Dagmar Niebur,NSF Workshop 11/03/03 1 Ekrem Gursoy, Huaiwei Liao and Dagmar Niebur Higher Order...

Dagmar Niebur, NSF Workshop 11/03/03 1

Ekrem Gursoy, Huaiwei Liao and Dagmar Niebur

Higher Order Statistical Techniques for Power System Signal Estimation and

Identification

*Center for Electric Power EngineeringDrexel University

Philadelphia, PA [email protected]

The authors gratefully acknowledge the support of the project from the National Science Foundation under Grant # 9985119


Motivation

Independent Component Analysis

Power System ApplicationsActive Load Profile EstimationReactive Load Profile EstimationHarmonic Source Profile EstimationPower System Topology Identification

Challenges

Conclusion

Outline


Power System Signal Processing

With advances in telecommunication and measurement technology (GPS, wireless, distributed sensors etc), data storage and computing power, power systems will be monitored more extensively and at a shorter time-scale

Current State of the Art in Traditional State, Load and Harmonic State EstimationSystem snapshots taken at fixed point in time t

Usually neglects of correlation of signals in time

Assumes Gaussian distribution of measurement deviations thus ignoring higher order statistics

Measurement deviations usually assumed to be entirely due to measurement noise

Requires complete knowledge of system topology and parameters

Requires accurate mathematical system model

Assumptions on availability of measurements at each relevant network metering point

Suitable for centralized estimation as opposed to local hierarchical estimation


Deregulation Independent power producers will engage in pool contracts with the load serving entities as well as bilateral contracts with individual industrial or residential customers

Challenge: How to forecast individual demand profiles?For many individual customers no historical hourly measurements exist Only aggregate substation measurements are usually availableFor spot-market pricing and energy trading only large industrial customers (>100 kW) are tariff metered on a half-hourly basis (Allera and Horsburgh, 1998)

Load profiles for smaller customer groups are estimated based onInterpretation of operating dataDerivation from half-hourly tariff metering data (when available)Estimation using statistical or simulation techniquesDirect monitoring of samples of customers

Objective: Establish daily customer daily active, reactive and harmonic load profiles using branch power or non-load bus voltage measurements without any knowledge of system topology or parameters

Study Case: Load Profile Estimation


Observed signals (party room sound) are linear mixtures of unknown statistically independent source signals (speakers)

Objective: Given xj(t), identify source signals sk(t) and mixing matrix A

for all t [0,T]

Tool: Independent Component Analysis

Motivation: Cocktail Party Problem

1 11 1 12 2 13 3

2 21 1 22 2 23 3

3 31 1 32 2 33 3

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

L t a S t a S t a S t

L t a s t a s t a S t

L t a S t a S t a S t


Independent Component Analysis for Load Profile Estimation

S1(t1…tm)

S2(t1…tm)

Sn(t1…tm)

Mixing

Matrix A

V1(t1…tm)

P1(t1…tm)

Demixing

Matrix W

V2(t1…tm)

P2(t1…tm)

Vn(t1…tm)

P1(t1…tm)

S1_es(t1…tm)

S2_es(t1…tm)

Sn_es(t1…tm)

Original Sources Sk(t1…tm)

Voltage Deviation orBranch Power Deviation Measurements Lk(t1…tm)

Estimate of Original Sources Sk_es(t1…tm)

Iterative Calculation of W

.

.

.

.

.

.

.

.

.


Problem Definition for Load Profile Separation

Assumption: Statistical independence and non-Gaussian distribution of fast-varying component of the individual loads

Linear superposition of load measurements

Objective: Estimating individual active, reactive or harmonic load profiles through split of the aggregate substation data into its different contributions with respect to active, reactive or harmonic power consumption.

Recognizing network topology and parameters from estimated mixing matrix.

Approach: Statistical technique called

Independent Component Analysis (ICA)

whose technical application is also known as

Blind Source Separation (BSS)


Independent Component Analysis Model

Assumptions:

1. Linear superposition (mixing) of original loads

Lj(ti) = Asj(ti) + ej

for all instants ti , i=1…T and residual Gaussian noise ej

2. The ensemble of loads sj(ti) are statistically independent at time ti

3. At most one of the loads is Gaussian distributed.

Given: Observed power or voltage measurements Lj(ti) RM at time ti, i=1…T

Unknown: Original individual active, reactive or harmonic loads sj(ti) RN

Mixing matrix A RM N

Goal: For M=N estimate original signals Sj_es(ti) = WLj(ti) Estimate de-mixing matrix W A-1

RN N


Independent Component AnalysisOptimization Problem

Goal: Estimate original signals Sj_es(ti) = WLj(ti) Ses = WL

Estimate de-mixing matrix W A-1 RN N

With less loads then time steps, i.e. (N<M) over-determined

Formulate as optimization problem with equality constraints

Optimization objective: Achieve independence of rows of Ses

Optimize J(wi) = E(G(wiTYj)2) for i = 1, ..., N,

subject to Sj_es= WLj

and wjTwj = jj for all columns wj of W

E denotes the statistical expectation

G is a non-linear “contrast” function that measures statistical independence.


Contrast Functions

Role of Contrast Functionto distinguish higher order statistics of components of Lto measure the degree of independence of the source signals.

Choices of G include (Hyvärinen, 1999):

1. Higher order statistical cumulants: Ex.: the 4th order kurtosis (Karhunen et al, 1998).

2. Minimal mutual information Ex.: Kullback-Leibler divergence (Haykin, 1998)

3. Maximum LikelihoodEx.: Maximum Entropy (Bell and Sejnowski, 1995)

4. Single unit general odd contrast function separator for supergaussian distributions

G(y) = -exp(-y2/2) (Hyvärinen, 1999)


PCA finds the directions of maximum variance

ICA finds the directions of maximum independence

Comparison of ICA and PCA


ICA Algorithm

1. Start with linear mixture of two uniform distributed random variables

2. Whiten mixture to remove correlation

3. Search projections to maximize non-gaussianity

4. Dewhiten to restore original signals & mixing matrix

1 1

2 2

5 10

10 2

l s

l s

Whitening

Maximize Nongaussianity

Restore


Linear Active Power Flow Model

Active bus power deviations: P=[P2, …, Pns]T

Angle deviations =[2, …, ns]T, where ns=# of busses.

Active branch power flow deviation PBranch = [P1_ij, ……, PN_i'j’]

Linearized nodal “DC flow” equations: P =B (1)

Branch flow for branch #k, k=1...N Pk_ij =Bij i - Bij j (2)

Branch flow matrix equation (3)

T is a rectangular matrix reflecting the imaginary part of the branch admittance (Im(Ybus) and the network topology (incidence matrix)

Active branch flow deviation measurements are linear mixtures of unknown active power load deviations

1

branch load P Tθ TB P

AL S


Linear Reactive Power Flow Model

1. Taylor expansion of power flow equations around operating point (V0,0)2. Weak coupling of P and V3. Constant power factor:

4. Linear reactive model:

At times ti

Voltage deviation measurements are linear mixtures of unknown reactive power load deviations

1 1

0 01

1

Q Q P

i it t for i T

f f fV V α Q Q

V δ δL S

A

L AS

0 0( ) P P α Q Q


Linear Harmonic Model

Linear circuit under non-sinusoidal conditions:

Ihj :phasor current at frequency h injected at node j and

Vhj: phasor voltage at frequency h at node j, j=1…N.

Yhi,j equivalent admittance at frequency h between nodes i and j.

Complex voltage measurements are linear mixtures of unknown complex currents

1 1 1 1 2 1 1

2 2 1 2 2 2 2

1 2

1

, , ,N

h h h h h

, , ,N

h h h h h

N N , N , N ,N N

h h h h h

I Y Y Y V

I Y Y Y V

I Y Y Y V

S LA


ICA Assumptions and Results

Assumptions:

1. Linear superposition (mixing) of original loads

Lj(ti) = Asj(ti) + ej

for all instants ti , i=1…T and residual Gaussian noise ej

2. The ensemble of loads sj(ti) are statistically independent at time ti

3. At most one of the loads is Gaussian distributed.

Results:

1. Estimated profiles in unknown order2. Estimated profiles up to unknown scaling factor


Independence and Gaussianity

1. Using a 4-point moving average low-pass filter M does not change mixing matrix:

2. We verified for zonal NYISO data that the fast components of L and S are statistically independent.

3. We verified for zonal NYISO data that L and S are super-gaussian distributed.

=> Aggregate load data is super-gaussian => Individual load data is most likely not gaussian.

slow slow

fast slow slow fast

L LM ASM AS

L L L AS AS AS


Eliminating Indeterminacy

ICA determines load profiles up to a re-ordering and scaling

1. Re-ordering through matching of estimated load shapes with historic ones by calculation of correlation coefficients.

2. Re-scaling as

Si_est(t) Estimated ith load shape corresponding to ith actual load,

ciSi_est(t)+bi Scaled ith estimated load approximation to ith actual load,

Wi Energy consumption of ith load from T0 to T1,

i Power factor for ith load.

Si,est_peak Peak of Si_est(t) ,

si,peakPeak of ith actual load, obtained from recording/ load forecasting,

N Number of actual reactive loads.

1

0

1

Ti i i i iT

i i _ est ,peak i i ,peak

c S _ est t b dt W

c S b S , i , ,N


Mixed Signals L Fast-varying

Component, Lf

Obtain Mixing Matrix and Estimated

Source Signals Q or P

Filtering

ICA

Estimated Load ProfilesRescaling &

Post-processing

Estimated Topology & Parameters

Prior Knowledge

Slowly-varying Component, Lslow

MeasureVoltage |V|

or Branch Power|P|

Load Profiles Separation Algorithm


PS Applications: Example 1Branch Active Power Flow as Mixed Signals

Goal:

Estimate active load profiles of bus 4, 5, 6, 9, 10, 11, 12, 13 and 14

from measurements of active power through branches ( circled ), 1-5, 4-5, 4-9, 6-12, 10-14, 13-14, 9-14 and 6-13.

IEEE 14 Bus System



4 8 12 16 20 24

26

47

4 8 12 16 20 24

5

7

4 8 12 16 20 24

21

29

4 8 12 16 20 24

8

9

4 8 12 16 20 24

2

3

4 8 12 16 20 24

3

6

4 8 12 16 20 24

7

13

4 8 12 16 20 24

7

14

Active Power Drawn by Load Bus

Bus 4 Bus 5

Bus 9 Bus 10

Bus 11

Bus 12

Bus 13

Bus 14

Source Signals: Active power drawn by load buses. Horizontal axis denotes time in hours, vertical axis denotes active power drawn by load bus in MW.

4 8 12 16 20 24

53

65

4 8 12 16 20 24-59

-47

4 8 12 16 20 24

10

13

4 8 12 16 20 24

3

4

4 8 12 16 20 24

-5

-3

4 8 12 16 20 24

3

6

4 8 12 16 20 24-1

3

8

4 8 12 16 20 24

8

14

Mixed Signals : Active Power Flow through Branches

1-5

4-5

4-9 6-12

10-11

13-14

9-14 6-13

Mixed signals: Active power flow through branches. The power flow injects the "From" end of branch.


0 4 8 12 16 20 24-1

0

1

0 4 8 12 16 20 24-1

0

1

0 4 8 12 16 20 24-1

0

1

0 4 8 12 16 20 24-1

0

1

0 4 8 12 16 20 24-1

0

1

0 4 8 12 16 20 24-1

0

1

0 4 8 12 16 20 24-1

0

1

0 4 8 12 16 20 24-1

0

1

0 4 8 12 16 20 24-1

0

1

0 4 8 12 16 20 24-1

0

1

0 4 8 12 16 20 24-1

0

1

0 4 8 12 16 20 24-1

0

1

0 4 8 12 16 20 24-1

0

1

0 4 8 12 16 20 24-1

0

1

0 4 8 12 16 20 24-1

0

1

0 4 8 12 16 20 24-1

0

1

Original Signals(Normalized) Reconstructed Signals(Normalized)



PS Applications: Example 2 Nodal voltage magnitude as Mixed Signals

Goal:

Estimate reactive load profiles of bus 4, 5, 9, 10, 11, 12, 13 and 14,

from measurements of voltage magnitude of these buses.


200 400 600 800 10001.0321.0341.0361.038

1.041.0421.044

200 400 600 800 10001.038

1.041.0421.0441.0461.048

200 400 600 800 1000

1.055

1.06

1.065

1.07

200 400 600 800 1000

1.05

1.055

1.06

200 400 600 800 1000

1.058

1.06

1.062

1.064

1.066

200 400 600 800 1000

1.062

1.064

1.066

1.068

200 400 600 800 1000

1.055

1.06

1.065

200 400 600 800 1000

1.04

1.05

1.06

Voltage magtitude as mixed signals when P,Q varying

PS Applications: Example 2 Nodal voltage magnitude as Mixed Signals

200 400 600 800 1000

0.2

0.4

0.6

0.8

1

200 400 600 800 10000.4

0.6

0.8

1

200 400 600 800 1000

0.6

0.8

1

200 400 600 800 1000

0.85

0.9

0.95

1

200 400 600 800 1000

0.2

0.4

0.6

0.8

1

200 400 600 800 1000

0.2

0.4

0.6

0.8

1

200 400 600 800 1000

0.2

0.4

0.6

0.8

1

200 400 600 800 1000

0.2

0.4

0.6

0.8

1

Original Load Profile of bus 4,5,9,10,11,12,13,14

Load profiles as source signals (active power and reactive power have same shape)

Nodal voltage magnitude as mixed signals


PS Applications: Example 2 Results

Estimated load profilesAbsolute percentage errors and mean percentage errors between normalized original signals and estimated ones

200 400 600 800 1000

-20

0

20

200 400 600 800 1000

-10

0

10

20

200 400 600 800 1000-20

-10

0

10

200 400 600 800 1000-10

0

10

200 400 600 800 1000

-10

0

10

20

200 400 600 800 1000

0

10

20

200 400 600 800 1000

-10

0

10

200 400 600 800 1000-10

0

10

20

Separated load profiles when P&Q varying, Vm as mixed signals

0

2

4

6

8

10

12

14

4 5 9 10 11 12 13 14

Bus #

Err

or

[%]

Max. Perc. Error [%]

Mean Perc. Error [%]


Qualitative Comparison of Smoothed Original and Estimated Signals

______: Original reactive profiles; ………: Estimated reactive profiles


System Parameter EstimationResults & AnalysisExample 2, Nodal voltage magnitude as mixed signals

Estimated de-mixing matrix-178.4518 61.9988 57.0370 -7.6979 4.7301 4.1592 11.4703 4.8742-615.6352 736.8732 -1.9885 8.0755 -24.6901 -20.8151 -56.5519 -11.8456 -1.3703 25.1752 -100.1960 20.4608 7.4773 0.2461 -11.0631 27.6159 -4.6217 9.3242 -196.1335 223.0159 -38.0283 2.7899 -3.2333 1.9740 0.6170 -9.6423 8.9327 -99.8930 112.5032 1.2090 4.9131 -1.4610 -28.3315 33.0683 16.1902 -8.9751 -2.5667 36.4178 -33.2520 -2.0882 14.4298 -23.3089 10.6519 -8.2563 6.0345 6.9161 -51.7313 22.5949 33.2886 -33.5190 14.7195 3.1457 0.5234 0.1404 15.8768 -61.8425

Corresponding B matrix (reduced) , 38.4851 -21.5786 -1.8500 0 0 0 0 0 -21.5786 35.2166 0 0 0 0 0 0 -1.8500 0 24.1502 -10.3654 0 0 0 -3.0291 0 0 -10.3654 14.7683 -4.4029 0 0 0 0 0 0 -4.4029 8.4970 0 0 0 0 0 0 0 0 5.4279 -2.2520 0 0 0 0 0 0 -2.2520 10.6697 -2.3150 0 0 -3.0291 0 0 0 -2.3150 5.3440

Used to estimate topology and parameters


PS Applications: Example 3Complex 5th Harmonic Voltage as Mixed Signals

3 similar harmonic sources (ASD) modeled as constant current source

Goal:

Estimate harmonic phasor current profiles of bus 3, 9 and 12

from measurements of harmonic voltage at bus 4, 6, 14.

1

2

3

5

6

12 11 10

9

4

1413

8

zz

z z z

7

G

G

ASD1

ASD2

ASD3

ASD Adjustable Speed Drive

G Generator

Harmonic voltagemeasurement


PS Applications: Example 3Complex 5th Harmonic Voltage as Mixed Signals

0 500 1000 15000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Estimated and Original Sources (Real, slow varying part)

0 500 1000 15000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Estimated and Original Sources (Imaginary, slow varying part)

0

0.5

1

1.5

2

2.5

3

3.5

HS1 HS2 HS3

Harmonic Source #

Err

or

[%] Mean Perc. Error

[%] Active Voltage

Mean Perc. Error [%]Reactive Voltage


Challenges

Investigate topology constraints for the mixing matrix

Investigate incomplete profiles and missing data

Identify topology and other parameters of electric power network

Develop and apply non-linear ICA for non-linear system identification and estimation

Study time-scales and real-time identification

Investigate minimum time windows for profile estimation, extend to state estimation

Bottleneck: Optimization procedure too slow for real-time computation


Conclusion

Feasibility study of application of Independent Component Analysis for active, reactive and harmonic load profile separation

Successful separations of load profiles from system measurements only without any prior knowledge about power electric network topology and branch impedances

Use of fast-varying component of load profiles to avoid statistical dependence between them

Estimating topology and parameters of electric power network by ICA separation process.

Dagmar Niebur,NSF Workshop 11/03/03 1 Ekrem Gursoy, Huaiwei Liao and Dagmar Niebur Higher Order...

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