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

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

Introduction Unsaturated Saturated conclusion Motivation Introduction Unsaturated Saturated conclusion

Introduction Unsaturated Saturated conclusion Problem description Introduction Unsaturated Saturated conclusion

Introduction Unsaturated Saturated conclusion Problem description Introduction Unsaturated Saturated conclusion

Introduction Unsaturated Saturated conclusion Model Introduction Unsaturated Saturated conclusion

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

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

Introduction Unsaturated Saturated conclusion Closed loop Introduction Unsaturated Saturated conclusion

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

Introduction Unsaturated Saturated conclusion Saturation Motivation: Introduction Unsaturated Saturated conclusion

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

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

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

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

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

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

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

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

Introduction Model Control conclusion