=
0.
Hence the optiniilm pipe diameter
do,
=
[(5j2)
E,/l;,]
'
'
or
11
,,,,
=
1,414
{i.cp~];(r,
Q'
kt,/[@
k3/2
+
Q,
k,
p/~,,,]}117,
(4.6
-
24)
wilere
cp
is
the load factor
as
the share of annual hours under rated load,
rl
the overall
efficieilcy of the
set
and gid up to the terriiinals of the consumer,
.A
the factor due
to
interest ra:c
17
over tlie
J
years ol'dt.prcci;ltiorl, see
(4.2
-
3),
1.
the loss
coefficient
of the pipe
as
a
fiinction of Reynolds nuniber. preliminarily
fised
as
a
ftinction of
d.
and
relative pipe
wall
roughness (see Fig.
4.6.11,
A,,
tlie electricity rate,
k,
the cost per
m3
of excavated
vo1111nc due to tlie pipe,
k,
the specific
pipe
cost per
k_g
incli~sivc of tran~port and erection
but without escavatinli,
cD
the
ratio of excavated cross sectional area to that of the
pipe
n
t12/J,
and
19
the maxiniu~n internal pressure.
From
thc
lsst
rclatiori
is
wen.
thilt
the
uptimt~m
pipe diamctcr decreases
with
iucrcasing internal
pressure
p.
Hence for
heads
abovu about
500
rn
the penstock diameter
is
staged according
to
the
requirements
of
the
last relation.
TIIC
economic dinnietcr of a pipc is also treated by
Sarkarin
[4.34]. A survey of pc!lstocks, already
in
usc, is presentcd by
~\fuhlentnnn
[4.35].
1;or
peak
load plants, with frcqucnt water hammer-induccd stress changcs in thc pens:ock walls.
riltig~ic cffccts have to be accounted for, before determining the wall thickness. Any resonailce of
nl:lchinc-induced pressure fluctuations with thc natural frequency of the pcnstock should
be
avoidcd.
This holds good espcci;llly in the case of an outdoor penstock. Here resonance may cause consider-
abl~ noise, undesirnble in a mountain rccrention area.
A
rather extreme remedy against this consists
in wrapping
a
sound-proof blanket around the penstock. Another rcmedy could be avoiding reso-
nance by
means of
a
change in the penstock support.
4.7.
Problems
of
optimization of electric power transmission
4.7.1.
Historical
survey
Long range power transmission has proved the most reliable rneans for nearly one
century of harnessing hydro power in sites far from the consumer. Thus electric
power
transmission is one of the most essential prerequisites for developing the potential in
remote areas especially in developillg countries.
Ilistorically, hydroelectricity started with a hydroelectric plant in Wisconsin in
1883.
By generating
DC
this could only supply its immediate neighbourhood. Power supply to consulners in places
far
from the plant started with high voltage
AC
power transmission in
1891.
This was for a 3-phase
AC
Table
4.7.1.
Survey of actual high voltage transmissions.
Plant year Power Next Distance Voltage in
kV
System
River supplied consumer In km (between poles
Country in
MW
Town in case of DC,
Continent
(final Country delta voltage
value) in
case of AC)
Cabora Bassa
Sambesi
Mozambique
Africa
Churchill
Falls
Labrador
Canada
North America
LG
2,
La Grandc
Quebcc
Canada
North America
Itaipu
Parani
Paraguay
Brazil
South America
Pretoria
Union
of
South
Africa
Hpdro-
Quebec
Montreal
Canada
Hydro-
Quebec
Montreal
Canada
S5o Paulo
Brazil
High tension
DC
transmission.
Converted by
thyristor
AC transformed
1
51220
and
2201735
trans~nission of
220
kW
aricl
15
!<V
over
a
distancc
cjf
175
k~n
bctweci~ :Icilbrunn upon Ncckar
and
I'r;in!;ft~rt on blain in Wcsi Gcrmany.
Mcanwhilc for reasons discussed later, also
DC
high voltagc
transmission
by
1ne;ins of thyristor
converters
has proved to
bc
succcssfi~l for
distances
up to
1500
kni, using voltages up to
1200
kv
[4.36
to
4.381.
The Table 4.7.1 shows the recent development in this scope.
4.7.2.
The
cost
of
1
%
loss
in
relation. to
that
of power trar~slr~issio~i
Aswme the power ~tation operates
h
hours annually during
z
years of depreciation.
Without interest and with an electricity rate of
k,,
the monetary equivalent of 1% of
1
kW
installed is
It
z
k,.
P.ssume the initial cost per installed
kW
to be
k,
and the cost
of
electric power transnlission in proportion to
k,
to
be
yk,.
Hence the cost ratio
of
1
%
output (lost or gained) io the initial cost of the power transmission is
K,.
=
10-2
h
z
kc,,'(?
k,).
The compilation of terms duc to initial cost in 2.2.2 shows that
y
may reach the order
0,l.
in a special example:
k.
=
0,05
US
S/kWh;
k,
=
500
USS/kW;
3'
=
0,l;
z
=
40 years;
It
=
5000 hours of annual operation. Hence
KT
=
2. Accounting
for interest
wo!!ld yield a somewhat lower figure.
The ex;in?ple ciemonstratcs, that the monetary equivalent of
1
O/o
loss may correspond to the initial
cost of power transmission. Hencs
in the foliowi~ig, the design of any componcn: of the power
transmission line
sho:;ld
be
subjected to thc direction, that thc initial or first cost plus the monetary
equivalent of the loss
duc to this element are made a minimum.
3.7.3.
The optimization of
the
conductor
In
the following the overall cost of the transmission line treated in the above sense is
assumed to be proportional to that of the conductor by itself. The real condl-~ctor, usually
a multitude of cables, each a bundle of distinct wires,
]laving different functions e.g.,
as
carriers or conductors, is assumed to be equivalent to one conductor of
a
certain material
with
a
circular cross sectional area (see Fig. 4.7.1.3
a).
Its conductivity may be distributed
unitormly over the whole cross sectional area (no skin effectj. Moreover
DC
is assumed
io pass through the line. Hence only the ohmic resistance is accounted for, when losses
are considered.
Fig.
4.7.1.
Means of power transmission.
1
Schematic design
conductor,
2
insulator:
a,
vertical;
b,
diagonal,
3
conductor
of adjacent towers in plane
:
a,
idealized;
b,
implemen
~hc material cost of such
a
conductor with
a
length L, diameter
d,
consisting ofn ltlaterial
,.ith a density
Q
and a price
k,
per unit mass amounts to
Assuming an admitted voltage drop AU, a current
I,
an electricity rate k,, a load factor
as the share of annual hours at rated load. a factor
a
from (4.2-3) due to interest rate
and
z
years of depreciation, the cash value of loss at the instant the operation starts, is
K,
=
8,76 AU
I
a
cp
k,. (3.7
-
2)
TO
account for the rise of power transmitted during the life of the plant, a mean power
p
transmitted is presumed the following consideration.
The power
P
kW transmitted by the line with its voltage available U
--
AU, needs a
current
1
=
lo3
P/[U (1
-
u)]
to transmit the power
P
under an admitted relative voltase
drop
tr
=
AU/U, set preliminarily. According to Ohm's law for a conductor of length
L
in
m,
a
diameter d in mm and a specific resistance
e0
in Rmm2/m, the drop in voltage
becomes as function of the current
I:
AU
=
I
go
L/[(Tc/~)~'].
Inserting
I
=
103 P/[U (1
-
~i)] and then AU into (4.7-2) yields
\Vith the abbreviations
A
=
(n/4)
Q
Lk,
;
B
=
8,76(4/n)
lo6
p2
Leo
cp
3
k,/[Uyl
-
u)']
the
resulting cost of cable from
(4.7-
1)
and (4.7-3) takes the form
K
=
K,
+
K,
=
A
d2
+
~d-~. Obviously the diameter minimizing the cost is obtained from
dKl2d
=
0, giving
an optimum
do,
=
(8,76
-
106)1/4(4/7r)112 [,'q
a
k,/(e k1)]'l4 {P/[U
(1
-
tr)])lI2. (4.7-4)
Strictly speaking the relative voltage drop
14
in the above relation AU
=
1iU
=
(4!nj
-
lo3
Peo L/[U d2 (1
-
u)] follows from' the quadratic equation with the solution
(d
now do,)
u
=
(1/2)(1
-
(1
-
4fl1I2)
zf,
(4.7
-
5)
where
f
=
(4/n)
-
lo3
P
e0
L/(u~ d:,). Hence (4.7-4) and (4.7- 5) have to be used for
the
determination of and do,. Example: U
=
10'
V;
P
=
lo6
kW;
tl
=
0,015;
(p
=
0,457,
z
=
25 years,
p
=
0;
a
=
25; material: Copper with
e0
=
0,016 lo-' Rm;
L
=
1,5
10Gm;
k,
=
1,5
US
$/kg;
e
=
8300 kg/m3; electricity rate: k,
=
0,025
US
S/k\Vh. From
(4.7-4):
do,
=
0,048
m
G
48 mm. (For its rather large breaking length, the usual real
able has a core of steel wires and a conductor of alumi~lum wires.) From
ti
=
0,015 the
voltage drop
AU
=
15 000
V.
With the resistance
R
=
13,3 R the power loss becomes
AP
=
(AU2/R)
=
17
.
103 kW. The cash value of loss is K2
=
34,2. lob
US
S.
The
initial cost is
K,
=
33,8
-
lo6
US
%.
Note that K2 and
K,
are of same order! Halving the
cable diameter would reduce the initial cost to 25%. The cost of capitalized loss then
Would increase to
K,
=
136,8
-
10'
US
$.
B.J~
in this case the current density would attain a value which would heat and lengthen
the
conductor in an inadmissible manner, since the cable then might lower its prescribed
minimum distance from the ground to a dangerously slnall value. Therefore any layout
has
10
be based on a maximum permissible electric current density, which is reached at
'he
end of the depreciation period.
According
to
Cnusse
the
above
optimization
is
also practised in the layout'of underground
cables
i4.3g~T~o
typa
of cconolnic current
densities
are to be considered, one
with
cables being continu-
nuhly loaded during 111c
z
yc:lrs of tlcp~.cciatior~, t11c o~licr w~th mblcs witl~
.I
li?c;i~.Iy rising
current
dcnsity In thc lirst
cast
1,7
tti
2.1
A'rnm2 arc atl~t~ittcd,
in
:hc .,cconJ
casc
!hc
finill
dcllsitizs ;Ire
5,1
to
6.3
~/nim' dc\!encl~ng on thc cxtrclnc iluctu;~tions of the pricc of cvppcr.
Thus the econonlic use
of
underground concepts requircs th;lt, thc mnxinlunl temper;lture
of
the
conductor renchcs tlic upper limit psrmissiblc from the thcrcial point of iriew at thc cnd of
the
depreciation pcriod.
4.7.4.
Tl~e
distance
of
adjacent
to~vers
and
cable
geometry
If the cable does not transmit any bending moment and if its centre line always makes
a small angle with the horizontal (see Fig. 4.7.1.1), equilibrium in the vertical y-direction
requires
tlzy/dx2
=
q/H,
where
q
is the load per unit length of cable,
H
the horizontal
puU
in the cable.
With
a,,
denoting the permissible tensile stress in the latter, an admissible breaking length
of cable may be defined by A,,
=
o,,/(~
g).
Thereby the above simplified differential
eqaation
(dy1tl.u
zz
0)
reads
Integration with respect to the coordinate system (Fig.
4.7.1) and the vertex of the cable
ctlrvc in the mid point of adjacent towers, whose suspension points have the same
altitude, yields the well known parabola as an approximation to
the catenary [4.44]:
y
=
x2/(2
.A,,).
The sagging of the conductor with the chord between its suspension points
(Fig. 4.7.1.1) due to
the two following towers having a distance
'(1'
results as
t-Issuming towers with a vertical distance
h,
of the suspension points underneath the
top
(Fig.
4.7.1.1) and a vertical length
hi
of insulators, the height
It,,,
of the towers is obtained
as a function of the minimum distance
b
of the conductor from the ground:
h,,,
=
a2i(8
A,,)
+
b
+
hi
+
h,.
The distance
b
(Fig.
4.7.1)
depends on vegetation, climate and the fact, that people may work for a
longer pei-iod underneath
a
cable of a definite voltage. Hencc
b
may be prescribed by legislation.
In
some regions it must be taken into account that the spark-over distance between conductor and
ground
may be diminished
by
fires, made regularily e.g., during the harvest, within fields of sugar
cane to expel rats and snakes but also to facilitate the harvesting.
From the last mentioned relation, the distance between neighbouring towers 'a', when
h,,,,
b,
A,,,
h,
and
hi
are given: follows as
a
=
[8
(h,,
-
b
-
h,
-
hi)
A,,]
'I2.
1
Obviously
'a'
increases with breaking length
A,,.
This figure usually is enlarged
by
stee,
inlay
(st) within the proper conducior from aluminum (al). The admissible breaking
length of such
a
cable resl~lts from the admissible tensile strengthes
a,,,,
and
a,,,
O$
aluminum and steel respectively with densities
Q,,
and
Q,,
respectively by means of relatioak
Aad
=
099
(muad
a,
+
st)/
[g
!)near
+
e,,)
I,
(4.7
-
8
where
m
is
the ratio of cross sectional areas of the aluminum part
A,,
and
the steel
pa
A,,,
0,9 is a safety factor see [4.45]. For m
=
6,
a,,,,
=
7
-
lo7
Pa,
a,,,,
=
45
-
lo7
P
ear
=
2,75
lo3
kgjrn3,
Q,,
=
7,8
lo3
kg/m3.
Moreover
h,,,,
-
h,
-
hi
=
40 m
b
=
10
m.
Hence from (3.7-8
a):
,dad
=
3285
m,
and from (4.7-8):
a
=
888 m.