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 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