The design problem

1. NRM Lec17 The design problem Andrea Castelletti Politecnico di Milano Niger Delta

2. Elements of the Design Problem A Design Problem is composed of: • The model of the system • The modelling time step • The project indicator (or objective) • Project time horizon • Design Scenario • Any other constraints

3. 1. The model of the system The global model of the system has to be identified (see Lecture 16)

4. 2. The modeling time step The modelling time step is selected (see Lecture 8). This can be also periodic (time-varying).

5. 3. The project indicator (objective) max Σ The project indicator is a functional of the system trajectories defined over the project time horizon Very often i is separable or can be reconducted to be so penalty step indicator

6. 3. The step indicator given weight Can be associated to a single component or Can result from the aggregation of the step indicators of n components Energy produced by the i-th plant Example:Vomano project Plant stress in the irrigation district or

7. 4. Time horizon

8. 4a. Finite time horizon Finite horizon: no matter what will happen after h. Important It can be easily defined only when the system will actually last for h steps. Otherwise, the penalty plays an important role.

9. apply m t+1/t(•) 4b. Receeding time horizon Receeding horizon: at each time step t the horizon [t ,t+h]is considered xtgiven t t +1 t +h Solution to the problem ut/t xt+1 known after one step t +1 t +h+1 ut+1/t+1

10. Advantages of the receeding horizon • The constant distance between the time the decision is taken and the end of the horizon makes the influence of the penalty on the decision trifling • By reformulating the control problem at each time step the new information gained in the interval [t, t+1) can be exploited. Any change in the system (e.g. changes in the demand following the out-of-service of a power plant) as well as new information on the disturbances can usefully be considered.

11. 4c. Infinite time horizon Infinite horizon with Total Discounted Costs (TDC): discounting the future with a rateg Average Expected Value (AEV) over infinite horizon:

12. t 6 0 1 2 3 4 5 up=1 +2 up= 2 +10 up= 3 +11 Time horizon may affect the decision Consider a deterministic automaton All the transitions are associated with zero cost but for the self-loops up must be selected as to maximize the benefit

13. TDC vs AEV - Infinite horizon with Total Discounted Costs (TDC) - Discounts the future: alternatives with good perfomance on the short term are favoured over the others, even if their performance tend to worsen as the time goes on. - Average Expected Value (AEV) over infinite horizon - Considers the long term behaviour only: worse perfomance in a even long, but finite, transient is accepted if good perfomance is ensured on the long term regime.

14. The set of the trajectories (either deterministic or stochastic) of the variables that are neither directly or indirectly affected by the alternative at hand and therefore do not depend upon the DM decision 5. Design scenario x0 is given x1 ,..., xh are recursively defined by the model of the system Are to be chosen Are part of the design scenario

15. Irrigation district 5. Design scenario Example: construction of a reservoir feeding an irrigation district • The design scenario includes: • agricoltural water demand (trajectory of) • prices of agricoltural products (trajectory of) • cost of the material used for constructing the dam • cost of the labour involved in the costruction works • flooding threshold in the coastline towns • inflow trajectory (or process) ! - If the alternatives foresee also the choice of the crop to be cultivated - If the market in which the product are sold are small and thus affected by the construction of the dam

16. 6. Any other constraint Flooding threshold

17. The Design problem Even with only one project indicator (full rationality) the problem can be particularly complex, because of 1. The existence of infinite alternatives; 2. The uncertainty of the effects induced by the presence of random disturbances; 3. The existence of recursive decisions. First let’s consider the simplest case hypothesis A. disturbances are deterministic hypothesis B. decisions are only planning ones Pure planning problem 17

18. The model of the system For the purpose of planning the model can be simplified

19. The global model of the system for planning The model of the system For the purpose of planning the model can be simplified Let’s assume the system is a periodic system of period T

20. Pure planning problem This is the more general formulation of the problem, accounting for both the transient and the steady state condition of the system. Often the DM is only interested in the steady state conditions

21. Reading IPWRM.Theory Ch. 8, 10