Systems, domains and causal networks Andrea Castelletti Politecnico di Milano NRML06

Systems, domains and causal networks

Andrea CastellettiPolitecnico di Milano

NRMNRML06L06

2

2. Conceptualisation

Sta

keh

old

ers

1. Reconnaissance

Defining Actions

(measures)

Identifying the Model

Defining Criteria and

Indicators

3Adriatic Sea

Fucino

VILLA VOMANO

PIAGANINI

PROVVIDENZA

CAMPOTOSTO

MONTORIO (M)

Gronda 1100 m.

PROVVIDENZA (P)

SAN GIACOMO (SG)Right interceptor 400 m.

Interceptors 1350 m.

Left interceptor 400 m.

Water works

Irrigation district(CBN)

S. LUCIA (SL)

Chiarino

Vomano

Physical scheme of the system

ComponentComponent: modelling elementary unit.

Every component has a specific function.

The model of the component must describe such a fuction.

Logical components are also allowed.

ComponentComponent: modelling elementary unit.

Every component has a specific function.

The model of the component must describe such a fuction.

Logical components are also allowed.

Choosing the components depends on:

• relevance of the component to the objective of the modelling exercise

• data availability

Choosing the components depends on:

• relevance of the component to the objective of the modelling exercise

• data availability

4

Identifying the Model

Definining the components and the system scheme

Identifying the models of the components Aggregated model

5Adriatic Sea

Fucino

VILLA VOMANO

PIAGANINI

PROVVIDENZA

CAMPOTOSTO

MONTORIO (M)

Interceptor 1100 m.

PROVVIDENZA (P)

SAN GIACOMO (SG)Right interceptor 400 m.

Interceptors 1350 m.

Left interceptor 400 m.

Water works

Irrigation District(CBN)

S. LUCIA (SL)

Chiarino

Vomano

6

Data analysis: time series provided by Enel

Campotosto: • level• aggregated daily flow rate the two intereceptors

Piaganini and Provvidenza: • level• daily flow rate from mass balance

Fucino

VILLA VOMANO

PIAGANINI

PROVVIDENZA

CAMPOTOSTO

MONTORIO (M)

Interceptor 1100 m.

PROVVIDENZA (P)

SAN GIACOMO (SG)

Right interceptor 400 m.

Interceptors 1350 m.

Left interceptors 400 m.

Waterworks

Irrigation district(CBN)

S. LUCIA (SL)

Chiarino

Vomano

During night-time without pumping

e.g. Provvidenza:

only aggregated flow dataonly aggregated flow data

7Adriatic Sea

Fucino

VILLA VOMANO

PIAGANINI

PROVVIDENZA

CAMPOTOSTO

MONTORIO (M)

Interceptor 1100 m.

PROVVIDENZA (P)

SAN GIACOMO (SG)Right interceptor 400 m.

Interceptors 1350 m.

Left interceptor 400 m.

Water works

Irrigation District(CBN)

S. LUCIA (SL)

Chiarino

Vomano

8Adriatic Sea

Fucino

VILLA VOMANO

PIAGANINI

PROVVIDENZA

CAMPOTOSTO

MONTORIO (M)

SAN GIACOMO (SG)

Irrigation district (CBN)

S. LUCIA (SL)

PROVVIDENZA (P)

Water works ???

Water works ???

9

Some difficulties in the scheme

1. PiaganiniPiaganini: there is no way to compute the indicator for the water works

?

Average water supply from hydropower reservoirsWW

10

How to solve them…

We need to fix a criterion for disaggregating the total inflow in the two single contributions of the interceptors. How?

Based on the surface and the morphological characteristics of the two catchments (regional analysis) we can assume a similar contribution from the two interceptors.

The hypothesis is validated using some flow rate measures locally available on the interceptors.

11

Some difficulties in the scheme

1. Piaganini: there is no way to compute the indicator for the water works

2. Campotosto: the contribution from the natural catchment is not accounted for.

12

Affluenti Campotosto

100 km2

Campotosto

13

Some difficulties in the scheme

1. Piaganini: there is no way to compute the indicator for the water works

2. Campotosto: the contribution from the natural catchment is not accounted for.

3. Provvidenza e Piaganini: understimation of snow-melting in spring and evaporation in summer.

14

Possible solutions

1. Piaganini: there is no way to compute the indicator for the water works

2. Campotosto: the contribution from the natural catchment is not accounted for.

3. Provvidenza e Piaganini: understimation of snow-melting in spring and evaporation in summer.

The daily inflow can be computed via mass balance using release and pumping data:

Provvidenza

Piaganini

Campotosto

15

Piaganini

P1i

ta

Snow melt is negligible

evaporation is NOT negligible

The estimate is reliable: we can use the new data obtained via mass balance (red) instead of those provided by Enel (blue).

The estimate is reliable: we can use the new data obtained via mass balance (red) instead of those provided by Enel (blue).

16

Provvidenza

Pr1ta

The estimate is not reliable.

Pumping is adding noise to data.

An understimation of evaporation is anyway evident in the data by Enel.

These data can be corrected by removing from them the evaporation that can be obtained from Piaganini, based on the many similarities between the two reservoirs.

The estimate is not reliable.

Pumping is adding noise to data.

An understimation of evaporation is anyway evident in the data by Enel.

These data can be corrected by removing from them the evaporation that can be obtained from Piaganini, based on the many similarities between the two reservoirs.

17

Some difficulties in the scheme

1. Piaganini: there is no way to compute the indicator for the water works

2. Campotosto: the contribution from the natural cacthment is not accounted for.

3. Provvidenza e Piaganini: understimation of snow-melting in spring and evaporation in summer.

18

Campotosto

this is impossible: at 40° max evap. 3m3/s

1Cta

Estimate is not reliable.

Oscillation are wider than in Provvidenza:

Pumping, but also the instrument precision (1cm) is amplifying the error

The contribution from the natural catchment is evident, but not easily quantifiable.

Inflow from Enel (blue) and from water balance (red) are not usable. What can we do?

Estimate is not reliable.

Oscillation are wider than in Provvidenza:

Pumping, but also the instrument precision (1cm) is amplifying the error

The contribution from the natural catchment is evident, but not easily quantifiable.

Inflow from Enel (blue) and from water balance (red) are not usable. What can we do?

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Natural inflow to Campotosto

Interceptors1350 mProvvidenza

Montorio

CAMPOTOSTOReservoir

Piaganini

Can we evaluate the significancy of the inflow contribution from the natural Campotosto’s catchment?

Water balance for the i-th year in Montorio

- Internal pumping to the system

- Error on the level negligible

From which

The valure for each year is obtained

The estimate is an annual value: how to move to a daily one?

The estimate is an annual value: how to move to a daily one?

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0,00

0,50

1,00

1,50

2,00

2,50

3,00

3,50

4,00

4,50

5,00

0,00 0,50 1,00 1,50 2,00 2,50 3,00

afflusso medio annuo gronde 1350 (mc/s)

affl

uss

o m

edio

an

nu

o c

om

ple

ssiv

o (

mc/

s )

Inflow estimate in Campotosto

evaporation

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Some difficulties in the scheme

1. Piaganini: there is no way to compute the indicator for the water works

2. Campotosto: the contribution from the natural cacthment is not accounted for.

3. Provvidenza e Piaganini: understimation of snow-melting in spring and evaporation in summer.

22Adriatic Sea

Fucino

VILLA VOMANO

PIAGANINI

PROVVIDENZA

CAMPOTOSTO

MONTORIO (M)

SAN GIACOMO (SG)

Irrigation District(CBN)

S. LUCIA (SL)

PROVVIDENZA (P)

Topological SchemeTopological Scheme

23

Identifying the Model

Definining the components and the system scheme

Identifying the models of the components Aggregated model

24Adriatic Sea

Fucino

VILLA VOMANO

PIAGANINI

PROVVIDENZA

CAMPOTOSTO

MONTORIO (M)

SAN GIACOMO (SG)

Irrigation District(CBN)

S. LUCIA (SL)

PROVVIDENZA (P)

25

Campotosto lake

Simplification

Let’s assume that only one criterion needs to be satisfied: flood reduction in the town of

Campotosto (on the lake shores)

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The (lake’s) domainThe whole set of quantities and information about the lake: inflow release level water characteristics biota algae ... batimetry topography stage-discharge function of the spillway ... Consorzio dell‘Adda (lake manager) Regione Lombardia (water authority) ...

(at)

(rt) (ht)

The domain is the first level of abstraction of reality. It does not require any assumption

about the mathematical relationships linking the variables.

It is not a representaton of reality, but a partition of knowledge.

The domain is the first level of abstraction of reality. It does not require any assumption

about the mathematical relationships linking the variables.

It is not a representaton of reality, but a partition of knowledge.

Models are a simplified representation of reality;

They should reproduce those features of the system that are important for the scope of the Project.

The first step to create a model is to select the essential variables within the domain.

Models are a simplified representation of reality;

They should reproduce those features of the system that are important for the scope of the Project.

The first step to create a model is to select the essential variables within the domain.

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The (lake’s) domainThe whole set of quantities and information about the lake: inflow release level water characteristics biota algae ... batimetry topography stage-discharge function of the spillway ... Consorzio dell‘Adda (lake manager) Regione Lombardia (water authority) ...

(at)

(rt) (ht)

1tt

th

1ta

An important convention

The subscript of a variable is the time instant at which it takes deterministically known value.

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Are the variables well defined?

Inflow at+1: total inflow in the interval [t,t+1)

It is better to divide it into:

t+1 = inflow from the natural catchment

wt = pumping from hydropower plant downstream

at+1t+1

wt Which unit of measurement? m3/s or m3 ?

Are the variables well defined?

YES, as long as we do not find errors: only falsification is possible.

It is very important that the domain is defined in strict collaboration with the concerned Stakeholders.

Sharing and agreeing on the assumptions made at this point is key to obtain a “trusted” model of the system.

It is very important that the domain is defined in strict collaboration with the concerned Stakeholders.

Sharing and agreeing on the assumptions made at this point is key to obtain a “trusted” model of the system.

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Identifying the model:the causal network

Is it a good representation of the real cause-effect relationships?

Release decision

30

1ts

Causal network of the lake

Is it a good model of reality?NO, evaporation is missing....

Loops are not allowed. An effect can not cause itself!!

Loops are not allowed. An effect can not cause itself!!

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- A priori: good sense, Analyst’s intuition

- A posteriori: accuracy of the model

identified starting from the network

How to check if the network is a good model?

Causal network of the lake

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input

input

input

Classification of the variables

state

output

control

disturbance

disturbance

deterministicdisturbance

randomdisturbance

internalvariables

The state is composed of all the variables that are necessary to describe the past history of the system, and, once these are known, the future evolution of the system is completely defined by the sole inputs.

1ts

33

The model structure

1ts

state transition function

output transformation function

set of the feasible controls

These two equations include all the information available in the network.

In the network the internal variables are explicitely considered.

These two equations include all the information available in the network.

In the network the internal variables are explicitely considered.

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In general: variables

state xt

ut control

up planning decision

wt deterministic disturbancet +1 random disturbance

From now on vectors will be in bold, e.g. xt is the state vector!

input

ouptut yt

with the following associated expressions Models are OBJECTS in the

computer-science meaning of the

word

Models are OBJECTS in the

computer-science meaning of the

word

output transformation function

proper model

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In general: structure

state transition function

time-varying modelModels interact with the outside only through inputs and ouputs. What happens inside is important only as far as it affects

the ouptuts.

Models interact with the outside only through inputs and ouputs. What happens inside is important only as far as it affects

the ouptuts.

This is a

DYNAMIC SYSTEM

improper model

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Not always systems are dynamic

Not always the state appears in the system dynamics.

E.g.: diversion dam

t+1 incoming flow

ut withdrawal decision

only the output transformation function

yt=ht(ut,t+1)

model

yt diverted flowthis is a non-dynamic

model

Time-varying is not a synonymus of dynamic!

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Simulation

simulation is aimed at computing

established a time horizon H (starting from time 0 and ending at time h)

giventhe initial state

the input trajectories

the state trajectories

the output trajectories

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Simulation

established a time horizon H (starting from time 0 and ending at time h)

giventhe initial state

the input trajectories

381

using the model recursively

20 h

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Conclusions

domain mental model causal network

Next step:

implicitly or explicitly define- the state transition function

- the output transformation function

How to classify model ?

with respect to

the nature

of their functions

the aumount of a priori information one has to know about the ongoing processes

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Bayesian Believe Networks Mechanistic models Empirical models Markov models

Readings

IPWRM.Theory Ch. 4

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