1. T.W. Scholten, C. de Persis, P. Tesi
Optimal Steady State Regulation of Distribution Networks with Input and Flow Constraints
T.W. Scholten, C. de Persis, P. Tesi

2. Outline Motivation Problem description Model Control goal 1
Introduction
Motivation
Problem description
Model
Unsaturated
Control goal 1
Controller design 1
Stability result
Saturated
Control goal 2
Controller design 2
Main result
Case study
Conclusion
Conclusions
Future work

3. Introduction Unsaturated Saturated conclusion
Motivation
Introduction Unsaturated Saturated conclusion

4. Introduction Unsaturated Saturated conclusion
Problem description
Introduction Unsaturated Saturated conclusion

5. Introduction Unsaturated Saturated conclusion
Problem description
Introduction Unsaturated Saturated conclusion

6. Introduction Unsaturated Saturated conclusion
Model
Introduction Unsaturated Saturated conclusion

7. Introduction Unsaturated Saturated conclusion
Control goal 1
Design distributed controllers
and
such that
(flow on the edges)
(input at the nodes)
where
Introduction Unsaturated Saturated conclusion

8. Introduction Unsaturated Saturated conclusion
Controller design
Flow on the edges
suitable gains.
Input on the nodes
Recall:
suitable gains.
Introduction Unsaturated Saturated conclusion

9. Introduction Unsaturated Saturated conclusion
Closed loop
Introduction Unsaturated Saturated conclusion

10. Introduction Unsaturated Saturated conclusion
Result
If:
undirected graph G is connected
there exists a of s.t.
then
Problem 1
Solved
Introduction Unsaturated Saturated conclusion

11. Introduction Unsaturated Saturated conclusion
Saturation
Motivation:
Introduction Unsaturated Saturated conclusion

12. Introduction Unsaturated Saturated conclusion
Control problem 2
Design distributed controllers
(flow on the edges)
(input at the nodes)
such that
given positive real (arbitrarily small) numbers
with
and
for all
Introduction Unsaturated Saturated conclusion

13. Introduction Unsaturated Saturated conclusion
Controller design
Flow on the edges
suitable gains.
Input on the nodes
suitable gains.
Introduction Unsaturated Saturated conclusion

14. Introduction Unsaturated Saturated conclusion
Closed loop system
Steady state deviation from optimum
Introduction Unsaturated Saturated conclusion

15. Introduction Unsaturated Saturated conclusion
Matching condition
Let
be the optimal steady state input and a
corresponding flowrate.
Then, the matching condition is satisfied if
Introduction Unsaturated Saturated conclusion

16. Introduction Unsaturated Saturated conclusion
Main result If
Matching condition is satisfied
There exists at least one pair of s.t.
The directed graph G is strongly connected
The directed graph G is balanced
and
Details skipped
Then Problem 2 is solved
Introduction Unsaturated Saturated conclusion

17. Introduction Unsaturated Saturated conclusion
Case study:
Discharge (volume)
Store (volume)
Unsaturated flows
Saturated flow
Saturated input
Optimal input
Transient behavior
Introduction Unsaturated Saturated conclusion

18. Introduction Unsaturated Saturated conclusion
Conclusions
Considered distribution network
Disturbances
Costs associated to input
Controller design
Flows on links
Input on nodes
Considered saturation of flows and input
(practical) stability results
Applied to district heating networks
Introduction Unsaturated Saturated conclusion

19. Introduction Unsaturated Saturated conclusion
Future work
Remove assumption of balanced graphs
Relax or remove bounds on
Relax restriction
More general model
Including pressures
Algebraic nodes (no storage)
Introduction Unsaturated Saturated conclusion

20. Introduction Model Control conclusion