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

2. 2 2. Conceptualisation Stakeholders 1. Reconnaissance Defining Actions (measures) Identifying the Model Defining Criteria and Indicators

3. 3 Schema fisicoSchema fisico Adriatic 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 Component Component: modelling elementary unit. Every component has a specific function. The model of the component must describe such a fuction. Logical components are also allowed. Component Component: 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. 4 Identifying the Model Definining the components and the system scheme Identifying the models of the components Aggregated model

5. 5 Schema fisicoSchema fisico Adriatic 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. 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. Water works Irrigation district (CBN) S. LUCIA (SL) Chiarino Vomano During night-time without pumping e.g. Provvidenza: only aggregated flow data

7. 7 Schema fisicoSchema fisico Adriatic 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

8. 8 Schema fisico (bacini)Schema fisico (bacini) Adriatic Sea Fucino VILLA VOMANO PIAGANINI PROVVIDENZA CAMPOTOSTO MONTORIO (M) SAN GIACOMO (SG) Irrigation district (CBN) S. LUCIA (SL) PROVVIDENZA (P) Water works ???

9. 9 Some difficulties in the scheme Piaganini 1. Piaganini: there is no way to compute the indicator for the water works ? Average water supply from hydropower reservoirs WW

10. 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. 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. 12 Affluenti Campotosto 100 km 2 Campotosto

13. 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. 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. 15 Piaganini 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).

16. 16 Provvidenza 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. 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. 18 Campotosto this is impossible: at 40° max evap. 3m 3 /s 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?

19. 19 Natural inflow to Campotosto Interceptors 1350 m Provvidenza Montorio CAMPOTOSTO Reservoir Piagani ni Can we evaluate the significancy of the inflow contribution from the natural Campotostos 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?

20. 20 Inflow estimate in Campotosto evaporation

21. 21 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.

22. 22 Schema fisicoSchema fisico Adriatic Sea Fucino VILLA VOMANO PIAGANINI PROVVIDENZA CAMPOTOSTO MONTORIO (M) SAN GIACOMO (SG) Irrigation District (CBN) S. LUCIA (SL) PROVVIDENZA (P) Topological Scheme

23. 23 Identifying the Model Definining the components and the system scheme Identifying the models of the components Aggregated model

24. 24 Schema fisicoSchema fisico Adriatic Sea Fucino VILLA VOMANO PIAGANINI PROVVIDENZA CAMPOTOSTO MONTORIO (M) SAN GIACOMO (SG) Irrigation District (CBN) S. LUCIA (SL) PROVVIDENZA (P)

25. 25 Campotosto lake Simplification Lets assume that only one criterion needs to be satisfied: flood reduction in the town of Campotosto (on the lake shores)

26. 26 The (lakes) domain The 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 dellAdda (lake manager) Regione Lombardia (water authority)... (a t ) (r t ) (h t ) 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.

27. 27 The (lakes) domain The 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 dellAdda (lake manager) Regione Lombardia (water authority)... (a t ) (r t ) (h t ) An important convention The subscript of a variable is the time instant at which it takes deterministically known value.

28. 28 Are the variables well defined? Inflow a t+1 : total inflow in the interval [t,t+1) It is better to divide it into: t+1 = inflow from the natural catchment w t = pumping from hydropower plant downstream a t+1 t+1 wtwt Which unit of measurement? m 3 /s or m 3 ? 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.

29. 29 Identifying the model: the causal network Is it a good representation of the real cause-effect relationships? Release decision

30. 30 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!!

31. 31 - A priori: good sense, Analysts 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

32. 32 input Classification of the variables state output control disturbance deterministic disturbance random disturbance internal variables 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.

33. 33 The model structure 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.

34. 34 In general: variables state x t u t control u p planning decision w t deterministic disturbance t +1 random disturbance From now on vectors will be in bold, e.g. x t is the state vector ! input ouptut y t

35. with the following associated expressions Models are OBJECTS in the computer-science meaning of the word output transformation function proper model 35 In general: structure state transition function time-varying model 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

36. 36 Not always systems are dynamic Not always the state appears in the system dynamics. E.g.: diversion dam t+1 incoming flow u t withdrawal decision only the output transformation function y t =h t (u t, t+1 ) model y t diverted flow this is a non-dynamic model Time-varying is not a synonymus of dynamic !

37. 37 Simulation simulation is aimed at computing established a time horizon H (starting from time 0 and ending at time h) given the initial state the input trajectories the state trajectories the output trajectories

38. 38 Simulation established a time horizon H (starting from time 0 and ending at time h) given the initial state the input trajectories 38 using the model recursively

39. 39 Conclusions domainmental 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

40. 40 Bayesian Believe Networks Mechanistic models Empirical models Markov models

41. Readings IPWRM.Theory Ch. 4 41