1. István Selek, PhD Post. Doctoral Researcher Systems Engineering Research Group University of Oulu, Oulu, Finland

2. Model Structure Conservation law (with respect to reservoirs) (Distrurbance) Prediction sub-model Auxiliary Component (required by the cost function) Hydrodynamic network model

3. Given a dynamic model of a water distribution system of the form Where, subject to

9. Find a policy

10. Solution proposal, 3 level hierarchial approach: 1.Optimal control decision u(k) is calculated 2.Optimal time distribution of flows for all actuators is calculated 3.Calculate actuator opreational rules using flow distribution

11. 1.Are there robust solutions ? 2.Is it possible to design a WDS which fulfills robust operational requirements ? 3.What are the necesary and sufficient conditions for the existence of robust solutions ?

12. The existence of effective target tube is a necessary and sufficient condition for robustness !

13. Constraints Putting these together

14. The calculation proceeds in two steps. Step 1

16. Find a policy which maintains the state trajectory within at time instant k, and minimizes

17. Receding Horizon Principle: each time instant k an open loop (closed loop) optimization problem is defined and solved on a finite time horizon [k, k+N] using a small number of lookahead steps. Findwhich minimizes Open Loop Feedback Control Findwhich minimizes Closed Loop Feedback Control The lookahead optimization will result in a sequence: Apply the first element of the sequence

19. Permutational Invariance: invariance of the state subject to control sequence permutations The control sequence The system’s dynamics Integrator form The distrurbance (demand) sequence

20. Permutational Invariance: invariance of the state subject to control sequence permutations

21. Tank dynamics Can be written as

22. Must be transformed at each time instant Control constraints Effective target tube (state space) Control Constraints Effective target tube Putting these together The a vector is calculated For which the following equation is satisfied for all admissible realizations of the disturbance

23. Cost (objective) function is not defined yet !

24. 8 direct (discrete) pumping stations 3 continuous pumping stations 8 tanks 6 (stochastic) consumption zones Water demand model Cost function (pumping costs)