Stochastic Estimation of Fundamental and Harmonic Signal Components

Chukwuemeka Aduba

Naval Surface Warfare Center, Philadelphia Division Philadelphia, Pennsylvania 19112, U.S.A.

Keywords:

Distortion, Ensemble, Harmonics, Kalman, Optimal Estimator, Power Quality, Power System, Sub-Optimal

Estimator.

Abstract:

The paper investigates the estimation of fundamental and harmonic components in power system signal using

stochastic estimator concept. The power system signal is approximated with a stochastic linear system model

where the phase and amplitude components are estimated using a Kalman ﬁlter (KF) and an Ensemble Kalman

ﬁlter (EnKF). The power system signal is modeled in both continuous and discrete form and then represented in

state-space approach. Simulation results show that EnKF estimates converge to KF estimates as the ensemble

size increases while reducing the computational complexity for highly-dimensional stochastic systems.

1 INTRODUCTION

Power systems have continued to evolve in recent his-

tory as designers and researchers propose different

power system architecture for commercial and mili-

tary shipboard applications. In shipboard application,

an overwhelming power system architecture proposal

has been one that integrates the propulsion (mobility)

with all mission and support load to form an inte-

grated power system (IPS) (McCoy, 2015). Further,

the IPS is improved with zonal electrical distribution

(ZED) model for increase ﬂexibility and reliability

as the system can source power from several direc-

tions without compromising performance due to un-

foreseen system faults. However, the integration of

propulsion and service or mission load still presents

power quality challenges in shipboard non-hybrid or

hybrid-based micro-grid. These micro-grids while

AC/DC in nature, are clusters of distributed genera-

tors, active and passive devices, energy storage sys-

tem and linear/nonlinear loads (Wang et al., 2019).

Similarly, the widespread utilization of power

electronics interfaces, proliferation of nonlinear loads

and power-electronics-based industrial load devices

have led to power quality concerns due to the result-

ing harmonic disturbances (Wang et al., 2014). Har-

monics are introduced in the power system as a re-

sult of nonlinear behavior of power-electronic inter-

faces and power-electronic-based industrial load to

sinusoidal current. This ﬂow of sinusoidal current

results in eventual non-sinusoidal periodic current.

The non-sinusoidal periodic current propagates and

interacts with the system impedance resulting in non-

https://orcid.org/0000-0001-6017-7287

sinusoidal periodic load voltage otherwise referred to

as voltage harmonics.

In frequency domain, harmonics are spectral com-

ponents of a distorted periodic signal whose frequen-

cies are integral multiples of the fundamental fre-

quency. These harmonics are undesirable in power

system networks due to the immediate and long-

term detrimental effects on power quality such as

degradation of electrical equipment, overheating of

transformers and malfunction of metering devices

(Farzanehrafat and Watson, 2013). To alleviate or

mitigate or possibly eliminate harmonic distortion, it

is imperative to be able detect, identify and classify

them. In this work, harmonic parameter estimation

for a linearly modeled system is explored to gain an

understanding into possible harmonic identiﬁcation

process.

Earlier studies of power system harmonics estima-

tion were reported in (Girgis et al., 1991), (Ma and

Girgis, 1996) while the more recent studies in har-

monics analysis and estimation have been described

in (Farzanehrafat and Watson, 2013), (Medina et al.,

2013) and (Wang et al., 2014). Girgis et al. (Girgis

et al., 1991) considered harmonics in power systems

based on optimal measurement scheme while the au-

thors in (Ma and Girgis, 1996) considered the identi-

ﬁcation and tracking of harmonic sources using KF.

In (Farzanehrafat and Watson, 2013), the researchers

investigated power quality state and three-phase state

transient state estimation. The authors in (Medina

et al., 2013), presented an overview of frequency-

domain, time-domain and hybrid frequency-time har-

monic analysis. In addition, the constraints and draw-

backs were highlighted for practical power network.

Aduba, C.

Stochastic Estimation of Fundamental and Harmonic Signal Components.

DOI: 10.5220/0012183400003543

In Proceedings of the 20th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2023) - Volume 2, pages 34-40

ISBN: 978-989-758-670-5; ISSN: 2184-2809

In (Wang et al., 2014), the modeling and anal-

ysis for harmonic stability problems in AC power

electronics-based power systems were performed to

gain insights into stability issues. Harmonic elimina-

tion or mitigation approaches have been through pas-

sive, active and hybrid ﬁltering involving integrated

passive/active ﬁlter combination. In addition, discrete

fourier transforms (DFT) and fast fourier transforms

(FFT) have been applied to harmonics ﬁltering in-

volving stationary signal. However, both transform

techniques exhibit poor results in time-varying fre-

quency systems (Girgis et al., 1991).

Traditionally, linear system ﬁltering application

has employed Kalman ﬁlters as optimal estimators.

EnKF has been proposed for signal processing of

harmonics in (Ray and Subudhi, 2012). The main

application for EnKF has been in geophysical sci-

ence for weather forecasting, hydrological model-

ing (Evensen, 1994) and more recently in medical

science such as epidemiology modeling (Lal et al.,

2021). EnKF is a random sampling implementation

of KF for high-dimensional, possibly nonlinear and

non-gaussian state estimation problem. It is related

to particle ﬁlter where the particle is the ensemble.

The starting point for EnKF is choosing a set of sam-

ple point (that represent an ensemble of the state esti-

mates) to capture the initial probability distribution of

the state. The ﬁnal states are estimated by assimilat-

ing the observation into the state model. For a tutorial

on EnKF including application areas, see (Evensen,

2009), (Roth et al., 2017).

The contributions in this study include formula-

tion of harmonic signal in continuous and discrete

fourier series form, detailed formalized equations for

the linear estimators and performance evaluation of

the estimators. The remainder of this paper is or-

ganized as follows: Section 2 models the harmonic

signal in both discrete and continuous form. In addi-

tion, the signal system is formulated in state space ap-

proach. Section 3 describes the KF and EnKF includ-

ing the recursive equations and the algorithm while

Section 4 illustrates a simulation example as a case

study. Finally, the conclusions including future direc-

tions are given in Section 5.

2 HARMONIC SIGNAL MODEL

2.1 Signal Model

The power system harmonic signal is modeled as a

distorted waveform in both continuous and discrete

form.

2.1.1 Continuous Harmonic Signal

Let the power system harmonic signal Y (t) at time

t be modeled as a distorted waveform in continuous

form as:

Y (t) =

∑

i=1

sin



t + θ



∑

i=1

(t),

(1)

where i = 1,2,... , N with N as the harmonic order.

The angular frequency, amplitude and phase angle of

the harmonic signal are given by w

, A

and θ

respec-

tively. The fundamental frequency is given as f with

= i2π f . For instance, w

= 2π f , w

= 4π f and

= 6π f . The ω

(t) is the additive noise component

with ω

= ω

= ... = ω

2.1.2 Discrete Harmonic Signal

Similarly, let the power system harmonic signal Y

k time step be modeled as a distorted waveform in

discrete form as:

∑

i=1

sin



+ θ



∑

i=1

(2)

where T

is the sampling time. The ω

is the additive

noise component. Similarly, the angular frequency,

amplitude and phase angle of the harmonic signal are

given by w

, A

and θ

similar to the continuous signal

case. The signal is nonlinear in phase and linear in

amplitude. In parametric form, the signal in (2) can be

converted to linear form as shown below by applying

trigonometric identities giving:

∑

i=1

sin(w

)cos(θ

) + A

cos(w

)sin(θ

)

∑

i=1

(3)

In compact form, the signal depicted in (3) can be

represented as:

= M

L +

∑

i=1

(4)

sin(w

) cos(w

) . . .. . .

... . .. sin(w

) cos(w

)

(5)

L =

cos(θ

) A

sin(θ

) A

cos(θ

) A

sin(θ

)

... . .. A

cos(θ

) A

sin(θ

)

(6)

Stochastic Estimation of Fundamental and Harmonic Signal Components

with L as the vector of in-phase and quadrature phase

components. The state equation (x

) in linear form

for the signal system (3) can be modeled as:







1 0 ... 0 0

0 1 ... 0 0

0 0 ... 1 0

0 0 ... 0 1













2N−1







(7)

where x

= A

cos(θ

), x

= A

sin(θ

) and x

2N−1

cos(θ

), x

= A

sin(θ

). The W

is the additive

noise component. In addition, the measurement equa-

tion (z

) in linear form for the signal system (3) can

be modeled as:







sin(w

)

cos(w

)

sin(w

)

cos(w

)













2N−1







(8)

where V

is the additive noise component.

3 ESTIMATOR MODEL

In this section, the estimators will be modeled based

on the harmonic signal structures in earlier section.

3.1 Kalman Filter

The Kalman ﬁlter is an optimal estimator for a linear

stochastic system corrupted by gaussian noise. The

details of the Kalman ﬁlter in discrete-time format are

given as follows. Let the linear state and measurement

dynamics be given as:

= Φ

k−1

+ w

k−1

= H

k−1

+ v

k−1

(9)

where Φ

k−1

is the state transition matrix, H

k−1

is the

observation matrix, x

∈ R

is the state vector, w

and

are gaussian random processes deﬁned on a proba-

bility space (Ω

,F,P) where Ω

is a nonempty set, F

is a σ-algebra of Ω

and P is a probability measure on

(Ω

,F). The initial state vector ˆx

= E(x

) has initial

covariance matrix P

= E



− ˆx

)(x

− ˆx

)



and z

is the initial output vector. The gaussian random pro-

cess w

has zero mean and covariance of E(w

) =

. Similarly, the gaussian random process v

has

zero mean and covariance of E(v

) = R

. The noise

processes w

, v

and x

are assumed uncorrelated.

The following Φ

k−1

are assumed to have appro-

priate dimensions. The sequential recursive computa-

tion steps for the Kalman estimator is summarized as

follows (Crassidis and Junkins, 2012):

ˆx

k/k−1

= Φ

k−1

ˆx

k−1/k−1

k/k−1

= Φ

k−1

k−1/k−1

k−1

+ Q

= P

k/k−1

k−1



k−1

k/k−1

k−1

+ R

k−1



−1

ˆx

k/k

= ˆx

k/k−1

+ K



− H

ˆx

k/k−1



k/k



I − K

k−1



k/k−1

(10)

The ﬁrst-two lines in (10) are the prediction or time

update steps while the last-three lines in (10) are the

analysis or measurement update steps. The analysis

steps adjust the prediction steps. In this sub-section,

the estimator described was envisaged for linear ap-

plication. However the nonlinear equivalent of (9) is

given as:

= F(x

k−1

) + w

k−1

= G(x

k−1

) + v

k−1

(11)

where F(.) and G(.) are nonlinear functions of state.

Remark: In general, extended Kalman Filter (EKF)

is applied for nonlinear gaussian systems estimation

while KF remains the optimal estimator for linear

gaussian system.

3.1.1 Algorithm 1

Data: Actual signal

Result: Amplitude and Phase

initialize the state;

initialize the covariance;

while for a ﬁxed output size N do

estimate the state;

estimate the covariance;

compute the ﬁlter gain;

estimate the output error;

update the state with measurement;

update the covariance with measurement;

end

Algorithm 1: KF Estimator Algorithm.

3.2 Ensemble Kalman Filter

The Ensemble Kalman ﬁlter is an estimator with ap-

plication to large-dimensional nonlinear system. For

large states, maintaining the state covariance matrix

can lead to computational instabilities, thus EnKF al-

lows sample covariance instead of state covariance to

be utilized in the process. In addition, the state distri-

bution in EnKF is represented by ensemble of a ﬁxed

size forecasted state estimates with random noise.

Let the ensemble Z

be represented as:



(1)

, x

(2)

, .. ., x

(N−2)

, x

(N−1)

, x

(N)



(12)

ICINCO 2023 - 20th International Conference on Informatics in Control, Automation and Robotics

where Z

is an n × N matrix, whose columns of R

are the samples from prior distribution. In terms of a

harmonic signal X



(1)

, x

(2)

, .. ., x

(2N−2)

, x

(2N−1)

, x

(2N)



(13)

with X

deﬁned as n × 2N matrix, whose column are

. As in equation (10), the sequential recursive com-

putation steps for the EnKF is summarized as follows

(Crassidis and Junkins, 2012), (Gillijns et al., 2006):

ˆx

(i)

k/k−1

= Φ

k−1

ˆx

k−1/k−1

+ w

(i)

k−1

¯x

k/k−1

∑

i=1

ˆx

(i)

k/k−1

(i)

k/k−1

= H

k−1

ˆx

(i)

k/k−1

¯z

k/k−1

∑

i=1

(i)

k/k−1



ˆx

(1)

k/k−1

− ¯x

k/k−1

,.. . , ˆx

(2N)

k/k−1

− ¯x

k/k−1





(1)

k/k−1

− ¯z

k/k−1

,.. . ,z

(2N)

k/k−1

− ¯z

k/k−1



2N − 1

)

2N − 1

)

(

)

−1

ˆz

(i)

k/k−1

= z

k/k−1

+ v

(i)

k/k

= ˆx

(i)

k/k−1

+ K



ˆz

(i)

k/k−1

− z

(i)

k/k−1



(14)

where the ensemble mean for the input and output are

given as ¯x

k/k−1

and ¯z

k/k−1

. The ensemble cross co-

variance and covariance are given as

and

The perturbed measured signal is ˆz

(i)

k/k−1

while the

measured signal is z

k/k−1

. The ﬁrst-four lines in (14)

are the prediction or time update steps while the last-

seven lines in (14) are the analysis or measurement

update steps. The analysis steps adjust the prediction

steps through the use of new measurement.

For additional signal parameter analysis, the am-

plitude A

i,k

and phase θ

i,k

at any time step k can be

computed following trigonometric identities as:

i,k



ˆx

2i−1,k

+ ˆx

2i,k



i,k

= arctan



ˆx

2i,k

ˆx

2i−1,k



(15)

where i = 1,2,3 ... , N − 2,N − 1, N.

Remark: For linear gaussian systems, EnKF esti-

mates should converge to KF estimates with increas-

ing ensemble size.

3.2.1 Algorithm 2

Data: Actual signal

Result: Amplitude and Phase

initialize the state;

initialize the covariance;

initialize the ensemble;

while for a ﬁxed output size N do

estimate the ensemble;

estimate the perturbed output;

estimate the cross covariance;

estimate the covariance;

compute the ﬁlter gain;

update the ensemble;

end

Algorithm 2: EnKF Estimator Algorithm.

4 SIMULATION EXAMPLE

Consider the following signal harmonic which con-

sists of sinusoidal continuous signal plus random

noise given as:

X(t) = 5

sin(2 × π × f

×t + 70

◦

)

+ 0.2 sin(2 × π × f

×t + 50

◦

)

+ 0.12 sin(2 × π × f

×t + 45

◦

)

+ 0.07 sin(2 × π × f

×t + 30

◦

)

+ 0.04 sin(2 × π × f

×t + 25

◦

)

+ 0.002µ(t),

(16)

where the gaussian random signal µ(t) has a mean of

zero with covariances R > 0, Q > 0. The problem is to

estimate the harmonic signal amplitudes and phases

using the KF and EnKF estimators. The following

parameters are deﬁned:

E(w

) = 0, E(w

) = Q

E(w

) = 0;( j ̸= k), E(v

) = 0,

E(v

) = R

, E(v

) = 0;( j ̸= k),

E(w

) = 0;( j∀k), f

= 3.0 kHz,

= 60 Hz, f

= 180 Hz, f

= 300 Hz,

= 420 Hz, f

= 540 Hz,

0/0

= 0.002I

10x10

= 5[0.34 0.93 0.12 0.15 0.08 0.08 0.06

0.03 0.03 0.01]

′

R = 3.6e − 3, Q = 3.6e − 3I

10x10

(17)

Figures 1, 2 and 3 show the amplitude and phase

estimation for Kalman ﬁlter-based algorithm while

Stochastic Estimation of Fundamental and Harmonic Signal Components

Figures 4, 5 and 6 show the corresponding amplitude

and phase estimation for the Ensemble Kalman ﬁlter-

based algorithm for the fundamental and harmonic

components of the given signal. For linear systems,

EnKF-based algorithm results converges to KF-based

algorithm results as expected for increased number of

ensembles (Roth et al., 2017).

Figures 7, 8 and 9 show the estimated signal for

both ﬁlters against the true signal and the signal fre-

quency spectrum. The frequencies can be estimated

by adjusting the formulation in (5) and (6) respec-

tively. Figures 10, 11 and 12, the mean square error

(MSE) for the signal estimation was captured for dif-

ferent ensemble sizes. From the MSE ﬁgures, it can

be observed that as the ensemble size increased, the

estimated signal mirrors the true signal.

0 0.01 0.02 0.03 0.04 0.05 0.06

Time (sec)

Amplitude (p.u)

Figure 1: Kalman Filter Amplitude Estimate for A

0 0.01 0.02 0.03 0.04 0.05 0.06

Time (sec)

Amplitude (p.u)

Figure 2: Kalman Filter Amplitude Estimate for A

5 CONCLUSION

In this paper, an application of Kalman ﬁlter and En-

semble Kalman ﬁlter estimator to power system har-

monic analysis was performed to get insights into sig-

0 0.01 0.02 0.03 0.04 0.05 0.06

Time (sec)

-80

-60

-40

-20

100

Phase Angle(degree)

Figure 3: Kalman Filter Phase Estimate for φ

,φ

0 0.01 0.02 0.03 0.04 0.05 0.06

Time (sec)

Amplitude (p.u)

Figure 4: Ensemble Kalman Filter Amplitude Estimate for

0 0.01 0.02 0.03 0.04 0.05 0.06

Time (sec)

0.5

1.5

2.5

Amplitude (p.u)

Figure 5: Ensemble Kalman Filter Amplitude Estimate for

nal detection in the presence of noise harmonics. A

representative signal of the distorted harmonic signal

was modeled and utilized in the simulation. The sim-

ulation study results show the effectiveness of the ﬁl-

ter algorithm. It was observed that the error covari-

ance parameter adjustments improved estimation re-

sults.

ICINCO 2023 - 20th International Conference on Informatics in Control, Automation and Robotics

0 0.01 0.02 0.03 0.04 0.05 0.06

Time (sec)

-100

-80

-60

-40

-20

100

Phase Angle(degree)

Figure 6: Ensemble Kalman Filter Phase Estimate for

,φ

0 0.01 0.02 0.03 0.04 0.05 0.06

Time (sec)

-8

-6

-4

-2

Amplitude (p.u)

Real

Estimated

Figure 7: Kalman Filter Signal of Actual (Real) and Esti-

mate.

0 0.01 0.02 0.03 0.04 0.05 0.06

Time (sec)

-8

-6

-4

-2

Amplitude (p.u)

Real

Estimated

Figure 8: Ensemble Kalman Filter Signal of Actual (Real)

and Estimate.

A further investigation is to consider time-varying

harmonic signal in amplitude and phase with addi-

tional estimators for evaluation. Another area of re-

search is analyzing captured data from a system im-

plementation to compare with simulation results.

0 100 200 300 400 500 600

Frequency (Hz)

Amplitude (p.u)

Figure 9: Single-Sided Amplitude Spectrum for the Actual

Signal.

0 0.01 0.02 0.03 0.04 0.05 0.06

Time (sec)

MSE

Figure 10: EnKF MSE for ﬁve (5) Ensembles.

0 0.01 0.02 0.03 0.04 0.05 0.06

Time (sec)

MSE

Figure 11: EnKF MSE for ten (10) Ensembles.

Stochastic Estimation of Fundamental and Harmonic Signal Components

0 0.01 0.02 0.03 0.04 0.05 0.06

Time (sec)

MSE

Figure 12: EnKF MSE for ﬁfty (50) Ensembles.

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