other components canals and diversions andrea castelletti politecnico di milano nrml13
TRANSCRIPT
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Other componentscanals and diversions
Andrea CastellettiPolitecnico di Milano
NRMNRML13L13
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Adriatic Sea
Fucino
VILLA VOMANO
PIAGANINI
PROVVIDENZA
CAMPOTOSTO
MONTORIO (M)
SAN GIACOMO (SG)
Irrigation District(CBN)
S. LUCIA (SL)
PROVVIDENZA (P)
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Canal
• the peak’s propagation velocity w is greater than the average velocity v;
• the difference (w-v) increases with the depth H of the stream.
space
inst
anta
neou
s flo
w
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• The peak time increases with the distance;
• The peak flow decreases with the distance;
• Hydrographs are a-symmetrical and widen;
Canal: storing effectsec. 1
sec. 2
sec. 3
control sections
elementary unit
time
inst
anta
neou
s flo
wsto
ring ef
fect
(flow buffe
ring)
storin
g effec
t
(flow buffe
ring)
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Example: the Po river
l1
l2
l3
l4
l5
tSEP OCT NOV
Hydrometric plots for 5 stations (li)
h
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Canal: causal network
qt+1v
q
t+?m
q
t+?m
at+1
qt+1v
at+1
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Canal: mechanistic model
qt+1v
q
t+1v =qt−τ +1
m + at+1
Travel time
x
t=qt
m qt−1m ... qt−τ +1
m Tstate
xt+1 =
01M0
00M0
..
..
..
..
..
..
00M1
00M0
xt +
10M0
qt+1m
q
t+1v = 0 0 ... 0 1 xt + at+1
internal representation
plug-flow
at+1
q
t+?m
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Canal: the delay τ
qt+1v
q
t−τ +1m
q
t+1v =qt−τ +1
m + at+1
If τ = 0 the system is a non-dynamic one: the state does not exist.
To reduce the computing time in solving the design problem, the more convenient solution would be to fix in a way that τ be equal to zero.
But how to determine τ ? ....
plug-flow
at+1
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Canal: how to determine τIf one is able to observe a flood wave ..
… but if this is not possible?
use the cross-correlogram
τ
τ
t
t
qtm
qtv
t
t qt
v
qtm
upstream
downstream
τ
computed using whitened series
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correlation ρcorrelation ρ ρxy is a statistics of x and y measuring
the strenghtness of the link between x
and y
if x = α y → |ρxy| = 1
Correlation
−1< xy <1if x = α y + εwhite → |ρxy| < 1
if x = εwhite → ρxy = 0
provides an estimate of
rxy=
(xt −μx)∑ (yt −μy)
(xt −μx)∑⎡⎣ ⎤⎦2
(yt −μy)∑⎡⎣ ⎤⎦2 xy
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yt yt+τ
τ( )
τ
(Self)correlogram
It measures the correlation of the pair
1
… separated by different time intervals …… separated by different time intervals …
ττ
Pairs of variablesPairs of variables
( yt , yt+τ ) as a function of τ :( yt , yt+τ ) as a function of τ :
… of which we are interested in the strenghtness of the link.
… of which we are interested in the strenghtness of the link.
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1ta 1ta
Canal: leakage
qt+1v
q
t−τ +1m
q
t+1v =qt−τ +1
m + at+1
If the leakage does not change with the time
q
t+1v =qt−τ +1
m −at+1
q
t+1v =(1−α) qt−τ +1
m
If the leakage changes with the time
In this way is never negative even for very small value of the entring flow.
qt+1v
plug-flow
−αqt−τ +1
m
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Detention areas
Structural interventions that create a storage upstream where part of the inflow is retained when the flow rate is partuclarly high. They can be of 3 types::
• detention areas
Produce a narrowing of the riverbed
Produce an increase in the canal section when the flow is above a given value
They can be modeled as the aggregation of two components:
a reservoir and a canal
• detention basin
• dry dams
ht+1c
hts
at+1 h
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Detention areas
qt+1m
qt+1v
canal at+1
reservoir
st
qt+1v =qt+1
m −a qt+1m ,q,st( )
st+1 =st + a qt+1m ,q,st( )
If travel times can be neglected
Recession phase
ht+1c ht
s
at+1 h
Concentration phase
ht+1c
hts
at+1 h
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Detention areas
qt+1m
qt+1v
canal at+1
reservoir
st
By assuming that:
• stage-discharge curve of the canal is linear
, vale a dire ; qt+1
m > q
• the reservoir is cylindirc, i.e. ; st
=βhts
• The stage-discharge curve between the canal and the reservois is linear in the difference of the levels
qt+1v =qt+1
m + a qt+1m ,q,st( )
st+1 =st −a qt+1m ,q,st( )
at+1 =a qt+1
m ,q,st( ) =
0 if st=0 e q
t+1
m≤q
γq
t+1
m−q
α−
st
β
⎛⎝⎜
⎞⎠⎟ otherwise
⎧
⎨⎪
⎩⎪
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The model of a planned canal
If the canal is going to be planned its model should include up.
Each value of up correspond to a different alternative.
Typical situation: the canal has to be sized
In that case up is the maximum flow conveyable into the canal
q
t+1v =min qt+1
m ,up{ }
up = 0 is the business as usual alternative: do not do nothing
!
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Step-indicator of a canal
A step indicator is often associated to the canal
g
t+1 =gt qt+1m( )
For example:
• the damage produced by floods along the canal shores
• the environmental cost due to low flow rates
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Diversion (dam): structure
A branch point is usually an artificial work called diversion dam.
back-flow profile
spillway crest
bank of the water course bank of the
water course
inlet
dam
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qmax
Diversion (dam)
Features:
• entirely or partly channels the flow into a diversion canal
• can be equipped with mobile parts (usually sluice gates) for regulating the channelled flow.
riverbed
canal
• the diversion canal flow rate (qmax) is limited thorugh a crest spillways.
A branch point is usually an artificial work called diversion dam.
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Diversion (dam): causal network
qt+1
m
qt+1d
ut
qt+1v
qt+1v
qt+1m
ut qt+1d
1mtq
1dtq
1vtq
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Diversion (dam): mechanistic model
Non-regulated diversion
Regulated diversion:
qt+1d =min ut ,qt+1
m ,qmax⎡⎣ ⎤⎦
qt+1v =qt+1
m −qt+1d
⎧⎨⎪
⎩⎪
qt+1d =min qt+1
m ,qmax⎡⎣ ⎤⎦
qt+1v =qt+1
m −qt+1d
⎧⎨⎪
⎩⎪
q
t+1d =min ut ,(qt+1
m −qtMEF )+ ,qmax
⎡⎣ ⎤⎦
… diversion with a MEF:
q
max
qt+1m
qt+1d
qt+1m
qt+1d
q
max
ut
qt+1d ≠0 ut > 0 and
(qt+1m −qt
MEF ) > 0
only if:
−qt
DMV( )+
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Adriatic Sea
Fucino
VILLA VOMANO
PIAGANINI
PROVVIDENZA
CAMPOTOSTO
MONTORIO (M)
SAN GIACOMO (SG)
Irrigation district(CBN)
S. LUCIA (SL)
PROVVIDENZA (P)
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Features of the reservoirs
4.954950 000Piaganini
5.5851 690 000Provvidenza
975.461.8217 000 000Campotosto
Ts [hours]
qmax [m3/sec]Vactive [m3]
T
s=
Vactive
qmaxtime for
emptying
3.530380 000V. Vomano
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Piaganini
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Adriatic Sea
Fucino
VILLA VOMANO
PIAGANINI
PROVVIDENZA
CAMPOTOSTO
MONTORIO (M)
SAN GIACOMO (SG)
Irrigation district(CBN)
S. LUCIA (SL)
PROVVIDENZA (P)
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26Adriatic Sea
VILLA VOMANO
PROVVIDENZA
(M)
(P)
(SG)
Irrigation district(SL)
PIAGANINI
CAMPOTOSTO
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27Adriatic Sea
VILLA VOMANO
PROVVIDENZA
(M)
(P)
(SG)
Irrigationdistrict(SL)
PIAGANINI
CAMPOTOSTO
Ppumping
SGpumping
Problems:
• only ENEL is interested in the internal water cycling;
• a daily modelling time step is too large to accurately describe the phenomenon.
Problems:
• only ENEL is interested in the internal water cycling;
• a daily modelling time step is too large to accurately describe the phenomenon.
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P
SG
M
SL
DMV Fucino
MEF Vomano
PIAGANINI
CAMPOTOSTO PROVVIDENZA
VILLA VOMANODistretto irriguo(CBN)
P_pomp
SG+P_pomp
Acquedotto del Ruzzo
DMV Montorio
Schema logico corretto
Advantages:
• only the minimun value of release and pumping are decided, while ENEL is let free to increase these value to cope with the availability/demand of the national grid.
Advantages:
• only the minimun value of release and pumping are decided, while ENEL is let free to increase these value to cope with the availability/demand of the national grid.
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Pumping:
u2 ≤pMAX
SG
u1 +u2 ≤pMAX
Pr
Hydroelectric constraints
P
SG
M
SL
MEF2 Fucino
MEF1 Vomano
Irrigation district(CBN)
P_pump
SG+P_pump
Ruzzo Water Works
MEF Montorio
u2
u1
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Confluence point
The model of a confluence point is a simple algebraic expression.
q
t+1v = qt+1
m,i
i=1
n
∑
qt+1m,1
qt+1m,2
qt+1m,3
qt+1v
Being i=1,...,n in coming canals, the model has the following form:
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P
SG
M
SL
MEF Fucino
MEF Vomano
PIAGANINI
CAMPOTOSTO PROVVIDENZA
VILLA VOMANOIrrigation district(CBN)
P_pump
SG+P_pump
Ruzzo water works
MEF Montorio
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Reading
IPWRM.Theory Ch. 5