Optimal Steady State Regulation of Distribution Networks with Input - - PowerPoint PPT Presentation

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Optimal Steady State Regulation of Distribution Networks with Input - - PowerPoint PPT Presentation

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Optimal Steady State Regulation of Distribution Networks with Input and Flow Constraints T.W. SCHOLTEN, C. DE PERSIS, P. TESI 1 Outline Introduction Unsaturated Saturated Conclusion Control goal 2 Motivation Control goal 1 Conclusions

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Optimal Steady State Regulation of Distribution Networks with Input and Flow Constraints

T.W. SCHOLTEN, C. DE PERSIS, P. TESI

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Outline

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

2

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Motivation

3 INTRODUCTION UNSATURATED SATURATED CONCLUSION

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Problem description

4 INTRODUCTION UNSATURATED SATURATED CONCLUSION

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Problem description

5 INTRODUCTION UNSATURATED SATURATED CONCLUSION

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Model

6 INTRODUCTION UNSATURATED SATURATED CONCLUSION

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Control goal 1

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Design distributed controllers and such that (flow on the edges) (input at the nodes) where

INTRODUCTION UNSATURATED SATURATED CONCLUSION

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Controller design

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suitable gains. suitable gains. Flow on the edges Input on the nodes

INTRODUCTION UNSATURATED SATURATED CONCLUSION

Recall:

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Closed loop

9 INTRODUCTION UNSATURATED SATURATED CONCLUSION

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Result

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1. undirected graph G is connected 2. there exists a of s.t. If: then Problem 1 Solved

INTRODUCTION UNSATURATED SATURATED CONCLUSION

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Saturation

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Motivation:

INTRODUCTION UNSATURATED SATURATED CONCLUSION

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Control problem 2

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and (flow on the edges) (input at the nodes) Design distributed controllers such that given positive real (arbitrarily small) numbers with for all

INTRODUCTION UNSATURATED SATURATED CONCLUSION

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Controller design

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suitable gains. suitable gains. Flow on the edges Input on the nodes

INTRODUCTION UNSATURATED SATURATED CONCLUSION

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Closed loop system

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Steady state deviation from optimum

INTRODUCTION UNSATURATED SATURATED CONCLUSION

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Matching condition

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Then, the matching condition is satisfied if Let be the optimal steady state input and a

INTRODUCTION UNSATURATED SATURATED CONCLUSION

corresponding flowrate.

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Main result

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and

INTRODUCTION UNSATURATED SATURATED CONCLUSION

1. Matching condition is satisfied 2. There exists at least one pair

  • f s.t.

3. The directed graph G is strongly connected 4. The directed graph G is balanced If Then Problem 2 is solved Details skipped

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Case study:

17 INTRODUCTION UNSATURATED SATURATED CONCLUSION

Store (volume) Discharge (volume) Unsaturated flows Saturated flow Optimal input Transient behavior

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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

18 INTRODUCTION UNSATURATED SATURATED CONCLUSION

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Future work

  • Remove assumption of balanced graphs
  • Relax or remove bounds on
  • Relax restriction
  • More general model
  • Including pressures
  • Algebraic nodes (no storage)

19 INTRODUCTION UNSATURATED SATURATED CONCLUSION

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INTRODUCTION MODEL CONTROL CONCLUSION 20