the reservoir: mechanistic model politecnico di milano nrmlec08 andrea castelletti
TRANSCRIPT
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The reservoir:mechanistic model
Politecnico di Milano
NRMNRMLec08Lec08
Andrea Castelletti
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Itaipu dam on the Parana river
Spillways in action
Dam
Penstocks and turbine hall
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Three Gorges dam (China)
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Typical Localization
Clan canyon damColorado river
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penstock
Longitudinal section
barrier
water surface level
storage
intake tower
intakes
minimum intake level
surface spillway
maximum storagebottom outlet
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Surface spillways
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Bottom outlet
Loch Lagghan damScozia
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surface spillway penstock
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Dam-gate structures
Gates: a) rising sector b) vertical rising c) radial
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Piave - S.Croce system
Hydropower reservoirs are often interconnected to form a network
Reservoir network
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Features of reservoirs
• the active (or live) storage;
• the global stage-discharge curve of the spillways;
• the stage-discharge curve of the intake tower;
By the management point of view a reservoir is characterized by:
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Stage-discharge curve (morning glory)
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Causal network
1ts 1t
r tu
1th
ts
1ta
st = storage volume at time t
at+1 = inflow volume in [t ,t+1)
rt+1 = effective release volume in [t , t+1)
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Causal network
tS
1tE1te
1ts 1t
a 1tr t
u
1th
ts
What is missed?
- evaporation
- r depends on a and e
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Mechanistic model
tS
1tE1te
1ts 1t
a 1tr t
u
1th
ts
What is missed?
- evaporation
- r depends on a and e
tt sSS surface
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Mechanistic model
tS
1tE1te
1ts 1t
a 1tr t
u
1th
ts
tt sSS surface
ttt sSeE 11 evaporation
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Mechanistic model
tS
1tE1te
1ts 1t
a 1tr t
u
1th
ts
ttt sSeE 11 evaporation
tt sSS surface
1111 ttttt rEass storage
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Mechanistic model
tS
1tE1te
1ts 1t
a 1tr t
u
1th
ts
ttt sSeE 11 evaporation
tt sSS surface
1111 ttttt rEass storage
tt shh level
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Mechanistic model
tS
1tE1te
1ts 1t
a 1tr t
u
1th
ts
ttt sSeE 11 evaporation
tt sSS surface
1111 ttttt rEass storage
tt shh level
111 ,,, tttttt EausRrrelease
11111 ,,, tttttttttt eausRsSeass
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Balance equation
Sean ttt 111 net inflow
11111 ,,, tttttttttt eausRsSeassbalance
111 ,, ttttttt nusRnss balance
111 ,, ttttttt nusRssnnet inflow estimator
Simplification: the storage is a cylinder S(st) = S
Pros
Cons
Using it when the storage is not actually a cylinder generates an error.
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Storage-level relationship
By inverting h(.) one obtains the value of the storage by measuring the level, i.e. the only measurable quantity
There exist a biunivocal relationship between the level measured in a point and the storage.
tt shh Implicit assumption:
the water surface is always horizzontal.
Example: if the storage is a cylinder
arbitary constant
A negative storage represents the missing volume required to bring the water surface up to the level corresponding to the zero storage.
infh s s S
infshSs
tt shh
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Storage-level relationship
By inverting h(.) one obtains the value of the storage by measuring the level, i.e. the only measurable quantity
There exist a biunivocal relationship between the level measured in a point and the storage.
tt shh Implicit assumption:
the water surface is always horizzontal.
tt shh
Non-cylindrical storage
batimetry of the reservoir (DEM)
The identification of h(.) can be performed in different ways, depending on which one of the following is known:
numerical computation point by point
interpolation tt sh ,historic time series
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Examples of storage-level relationships
Campotosto reservoir (Italy)
60,000
80,000
100,000
120,000
140,000
160,000
180,000
200,000
220,000
1304 1306 1308 1310 1312 1314 1316 1318Level [m a.s.l.]
Sto
rag
e[1
03m
3]
Historic series
Regression line
Campotosto reservoir (Italy)
60,000
80,000
100,000
120,000
140,000
160,000
180,000
200,000
220,000
1304 1306 1308 1310 1312 1314 1316 1318Level [m a.s.l.]
Sto
rag
e[1
03m
3]
Historic series
Regression line
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Examples of storage-level relationships
Piaganini reservoir (Italy)
0
100
200
300
400
500
600
700
800
387 388 389 390 391 392 393 394 395 396 397Level [m a.s.l.]
Sto
rag
e[1
03
m3]
Historic series
Piaganini reservoir (Italy)
0
100
200
300
400
500
600
700
800
387 388 389 390 391 392 393 394 395 396 397Level [m a.s.l.]
Sto
rag
e[1
03
m3]
Historic series
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Examples of storage-level relationships
Piaganini reservoir (Italy)
0
100
200
300
400
500
600
700
800
387 388 389 390 391 392 393 394 395 396 397Level [m a.s.l.]
Sro
rag
e[1
03
m3]
Data from 1988 to 1992
Data from 1993 to 2001
Data of February 19933rd February 1993
Piaganini reservoir (Italy)
0
100
200
300
400
500
600
700
800
387 388 389 390 391 392 393 394 395 396 397Level [m a.s.l.]
Sro
rag
e[1
03
m3]
Data from 1988 to 1992
Data from 1993 to 2001
Data of February 19933rd February 1993
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Surface-storage relationship t tS S s
Can be determined with similar techniques.
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Continuous-time model of reservoir
( ) , , ,ds t
a t i t s t e t S s t r t s t p tdt
s(t) = storage volume at time t [m3]
a(t) = inflow rate at time t [m3/s]
e(t) = evaporation per surface area unit at time t [m/s]
i(t,s(t)) = infiltration [m3/s]
S(s(t)) = surface area [m2] r(t,s(t),p(t)) = release when the dam gate are open of p [m3/s]
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It depends on the storage-discharge functions and the
position p of the intake sluice gates
It depends on the storage-discharge functions and the
position p of the intake sluice gates
( ) , , ,ds t
a t i t s t e t S s t r t s t p tdt
Simplifications
Cylindric reservoir n(t) = a(t)-e(t)S net inflow
i = 0 almost always true, at least in artificial reservoirs
n(t)
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Instantaneous storage-discharge relationships
• s min , s max : bounds of the regulation range• s* : storage at wich spillways are activated
s min s max s*
spillway
open gates
maximum release
minimum release
s(t)
limited
min,norN
maxN
max,norN
minN
max, maximum allowed release norN min, minimum allowed release norN
( )r t
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Example of storage-discharge relationship
flow
ra
te [
m3/s
]
storage [Mm3]
N max (•)
N min (•)
~
~
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Modelling for managing
The continuos-time model can not be used for managing:
• decision are taken in discrete time instants• data are not always collected continuously
discretize
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st+1 = st +nt+1 -rt+1
nt+1 = net inflow volume in [t ,t+1)
+nt+1
Discrete model of a reservoir
t t +1
nt+1= net inflow volume in [t , t+1)
nt+1
we assume it uniformly distributed
st = storage volume at time t
st
rt+1 = volume actually released in [t , t+1)
-rt+1
( )n t
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The release function
rt+1 = Rt (st ,nt+1 ,ut)
1
1max,, ,t t t
tnor
t
V s n N s d
Maximum dischargeable
volume in [t , t+1)
1
1min,, ,t t t
tnor
t
v s n N s d
Minimum dischargeable
volume in [t , t+1)
rt+1
nt+1
ut
ut
rt+1given st & ut
given st, & nt+1
Vt
vt
45°
release decision
with
min,( ) ( ) ( , ( ))
( ) [ , 1)
n r
t
os n s
s t s t t
N
max,( ) ( ) ( , ( ))
( ) [ , 1)
nor
t
s n s
s t s t t
N
with
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An example of minimum and maximum release (Campotosto, Italy)
n = 50 m3/s
n = 0 m3/s
flow
ra
te [
m3/s
]
storage [Mm3]
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Set of feasible controls U(st)
feasible control
U(st)flo
w r
ate
[m
3/s
]
storage [Mm3]
depends on the inflow!
1 1min( ) : ( , ) ( ma, x )t t tt tt t t t tU n ns u v s u V s 1 1min( ) : ( , ) ( ma, x )t t tt tt t t t tU n ns u v s u V s
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ut
st given
Vt (st,min{nt+1})
vt(st,min{nt+1})
rt+1
45°
Vt (st,max{nt+1})
vt(st,max{nt+1})
U(st)
Set of feasible controls U(st)
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Remarks
If nt+1 is known
Alternative ways of formulating the mass balance equation
rt+1 = ut
1 1
1
t t
t t
n rh h
S S
ht = level at time t rt+1 = Rt(st ,nt+1 ,ut) = Rt (ht ,nt+1 ,ut) actual release in [t , t +1)
1 1 1t t t th h n r nt+1= net inflow in terms of level
rt+1= actual release in terms of level
vt(st ,nt+1) rt+1 Vt(st ,nt+1)
vt(st ,nt+1) ut Vt(st ,n t+1)
constraint already included in Rt(•)
NO releases of interest)( ttt sUu
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CONCLUSIONS
1 1 1 1 1
1 1 1
1 1
( , , , )
( , , , )
( )
t t t t t t t t t t
t t t t t t
t t
t t
t t t
t t
s s a e S s R s u a e
r R s u a e
h h s
S S s
E e S s
u U s
Model of a reservoir in operation
outputs
feasible controls
state transition
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CONCLUSIONS
1 1 1 11
1 1 1
1 1
( , , , , ) if
0 if
( ,
>0
=0
, , , )
( , )
t t t t t t t t tt
t t t t t t
p p
p
p
p
p
t t
t t
t t t
t t
p
s a e S s R s u a es
r R s u a e
h h
u u
u
u
u
s
S S
u
s
E e S s
u U s
U
Model of a reservoir to be constructed
1 1( , , , , )t t t tp
tR u a es u
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Natural lakeh
e
s
a r
hmin
n t a t e t
r
s
(s - smin)
0 if ssmin
N(s) =
s t n t r t
N(s(t))
net or effective inflow
Stage-discharge function( ) ( ( ))r t N s t
smin
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min min
1 +
s s s s e n
t
e d
t
s t n t N s t (s - smin)
Remark: s(t+1) depends on s(t) only if = .
T is the so-called time constant of the reservor.
Linearization and time constant
r
ssmin
t t+1
Linear continuos system
( )s t As t Bn t
0
(0) t
A tAts t e s n e d
Lagrange formula
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Remark: s(t+1) depends on s(t) only if = .
T is the so-called time constant of the reservor.
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min min
1 +
s s s s e n
t
e d
t
s t n t N s t (s - smin)
Linearization and time constant
r
ssmin
t t+1
Meaning of T
By assuming =T=1/ one gets
T is the time required for the storage to reach 1/3 of its initial value.
1min min1s t s s t s e
min min
1 +
s s s s e n
t
e d
t
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min min
1 +
s s s s e n
t
e d
t
s t n t N s t (s - smin)
Linearization and time constant
r
ssmin
t t+1
An accurate modelling does require Shannon’s or Sampling theorem
An accurate modelling does require Shannon’s or Sampling theorem
Remark: s(t+1) depends on s(t) only if = .
T is the so-called time constant of the reservor.
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LAKE S T=
[km2] [daysi]
Maggiore 212.0 7.4Lugano 48.9 8.7Varese 15.0 34.7Alserio 1.5 8.0Pusiano 5.2 15.0Como 146.0 7.7Iseo 61.0 7.8Garda 370.0 86.6
The modelling time-step for lakes with T = 8 is about 1 day.
For ll these lakes the catchment areas is relatively small compared to the lake surface.
The outlet mouth has not yet reached an equilibrium condition.
For most of the lakes T is nearly 8 days.
Time constant and the Lombardy lakes
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dB1/T
• Incoming waves whose frequency is smaller than 1/T are not smoothed.
E.g.: flood waves from snow melt.
• Waves with frequency greater than 1/T are smoothed.E.g.: flood waves from storms.
Bode diagram
Buffering effecti.e. low-pass filtering
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The modelling time-step depends upon the state
Non-linear model: T is not defined
It would be useful to have models with a time-step that changes with s, however this is not possible with the algorithms nowadays available.
Solution: use models with different at different time steps.
changes with s
T changes with the value s around which the system is linearized
Linearization of the system
To be sure that the system is well
represented by the model: 0,1* T
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r
h
Comparison between two lakes
min
nh h
2 >1 12h h
12h t h t
Average level
2
2
1
10 0h h
.
min n t
h h hS S
1) ( )
2) 0
n t n.h
hmax
h
t
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Lake shore inhabitants
*
minmax 0t
n Ph t h h
S
Downstream users *
t
TPr t n e
S
happy with lake 2 ( T small)
Happy with lake 1 ( T big )
Comparison between two lakes:impulsive flood
CONFLICTCONFLICTh
t
r
t
*
min 0t
Tn Ph t h e t
S
Average levelAverage level*
min
nh
*
min
nh
Impulse response Impulse response
t
TPe
S
t
TPe
S
n
t
n*
P
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49
Lake shore population big
Downstream users small
Which compromise ?
Natural lake Regulated lake
Different stage-discharge
functions in different time
instants
Natural regime curve
Natural curve
Curves for different gate positions
r
h
Lake regulation
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Downstream users
Months
t
t
Lake popul.
h(t)
r
Lake regulation
t
t
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Readings
IPWRM.Theory Ch.5