1

Miguel Vélez-Reyes1 and Aleksandar Stankovic2

1University of Puerto Rico-Mayagüez2Northeastern University

Challenges in Modeling and Identification of Power Systems: Dealing with Ill Conditioning, High Dimensionality, and other Challenges.

2

Need for System Identification and Parameter Estimation

Operation of power systems near its limits leads to the need of having accurate models for

Operation Stability Analysis Control Power flow Reliability

3

Motivation Model selection in many areas of engineering is

typically based on physical principles. Many of these models have a relatively large number of

parameters. Physical restrictions may not allow data collection for

parameter estimation based on experimental design. Measurements may not be rich enough to adequately

reflect the individual effects of all parameters. Distributed sensor networks allow us the observation of

spatio/temporal characteristics for power systems and other geographically distributed systems.

A vital issue in establishing utility of a model is whether parameters can be reliably identified from available experimental data.

4

Problem of Interest #1 What model structures will give

us useful descriptions that will take full advantage of available data and be useful for the intended application

Gray Box models Physical + data driven modeling

5

Dynamical Equivalents using NN

Consequences of deregulation – power systems that are more stressed, with participants less inclined to cooperate in overall power system control.

Increased reliance on 1) dynamical equivalents and 2) communication and computer technology to maintain stability.

 Our research direction – generation and validation of dynamic equivalents directly from real-time measurements (and from locally available prior information)

Stankovic and Saric, ‘03

6

New England/New York test power system

Retained subsystem

1

G153

G8

60

2625 28 29

61

G9

2427

30

9

8

3

18

17

2

4 14

15

21

16

57

19

20

G5

56

G4

58

G6

22

59

G7

23

1311

12

5

6

754

G255

G3

10

65

G13

37

G14

6640 48 47

41

G10

62

31 G1

1

63

32

33

38

46

G15

67

4249

36

64

G12

34

35

43

68

G16

52

50

51 45

44

39

ANN-based reduced-ordersubsystem

Interconnection and measu-rement points subsystem

7

Bottleneck ANN Interface data

Interface data

8

Recurrent ANN

9

0 1 2 3 4 5 6 7 8 90.991

0.992

0.993

0.994

0.995

0.996

0.997

0.998

0.999

1

1.001

Gen

erat

or s

peed

(pu)

Time (sec)

Initial (DYNRED)Gray box (bottleneck ANN)Gray box (recurrent ANN)

0 1 2 3 4 5 6 7 8 9-2

-1.5

-1

-0.5

0

0.5

Act

ive

pow

er (p

u)

Time (sec)

Original systemGray box (bottleneck ANN)Gray box (recurrent ANN)

Equivalent generator speed in reduced subsystem.

Active power flow in interconnection 98.

Application

10

Problem of Interest #2 Available data for parameter estimation

does not contain enough information about model parameters

Overparametrized models Poor quality of data

The resulting parameter estimation problem is ill-conditioned

Given a data set and a model for the system of interest what parameters can be reliably estimated from the data

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Parameter Estimation (PE)

Parameter

estimation

algorithm

Input

Data

Parameter

estimate

A priori

information

Estimation as an optimization problem

)(αV : is the measurementof the model fit

α : is the parameter estimate

A : is the allowable parameter set

)αV(argminαAα

ˆˆ

12

Objectives

Develop metrics to diagnose ill- conditioning in a parameter estimation problem

Study how different formulations of the parameter estimation problem can be used to overcome ill-conditioning

Apply these techniques to parameter estimation problems

13

Another Linear Example

Third column is noise plus columns 2 and 4 and b is the first column minus half of the second plus 0.01 times the third and fourth columns

A b

0.4919

, =

0.4249

- 0.1981

0.6291

- 0.7905

01112 0 5116 0 6234

0 5050 0 6239 0 5644 0 0589

0 5728 0 0843 0 6640 0 7480

0 4181 0 7689 0 9893 0 2200

. . .

. . . .

. . . .

. . . .

\ \

A b A b e

1.0000

- 0.5000

0.0100

0.0100

, =

0.8376

45.1105

- 45.5631

45.5513

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Quantifying Ill-Conditioning using Sensitivity Functions

Norm condition number

where J is the Jacobian of F at d. Componentwise condition number

dJd

ˆS

ˆ

d

dd

i

i

ˆ

d fˆ

S i

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Condition Numbers Norm Condition Number

Componentwise condition numbersParameter

i q Ai / / i

1 1.1182 1 1.11812 2.2722103 1.0161103 2.23623 1.1361105 1.0161103 111.81234 1.1361105 1.0161103 111.8123

A bA A

, 1

= 1.7599 103

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Parameter Identifiability Model identifiability

Local identifiability if Jacobian is full rank

)(y)(y **

)(y

)(J

17

Nearness to Unidentifiability Clossedness to an unidentifiable

system is given by

Information about the rank structure of J() is contained in its SVD.

Jcond

1Jd

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SVD Analysis Jacobian for linear system is given by A Singular Values of A

1.7320 > 1.0000 > 0.9999 >>> 0.006 Numerical rank of A is equal to 3

Only three degrees of freedom Estimate only 3 parameters: Parameter subset

selection which ones? Parameters associated with the three

most independent columns of A will give the best solution from the sensitivity point of view (i.e. 1,2,4)

Bias versus sensitivity tradeoff

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Reduced-Order Problemwith 3=0

([ :,[ \

([ :,[ \

.

.

.

1 1

1 1

0 8376

0 4514

0 0106

2 4]) 2 4]

1.0000

- 0.4900

0.0200

,

2 4]) = 2 4]

A b

A b e

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Condition Numbers for Reduced-Order Problem Normed condition number

Componentwise Condition NumbersComponent

i q Ai / / i

1 1.1094 0.9922 1.11812 2.2188 0.9922 2.23624 110.9339 0.9922 111.8079

A bA A

, 1

0= .9923

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)(ˆ)( αyyαr

Noise

TunningMechanism

Output Error

++

+

-

u

y

)u(t)D()x(t)C(y(t)

)u(t)B()x(t)A((t)x

)(αy

SynchronousGenerator

Synchronous Motor Parameter Estimation

Fig. 1: Output Error Formulation

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Synchronous Machine

)ΔVE(x

1Δi

)ΔVE(x

1Δi

ΔVxT

xxΔE

xT

x

dt

Ed

ΔVT

kΔV

xT

xx

xT

xxE

xT

xx

xT

xx

T

1ΔE

TT

1

dt

Ed

ΔVT

kΔV

xT

xxΔE

xT

xxΔE

T

1

dt

Ed

d"d"

qq

q"q"

dd

d"q

"qo

"qq"

d"q

"qo

q"d

fd'do

q"d

"do

"d

'd

"d

'do

'dd"

q"d

"do

"d

'd

"d

'do

'dd

"do

'q'

do"do

"q

fd'do

q"d

'do

'dd"

q"d

'do

'dd'

q'do

'q

Where:

,T"

qo"qq

"do

'do

"d

'dd TxxkTTxxxα

fd

md

R

xk

Linearized Model

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Simulated Waveforms

Fig. 2 Field Voltage

Fig.3 Direct and Quadrature Voltages

Fig. 4 Direct and Quadrature Currents

Simulated waveforms with 0.5% of added noise

0.01

0 2 4 6 8 10 12 -0.03

-0.01

0

Time/sec

0 2 4 6 8 10 12

-0.02

0

0.02

0.03

0.04

0.05

0.06

0.07

q

ΔV

ΔV d

0 2 4 6 8 10 12 -3.5

-3 -2.5

2

-1.5 1

- 0.5 0

0.5 x 10 -4

Time/Sec

fdΔV

-0.15

-0.1

-0.05

0 2 4 6 8 10

12 -0.2

0

0.05

0.15

0.1

Time/sec

qi

di

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PE and CA for Synchronous Generator

Parameter Nominal Estimate (%) Error Condition Number

xd 1.414 1.101 22.07 568

x’d 0.333 0.483 45.109 1.07*103

x”d 0.208 0.267 28.393 611

T’do 5.85 -5.092 187.048 4.52*103

T”do 0.194 0.108 44.148 893

k 1552 -703.757 145.345 3.73*103

xq 1.302 1.457 11.94 3.724

x”q 0.396 0.477 20.474 47.229

T”qo 0.955 1.0904 14.173 34.865

Table 1: Full order Parameter Estimation and Their Condition

Numbers

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SVD Analysis Information about the rank structure of J() is

contained in its SVD. Singular Values of J

9.3225 > 2.2216 > 1.0955 > 0.8187 > 0.3547 > 0.1587 >

0.1296 >> 5.8846*10-3 >>> 8.0444*10-7

Numerical rank of J is equal to 7

Only seven degrees of freedomEstimate only 7 parameters

Bias versus sensitivity tradeoff

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Dealing with Ill-Conditioning

Regularization techniques Subset Selection (SS)

2

2

2

2)(ˆˆ *2

TIKH ααλ αyyargminα

Tikhonov Regularization (TR)

*22

T2

T1

ααs.tαyyargminα

)α(αα

2

2)(ˆˆ

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Subset Selection

From SVD analysis we know that there are two parameters to be fixed

Subset selection analysis gives the idea of which parameters to fix

Chose subset which results in best conditioning for the Jacobian.

x’d,k – Our Result xd, k – Our Result T’do, k – Selected by Burth (1999)

[1] M. Burth, G.C. Verghese, M. Vélez-Reyes “Subset Selection for Improved

Parameter Estimation in on-line Identification of a Synchronous Machine”,

IEEE Trans. on Power System. Vol14 1 pp. 218-225 March 1999

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Error Percentages of PE with Regularization

Regularization Method

FullOrder

Subset Selection Tikhonov

xd, k x’d, k T’do ,k -

%Error %Error %Error %Error

xd - 0.638 0.038 10.89 22.07

x’d 0.28 - 0.118 3.55 45.11

x”d 13.885 11.389 5.664 3.42 28.39

T’do 0.539 0.896 - 25.04 187.05

T”do 3.67 12.133 6.594 11.16 44.148

k - - - 25 145.35

xq 0.13 0.0624 0.007 0.309 11.94

x”q 0.538 0.555 0.04 0.241 20.47

T”qo 0.212 0.0505 0.0124 0.37 14.17

Table 2: Error Percentages of Estimated Parameters

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Conditioning Analysis with Regularization

Table 4: Condition Numbers with Regularization

Condition Numbers

Subset Selection Tikhonov

Full order

xd ,k x’d , k T’do , k -

xd - 3.311 2.731 161.86 568

x’d6.178 - 10.792 278.09 1.07*103

x”d99.82

6106.62 95.164 379.3 611

T’do4.156 8.80 - 113.88 4.52*103

T”do117 121.701 113 117.05 893

k - - - 10.499 3.73*103

xq3.724 3.724 3.724 3.7324 3.724

x”q47.22

2 47.222 47.222 47.02 47.229

T”qo34.86 34.86 34.86 35.073 34.865

α

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Comparison

0.37-35.073 -T’do

0.241-47.02 -k

0.3090.01243.7324 2.731xd

250.0410.499 113T”do

11.160.007117.05 95.164x”d

25.046.594113.88 10.792x’d

3.425.664379.3 47.229x”q

3.550.118278.09 34.864T’qo

10.890.0384161.86 3.724xd

NoneT’do,kNoneT’do,kfixed

TRSSTRSS

% ErrorCondition Number

Methodologies

Subset selection is the best but requires good prior estimates.

In Tikhonov Regularization some initial guess is required but estimate all parameters.

Table 5: Result of Comparisons

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Comments

Local sensitivity analysis provide reasonable metrics to determine the parameter conditioning.

Parameters associated with the direct axis were the worst conditioned

Parameter estimated error were reduced by using regularization

Sensitivity was significantly reduced by regularization

Different formulations to deal with ill-conditioning give different results with different information requirements.

Best Performance overall was for subset selection Initial Guess Number of parameters to estimate

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Some Challenges WE NEED NEW ALGORITHMS

THAT CAN TACKLE NONLINEAR, COMPLEX AND LARGE-SCALE PROBLEMS

WE NEED INTEGRATED APPROACHES FOR DATA MANAGEMENT, MODELING, SOLUTION, ANALYSIS AND VISUALIZATION

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Conclusions

“...While it is hard to predict the future, creating it is much easier…”

“...While it is hard to predict the future, creating it is much easier…”

We have an exciting opportunity to shape an integrated approach to merging models and data

G.J. McRae, MIT Course 10

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Acknowledgements Alex’s work ECS-9820977 Miguel’s work ECS-9702860