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Lecture 2
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Lecture 2: Stability Criteria
S.D. SudhoffEnergy Sources Analysis ConsortiumESAC DC Stability Toolbox Tutorial
January 4, 2002Version 2.1
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Lecture 2
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Lecture 2 Outline
• Comparison of Stability Criteria• Design Specifications From Arbitrary
Stability Criteria• Generalized Impedance / Admittance
Concepts
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Stability Factoid
The source load system is stable provided that the evaluationof along the Nyquist contour does not encircle -1lsYZ
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Stability Criteria
OpposingArgumentCriteria
ESACCriteria
PM
GMPMCriteria
1/GM
ImaginaryAxis
Unit Circle
Real Axis
Middlebrook Criteria
[2] S.D. Sudhoff, “Admittance Space Based Stability Specification,” Proceedings of the 1998 ONR -Drexel-NSWC Workshop on Electric Shipboard System Modeling, Simulation and Control, June 22-23, 1998, Philadelphia, PA, USA
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Purpose of Stability Criteria
• Primary– Basis for calculating load admittance spec from
source impedance; or source impedance spec from load admittance
• Secondary– Check of stability
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Comparison of Stability Criteria
• Cost of resulting design• Amenability to arbitrary component
grouping• Amenability to formulation of design
specification
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Cost of Resulting Design
OpposingArgumentCriteria
ESACCriteria
PM
GMPMCriteria
1/GM
ImaginaryAxis
Unit Circle
Real Axis
Middlebrook Criteria
[2] S.D. Sudhoff, “Admittance Space Based Stability Specification,” Proceedings of the 1998 ONR -Drexel-NSWC Workshop on Electric Shipboard System Modeling, Simulation and Control, June 22-23, 1998, Philadelphia, PA, USA
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Cost of Resulting Design
• Middlebrook (Highest)• Opposing Argument• Gain/Phase • ESAC (Lowest)
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Grouping: Case Study 1
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Grouping: Case Study 1 -Nyquist Plane Results
ImaginaryAxis
Nyqusis ContourStable Case
Nyquist ContourUnstable Case
Real Axis
ESAC Criteria
GMPMCriiteria
-3
-1.5
1.5
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Grouping: Case Study 2
+_ C - R
Source Load
Vs
r L
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Grouping: Case Study 2 Nyquist Plane Results
GMPMCriteria
ESACCriteria
Real Axis
Nyquist Contour,Unstable Case
Nyquist Contour,Stable Case
ImaginaryAxis
Arcs at |s| =
3
3
-3
-3
400π
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Grouping: Summary
• ESAC Criteria much less sensitive to grouping than other proposed criteria
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Design Specification: Middlebrook
• Suppose known• Design specification on load becomes
• Alternately, could come up with specification on load impedance
sZ
||1||
sl ZGM
Y <
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Design Specification: Gain and Phase Margin Criteria
• Design specification based on
GMZY ll
1|||| <
( )( )PMsZsY
PMsZsY
osl
osl
+−≥∠+∠
−≤∠+∠
∠
∠
180)()(
and180)()(
))Im()(Re( xjxanglex +≡∠
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Design Specification: ESAC Criteria
Construction of a load admittance specification at a point
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Design Specification: ESAC Criteria
Construction of load admittance constraint at a frequency
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Design Specification: ESAC Criteria
60
40
20
0
-20
-40300
200
1000
0-2
24
log of frequency, Hzphase, degrees
mag
nitu
de, d
B
Inflection
Stability Constraint in Admittance Space
Load Admittance,Unstable Case
Load Admittance,Stable Case
3-D stability constraint in admittance space
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Comments on Stability Criteria
• Design Cost (Highest to Lowest)– Middlebrook, Opposing Argument, GMPM,
ESAC• Component Grouping
– ESAC criteria much less sensitive to grouping• Translation to Design Specification
– Middlebrook most readily used– GMPM not bad– ESAC requires toolbox
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Dealing with Reality:Example System
SM IMTurbineMechanical
Load
IMControls
Vol. Reg./Exciter
iabcsvdci
vdcsvr
ωrm,imTe,desvdcs
*
3 - Uncontrolled
Rectifier
φ 3 - Fully C ontrolled
Inverter
φLCFilter
C apacitiv eFilter
Tie Line
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Generalized Source Impedance
• Speed: 0.9-1.1 p.u.• Power: 0-1.1 p.u.• Voltage: 0.95-1.05 p.u.• Number of Plants Considered: 125
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Generalized Source Impedance
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Generalized Load Admittance and Load Admittance Constraint
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Generalized Source Impedance and Source Impedance Constraint
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Measured Performance
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Mitigation: The Nonlinear Stabilizing Control Architecture (NSCA)
[3] S.D. Sudhoff, K.A. Corzine, S.F. Glover, H.J. Hegner, and H.N. Robey, “DC Link Stabilized Field Oriented Control of Electric Propulsion Systems,” IEEE Transactions on Energy Conversion, Vol. 13, No. 1, March 1998.
[4] S.D. Sudhoff, “Control of Power Electronics Based Systems” Proceedings of the 1998 ONR -Drexel-NSWC Workshop on Electric Shipboard System Modeling, Simulation and Control, June 22-23, 1998, Philadelphia, PA, USA
[5] S.D. Sudhoff, S.F. Glover, “Nonlinear Stabilizing Control for Power Electronic Based Systems,” U.S. Patent No. 6,051,941, April 18, 2000. International Patents Applied For.
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Generalized Load Admittance and Constraint with NSCA
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Generalized Source Impedance and Constraint with NSCA
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Measured Performance with NSCA
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Conclusions
• ESAC Criteria – Leads to Less Expensive / Higher Performance Designs– Facilitates Modularity in Design Process
• 3-Dimensional Admittance/Impedance Space Approach– Allows ESAC (and Arbitrary) Stability to Be Used– Facilitates Specification of Source Given Load– Facilitates Specification of Load Given Source