Other componentscanals and diversions

Andrea CastellettiPolitecnico di Milano

NRMNRML13L13

2

Adriatic Sea

Fucino

VILLA VOMANO

PIAGANINI

PROVVIDENZA

CAMPOTOSTO

MONTORIO (M)

SAN GIACOMO (SG)

Irrigation District(CBN)

S. LUCIA (SL)

PROVVIDENZA (P)

3

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

4

• 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)

5

Example: the Po river

l1

l2

l3

l4

l5

tSEP OCT NOV

Hydrometric plots for 5 stations (li)

h

6

Canal: causal network

qt+1v

q

t+?m

q

t+?m

at+1

qt+1v

at+1

7

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

8

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

9

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

10

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

11

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.

12

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

13

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

14

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

15

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

⎧

⎨⎪

⎩⎪

16

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

!

17

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

18

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

19

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.

20

Diversion (dam): causal network

qt+1

m

qt+1d

ut

qt+1v

qt+1v

qt+1m

ut qt+1d

1mtq

1dtq

1vtq

21

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( )+

22

Adriatic Sea

Fucino

VILLA VOMANO

PIAGANINI

PROVVIDENZA

CAMPOTOSTO

MONTORIO (M)

SAN GIACOMO (SG)

Irrigation district(CBN)

S. LUCIA (SL)

PROVVIDENZA (P)

23

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

24

Piaganini

25

Adriatic Sea

Fucino

VILLA VOMANO

PIAGANINI

PROVVIDENZA

CAMPOTOSTO

MONTORIO (M)

SAN GIACOMO (SG)

Irrigation district(CBN)

S. LUCIA (SL)

PROVVIDENZA (P)

26Adriatic Sea

VILLA VOMANO

PROVVIDENZA

(M)

(P)

(SG)

Irrigation district(SL)

PIAGANINI

CAMPOTOSTO

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.

28

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.

29

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

30

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:

31

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

32

Reading

IPWRM.Theory Ch. 5