The reservoir: mechanistic model

1. NRM Lec08 The reservoir:mechanistic model Andrea Castelletti Politecnico di Milano

2. Penstocks and turbine hall Itaipu dam on the Parana river Dam Spillways in action

3. Three Gorges dam (China)

4. Typical Localization Clan canyon dam Colorado river

5. Longitudinal section intake tower surface spillway intakes maximum storage bottom outlet water surface level minimum intake level penstock storage barrier

6. Surface spillways

7. Bottom outlet Loch Lagghan dam Scozia

8. penstock • surface spillway

9. Gates: a) rising sector b) vertical rising c) radial Dam-gate structures

10. Piave - S.Croce system Reservoir network Hydropower reservoirs are often interconnected to form a network

11. Features of reservoirs By the management point of view a reservoir is characterized by: • the active (or live) storage; • the global stage-discharge curve of the spillways; • the stage-discharge curve of the intake tower;

12. Stage-discharge curve (morning glory)

13. Causal network st = storage volume at timet at+1 = inflow volume in [t ,t+1) rt+1 = effective release volume in [t , t+1)

14. Causal network What is missed? - evaporation - r depends ona and e

15. surface Mechanistic model What is missed? - evaporation - r depends ona and e

16. surface evaporation Mechanistic model

17. surface evaporation storage Mechanistic model

18. surface evaporation storage level Mechanistic model

19. surface evaporation storage level release Mechanistic model

20. balance net inflow balance net inflow estimator Cons Using it when the storage is not actually a cylinder generates an error. Balance equation Simplification: the storage is a cylinderS(st) = S Pros

21. Implicit assumption: the water surface is always horizzontal. arbitary constant Storage-level relationship There exist a biunivocal relationship between the level measured in a point and the storage. By inverting h(.) one obtains the value of the storage by measuring the level, i.e. the only measurable quantity Example:if the storage is a cylinder A negative storage represents the missing volume required to bring the water surface up to the level corresponding to the zero storage.

22. Implicit assumption: the water surface is always horizzontal. historic time series Storage-level relationship There exist a biunivocal relationship between the level measured in a point and the storage. By inverting h(.) one obtains the value of the storage by measuring the level, i.e. the only measurable quantity Non-cylindrical storage The identification of h(.) can be performed in different ways, depending on which one of the following is known: numerical computation point by point batimetry of the reservoir (DEM) interpolation

23. Examples of storage-level relationships Relazione invaso-quota Campotosto

24. Examples of storage-level relationships Relazione invaso-quota Piaganini

25. Examples of storage-level relationships Ricalibrazione scala

26. Surface-storage relationship Can be determined with similar techniques.

27. Continuous-time model of reservoir s(t) = storage volume at time t [m3] a(t) = inflow rate at time t [m3/s] i(t,s(t)) = infiltration [m3/s] e(t) = evaporation per surface area unit at time t [m/s] S(s(t)) = surface area [m2] r(t,s(t),p(t)) = release when the dam gate are open of p [m3/s]

28. Simplifications n(t) It depends on the storage-discharge functions and the position p of the intake sluice gates • i = 0almost always true, at least in artificial reservoirs • Cylindric reservoir n(t) = a(t)-e(t)S net inflow

29. Instantaneous storage-discharge relationships maximum release open gates  minimum release limited spillway  smin smax s* s(t) • s* :storage at wich spillways are activated • smin , smax:bounds of the regulation range

30. Example of storage-discharge relationship ~ N max (•) flow rate [m3/s] ~ N min (•) storage [Mm3]

31. 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

32. st +nt+1 -rt+1 st = storage volume at time t nt+1 = net inflow volume in [t ,t+1) rt+1 = volume actually released in [t , t+1) t t +1 Discrete model of a reservoir st+1 = st +nt+1 -rt+1 nt+1= net inflow volume in [t , t+1) we assume it uniformly distributed nt+1

33. release decision with with 45° The release function rt+1 = Rt (st,nt+1 ,ut) Minimum dischargeable volume in [t , t+1) Maximum dischargeable volume in [t , t+1) givenst, &nt+1 givenst & ut rt+1 rt+1 Vt ut vt nt+1 ut

34. n = 50 m3/s n = 0 m3/s An example of minimum and maximum release (Campotosto, Italy) flow rate [m3/s] storage [Mm3]

35. depends on the inflow! Set of feasible controls U(st) feasible controlU(st) flow rate [m3/s] NO storage [Mm3]

36. 45° Set of feasible controls U(st) stgiven rt+1 Vt (st,max{nt+1}) Vt (st,min{nt+1}) vt(st,max{nt+1}) vt(st,min{nt+1}) ut U(st)

37. releases of interest Remarks • vt(st ,nt+1)rt+1  Vt(st,nt+1) constraint already included inRt(•) NO • vt(st,nt+1)ut  Vt(st,nt+1) • Ifnt+1is known rt+1 = ut • Alternative ways of formulating the mass balance equation ht =level at timet rt+1 = Rt(st ,nt+1 ,ut) = Rt (ht ,nt+1 ,ut) actual release in [t , t +1) nt+1= net inflow in terms of level rt+1= actual release in terms of level

38. CONCLUSIONS Model of a reservoir in operation state transition outputs feasible controls

39. CONCLUSIONS Model of a reservoir to be constructed

40. r a(s - smin)b 0 if ssmin N(s) = s smin Natural lake e h r a s hmin net or effective inflow Stage-discharge function N(s(t))

41. Linearization and time constant r a smin s D Linear continuos system tt+1 Lagrange formula a(s - smin) Remark:s(t+1) depends ons(t) only ifD<< T = a-1. T is the so-called time constant of the reservor.

42. Linearization and time constant r a smin s D Meaning of T tt+1 a(s - smin) By assumingD=T=1/aone gets Remark:s(t+1) depends ons(t) only ifD<< T = a-1. T is the so-called time constant of the reservor. T is the time required for the storage to reach 1/3 of its initial value.

43. Linearization and time constant r a smin s D tt+1 a(s - smin) Remark:s(t+1) depends ons(t) only ifD<< T = a-1. T is the so-called time constant of the reservor. An accurate modelling does require D<<T. Shannon’s or Sampling theorem

44. Time constant and the Lombardy lakes For most of the lakes T is nearly 8 days. LAKEST= a-1 [km2] [daysi] Maggiore 212.0 7.4 Lugano 48.9 8.7 Varese 15.0 34.7 Alserio 1.5 8.0 Pusiano 5.2 15.0 Como 146.0 7.7 Iseo 61.0 7.8 Garda 370.0 86.6 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. The modelling time-step for lakes with T = 78 is about 1 day.

45. Buffering effecti.e. low-pass filtering dB 1/T w Bode diagram • 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.

46. a a The modelling time-step depends upon the state To be sure that the system is well represented by the model: D 0,1*T Non-linear model: T is not defined Linearization of the system T changes with the values around which the system is linearized D changes with s It would be useful to have models with a time-step D that changes with s, however this is not possible with the algorithms nowadays available. Solution: use models with different D at different time steps.

47. Comparison between two lakes a2 >a1 r a2 a1 h h hmax t Average level

48. n Comparison between two lakes:impulsive flood P n* Average level t Impulse response Lake shore inhabitants Downstream users h r t t CONFLICT happy with lake 2 ( T small) Happy with lake 1 ( T big )

49. Lake regulation Lake shore population a big Downstream users a small Which compromise ? Natural lake Regulated lake r Natural regime curve Natural curve Curves for different gate positions h Different stage-discharge functions in different time instants

50. Lake regulation at r Downstream users at bt Lake popul. bt h(t) Months