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

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

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

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

The Equivalent Circuit Representation in Synchronous Reference Frame is assumed

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

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

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

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.

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:

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

Local optimal path at the origin

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

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

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

At 21000RPM

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