1. A fast current response control strategy for flywheel peak power capability under DC bus voltage constraint L. Xu and S. Li Department of Electrical Engineering The Ohio State University Grainger Center for Electric Machinery and Electromechanics University of Illinois at Urbana-Champaign Dec. 2001

2. Presentation Outline Introduction Problem Formulation
Prerequisite – Case of Disk Voltage Constraint
Feedback Time-Optimal Design under Hexagonal Voltage Constraint
Application in Flywheel Energy Storage Systems
Conclusion

3. Introduction Literature Review: Motivations:
General concept of minimum-time current transition at DC bus voltage constraint, Choi & Sul [2].
PMSM application, torque patching and current regulator conditioning, Xu [3], [4].
Motivations:
Peak power delivery of flywheels as energy storage devices
Disk constraint V.S. Hexagonal constraint
Feedback solution is preferable

4. II. Problem Formulation
Efficient DC bus utilization for high speed PMSM operation for fast peak power delivery
Synchronous reference frame model of PMSM,
Denote
Then with stator resistance neglected,
Now define the state as:
Then, where

5. The Equivalent Circuit Representation in Synchronous Reference Frame
is assumed

6. In stationary reference frame:
Voltage Constraints
In stationary reference frame:
Voltage Constraint:
Case of Disk Voltage Constraint
Hexagonal Voltage Constraint

7. III. Prerequisite – Case of Disk Voltage Constraint
Geometrical explanation
Given,
Solution,

8. IV. Feedback Time-Optimal Design under Hexagonal Voltage Constraint
Dynamic equation:
Define the Hamiltonian:
By Pontryagin’s maximum principle, necessary conditions:

9. Some Theoretical Implications
Assumption: consider the regulator problem:
System is “normal”, i.e.,
are all controllable,
so, the optimal control is unique and is determined by the necessary conditions.
The co-state is a rotating vector.

10. Under the hexagonal voltage constraint, solutions to are
almost everywhere in time t.
Due to the nature of maximization problem and the special form of the co-state:

11. With a constant voltage input ,
solution to :
is actually an angular transformation of a clockwise angle

12. Local optimal path at the origin

13. Construction of a global feedback switching diagram
For autonomous system, theoretically we can integrate backwards to find the solution
Our case is very special:
The co-state is a rotating vector.
The maximization problem is:
So, sequencing and voltage vector impress
Compare with the solution to the case of the disk voltage constraint

14. Feedback Switching Diagram under the Hexagonal Constraint
Consider the case where
General case can be similarly treated
The example

15. Applications in Flywheel Energy Storage Systems
10kw flywheel energy storage system
PMSM parameters:

16. At 21000RPM

18. V. Conclusion
New current control for flywheel energy storage applications
Solved the feedback control design problem of the time-optimal current transition
Reduced computational requirements in practical implementations
Laboratory implementation is under way