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
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Presentation Outline
1.
2.
3.
4.
5.
6.
Introduction
Problem Formulation
Prerequisite – Case of Disk Voltage
Constraint
Feedback Time-Optimal Design under
Hexagonal Voltage Constraint
Application in Flywheel Energy Storage
Systems
Conclusion
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I.
Introduction
Literature Review:
 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 vd2  vq2  V02 V.S. Hexagonal constraint
 Feedback solution is preferable
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II. Problem Formulation
 Efficient DC bus utilization for high speed PMSM
operation for fast peak power delivery
Synchronous reference frame model of PMSM,
λ
~
i

i

Denote
L
Then with stator resistance neglected, Ld ~id  Lq e iq  vd
f
d
d
d
~
Ld iq   Ld e id  v q
Now define the state as:
~
x  [ Ld id , Lq iq ]T
v  [vd , vq ]T
Then, x  Ax  v where
 0
A
 e
e 
0

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The Equivalent Circuit Representation in
Synchronous Reference Frame
Synchronous
Reference Frame
is assumed
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Voltage Constraints
In stationary reference
frame:
Voltage Constraint:
 Case of Disk Voltage
Constraint
vd2  vq2  V02
 Hexagonal Voltage
Constraint
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III. Prerequisite – Case of Disk Voltage
Constraint
 Given,
 Geometrical explanation
~
x(0)  [ Ld id (0), Lq iq (0)]T
~*
x( t f )  [ Ld I d , Lq I q* ]T
 Solution,
 cos( e t ) sin( e t ) 
v  V0 
0

 sin( e t ) cos( e t )
 cos(e t f ) sin( e t f ) 
x (t f )  
( x (0)  V0 t f  0 )

 sin( e t f ) cos(e t f )
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IV. Feedback Time-Optimal Design under
Hexagonal Voltage Constraint
Dynamic equation: x  Ax  v
Define the Hamiltonian: H (, x, v )    ( Ax  v )
By Pontryagin’s maximum principle, necessary
conditions:
H
*
 
x
 Ax*  v *


*
H

  * A
x
H ( , x , v )  max {  ( Ax  v)}
*
*
*
uHexagon
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Some Theoretical Implications
 Assumption: consider the regulator problem: x(t f )  0
 System is “normal”, i.e.,
( A, [1,0]T )
( A, [0,1]T )
are all controllable,
so, the optimal control is unique and is determined
by the necessary conditions.
 The co-state is a rotating vector.
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 Under the hexagonal voltage constraint,
solutions to v (t ) are
v (t )  {Vi ; i  1,...,6}
*
almost everywhere in time t.
 Due to the nature of maximization problem
and the special form of the co-state:
cos(e t )  sin( e t )
 (t )   (0) 

sin(

t
)
cos(

t
)
e
e


*
*
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 With a constant voltage input v  Vi,
solution to x  Ax  v :
x (t )  e At ( x (0)  A1Vi )  A 1Vi
e
At
 cos(e t ) sin( e t ) 


 sin( e t ) cos(e t )
At
e is actually an angular transformation of
a clockwise angle    et
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 Local optimal path at the origin
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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:
H ( * , x* , v* )  max ( *  v)   * Ax*
vHexagon
v* (t )  {Vi ; i  1,...,6}
So, sequencing and 60 voltage vector impress
Compare with the solution to the case of the disk
voltage constraint
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Feedback Switching Diagram under the
Hexagonal Constraint
 Consider the
case where
x(t f )  0
 General case
can be similarly
treated
 The example
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Applications in Flywheel Energy
Storage Systems
 10kw flywheel energy
storage system
 PMSM parameters:
Ld  Lq  0.95mH
 f  0.0425Wb
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 At 21000RPM
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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
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