Raabe J. Hydro power - the design, use, and function of hydromechanical, hydraulic, and electrical еquipment
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4.3.5.
The
cost
of
ground
excavation, power house and dam
According to the assumptions. thc width of the river at site h, and the width of spillway
required for spillage of flood flow b,, diffcr by
a
figure b,,
=
b
-
b,,, corresponding to
the maximum length of power house located across the river. The plant considered needs
a
power house of a length h, smallcr than b,,. Hence the calculation is based on the
bp,n
=
b
-
bSp
(4.3
-
9)
In the general case of
a
Francis turbine with a runner throat diameter Dl (see Fig. 10.3.181,
the excavation of ground for the substructure of the power house has
a
depth
d,=k2D,- h,-dR,alength
L=
k,D,andawidthpersetb= k,D
=
k,(L>21L)1)Dl,kl
being defined after (4.2-
5),
h,
being the suction head,
d,
the depth of river bed underneath
the tail water level,
(D2/D,) a figure given from the head
H.
With a factor
rl/
<
1,
due to
the lateral variation of depth, the volume excavated of
i
sets beconles Ve
=
I)
k
k (D2/D
,)
!
3
-
i
D:
(k, Dl
-
11,
-
d,).
Inserting here Dl from (4.3-4), with k, being the
specific
cost per
volume excavated, the excavstion cost of
i
sets becomes
K,,
=
k,, Ve
=
E(k2 C
i-'I2
-
h,
-
d~)
,
(4.3-
1
1)
Since the task of the spillway is not affected by the number
i
of sets, which is the \rariab!e
of interest, the cost of the spillway is a constant, hence set aside.
The cost of the power house superstructure from (4.2-5)
may
be reduced to
the
throat
diameter
Dl by D
=
(D,/I),) Dl, where the diameter ratio D,il), is given by the head.
Hence
Kp
=
ik,k, k, (D,/D1)' Di
[H,
+
k, (D2/11,) Dl]
1;
Putting here
U,
from
(4.3-4)
gives,
Kp
=
:!
k, k,!, (D,/D,)' C2
[H,
+
k6(Dz/Dl)
C
i-"-].
From this only the
cost
term,
that varies with
i,
IS
taken from now on as cost due to the power house superstructure,
namely,
K,
=
I.',--
'12
7
(4.3
-
1
3)
The cost of the proper dam section (if any at all)
KD
is valid for a dam of length
h,,
resulting from b,
=
b
-
b,,
-
b,, in which b is the width of the river valley, b,, that of
spillway and
b, that occupied by the length of the power house. According to
(3.2-8)
bp
=
ik,
D.
Reducing
L)
to the throat diarneter by the diameter ratio D21D, which
is
known from
I-I
and putting
D,
from (4.3-4) gives the length of the power house as
b~
=
kl(D2/D1) Ci1I2.
Introducing the cross sectional area of the dam as
x
H2
with
x
as a forin factor, the
volume of the proper dam section is VD
=
x
H'
bD
=
x
H2
[b
-
bsy
-
kl (D2/D1) C ill2],
H
denoting the rated head. With the specific cost k, per m3 of the proper dam section, the
cost
of
this section becomes
Strictly
spcalting.
the bnrr;tg;ce c,~t~ be subdiviclcd into the proper barrage portion and into
a
p:lrt. whicll corrcspond~
in
ils
base to tl?c suhstructi~rc of tllc power hocsc (for heads
bclow
20
to
-10
m), or
whicli
contains tlic
pourer
housc (whcn
it
is within
the
d;~m), or
fi~;nlly, is
supported
dowiistrcarn
by
tfx
povJcr housc, but also pcrfor'lted by the pen-
stocks or
prcssure shafts. Attributing these elcmcnts to thc upper part of the power house,
for simplicity
it
is
assumed
here that the cost of this
dam
portion linked to the power
house is
chargcd to the cost of the powcr house.
'
4.3.6.
The
cost
of fabrication
and
erection of sets
The cost for f;~brication, transport and assembly
of
one set contains
a
term which is
proportional to Dn, where
n
=
2,3 to 2,5. This shows that part of this procedure is
proportion~~l to the face to be machined (D~) and another part is proportional to the
vc;lunie of inaterial to be bought and transported (D~). Introducing an empirical cost
factor
k,.
the hcad-linkcd diameter ratio D2/Dl and the throat diameter
D,
from (4.3-4)
give one cost term for
i
sets as
I,
=
ks(D2/Dl)"
C"
i-"I2
[4.6].
To this
a
tern: has to be added, proportional to thc number of units, namely
1,
=
k,,
i,
where
k,,
again
is
nn empirical cost factor. This term accounts for the fact that at least
;I
portior? of thc cost
due
to the machincs increases in consequence of the number of
accessories
!i.e., piping, valves, bearings, control mechanism, c:lsin_gs needed to encase the
pit,
scr~ll casing 2nd dralt tube wall) and
!he
number of large picces to be transported
and
to be rnacllined on nlachine tools, e.g. large boring mills or lathes, only av:iilable in
!irnited number and due to each of the
i
sets. Therefore the cosi of fabrication aid crection
of tile
i
scts is
4.3.7.
The
cash value
of
energy
loss
during
useful life.
I;
1s assumed that thc dinier?sionless loss here is composed
of
a portion
hb
independent
of the
size
D
of the set and another portion decreasing according to the known fi~nctional
dependence of loss and Reynolds number of the machine (see
Ccp.
9.3)
by
n
D-
'Irn
with
rising size
D
of set. Here
111
is about
5
and
'a'
about
0,3
to
0,7
accordiilg to tile type
of
machine. Reducing
I,
to the throat diameter
Dl
by thz head-!inked diameter ratio D,/D,
and Dl to
i
by
(4.3
-4)
gives thz dimensionless loss
The capitalized cash
vlllue of energy loss during the period of
z
years, for depreciation at
an
interest rate
p,
with the electricity rate
k,
per
kWh
for
a
hydro power station with
an
annual
oi~tput
Win kwh at the instant
when
power production starts, is with the interest-
linked
h~;;rr.ction
ct
from (4.2-3)
The annual work
I.V
produced follows from the rated discharge Q, and head
H,
the load
factor
q,
as the share
of
annuai horlrs undcr rated load, the
mean
resulting ei7iciency of
the generator and the grid cficicncy
II,,
(the ratio of consunled to generated output) by
1~
=
8776
o
v
11
Qr
g1-I
Vgr
.
4.3.8.
The resulting cost as
a
function of the nl~rnber of sets
i
For
m
=
5
and
II
=
2,4 the resulting first cost together with cash value of energy loss
during depreciation
K,,,
=
K,,
+
Kp
+
KD
+
Kb
+
1
is as function of set number
K,,,
=
E
(k,
Ci-'s5
-
h,
-
dR)
+
F
i-'p5
+
S
i-lv2
+
ksl
i
+
G
(b
-
b,,
-
~i'.~)
+
J
(hb
+
LiO.').
(4.3
-
23)
From this it is seen that the resulting cost of a plant with given rated head and discharge
decreases
with
a rising number of sets
i
in terms of: a) ground excavation due to depth
of the draft tube
k,
CE,
b)
the upper part of the power house
(I;),
c) the fabrication,
transportation and assembly of the sets
(S),
d) the dam part due to the power house
(G).
Contrary to the above the resulting cost increases with a rising number of sets
i:
e)
i11
consequence of the accessories and the processing due to each set
(k,,),
f) in consequence
of the then higher cash value of energy loss
(JL).
The optimum number of sets corre-
sponds to the minimum of
K,,,,
determined graphically or analytically and expressed in
integers. The usually observed predominance of the term due to e) makes
it
advisable to
set the
nr~mber of sets as small as possible. This has to be done within the frame of given
facilities due to the transportation and machining of the largest parts
everltually on site.
Cost estimates are discussed in
[4.13].
4.3.9.
Reasons
for
setting
i.knit
size
as
large as possible
In some cases the assumption (4.3-
10)
of sufficient space at site for the required pourer
house is not satisfied. With regard to growing peak load
demand in future and the profit
to
be
made by rising electricity rate for
peak
load, the power installed should often be as
large as possible. This raises the question of maximum power capacity that could be
installed under a given rated head
11
within a power house of a certain length
b,,
in a
valley of given width at a certain site.
According to (4.2-8) the length of a power house with
i
sets
of
the same diameter becomes
b,,
=
ik,
D.
With
P,,
as the head-linked unit power,
i
can he expressed by
yH,
D, and
the resulting power
installed
P,,,
as follows
i
=
P,,,/(P,,
D~
(LJH)~'~).
Hence the resulting
power as function of the runner diameter:
From this it is seen that for a given width of valley and hence given length of power house
across
it
b,,,, the installed capacity increases with the runner diameter. Escaping this
constraint in the case of given D and b,,, consists in splitting
the power house into
underground locations of the valley slopes, provided the geology and topography allow
this (Cabora Bassa).
Another solution is to arrange
thc power house at the
foot
of a forebay dam
(Fig.
4.3.1
d).
This has been done, e.g. in Grand Coulee
111
[4.14].
TIIC
spccific cost
of
cxc;~vation per unit o~ltput: The cost
of
eucn-;.?tins the power house
with
h,,
-
h,,,.
using
illc
I~c:ttl-li~lkc~.l di:~meter ratio Il,/D, nntl
the consiclerations
yicltiillg
(4.3-
1
I)
Key
=
I)
kt.,
k,(Dl/!12)
Dh,,,,[k, (Dl/D2)
1)
-
It,
-
dR].
Substitilting here
h,,,,
D
from
(4.3-24)
the specific cost
of
excavation per unit output is
Obviously
K:
rises with the runner diameter.
-
The spccific cost of
sets
due to fabric:\tion and erection per ui~it output: According to
the considerations
leading to
(4.3-
19), the cost for
the
L~brication and ercction of
i
sets
is
I
=
(k,/k,)ik,
D
D
Dm-'
+
(ksl/k,)ik, DDD-'. Substituting now from
(4.2-8)
i
k,
D
=
b,,
and
then expressing this from
(4.3-24)
gives the following specific cost per
urli
t
power
Sincc
11
-
2,3
to
2,5
and since the second term in the numerator us~l:illy is much bigger
than
the first, it is seen that the specific cost of fabrication and erection per unit power
installed falls with
a
rising runner diameter and decreasing number of sets
i.
The rexson for this results from the rising number of accessories, plankings and large parts, to be
ni;tchir!:d
on
mackinc tools, available in limited number, and the more complicuted risscmbly, repair
2nd
inrpcc~ion
duc
to a rising number of sets. Here thc feasibility of mass production vanishes.
In the important
<peed
range, a limiting alternator design shows
a
drop
of
output
Pa
vs speed
o
(Cap.
11.4)
about
Pa
-
Using
o,
-
D-'
and assumin2 the spiral casing to determine the lateral
disrun.:~
ef
adjacent
$et:,
the capacity of
a
power
house of length
b,,
bccomcs
c,,
-
D
b,,.
Hence
fcr
a
mnvimum capacity the alternator also requires
a
maxi~num possiblc
diameter
D.
Ob\
iott:;ly the size
of
a set should be set with respect to the power instclled in
tltc
grid. The arguments
iil fi~vour of
3
mi~imum nurntcr
of
sets ire also limited by the load gauge of access roads to the site,
by the
cu?aci:y
of cr~rlcs on site
and
in the workshop and by the
load
cap:~city of bridges to be
passcti. Such obstacles rvist niainly in rcmote
and
;nountainous
areas.
They can be ovcrcolnc to
a
certain degrce by applying fabrication of large
pieces
on site by
mcdern
welding toct~niqucs
(2.21).
4.4.
The
optimum
coefficient
of
peripheral
blade
speed
Ku
4.4.1.
Illtroduction
to
the
problem
From the similarity laws (Cap.
9.2)
it is known that the type number
n,
(here used in its
nondimensional form
n,
=
to
Q112/(gH)314) rises ro~ighly with the
1,5th
power of the blade
speed coefficient
Ku
=
ZI;(Z~H)''~ at the runner tip diameter
D
of
s
reaction turbine.
To
lixnit the submereence
n,
depends on the rated head
H
according to Tables
9.2.1
or
10.3.1.
It is known, that different vr?lues of
Ku
may be assigned to
a
certain type number
n,.
This
depends on the purpose
of
the machine (pump, pump turbine
or
turbine) but also on the
rotor blade form
choscn. For example backward curved vanes also applied in turbines
sive a higher
Ku.
Thcrefore the problem will b;c reflected by the following question: What
value
of
KI~
would yield the lowest resuliing cost when the rated head
H
iind discharge
of
the turbine are known and the rated point coincides with the bep with vanishing whirl
c,,
-
0
past the runner? Strictly speaking the answer should be given so as to minimize
tile
initial cost and the capitalized cash value of the loss during thc useful lifc of the
turbine, where the loss is split into its components.
This problem was treated by the author for a
FT
[4.15]. In the following an attenipt is
to determine
Ku by limiting the economic aspect to the following items: a) cost
K,
due to fabrication, crection and assembly of the turbine, see (4.2-13); b) cost due to
superstructure
of power house
K,,
see (4.2-5) of one set; c) the same due to excavation,
see
(4.2- 18); d) cost due to capitalized loss, see (4.3-20-22).
-
Simplifications in the cost terms: Assuming the term
H,
+
k,
D in (4.2-5) to be
proportional to D, the cost of the power house superstructure per set is now
k,,
being a specific cost factor per unit volume.
The volume excavated per turbine has, (Fig. 4.3.1 b) and
(10.3.18), the streamwise length
k,
Dl, see (4.2-7), the depth underneath the bed
of tailwater, see (4.2-6),
d,
=
(k,
Dl
-
h,
-
d,)
$,
(d,
being the depth of tailwater bed below the lowest tailwater level,
11,'the suction head) and a width b,
=
bl D, see (4.2-8). Using the head-linked diameter
.
ratio Dl/D2 and
cf,
=
$
k,
k, D2/Dl, this value may be expressed by the rotor tip dial~leter
D (D
=
D,) as follows
I/,
=
@
D2 [k, (Dl/D2) D
-
h,
-
d,], Dl being the throat diameter.
Expressing the suction head (Cap. 8.2.) by
h,
=
B
-
NPSH
with
B
from (4.2-9), and
splitting
NPSH
according to (4.2-10) (see also Cap. 8.2), the excavation cost reads
where
Kc,,
=
cl1,,/~, Ku,
=
1i1/J2SH, and
ke,
is a specific cost factor per znit
volume.
-
Steps, to reduce the runner diameter D to the coeficient Ktc:
a)
Rated head
H
and rated discharge Q are given at the bep.
b)
The nondimensional type number
n,
results from the working data
H
and
Q
in a)
according to the state of the art, Cap. 10.3, especially with respect to
n,(H) (see
Table
9.2.1).
c)
With
n,
known, the geometry of the runner and distributor with their essential features
is also known,
i.e.: b,/D,, D,/D, in which D, is the runner tip diameter at inlet (see
Table
9.2.1), b, the inlct breadth of the runner, Dl the runner throat diameter. In the
following for
convenience D,
=
D (without ifidex).
d)
With the reservations made in Cap. 4.3.1, the angular velocity
o
of the set results from
the type number
n,,
the head
H
and the rated flow by
w
=
n,(y~)~/~/Q"~.
With
o
known, the diameter
D
as a function of Ku follows via the relation
D
=
2
u/o
as
e)
From the known relation between n,,
Ku
and Kc,,,, (Cap. 9.2) at b,/D given
I1,=
2'14
n1I2
(h2/D2)lI2
KU
KC:,';
it is also possible to exprcss Kc,, in terms of the non-
dimensional
runner inlet breadth b,!D,, given according to
c)
and hence
as
ii
function of the
type
number
n,
and the speed coeflicient
Ktr
Kc,,
-
(r1,/l2)~
Kum2.
.
(4.4
-
6)
By
means
of
continuity and assuming constant mcridional velocity rcross the
flow
passages at runner entrance and exit and introducing the figure
;C
defined
below
ar,d also
known
from c), the coeflicient of mericlional velocity at the runner exit
Kc,,
may
be
expressed in terms of that at the runner inlet
Kc,,
as
(b,
=
rotor exit breadth)
Kc,,
=
x
Kcm2,
x
=
4 (D2lDl) (b2/b,)
-
4.4.3.
Expressing
the
cost terms as a function
of
Ku
-
The fabrication and erection cost of one set: They are with respect to
(4.2-13)
and
(4.4-3)
and
(4.4-4)
The cost of the power house superstructure of one set is according to
(4.4-1)
and
(4.4-3)
and
(4.4-4)
The cost of excavation of one set becomes from
(4.4-
2)
with
Kc,,
from
(4.4-
6),
(4.4-
7),
(4.4-3)
and
Ku,
=
(Dl/D2) Ku
K,,
=
I;,,
@
J2
Ku2
[k,
(D1/D2)
S
Ku
+
E,
Ku-'
+
E2 Ku2
-
B
-
dR],
where
(4.4
-
1
1)
-
The cost
cf
capitalized loss during the depreciation period: Following the consider-
ations leading to
(4.3-20),
the cash value of loss during the depreciation period at the
instant of starting energy production becomes with respect to
(4.3-4)
and with
D
from
-
(4.4-3)
K',
=
J(hb
+
a6-11m~u-'/m).
(4.4- 14)
4.4.4.
The
resulting cost of the
turbine during
its
useful
life
Adding the cost terms
Kc K,,
Kex
and
K:,
the
resulting cost
of
a
set during its depreci-
ation becomes as a function of
Ku
=
x
K,,,=J~~+C,X~+C,X~+-C~~"-C,X~+C-,~-~+C
-,,,
,x-'Im,
*
where
(4.4-
15)
2
The cost
K,,
has a constant term due to a portion of the capitalized loss
J
ht.
It contains
the term C, due to excavation, C, due to fabrication and erection. both rising with
Klr,
cn
due to fabrication and erection all rising with
Ku.
Then
K,,
contni~ls the terms
C;
and
C-2
both due to excavation and C-,,, due to capitalized loss all decreasing with Kir.
Hence
dK,,,/dx
=
0.
(4.4
-
22)
From
this follows an optimum of the speed coeficicnt
s
=
x,,
or
Ktr
=
Ku,,
.
According to experience this can be obtained rather easily after a first estimate by
Newton's approximation, e.g. for a
FT
with a head of 300
m
this is about
Ku
=
0,72.
More details are given in reference
I4.151.
4.5.
Problems
due
to
the
layout of
the reservoir
4.5.1.
Basic considerations concerning
a
reservoir
4.5.1.1.
Introduction
A
reservoir or basin as
a
water storing device
is
the necessary counterpart of a dam or
other damming devices such as dikes.
It
is the prerequisite of any peak load
and
11er.c~
storage or pumped storage plant, but
it
is also necessary for any run-of-river plant. As
such it may be arranged in a chain of river power
p!ants and thus be used simultaneously
as head and tailwater basin.
In
the following a reservoir in a narrover sense is understood as a basin used as head
water with a large, live storage volume and hence a considerable change in
levcl usually
generating
a
head in a range of at least 20
m.
Excluding the rare cases of reservoirs in natural caves or in abandoned mines, the
reservoir considered is located in
a
valley, usually that of
a
river. Its erection depends on
the tcpography and the geology of the neighbourhood, the water tightness of the ground,
the
economic facility of building dams as an enclosure.
Excluding
a
pump storage plant, any reservoir also needs
a
catchment area with an
adcquate precipitation rate.
The
bcnefits of a reservoir
in
its
widest sense can be briefly listed
as
~OIIOWS:
a)
Prevention
of floods espccislly
in
mountainous and monsoon areas.
b)
Irrigation especially in
rural
areas
with
irregular precipitation.
c)
Power generation especially for peak load demand or
in
a pumped storage plant.
Eqrlalizatiorl and regulation of run-off into a certain river.
Improvemcnt of navigation.
r)
Tourism.
t)
Fisheries.
?)
climatic
bencfits through damping temperature fluctuations.
'1
Social and cultural changes by enlarging social communications.
4.5.1.2.
~ss11111j):ioris
for
basic
relations
of
iilyout
In
111e followit;$
;I
rcservoir is co~~bidcrctl fillcd by
a
rivcr wit11
;I
1;irgc c;~lcii!ne~ll i1rt.a aiid
i1
know11 hydrc>grnl;Ii
Q(r)
Ibr the averagc year, which
ib
considcrccl as lhc opcrnti:~nal
cycle of the reservoir. Cilins by prcciyit;ltion on thc surhccs
of
thc rcservoir and losses
by seepage
act1 evaporation are
neglected.
TIie cycle for thc operation is given by thc fact, that thc altitude
11
of its level returns to
its initial value
/I,
after the period considercd. This value coincides with the deacl storage
of the reservoir
iletcrmining the minimum head of the turbines supplied by it. Thc turbine
type depends also on the maximum of this level, due to the ~naxi~num head. This results
from
tile topography of the site, the crest height of the bnrrrtfc and the flcjod storage
capacity eventually needed at the
instant of
:I,,,
.
The topogray;hy of thc reservoir also
gives
its
surface area
A
as a function of the altitude It of its level. Thus
A
-7
A
(11).
Far irrigatioli
znd
oiher purposes a minimum
flow
target releasc
Q,,
=
Q,(t)
must
be
delivered as
a
function of time
t
into the river downstreatn of the reservoir. Also
adequately timed, with respect, to power demands
P(t) (especially for peak load), to the
target release
Q,
(r), to thc hydrograph
Q
(t), and to the basin's geometry
A
(II), a flow rate
for power
gencration has to be delivered by thc reservoir:
Q,.
=
Qp(t).
Downstream of the reservoir a gauging station monitors the rivcr for
permissible
lebel.
This may be reached by appropriate regulating devices. TvIeasui-in? any altitude from this
level, specifically
the tail water level of the barrage can be assumed to follow a "head on
weir" flow
relation
in whicl~
k,
is a constant.
A
power station
may
he
at
the base of the barrage, receiving its discharge
Qp(t)
by a duct
separate from that for
QR
(t). This power sclpply rnay have
a
head lobs
k,
Qi(r). Then the
1;ct head is obtained from the instantaneous altitude h(t)
of
the reservoir as a function of
H (t)
=
h
(t)
-
k2
(Qp
(t)
+
QR
(t))2'3
-
Q;
(t)
4.5.1.3.
I'ile
basic
relation
for
the
layout
and
its
problems
LJnder the assumptions made,
h
(t) results from continuity
Counting the time
t
from the instant at which
h
=
]I,,
it follows that
The
turbine output required may
be
given as a function of time as
P
=
P
(I).
P
(I) depends
on the net head
H
(rj, the power-linked flow rate
Qp(t)
and tile effiuretlcy
11
of the turbines.
Joint regulation of all the sets
may
facilitate operation of
all
the
sets
around their
bep.
Thus
efficiency depends only on the head according to
?I
=
r;
(H).
Hence the power-linked
flow
QP
(t)
=
P
(?)I
[Q
6'
H
(t)
11
(H (t))l,
.
(4.5
-
5)
where H(r) results from (4.5-2) and (4.5-4). One problem consists in fixing the live
hm,,
storage volume according to
AV
=
1
A(h)dll.
If flood storage is also needed and the
ho
flood is considered, as usual, as a random elTect not belonging to the average year, the
level
/I,,,,,
rnust also allow for additional flood storage whose volume then may be added
as a benefit to
Q,
or (and) Qp.
The so!utioil of
d
1'
for a given geometric parameter
A
(11)
of the valley, a given hydro-
Q
(t),
kz and k,, can be reached only by trial and error.
The inverse problem of setting Qn
(t)
and
P
(t)
or Qp(t) for a given live storage volume
d
V
can also be reached only by trial and error.
Therefore in the following
section an approximate treatment of the problem will be
discussed. In this treatment the figure Q
(r)
-
Qn
(t)
is replaced by a power flow duration
curve
Qp(t) of simple form. The function
A(h)
is replaced by a mean surface area of the
reservoir, H by a mean head
H,,
and the power requirement
P
(r)
by
a
simple sinusoidal
power duration curve of daily peak load demand.
The optimization and adaptation of a reservoir has been treated in detail by
Serls
[4.16;
4.171.
4.5.2.
An approach to the layout
of
a peak load storage plant
4.5.2.1.
Basic relations of the'reservoir
and
the river feeding
it
Instead of the compatibility of the momcntasy level of reservoir
h(t),
and the power-
linked flow rate
Qp(t) within the permissible limits of the level and the momentarily
available head, it is now assumed that at constant
efiiciency, the stored electric potential
energy within one storing cycle equals the electric energy demand, which is generated
during this time interval.
The live storage volume is approximated by the minimum and the maximum
head
Hmin
and
H,,,,,
respectively, during the storing cycle and the average surface area
of
the
reservoir
A,
by
The electric energy of the stored volume is approximately with an overall efficiency
q,
and
a mean head H,
=
(H
,,,,,
+
Hmin)/2
E,,
=
Q
A
VgH,
'lm
=
(112) q,,
8
e
A,,
Hi.,
(1
-
x2).
(4.5
-
8)
It
is assumed that the reservoir has to supply this energy during a definite time interval
11.
e.g. one year.
A
river used to fill the reservoir may have the following flow duration curve for the
average year
in
which Qm and Q, are the maximum and minimum "flow remaining" respectively within
'he
average year. The "flow remaining" means here the surplus of flow over the legally
Prescribed minimum flow-off, the so called minimum target release.
The
live storage volume
d
V
of the reservoir has to be released into the river during the
"me
illterval
t,
(now one year). Expressing the surplus of seepage and evaporation over
'
!
rainfall (see
14.18;
4.191)
by
the tcrm
E:,
continuity gives
3
Qdt
=
V,.
-I-
d
V.
Inserting
(4.5--9)
arid
(4.5-6)
results in
o
t
(Qm
+
2
Q0)i3
=
'
+
A,,,
H,,,,
(1
-
XI.
(4.5-
lo)
4.5.2.2.
Load
demand
;ind
its
balance wit11 energy stored
The demand of the consumer may be approximated by the following periodical function
over the daily time
intzrvnl
T
and with
P,,,
as peak load:
P
=
P,,,
[I
-
cos
(2n t/T)]/2.
Hence the ilaily energy consumption
Ed
=
P,
T/2.
Imagine this is needed during 11 days
of the average year then the annual power demand becomes
Coiltrary to
a
run-of-river plant, delivering base load, the power capacity
P,,
is indepen-
dent of the
"flow
remaining" offered by the river. The energy
E,,,
from
(4.5-8)
stored
during
the period
t,
has to be equated to the simultaneous energy deinand
E,
(4.5-11).
Hence
e
LI
A,
Hiax
(1
-
x2)
=
h
Pm
T.
(4.5- 12)
The
elimination of the mean area
A,
from
(4.5-
10)
results in the peak load as a f~~nction
of the river
parameters
Q,,,
Q,,
the
max
head
H,,,,,
the head ratio
X,
the time parameters
i,,
h
and the surplus
V,
of seepage and evaporation over precipitation:
H,,,,,
is eiven
by
the topography,
Q,,,
Q, by the flow duration curve of thz river, while
q,
and
1
=
H,,,i,~H,,,,
depend mainly on the design of the turbine. Hence a given
Hn,i,/Hmx
indicates the
type
of turbine to be installed.
When
the
annual consu~nption period
h
T
and the peak load are prescribed by the
consumers then only the turbine design conditioned by the head ratio
;(
may help to
satisfy
the
last relation.
4.5.2.3.
Storage volume required,
also
for
pumped
storage
Squaring
(4.5--13)
and substituting
H&,
from
(4.5- 12)
gives the peak load as
a
function
of the hydrological
dzta Q,, Q,, the time parameters
h,
t,,
the head ratio
2
and the
surface
A,,
of reservoir required:
Pm
=
e
s
vm
[(Qm
+
2
QO)
t1/3
-
K12
(1
+
~)/[(1
-
X)~~,~TI.
(4.5-
14)
Changing the places of
Pm
and
A,
and then multiplying by the head range
Hma,(l
-
X)
yields the live storage volume
AV
4
The lasi relation could also be applied to a mere pumped storage scheme, whose livei
storage volume circulates e.g., daily. Her.e the values Qo and
Qm
may vanish, as only
a
srnall water source is needed to fill the system and cover seepage and evaporatio
Remember that
'V,
in the above relation was the surplus of seepage and evaporation ov
rainfall. Hence
V;.
Can be replaced
by
a
negative live storage volume, if the latter
imagined to be filled by precipitation only. Hence
A
V
=
-
V,.