International Conference on Electrical, Electronics and Optimization Techniques (ICEEOT)-2016
Harmonic Detection Using Kalman Filter
Vidit A. Desai
Assistant professor, Electrical Department
Vadodara Institute of Engineering, Vadodara
desai.vidit30@gmail.com

Abstract—Now a days, because of utilization of highly non-linear
load and power electronic devices, harmonics are generated in
the power system. These harmonics create stress on electrical
devices connected to the power system. These harmonics also
cause failure of devicesand/or create disturbances in
performance of devices. Hence this sets a requirement of
harmonic filters. However, before designing a filter, the process
of harmonic analysis is essential. This paper describes the design
of an Extended Kalman filter for PWM converter for harmonic
detection. Measurement of magnitudes and frequencies of
fundamental and harmonic components present in system voltage
and current waveformsare detected in this paper. The result of
this filter are compared with the results of Fast Fourier
transform (FFT) function in MATLAB. Simulation results show
that the Extended Kalman filter is more accurate and requires
less time for harmonic detection as compared to others.
Index Terms—Extended Kalman Filter (EKF), Discrete Fourier
transform (DFT), Fast Fourier transform (FFT), PWM
converter, Harmonic detection, Noise
I. INTRODUCTION
T
he need for increased power system efficiency reliability
and security has lead to introduction of wide use of
nonlinear devices and loads. The nonlinear loads such as
power converters, adjustable speed drives, and uninterruptible
power supply (UPS) lead to harmonic pollution. The switching
devices are most often pulse width modulationbased and they
tend to further aggravate the noise and perturbations
introduced in the power system. Sudden changes in load,
switching transients tend to deviate the magnitude and
frequency of fundamental and harmonic components from
their nominal values. Therefore, it is becomes imperative to
look for novel and efficient methods for accurately
determining harmonic components present in the measured
power signals thereby addressing the power quality issues.
Of many detection techniques proposed for harmonic
detection in the past, Fast Fourier transform (FFT) and
Discrete Fourier transform (DFT) have been widely used
because of their simplicity and effective computational
abilities. However, the deviation of harmonic parameters like
magnitude, frequency and phase etc., may lead estimation
errors which create leakage effect in both the techniques. To
overcome this leakage effect, the authors in [1], proposed a
multi-spectrum-line interpolation correction algorithm
(MICA) to correct both long-range and short-range leakage
effect. However, MICA also tends suffers from problem of
Sandhya Rathore
Professor, Electrical Department
SCET, Surat
sandhya.rathore@scet.ac.in
synchronous deviation.The drawback of synchronous
deviation has been overcome by [2] using a phase difference
algorithm. On the similar lines, the authors in [3] proposed a
new algorithm for harmonic analysis. Yet another team of
researchers proposed an algorithm that utilizes a NUTTALL
window to decrease the long-range leakage. Therein three
spectrum lines are chosen to reduce the short-range leakage by
interpolation, refer [4] for greater details. A novel algorithm is
introduced by Varaprasadetal. [5] to transform asynchronous
sampling to synchronous sampling with customized
mathematical formulae. All these algorithms are difficult to
apply to some cosine windows with high-order or certain
coefficient.
Because of demerits of DFT and FFT technique, the adaptive
linear combiner (ADALINE) method became more popular
for harmonic assessment in the time domain. Artificial Neural
Networks have been applied for harmonic detectionas well [6].
An alternative method using neural network algorithm is used
for harmonic detection in noisy environments can be found in
[7]. An adaptive linear neuron (ADALINE) is used for online
estimation of fundamental component and selected harmonic
content of a distorted signal [8]. An effective procedure based
on the radial-basis-function neural network is proposed to
detect the harmonic amplitudes of the measured signal [9].For
the ADALINE filters, however, the results are easily affected
when the signals contain harmonic components that are not
included in the ADALINE structure. This problem can be
overcome by introducing higher order harmonic filters but
convergence of Fourier coefficient has to be compromised.
Kalman filter is yet another very populartechnique because it
accurately detects harmonic components under noisy
condition. Kalman filter have been utilized for estimation of
power system voltage, current and frequency in a digital AVR
[10]. A novel Kalman filter is also used in three-phase power
systems [11]. Online estimation of signal parameters is also
done using Kalman filter in the presence of harmonic and
noise distortion, refer [12]. An approach for decomposition of
harmonic voltage and current using Kalman filter combined
with a frequency detection algorithm can be found in [13].
In this paper, unlike ADALINE, DFT and FFT, we employ
EKF filter to find out the harmonic components in terms of
their magnitudes and frequencies to very high degree of
accuracy. This paper represents designing of an Extended
Kalman filter for detection of harmonic components upto 25th
order in ac-to-dc-to-ac PWM converter.
The organization of this paper is as follows. Section II
presents the formulation of Kalman filter while section III
describes the harmonic detection process of Kalman filter.
Experimental and simulation results are represented in section
IV and the conclusion is made in Section V.
II. FORMULATION OF KALMAN FILTER
III. HARMONIC DETECTION PROCESS USINF KALMAN FILTER
The dynamics and measurements of non-linear control system
under consideration may be represented as follows:
The analog multi-frequency signal is not exactly periodic
because amplitudes, frequencies and phases are continuously
changes slowly over a time. So for that purpose we first take a
periodic signal 𝑦(𝑡) with a zero dc component and Fourier
series representation of that signal as shown below
𝑥𝑘 = 𝑓𝑘−1 (𝑥𝑘−1 , 𝑢𝑘−1 , 𝑤𝑘−1 )
𝑦𝑘 = ℎ𝑘 (𝑥𝑘 , 𝑣𝑘 )
where, 𝑤𝑘 and 𝑣𝑘 are the process and observation noises which
are both assumed to be zero mean multivariate Gaussian noise
with covariance 𝑄𝑘 and 𝑅𝑘 𝑤𝑘 ~ (0, 𝑄𝑘 ) and 𝑣𝑘 ~ (0, 𝑅𝑘 )
respectively.
The functions 𝑓 and ℎ are the process and measurement
dynamics respectively. Extended Kalman filter uses the first
order Taylor expansion of the process and measure dynamics
by evaluating the Jacobian matrix. The Jacobians are required
to be evaluated at each prediction step. These matrices are
used in the EKF equations. This process actually linearizes the
non-linear function around the present estimate.
But before following above pattern, the filter can be initialized
as follows:
𝑥̂0 + = 𝐸(𝑥0 )
𝑇
𝑃0+ = 𝐸[(𝑥0 − 𝑥̂0 + )(𝑥0 − 𝑥̂0 + ) ]
Before calculating present state and covariance matrix using
time update equation, compute partial derivative matrices as
follows:
𝜕𝑓𝑘−1
│
𝜕𝑥
𝑥̂𝑘−1 +
𝜕𝑓𝑘−1
=
│
𝜕𝑤 𝑥̂𝑘−1+
𝐹𝑘−1 =
𝐿𝑘−1
And from partial derivative matrices, present state and
covariance matrix can be represented as follows:
+
𝑇
𝑃𝑘− = 𝐹𝑘−1 𝑃𝑘−1
𝐹𝑘−1
+𝐿𝑘−1 𝑄𝑘−1 𝐿𝑇𝑘−1
+
𝑥̃𝑘− = 𝑓𝑘−1 (𝑥̂𝑘−1 ,𝑢𝑘−1 , 0)
Once again, before calculating updated state and covariance
matrix using measurement update equation, compute partial
derivative matrices as follows:
𝜕ℎ𝑘
│
𝜕𝑥 𝑥̂𝑘 −
𝜕ℎ𝑘
𝑀𝑘 =
│
𝜕𝑣 𝑥̂𝑘 −
𝐻𝑘 =
𝑦(𝑡) = ∑∞𝑘=1 𝑟𝑘 sin(𝑘𝑤𝑓 𝑡 + 𝛷𝑘 ), 𝑡 = 0,1,2, ….
As the signal 𝑦(𝑡) is not exactly periodic but parameters
amplitude 𝑟𝑘 , frequency 𝑤𝑓 and phases 𝛷𝑘 are slowly time
varying.
So, we can state them as follows:
𝑤𝑓 = 𝑤𝑓 (𝑡)
𝑟 = 𝑟𝑘 (𝑡)
𝛷𝑘 = 𝛷𝑘 (𝑡)
Assume that the signal 𝑦(𝑡) is corrupted by white Gaussian
noise. So the measurement can be given as follows:
𝑧(𝑡) = 𝑦(𝑡) + 𝑣(𝑡)
Now the task is to estimate 𝑟1 (t)..., 𝑟𝑚 (t), 𝑤𝑓1 (t)…𝑤𝑓𝑚 (t)from
the measurements where 𝑚 represent the number of the
significant harmonic components present in a signal. The
parameters are only estimated up to 𝑚𝑡ℎ orderharmonics and
higher harmonics are assumed to be negligible. So total 2𝑚 of
parameters areto be estimated.
State space representation of a signal can represented as
follows:
𝑥(𝑡 + 1) = 𝐴𝑥(𝑡) + 𝑔(𝑡)
𝑧(𝑡) = ℎ(𝑥(𝑡)) + 𝑣(𝑡)
= 𝑦(𝑡) + 𝑣(𝑡)
Where,
𝑥(𝑡) = [𝑟1 (𝑡) , 𝑟2 (𝑡) . . 𝑟𝑚 (𝑡) , 𝑤𝑓1 (𝑡), 𝑤𝑓2 (𝑡) … 𝑤𝑓𝑚 (𝑡) ]𝑇
ℎ(𝑥(𝑡)) = ∑∞𝑘=1 𝑟𝑘 sin(𝑘𝑤𝑓 𝑡 + 𝛷𝑘 ),and
𝐼𝑚 0 0
𝐴= [ 0 1 0 ] Where, 𝐼𝑚 is an identity matrix of 𝑚𝑡ℎ order.
0 0 𝐼𝑚
And 𝑔(𝑡) is a white Gaussian noise with a zero mean and has
a variance represented as follows:
𝐸[𝑔(𝑡)𝑔(𝑡)𝑇 ] = 𝑄
Now by using Jacobian elements, updated state and covariance
matrix and Kalman gain can be represented as follows:
The observation noise 𝑣(𝑡) is also a white Gaussian noise with
zero mean and has a variance represented as follows:
𝐾𝑘 = 𝑃𝑘− 𝐻𝑘𝑇 (𝐻𝑘 𝑃𝑘− 𝐻𝑘𝑇 + 𝑀𝑘 𝑅𝑘 𝑀𝑘𝑇 )−1
𝑥̂𝑘 + = 𝑥̂𝑘 − + 𝐾𝑘 (𝑦𝑘 − 𝐻𝑘 𝑥̂𝑘 − − 𝑍𝑘 )
= 𝑥̂𝑘 − + 𝐾𝑘 [𝑦𝑘 − ℎ𝑘 𝑥̂𝑘 − , 0)]
+
𝑃𝑘 = (𝐼 − 𝐾𝑘 𝐻𝑘 )𝑃𝑘−
𝐸[𝑣(𝑡)𝑣(𝑡)𝑇 ] = 𝑅
which is uncorrelated with 𝑔(𝑡).The same can be represented
as follows:
𝐸[𝑔(𝑡)𝑣(𝑡)] = 0
Throughout the paper operator Eand 𝑥̂has been used to denote
expectation operation over a noisy variable. Furthermore, we
have process noise variance 𝑄 matrix which is diagonal in
nature. From the equation of 𝑥(𝑡 + 1) in the state space
representation, we can conclude that the amplitude of
harmonic components evolve randomly over a period of time.
Also the same argument is true for the fundamental frequency
of the signal. The rate of the random walk can be determined
by diagonal 𝑄 matrix. A Kalman filter will be applied for
estimating 𝑥̂(𝑡/𝑡) or 𝑥̂(𝑡/𝑡 − 1) of 𝑥(𝑡) from the
measurement 𝑧(𝑡) . Here 𝑥̂(𝑡/𝑡) denotes estimation of 𝑥(𝑡)
with given measurements at time 𝑡 and 𝑥̂(𝑡/𝑡 − 1) denotes
estimation of 𝑥(𝑡) with given measurements at time 𝑡 − 1.
𝑥̂(𝑡/𝑡) = 𝑥̂(𝑡/𝑡 − 1) + 𝐺(𝑡)[𝑧(𝑡) − ℎ(𝑥̂(𝑡/𝑡 − 1))]
𝑥̂(𝑡 + 1/𝑡) = 𝐴𝑥̂(𝑡/𝑡)
𝐺(𝑡) = 𝑃(𝑡)𝐻 𝑇 (𝑡)(𝐻(𝑡)𝑃(𝑡)𝐻𝑇 (𝑡) + 𝑅)−1
𝑃(𝑡 + 1) = 𝛷[𝑃(𝑡) − 𝐺(𝑡)𝐻(𝑡)𝑃(𝑡)]𝛷 𝑇 + 𝑄
Where, 𝐻(𝑡)is the Jacobian of ℎ(𝑡) that can be represented as
follows:
𝜕ℎ(𝑥̂(𝑡/𝑡−1))
𝐻(𝑡) =
̂(𝑡/𝑡−1)
𝜕𝑥
𝐻(𝑡) = [sin(𝑤𝑓 𝑡 + 𝛷𝑘 ) … sin(+ 𝛷𝑘 ) 𝑟̂1 cos(𝑤𝑓 +
𝛷𝑘 ) … 𝑟̂𝑘 cos(𝑤𝑓 𝑡𝛷𝑘 )]
And the initial values for state and covariance are represented
as follows:
Fig. 1: Voltage and Current waveform for harmonic Analysis
Figure 1 shows the waveforms of voltage and current
waveforms which further serve as the input to the EKF
method. It is evitable that the waveforms are highly nonlinear
and distorted owing to stepped wave shape.
𝑥̂(0) = 𝐸[𝑥(0)] = 𝑥̃(0)
𝑃(0) = 𝐸[(𝑥(0) − 𝑥̃(0))(𝑥(0) − 𝑥̃(0))𝑇 ]
IV. EXPERIMENT AND SIMULATION RESULTS
IV-A. SIMULATION RESULT FOR FFT TOOL IN MATLAB
For the sake of simulation, a three-phase source is connected
to the three-phase load with series connection of three-phase
step down transformer and three phase ac-to-dc-to-ac PWM
converter between source and load and distorted voltage and
current waveforms are taken for harmonic analysis and detect
magnitudes and frequencies of fundamental as well as
harmonic components present in voltage and current
waveforms. The same is presented in form of a table:
Threephase system
Load
Transformer
PWM IGBT Inverter
𝑉𝑠 = 25kV (RMS),
Frequency= 50HZ,
VA rating= 10MVA
X/R ratio = 5
𝑉𝐿 = 380V (RMS),
Avtive Power = 50KW
Voltage ratio= 25KV/415V,
VA rating= 45KVA
Snubber Resistance= 5000
Snubber Capacitance= inf.
Ron= 1 × 10−3 Ohms
Table 1: Simulated System Parameters
Fig. 2: FFT analysis of Voltage waveform
FFT is often used as a standard tool to analyses the total
harmonic content present in the distorted current and voltage
waveforms. We use this standard tool as a relative measure to
check the superiority and efficacy of EKF technique.
Figure 2 and 3 show the FFT of current and voltage waveform
respectively.
Fig. 3: FFT analysis of Current waveform
Fig. 6: Magnitude and Frequency of 5𝑡ℎ component of
load Voltage
IV-B. SIMULATION RESULT FOR KALMAN FILTER METHODOLOGY
Next we demonstratethe nonlinear Kalman filter which
handles all this white noise present in the system considerably
well.
Fig. 7: Magnitude and Frequency of 7𝑡ℎ component of
load Voltage
Fig. 4: Magnitude and Frequency of fundamental component
of load Voltage
Fig. 8: Magnitude and Frequency of 9𝑡ℎ component of
load Voltage
Fig. 5: Magnitude and Frequency of 3rdcomponent of
load Voltage
load Current
Fig. 9: Magnitude and Frequency of fundamental component
of loadCurrent
Fig. 13: Magnitude and Frequency of 9𝑡ℎ component of
load Current
The waveforms of fundamental component as well as various
harmonics namely third order upto ninth order are illustrated
in figures 4 to 13. The Kalman filter is able to process noise
in a better manner as compared to the conventional
techniques. The table shown below illustrates the performance
of KF technique adopted in the paper with FFT tool in Matlab.
Fig. 10: Magnitude and Frequency of 3𝑟𝑑 component of
load Current
Magnitude
Order of
Harmonic
Fig. 11: Magnitude and Frequency of 5𝑡ℎ component of
load Current
Frequency
1
FFT
583.70
583.70-583.71
FFT
50
3
1.40
1.40-1.41
150
5
114.99
114.19-115.00
250
7
85.16
85.16-85.17
350
9
1.40
1.40-1.41
450
11
52.65
52.65-52.66
550
13
45.70
650
15
1.40
45.70545.715
1.40-1.41
17
33.74
33.74-33.75
850
19
31.58
31.58-31.59
950
EKF
750
EKF
50.000050.0001
150.000150.0001
250.000250.0001
350.000350.0001
450.000450.0001
550.000550.0001
650.000650.0001
750.000750.0001
850.000850.0001
950.000950.0001
Table 2: Comparison of EKF and FFT for load Voltage
Fig. 12: Magnitude and Frequency of 7𝑡ℎ component of
Magnitude
Order of
Harmonic
Frequency
1
FFT
116.40
116.40-116.41
FFT
50
3
0.49
0.49-0.50
150
5
23.27
23.27-23.28
250
7
16.74
16.74-16.75
350
9
0.49
0.49-0.50
450
11
10.81
10.81-10.82
550
13
8.85
8.85-8.86
650
15
0.49
0.49-0.50
750
17
7.02
7.02-7.03
850
19
6.02
6.02-6.03
950
EKF
EKF
50.000050.0001
150.000150.0001
250.000250.0001
350.000350.0001
450.000450.0001
550.000550.0001
650.000650.0001
750.000750.0001
850.000850.0001
950.000950.0001
Table 3: Comparison of EKF and FFT for load Current
V. CONCLUSION
This paper represents aspects,design and characteristics of
Extended Kalman filter. EKF isapplied to the PWM converter
for harmonic detection in distorted voltage and current
waveforms
From simulation results of Kalman filter and comparison
table, it is found that Kalman filter technique is very accurate
and fast when estimating the amplitude and frequency of
voltage and current waveforms. The algorithm provides very
good results even for the waveforms contaminated with
significant noise or harmonics and the filter is adaptive with
respect to system variations of harmonic parameters. The
drawbacks of the Kalman filter related to its sensitivity to the
disturbances and low noise to signal ratio. Also if the number
of harmonic components increased for detection process then
Kalman filter becomes slower and complexity is increased
because of linearization.
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