Efficient Online Estimation of ElectromechanicalModes in Large Power Systems

F. J. De Marco, J. A. Apolinario Jr. and P. C. PellandaInstituto Militar de Engenharia

Praca General Tiburcio, 80, Praia Vermelha, 22290-270

Rio de Janeiro, RJ, Brazil

Emails: fernandojdemarco@gmail.com, apolin@ime.eb.br, pcpellanda@ieee.org

N. MartinsCEPEL

P.O. Box 68007, 20001-970

Rio de Janeiro, RJ, Brazil

Email: nelson@cepel.br

Abstract—This paper investigates the performance of a fastconverging adaptive filter, the Recursive Least Squares algorithmbased on the Inverse QR Decomposition (IQRD-RLS), with anexact initialization procedure, for the online estimation of low-damped electromechanical modes in a power system. In thisapproach, the modes are tracked from ambient data, once itis assumed that load variations constantly excite the electrome-chanical dynamics as a nearly white noise input. Monte Carlolinear simulations are run on the full Brazilian InterconnectedPower System model to generate power system ambient data. Theperformance of the IQRD-RLS algorithm is compared to that ofthe Least Mean Squares (LMS) algorithm when estimating theslowest interarea mode in the system.

I. INTRODUCTION

Power systems exhibit electromechanical modes of differ-

ent nature (from intraplant to interarea), whose frequencies

typically range from 0.2 to 2.5 Hz. Some of these modes

may show insufficient damping and their accurate online

estimation constitutes a critical element in the development of

situational awareness tools regarding oscillations to empower

modern control centers [1]. Power system modal analysis can

be accomplished by using two basic approaches: applying

eigenanalysis to a small-signal model or optimally fitting a

linear model to a measured system response [2].

Different measurement-based methods have been applied

to the three types of measured data: ambient data, ringdown

signals and probing responses. The adaptive filtering approach

allows having a near real-time estimate of the dominant system

mode characteristics based on measured ambient data, which

can be collected from synchrophasor measurements captured

by existing monitors that have been strategically placed in the

power system. The basic assumption for this approach is that

there are practically continuous random changes in the system

loads that excite slightly the electromechanical dynamics of

the system causing ripple-like disturbances which are known

as ambient noise, in the measurements of voltage, current,

and power signals. Assuming the random variations are white,

the electromechanical modal frequencies and damping are

estimated from the spectral content of the ambient noise [3].

This work was motivated by the IEEE Task Force Report on

Identification of Electromechanical Modes in Power Systems

released in June 2012 [4], which describes only a very simple

adaptive filtering method to estimate electromechanical modes

from ambient data, which is the Least Mean Squares (LMS)

algorithm. The use of the standard LMS algorithm, the LMS

with adaptive step-size, and a combination of them have

been investigated in [5]-[9]. Although the LMS technique

offers computational simplicity, its performance, apart from

experiencing initialization problems, strongly depends on the

correlation of the input signal. This paper investigates the

implementation of an adaptive filter using the Recursive Least

Squares algorithm based on the Inverse QR Decomposition

(IQRD-RLS). RLS algorithms are known to have fast conver-

gence even when the eigenvalue spread of the input signal

correlation matrix is large.

In [10], the application of the conventional RLS algorithm to

the identification of power system modes based on measure-

ment data is studied. To make the estimation less sensitive

to the large deviation from the assumed noise models (due

to outliers or ringdown data) than the conventional quadratic

criterion, [10] proposes to change the loss function when

large prediction errors are detected, and then use a Newton-

Raphson-type method to solve recursively the Least Squares

(LS) problem. This paper utilizes an improved RLS algo-

rithm employing the QR decomposition to triangularize the

input data matrix, leading to better numerical behavior [11].

Also, the Inverse QRD-RLS algorithm (IQRD-RLS) [12] pro-

vides the filter’s coefficient vector at every iteration, allowing

a direct mode estimation instead of an additional (back-

substitution) routine to compute these weights (as needed by

the conventional QRD-RLS algorithm). Aiming at a faster

convergence without the odds of a soft initialization, we have

employed an exact initialization scheme. We note that the

numerical robustness comes with no extra computational com-

plexity when compared to the conventional RLS algorithm,

i.e., O[N2] multiplications, where N is the order of the filter.

To generate power system ambient data, Monte Carlo lin-

ear simulations, assuming a random step disturbance vector

applied to a set of load buses, are run on the full Brazilian

Interconnected Power System (BIPS) model, released by the

Brazilian System Operator (ONS) in June 2011 [13]. The

978-1-4673-4900-0/13/$31.00 c© 2013 IEEE

IQRD-RLS and the LMS algorithms are then applied to the

simulation results of each Monte Carlo iteration and the

statistics are calculated from this set of simulation results. The

performances of these adaptive algorithms are compared when

estimating the North-South mode of the system.

II. ADAPTIVE ALGORITHMS FOR MODE ESTIMATION

This section presents the topology of the adaptive filter

used to estimate the electromechanical modes, along with the

algorithms to adapt its coefficients used in the study case of

this paper.

A. Whitening filter

The assumption that ambient noise in a power system is

caused by an underlying white noise driving function allows

the use of a whitening filter approach for electromechanical

modes estimation [3]. The input samples are unknown in

practice, and the ambient output of the power system can be

processed by a whitening filter, as shown in Fig. 1, in which

additive noise is added to the power system output.

+

WhiteWhite

Additive

noise

noise

noise, e(k)

Sampled

data, u(k)G(z) =

1

D(z)F (z)

WHITENINGFILTER

POWERSYSTEM

Fig. 1: Whitening filter processing the power system output.

In order to whiten the data as shown in Fig. 1, the denomi-

nator polynomial D(z) of the power system transfer function

G(z) is approximated with a high order FIR (Finite Impulse

Response) filter in the z-domain, described by the transfer

function

F (z) = 1− w0z−1 − w1z

−2 − ...− wNz−(N+1). (1)

The whitening filter is conceived as the signal predictor

whose block diagram is shown in Fig. 2. For each sample

k, the adaptive algorithm calculates the weights vector w(k),whose elements are the coefficients of the Nth-order polyno-

mial

W (z) = w0 + w1z−1 + ...+ wNz−N . (2)

The weights vector can be initialized as a zero vector (cold

start), through a block processing approach [5] or using the

results of previous runs.

+

−

ADAPTIVE

ALGORITHM

u(k) x(k)

e(k)

y(k)

d(k)

W (z)z−1

Σ

Fig. 2: Linear predictor used for obtaining the whitening filter.

The zi roots of F (z) are calculated and the s-domain poles

si are obtained through the impulse invariance transformation

as si = ln(zi)/T , where T is the sampling period. The

dominant modes (true system modes) within a specified region

of the s-plane are identified as those with highest pseudo-

energy in the signal, calculated as proposed in [2].

B. The Least Mean Square (LMS) algorithm

The objective function used by the LMS algorithm is an

instantaneous estimate of the mean-square error defined as

ξ(k) = E[e2(k)], (3)

where, as illustrated in Fig. 2, e(k) = d(k) − y(k) is the

prediction error, d(k) = u(k) is the reference signal, and

x(k) = u(k − 1) the input signal of the adaptive filter. The

updating equation of the LMS algorithm is given as [11]

w(k + 1) = w(k) + µe(k)x(k), (4)

where x(k) = [x(k)x(k − 1) · · ·x(k − N)]T , e(k) = d(k) −xT (k)w(k) and the convergence factor µ should be chosen in

a range to guarantee convergence. The convergence speed of

the LMS is known to be strongly dependent on the eigenvalue

spread of the input signal autocorrelation matrix.

C. The IQRD-RLS algorithm

The RLS algorithm seeks to minimize the deterministic

objective function

ξd(k) =k

∑

i=0

λk−i[d(i)− xT (i)w(k)]2, (5)

where λ is a forgetting factor chosen in the range 0 ≪ λ ≤ 1.

Let the input data matrix and the desired response vector,

respectively, be

X(k) = [x(k) λ1/2x(k − 1) . . . λk/2x(0)]T , and (6)

d(k) = [d(k) λ1/2d(k − 1) . . . λk/2d(0)]T . (7)

The optimal Least Squares (LS) solution is obtained when the

coefficient vector satisfies

w(k) = R−1(k)p(k), (8)

where R(k) = XT (k)X(k) is the input-data deterministic au-

tocorrelation matrix, and p(k) = XTd(k) is the deterministic

cross-correlation vector. The RLS problem may be solved by

means of a QR decomposition, which is numerically well-

conditioned and is based on the following triangularization of

the input data matrix

Q(k)X(k) =

[

0

U(k)

]

. (9)

Matrix U(k) has a triangular structure obtained through matrix

Q(k); the later, in the recursive version of this class of

algorithms, can be expressed in a fixed dimension product of

Givens rotations partitioned as

Qθ(k) =

[

γ(k) gT (k)f(k) E(k)

]

. (10)

The IQRD-RLS algorithm, by updating the inverse of the

Cholesky factor U(k), allows the calculation of the weights

vector without back-substitution using the updating equation

w(k) = w(k − 1)− γ(k)e(k)u(k), (11)

where vector u(k) = λ−1/2U−1(k − 1)g(k) is obtained as

a by-product of the updating process from U−1(k − 1) to

U−1(k). See [12] for more details.

III. SIMULATION DATA

Ambient power system data is generated using the dynamic

data base of the BIPS, whose single-line diagram is shown in

Fig. 3. The linearized power system model of about 4,600

buses, 210 generating plants and 4,400 state variables, is

generated using CEPEL’s software PacDyn, and time-domain

simulations are run in Matlab. The system comprises five

interconnected geographical regions, being its slowest elec-

tromechanical mode between the Northern and Southern areas,

whose frequency ranges between 0.20 and 0.40 Hz [14].

Fig. 3: Brazilian Interconnected Power System (BIPS) with

eight candidate lines for active power measurement.

Defining λs as the dominant mode (pole) in the signal,

the mean values of its frequency (fd = ℑm{λs}/2π) and

damping ratio (ξ = −ℜe{λs}/|λs|) estimates are calculated

using the Monte Carlo method with 100 independent linear

simulations. For these trials, random variations of active power

are independently injected to the 40 largest system loads

in the system, which represent 10 % of the total demand.

Measurement noise is added to the system outputs, so that the

signal to noise ratio is 20 dB. The mean value (DC) is removed

from the simulated data, which is then low-pass filtered using

a 6th order Butterworth filter with a cutoff frequency of 2 Hz;

data is then resampled at 10 Hz rate. The real power flow

on the line Gurupi-Miracema 500 kV (signal #1 in Fig. 3)

is selected as the input to the adaptive algorithms tested in

Section IV due to the fact that it has high observability of the

North-South mode; this can be seen from the power spectrum

density of the measured active power shown in Fig. 4.

0 1 2 3 4 50

2

4

6

Frequency (Hz)

Po

we

r S

pe

ctr

um

De

nsity

Fig. 4: Power spectrum density of the active power measured

at line #1.

IV. MODE ESTIMATION RESULTS

In order to compare the performances of the adaptive

algorithms, the North-South mode of the BIPS is estimated by

applying the IQRD-RLS and the LMS algorithms to the set of

100 Monte Carlo simulations. Typical filter orders range from

20 to 30 [4], [5]; we have used a filter of order 20. The North-

South mode obtained from the linear system eigensolution is

λNS = -0.4022 + j2.4300, while the mean value of the LS

solutions for the 100 simulations is λLS = -0.3390 + j2.2974.

This bias is attributed to the use of an all-pole model instead

of an autoregressive-moving average (ARMA) model for the

power system [4], [5]. The condition number (eigenvalue

spread) of the measured signal autocorrelation matrix for the

first simulation is 7.6e+5.

The mean estimates of frequency and damping ratio cal-

culated with the LMS algorithm are shown in Fig. 5 for

µ1 = 3.6e−4 and µ2 = 1.98e−4, starting from cold start. Fig. 5

also presents the results obtained with µ3 = µ1/10, starting

from suboptimal solutions with a mean offset of 0.02 Hz in

frequency and 4.4 % in damping; these solutions are obtained

from previous 5-minutes LMS runs with µ1. Note that when

µ is increased, higher speed of convergence is achieved. Note

also that the initialization of the weights vector is critical.

0 5 10 150.3

0.35

0.4

0.45

0.5

Time (minutes)

Me

an

fre

qu

en

cy (

Hz)

0 5 10 150.1

0.15

0.2

0.25

Time (minutes)

Me

an

da

mp

ing

(p

u)

Cold start, µ1 = 3.60e−004

Cold start, µ2 = 1.98e−004

Start with offset, µ3 = 3.60e−005

Mean LS solution

Fig. 5: Mean frequency and damping ratio LMS estimates for

cold start, and starting from a solution with offset.

The mean frequency and damping estimates obtained with

the LMS algorithm are compared in Fig. 6 with the estimates

of the IQRD-RLS algorithm for the two forgetting factors

λ = 1−α that result from considering α1 = 7.5e−4 and

α2 = α1/75. The exact initialization procedure [12] is used

to calculate the tap weights in the first N + 1 iterations, so

the influence of the initialization on the speed of convergence

of the IQRD-RLS algorithm is minimized. As it can be seen,

the speed of convergence of the IQRD-RLS algorithm depends

much less on λ (or α) than what the LMS algorithm depends

on µ.

0 5 10 150.3

0.35

0.4

0.45

0.5

Time (minutes)

Me

an

fre

qu

en

cy (

Hz)

0 5 10 150.1

0.15

0.2

0.25

Time (minutes)

Me

an

da

mp

ing

(p

u)

LMS (Cold start), µ1 = 3.60e−004

LMS (Cold start), µ2 = 1.98e−004

IQRD, α1 = 7.50e−004

IQRD, α2 = 1.00e−005

Mean LS solution

Fig. 6: Mean values of the estimates versus time calculated

with the LMS and the IQRD-RLS algorithms.

The variances of the estimates as a function of time are

presented in Fig. 7 for both algorithms. These variances

increase as the forgetting factor is reduced (i.e., as α is

increased) in the IQRD-RLS algorithm and as µ is increased

in the LMS algorithm.

0 5 10 150

1

2

x 10−4 Comparison of the variances of the estimates

Time (minutes)

Va

ria

nce

of

fre

qu

en

cy

0 5 10 150

0.5

1

1.5

2

2.5x 10

−3

Time (minutes)

Va

ria

nce

of

da

mp

ing

LMS (Cold start), µ

1 = 3.60e−004

LMS (Cold start), µ2 = 1.98e−004

IQRD, α1 = 7.50e−004

IQRD, α2 = 1.00e−005

Fig. 7: Variances of the estimates versus time calculated with

the LMS and the IQRD-RLS algorithms.

V. CONCLUSION

This paper shows that the IQRD-RLS algorithm presents

a better performance in terms of time of convergence and of

variance in the ensemble of the mode estimates than the LMS

algorithm. Selection of the step-size parameter and an initial

weight vector close to the optimum weight vector is critical

in using the LMS algorithm, which is known to be affected

by the high condition number (>>1) of the power system

output signals [11]. On the other hand, the exact initialization

procedure makes the IQRD-RLS algorithm independent of the

selection of an initial weights vector (when we do not have

an offset for a proper soft start), while the speed of algorithm

convergence is not much sensitive to the chosen forgetting

factor (typically near 1). The excellent performance of RLS

algorithms when working in time-varying environments [11]

shall be further explored when tracking non-stationary modes.

ACKNOWLEDGMENT

The authors thank the Brazilian Agencies CNPq (contracts

471230/2011-1 and 309846/2011-0) and CAPES for partially

funding this work.

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