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COMPEL - The international journal for computation and
mathematics in electrical and electronic engineeringSmart VRP-NLMS algorit hm for est imat ion of power syst em f requency
Seyed Reza Aali, Mohammad Reza Besmi, Mohammad Hosein Kazemi,
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To cite this document:Seyed Reza Aali, Mohammad Reza Besmi, Mohammad Hosein Kazemi, (2018) "Smart VRP-NLMS algorithm for estimation of power system frequency", COMPEL - The international journalfor computation and mathematics in electrical and electronic engineering, https://doi.org/10.1108/COMPEL-06-2018-0263Permanent l ink t o t his document :https://doi.org/10.1108/COMPEL-06-2018-0263
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Smart VRP-NLMS algorithmfor estimation of power
system frequencySeyed Reza Aali, Mohammad Reza Besmi and
Mohammad Hosein KazemiElectrical and Electronic Engineering, Shahed University, Tehran, Iran
Abstract
Purpose – The purpose of this paper is to study variation regularization with a positive sequenceextraction-normalized least mean square (VRP-NLMS) algorithm for frequency estimation in a three-phaseelectrical distribution system. A simulation test is provided to validate the performance and convergence rateof the proposed estimation algorithm.
Design/methodology/approach – Least mean square (LMS) algorithms for frequency estimationencounter problems when voltage contains unbalance, sags and harmonic distortion. The convergence rate ofthe LMS algorithm is sensitive to the adjustment of the step-size parameter used in the update equation. Thispaper proposes VRP-NLMS algorithm for frequency estimation in a power system. Regularization parameteris variable in the NLMS algorithm to adjust step-size parameter. Delayed signal cancellation (DSC) operatorsuppresses harmonics and negative sequence component of the voltage vector in a two-phase Î 6 Î2 plane.The DSC part is placed in front of the NLMS algorithm as a pre-filter and a positive sequence of the gridvoltage is extracted.
Findings – By adapting of the step-size parameter, speed and accuracy of the LMS algorithm are improved.The DSC operator is augmented to the NLMS algorithm for more improvement of the performance of thisadaptive filter. Simulation results validate that the proposed VRP-NLMS algorithm has a less misalignment ofperformance with more convergence rate.
Originality/value – This paper is a theoretical support to simulated system performance.
Keywords Adaptive filtering, Delayed signal cancellation (DSC), Frequency estimation, NLMS,Variable regularization
Paper type Research paper
Nomenclature
h = Step-size parameter;« = Regularization parameter;DT = Sampling interval;Dv g = Deviation of the grid frequency;ek = Error signal;
f̂ kð Þ = Estimated frequency;Jk = Cost function;f = Phase of voltage;T = Fundamental period;k = Time instant;THD = Total harmonic distortion;
Estimation ofpower system
frequency
Received 28 June 2018Revised 20 August 2018
Accepted 24 September 2018
COMPEL - The internationaljournal for computation andmathematics in electrical and
electronic engineering© EmeraldPublishingLimited
0332-1649DOI 10.1108/COMPEL-06-2018-0263
The current issue and full text archive of this journal is available on Emerald Insight at:www.emeraldinsight.com/0332-1649.htm
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m = Step-size coefficient;V̂ k = Estimated voltage;v = Angular frequency;= :ð Þ = Imaginary part of a complex-valued number;< :ð Þ = Real part of a complex-valued number;
j jVk jj2 = L2-norm of Vk ;
j jf̂ jj2 = L2-norm of f̂ ;DSC = Delayed signal cancellation;FFPS = Fundamental frequency positive sequence;FRP = Fixed regularization with a positive-sequence-extraction;LMS = Least mean square;NLMS = Normalized LMS;NMSD = Normalized mean square deviation;PLL = Phase locked loop; andVR = Variable regularization.
1. IntroductionFrequency is an important factor in power quality control, protection and synchronizing ofpower converters connected to the grid. Also, an accurate estimation of system frequency isessential for protection, dynamic control and power system stabilization (Chilipi et al., 2016;He et al., 2017; Li et al., 2017; Smith et al., 2009). Several strategies such as the discreteFourier transform -based (Xia et al., 2017), a phase-locked loop (PLL) (Ciobotaru et al., 2010),frequency-locked loop (FLL) (Bei and Wang, 2016), Kalman filters (Dini and Mandic, 2013)and Clark’s transformation-based frequency measurement (Dash et al., 2000) have beenpresented for frequency estimation in power systems.
Adaptive filtering is an effective technique in many applications such as systemidentification, channel equalization, network and acoustic echo cancellation and active noisecontrol. Adaptive filters are established based on minimizing the mean square error andhave a simple configuration, less computation, more efficiency and stability and robustnessin the presence of noise and harmonic distortions (Chellappa and Theodoridis, 2014; Sayed,2003; Simon, 2002). Adaptive filters such as least mean squares (LMS) method (Xia andMandic, 2012), normalized LMS (NLMS) (Sayed, 2003) are prevalent for DSP-based powerconverters because voltage components calculated via Clark’s transformation are alreadyavailable for synchronizing of network current or power controller loops. Most protectionand control applications in power systems need to an accurate and fast estimation of thesystem frequency. Disturbances including voltage sag, harmonics and unbalance distortionare the major difficulties of frequency estimation (Arablouei et al., 2015; Wang and Li, 2011a,2011b). Therefore, the conventional LMS and NLMS algorithms will be failed against thesedisturbances and cannot track the frequency deviations accurately. The performance of theNLMS tends to deteriorate in the presence of negative sequence components in grid voltage.
For overcoming the aforementioned problem, an additional filtering stage is usuallyadded, either before or within the PLL loop. Among filtering strategies, the delayed signalcancellation (DSC) operator is more popular because it can be easily tailored to different gridscenarios. The DSC operator eliminates all orders of harmonics, including DC offset andnegative sequence components in highly distorted voltage. The DSC operator extracts thefundamental frequency positive sequence (FFPS) component of the grid voltage. The DSCoperator(s) can be used either as an in-loop filter in the PLL structure or as a pre-filter for thePLL input (Golestan et al., 2017; Golestan et al., 2015; Huang and Rajashekara, 2017; Subudhiet al., 2009). The DSC filter has a low computational burden for the implementation and
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ensures a fast dynamic response and high filtering capability. A hybrid adaptive/non-adaptive DSC operators-based PLL for the accurate and fast extraction of the grid voltageparameters was proposed in Golestan et al (2017).
In this paper, a combination of a DSC operator with a variable regularization (VR)-NLMSalgorithm is presented for fundamental frequency estimation of the power system. The DSCpart with a simple complexity is placed in front of the NLMS algorithm as a pre-filter, and apositive sequence of the grid voltage is extracted. The normalized part of NLMS is alsochanged. The DSC part and VR of the proposed NLMS algorithm change the performanceand convergence rate of the adaptive filter.
This paper is organized as follows: Section 2 reviews the NLMS estimation algorithm.The VR-NLMS algorithm is presented in Section 3. Section 4 analyses the stability of theproposed algorithm. The DSC as a pre-filter in the proposed algorithm is stated in Section5. Section 6 presents simulation results for frequency estimation under distortedconditions, including voltage sags, unbalanced and higher-order harmonics. Finally,Section 7 concludes the paper.
2. Review of normalized least mean squares algorithmThe discretized balanced voltages of a three-phase power system can be represented as:
va;k
vb;k
vc;k
2
6
6
4
3
7
7
5
¼
Va cos vkDT þ fð Þ
Vb cos vkDT þ f � 2p3
� �
Vc cos vkDT þ f þ 2p3
� �
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
(1)
where va;k, vb;k and vc;k are three-phase voltage at a time instant k. Va, Vb and Vc aremaximal values of the phase voltages. DTis the sampling interval, f is the phase of thefundamental component, v ¼ 2p f is the angular frequency of the voltage signal and f is thesignal frequency.
The three-phase system can be transformed into a two-phase complex using Clarke’stransformation into a zero sequence v0, and the complex and orthogonal va and vb accordingto the following equation:
v0;k
va;k
vb ;k
2
6
6
4
3
7
7
5
¼ffiffiffi
23
r
ffiffiffi
2p
2
ffiffiffi
2p
2
ffiffiffi
2p
2
1 � 12
� 12
0
ffiffiffi
3p
2�
ffiffiffi
3p
2
2
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
5
va;k
vb;k
vc;k
2
6
6
4
3
7
7
5
(2)
The zero-sequence component v0;k is not necessary for analysis, and zero-sequence is inmany cases blocked by the transformers; complex voltage is defined by:
Vk ¼ va;k þ jvb ;k (3)
where j ¼ffiffiffiffiffiffi
�1p
.
Estimation ofpower system
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Underlined letter _ distinct the notation of vectors and scalars (Farhang-Boroujeny, 2013;Mandic and Goh, 2009; Manolakis et al., 2000). The complex voltage can also be writtenusing polar coordinates as:
Vk ¼ Ae jvkDTþfð Þ (4)
whereA is the estimated amplitude of the complex vector, v is estimated angular frequencyof the voltage.
A new vector value can be expressed in the recursive method using the previous stepvalue. Assuming a constant signal frequency between two signal sample instants results in
Vk ¼ Vk�1 ejvkDT (5)
This equation is the foundation of the complex recursive LMS algorithm for frequencyestimation.
The complex LMS (CLMS) is a natural extension of the real LMS and work on complexnonlinear architectures (Farhang-Boroujeny, 2013; Mandic and Goh, 2009). The CLMSerror signal is calculated by using the measured Vk and estimated V̂ k voltage vector in kstep as:
ek ¼ Vk � V̂ k (6)
where ek is the error between input vector and combiner output.In addition, complex weight vectors will be Wk�1 ¼ ejv̂ kDT , and combiner output
(complex voltage vector) is estimated by:
V̂ k ¼ Wk�1 : V̂ k�1 (7)
The weight vector of CLMS is calculated recursively using
Wk ¼ Wk�1 þ mek�1V*k�1 (8)
where m is the algorithm’s step size (coefficient) that controls the convergence rate.Superscript * for the complex-conjugate of the vector. The small value of m leads to a slowconvergence rate. The large value of m causes the algorithm to degrade the performance ofthe filter.
Therefore, selecting a step size is very important. The LMS algorithm has a simplestructure and easy implementation with low computational complexity, but thedisadvantages of LMS equations (9)-(10) are a fixed step size for every iteration. Finally, theinstantaneous frequency fk can be estimated as (Farhang-Boroujeny, 2013; Mandic and Goh,2009; Manolakis et al., 2000):
fk ¼1
2pDTsin�1 = Wk
� �� �
(9)
where = :ð Þ represents the imaginary part of a complex-valued number. In NLMS, the stepsize is proportional to the inverse of total energy for the instantaneous values of the inputvectorVk. The step size of the NLMS algorithm is updated by the following equation (18):
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m k ¼m
j jVk j j22(10)
where m k is time-varying (0 < m k < 2) and m is selected constant value.To avoid division by 0 for small values of input j jVk jj2 (Farhang-Boroujeny, 2013), a
small “regularization” parameter « is added to NLMS structure:
m k ¼m
j jVk j j22 þ «(11)
3. The variable regularization–normalized least mean squares algorithmThe constant regularization parameter « may lead to a low convergence rate of the NLMSalgorithm. Parameter « can be updated based on the gradient adaptive method (Jenkinset al., 2012) as:
« k ¼ « k�1 � h@Jk
@« k�1(12)
where Jk is the cost function (Jk ¼ 12 e
2k ¼ 1
2 eke*k ), and h is step size parameter.
Computing the gradient:
@Jk@« k�1
¼ 12
ek@ek
*
@« k�1þ ek
* @ek
« k�1
!
¼ < ek* @ek
@« k�1
� �
(13)
where< implies the real component:
@ek
@« k�1¼ �Vk
T@Wk
@« k�1(14)
and
@Wk
@« k�1¼ �
m ek�1Vk�1*
jjVk�1 j j22 þ « k�1
� �2 (15)
Using the equations (13), (14) and (15) in equation (12), results in:
« k ¼ « k�1 � hm< ek
*ek�1VkTVk�1
*� �
jjVk�1 j j22 þ « k�1
� �2 (16)
The detail steps of this VR-NLMS algorithm are shown in Figure 1.In addition, the VR-NLMS algorithm may be concluded according to Frequency
estimation using the VR-NLMS algorithm:
Estimation ofpower system
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Parameters:
Coefficient h ; mf gInitialization:
«0 ¼ 0:001W0 ¼ 0
Available data:
Three-phase voltage signal va;k; vb ;k; vc;kf gAlgorithm:
For k ¼ 1;2;3; :::;N do
ConstructVk using (3)
ek ¼ Vk � V̂ k
« k ¼ « k�1 � hm< ek
*ek�1VkTVk�1
*� �
jjVk�1 j j22þ« k�1
� �2
m k ¼ m
j jVk j j22þ« k
Wk ¼ Wk�1 þ m kek�1V*k�1
V̂ k ¼ Wk�1 :V̂ k�1
f̂ k ¼ 12pDT
sin�1 = Wk
� �� �
End
Output:
Estimated frequency f̂ k
4. Stability analysis of the variable regularization–normalized least mean
squares algorithmThe theoretical stability analysis of the proposed algorithm is presented in this section. Inthe VR-NLMS algorithm, the weight vector Wk is updated according to the followingequation:
Figure 1.
Block diagram of theVR-NLMS
Start
Read input vector Vk
Initialize coefficient, weights
& regulation parameter
Compute combiner output
from input vector & weight
Compute estimation error
between input vector &
combiner output
Is the estimation-
error zero?
Compute estimated
frequency
End
Update regulation
parameter
Update step size
Update weights
yesno
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Wk ¼ Wk�1 þm
j jVk j j22 þ « k
ek�1V*k�1 (17)
The functionVLy ¼ j ek j is considered for stability analysis of this algorithm, andwe have:
DVLy ¼ jek j � jek�1 j ¼ jd � VTk Wk j � jek�1 j
¼ jd � VTk fWk�1 þ
m
jjVk j j22 þ « k
ek�1V*k�1gj � jek�1 j (18)
where d is the desire signal.Then,
DVLy ¼ ja � VTk
m
jjVk j j22 þ « k
ek�1V*k�1j � jek�1j (19)
where a ¼ d � VTk Wk�1 .
On the other hand, we have (Mengüç andAcir, 2015; Zhang et al., 2017):
m ek�1 ¼ a� k jek�1 j signðaÞ; 0#k < 1 (20)
Therefore, equation (19) can be rewritten as:
DVLy ¼ ja � VTk V*
k�11
jVk j j22 þ « k
a � k jek�1 j sign að Þ� �j � jek�1 j
� ja� kVkk21
j jVk j j22 þ « k
a � k jek�1 j sign að Þ� �j � jek�1 j# ja
� a � k jek�1 j sign að Þ� �j � jek�1 j (21)
where j jVk jj2 is L2-norm ofVk that has a positive value and implies:
DVLy ¼ jk jek�1 j signðaÞj � jek�1 j ¼ � 1� kð Þjek�1 j ) DVLy < 0 (22)
The result, DVLy < 0, means that the error ek converges to zero, which proves the stabilityof the proposed estimation algorithm.
5. Variable regularization with a positive sequence extraction–normalized
least mean squares algorithmWhen dealing with signals that are unknown and operate in a wide range, applying a fixedregularization parameter « may degrade the NLMS algorithm convergence andperformance. To solve these problems, the regularization parameter « should be variable.
The fixed regularization with a positive sequence extraction (FRP)-NLMS operateswith a constant parameter of regularization (« k ¼ « ) and DSC pre-filtering. Thisregularization parameter has small positive value to avoid the normalization with thedivision to zero.
Figure 2 shows the structure of the proposed strategy for the estimation of thefundamental frequency. The structure of the proposed variable regularization with apositive sequence extraction (VRP)-NLMS consists of three parts; Clarke’s transformation,VR-NLMS algorithm and ab � DSC part. Three-phase voltage is converted into a two-phase complex using Clarke’s transformation into a zero-sequence v0 and the direct (va) and
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quadrature-axis (vb ) components according to equation (2). The VRP-NLMS algorithmoperates as explained in frequency estimation using the VR-NLMS algorithm described inSection 3. The ab � DSC part of the ab -frame (complex) eliminates harmonic componentsby summing with their delayed versions. The DSC operator extracts FFPS component in thestationary reference frame according to the following equations (16, 26- 28):
Vþab ;1
tð Þ ¼ 12
Vab tð Þ þ ej2pn Vab t � T=nð Þ
� �
(23)
where Vab tð Þ ¼ Va tð Þ þ jVb tð Þ is the voltage vector in the ab -frame, Vþab ;1 tð Þ ¼
Vþa;1 tð Þ þ j Vþ
b ;1 tð Þ is the separated FFPS-voltage vector, T = 1/f is the nominal value of thevoltage fundamental period and n is the delay coefficient.
The Laplace domain of theab � DSC operator is defined as:
DSCn sð Þ ¼Vþ
ab ;1 sð ÞVab sð Þ ¼ 1þ e
j2pn e�
Tns
2(24)
whichDSCn operator determines nth disturbance components of the voltage. Substitutings ¼ jvin theDSC operator implies that:
DSCn jvð Þ ¼ j cos vT
2n� p
n
� �
j/� vT
2n� p
n
� �
¼ j cos ph
n� p
n
� �
j/� vT
2n� p
n
� � (25)
Figure 3 shows the definition of the DSC block with nth time delay factor in a time domain.The DSCn block can eliminate certain sets of harmonics, for example, n = 4, jDSC4 hð Þj ¼ 0satisfies for h ¼ 4k � 1 which means DSC4 will eliminate all 4k � 1(k ¼ 61;62;63;:::)order harmonics in the grid voltage. Also, for n = 12, jDSC12 hð Þj ¼ 0 satisfies forh ¼ 12kþ 1 andDSC12 eliminate all 12kþ 1 order harmonics in the grid voltage.
Most often a single DSC operator is not enough to eliminate all harmonic components.Therefore, several DSCoperators hybrid series form with special delay factors are used.Equation (26) describes the ab � DSC operator in s-domain where m is the number ofcascaded ab � DSC operators:
DCSn1;n2 ;:::nm sð Þ ¼DCSn1 sð Þ:DCSn2 sð Þ::: :DCSnm sð Þ
(26)
Figure 2.
Schematic diagram ofDSC operator-basedVRP-NLMS
Va
Vb
Vc
Vα
Vβ
estimated
frequency
DSCn1 DSCn2...
DSCnm...
Vα,1+
Vβ,1+
αβ
abcDSCn1,n2,…,nm
VR- NLMS
Algorithm
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The multiple cascaded-DSC operators extract the fundamental-frequency or positive-sequence component effectively (Zhang et al., 2017; Wang and Li, 2011a, 2011b).
6. Simulation results and discussionsFigure 4 shows the frequency response of the DSC4;4;12;12;24;24 operator (Golestan et al., 2017)in cascade together. The frequency response of the filter is approximately flat about 50 Hzand separate fundamental-frequency positive-sequence component of the voltage from thenegative sequence and harmonics.
Replacing s ¼ jv g into equation (25), where v g is the fundamental frequency, gives thephase andmagnitude of the DSC operator at the fundamental frequency as:
/DSC4;4;12;12;24;24 jv g
� �
¼ �X
n¼4;4;12;12;24;24
v gT
2n� p
n
� �
(27)
jDSC4;4;12;12;24;24 jv g
� �
j ¼Y
n¼4;4;12;12;24;24
j cos v gT
2n� p
n
� �
j (28)
Defining v g ¼ v 0 þ Dv g , where v 0 ¼ 2p
Tis the nominal value of the frequency and Dv g
denotes the deviation of the grid frequency from its nominal valueand replacing it intoequations (27) and (28) obtains:
/DSC4;4;12;12;24;24 jv g
� �
¼ �X
n¼4;4;12;12;24;24
Dv gT
2n
� �
¼ �kwDv g (29)
jDSC4;4;12;12;24;24 jv g
� �
j ¼Y
n¼4;4;12;12;24;24
j cos Dv gT
2n
� �
j
�Y
n¼4;4;12;12;24;24
1� Dv gT
2n
� �2
=2
!
� 1� ka Dv gð Þ2 (30)
Figure 3.
Time domainimplementation of the
DSCn operator
+
+
-
+
+
+
T/n delay
Nonadaptive DSCn operator
a = cos(2π/n)
a’ = sin(2π/n)
a
a
a’
a’
0.5
0.5
Vα
Vβ
Vα,1+
Vβ,1+
Estimation ofpower system
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where kw ¼ T2
X
n¼4;4;12;12;24;24
1n
!
and ka ¼ T2
8
X
n¼ 4;4;12;12;24;24
1n2
!
.
Therefore, equations (29), (30) show the phase and amplitude of DSC around thefundamental frequency (50 Hz). The DSC4;4;12;12;24;24 operator eliminates symmetrical andasymmetrical harmonics in grid voltage.
Simulations were performed in theMatlab programing environment with a sampling rate of20 kHz, and the step-size m of all NLMS algorithms was set to be 0.01. Also, the step-sizeparameter of h was set to be 0.1 in all simulations. The electrical system consists of a 25-kVvoltage source with 100MVA short-circuit level, and 100 kVA three-phase power transformers,transmission lines 30 km in length that feeds load of 33 kW. Power transformers have a starconnection in the high voltage winding and a star connection in the low voltage winding.
6.1 Frequency estimation under unbalance voltageType C voltage sags are one kind of unsymmetrical voltage sags that originate from line-to-line fault. Figure 5 shows a phasor view of Type C voltage sags. The balanced systembecame unbalanced after the fault. The fault occurs in the vicinity of the load before
Figure 4.
Frequency response(a) magnitude; and(b) phase of theDSC4;4;12;12;24;24
operators
–650 –550 –450 –350 –250 –150 –50 50 150 250 350 450 550 650
0
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–650 –550 –450 –350 –250 –150 –50 50 150 250 350 450 550 650
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–150
–100
–50
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100
150
200
Ph
ase
(deg
)
Ferquency (Hz)
(a)
(b)
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transformer with a resistance of 2X, and it is caused about a 20 per cent drop and 15° phaseangles offset for phases Vb and Vc. The real-imaginary curves convert from circularity(balance) to non-circularity of Clarke’s voltage in unbalanced conditions.
The DSC pre-filter suppresses the asymmetric component of the measured voltage in thephasor domain and non-circularity of Clarke’s voltage is changed to circularity voltage asVR-NLMS input.
The negative-sequence and zero-sequence components illustrate the imbalance betweenthe three voltages; the zero-sequence is in many cases blocked by the transformers. Avoltage dip is characterized by a relationship between the positive-sequence voltage and thenegative-sequence voltage. The zero-sequence component is not used in the estimation offrequency in the NLMS algorithm. Therefore, the only zero-sequence component of thevoltage has no effect on the proposed-estimation algorithm.
In addition, the hybrid DSC operator with PLL (DSC-PLL) presented in Golestan et al(2017) is compared with the proposed algorithm. The DSC operator extracts the positivecomponent of the voltage. This positive voltage is given to NLMS with variable regulationin VRP-NLMS. The estimator’s performance in the presence of Type C voltage sags iscarried out with the simulation presented in Figure 6, and the numeric results are exposed inTable I. The sag is occurring in t = 1 to 1.2 sec. The simulation is divided into two zones: inZone 1 and Zone 2. Zone 1 is extended from t= 1 to 1.2 sec, and Zone 2 is the time larger than1.2 sec. The positive sequence of voltage is separated by DSC in FRP and VRP-NLMS.Therefore, these algorithms are stable where the FR-NLMS and VR-NLMS estimatorscannot handle the disturbance in steady state, exhibiting an error greater than 10 Hz. TheFRP and VRP-NLMS track the reference frequency under distortions. The VRP-NLMS hasshorter settling times with minimum error. Hence, time-varying regularization of the NLMSshows a better dynamic response and a smart operation under Type C voltage sags.
The normalized mean square deviation (NMSD) is used as an index for qualityevaluation of the adaptive filter. It evaluates the accuracy and convergence rate which isdefined as (Sayed, 2003):
NMSD ¼ 10log10j jf � f̂ j j22
j jf̂ jj22
!
(31)
NMSD is used to show the performance of the estimation with respect to the convergencerate and steady-state misalignment. The NMSD curves for this sort of voltage sags areshown in Figure 7 with different NLMS algorithms and DSC-PLL during the first 1.5 sec.
Figure 5.
Steady-statetrajectory in an alpha-beta plane (right) andthe phasor (left) views
of Type Cvoltage sags–1.5 –1 –0.5 0 0.5 1 1.5
–1.5
–1
–0.5
0
0.5
1
1.5
V
V
Type C
DSC output
Type C
Vc
Vb
Va
α
β
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Initial transient condition effects on NMSD curves before first 1 sec. The three-phase voltagearrived in the steady-state condition in the vicinity of 0.8 sec, and fault occurs at t = 1 to 1.2sec. The VRP-NLMS algorithm exhibits the best performance among the others in transientconditions.
If the voltage sags of Type C pass through a delta/y transformer, it will be converted toType D voltage sags. The phasor view of Type D voltage sags, having about 20 per centvoltage drop at the line Va and a 10 per cent voltage drop on both Vb andVc with a 5° phaseangle offset as illustrated in Figure 8. This fault also leads balanced system to anunbalanced system. The non-circularity of Clarke’s voltage is changed to circularity voltage.The DSC output fed VR-NLMS and PLL to improve the operation of these measurements.
Figure 9, Table II and Figure 10 indicate frequency estimation and NMSD curves forType D voltage sags with different methods. The FR-NLMS and VR-NLMS have theworst performance with the settling times and with the maximum errors. The DSC-PLLhas a slow convergence rate with a bad performance ratio than the FRP and VRP-NLMS.Hence, it is observed that the proposed algorithm provides a high-efficient estimationwith the best performance and low steady-state misalignment compared with the otheralgorithms.
Figure 6.
Frequency estimationfor Type C voltagesags with differentNLMS algorithmsand DSC-PLL 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3
35
40
45
50
55
60
65
70
Time (sec)
Fre
qu
en
cy (
Hz)
FR- NLMS
VR- NLMS
FRP- NLMS
VRP- NLMS
DSC- PLL
Zone 2Zone 1
Table I.
Measurements fromFigure 6
Settling time (ms)Maximum error (Hz)
Transient Steady state
Zone 1FR-NLMS 1 9.92 9.89VR-NLMS 1 9.31 9.29FRP-NLMS 41 3.51 0VRP-NLMS 33 3.05 0DSC PLL 162 4.77 0
Zone 2 0FR-NLMS 45 10.47 0VR-NLMS 37 9.45 0FRP-NLMS 40 3.72 0VRP-NLMS 34 3.16 0DSC PLL 295 4.77 0
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6.2 Frequency estimation under harmonic voltageThe 6-pulse delta/y and delta/delta connected rectifiers as nonlinear loads are connected tothe tested distribution system at t = 1 to 1.2 sec. The grid voltage is polluted with harmonicsand total harmonic distortion (THD) of the voltage is 13.12 per cent. Figure 11 showsgeometric and the phasor views of voltage harmonics under three-phase to ground fault(balance). Also, the real–imaginary plots illustrate the non-circularity of Clarke’s voltage inharmonic condition.
Frequency estimation, numeric results and NMSD curves under voltage harmonics withdifferent NLMS algorithms and DSC-PLL is shown in Figure 12, Table III and Figure 13,respectively. The simulation is divided into two zones: in Zone 1, the voltage isinstantaneously polluted, and in Zone 2, the voltage recovers from the disturbanceinstantaneously. In Zone 1, the FR-NLMS and the VR-NLMS have steady-state errorsgreater than 4 Hz. As such, the settling times for these estimators are infinite. In Zone 1, thesteady-state error of the FRP-NLMS, VRP-NLMS and DSC PLL are zero because of theoperation of the DSC.
Also, the FR-NLMS and VR-NLMS estimators have the greater transient error. TheVRP-NLMS and the FRP-NLMS strategies have similar responses to the lower transient
Figure 7.
NMSD curves forType C voltage sagswith different NLMS
algorithms andDSC-PLL
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.5– 40
–35
–30
–25
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–10
–5
0
Time (sec)
NM
SD
(d
B)
FR-NLMS
VR-NLMS
FRP-NLMS
VRP-NLMS
DSC- PLL
Figure 8.
Steady-statetrajectory in thealpha-beta plane
(right) and the phasor(left) views of Type
D-unbalancedvoltage sags
–1.5 –1 –0.5 0 0.5 1 1.5–1.5
–1
–0.5
0
0.5
1
1.5
Vα
Vβ Type D
DSC output
Type D
Va
Vc
Vb
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Figure 9.
Frequency estimationfor Type D voltagesags with differentNLMS algorithmsand DSC-PLL 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3
35
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65
Time (sec)
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FRP- NLMS
VRP- NLMS
DSC- PLL
Zone 1 Zone 2
Table II.
Measurementsfrom Figure 9
Settling time (ms)Maximum error (Hz)
Transient Steady state
Zone 1FR-NLMS 1 10.05 10.03VR-NLMS 1 9.25 9.22FRP-NLMS 24 3.43 0VRP-NLMS 21 3.75 0DSC PLL 110 5.62 0
Zone 2 0FR-NLMS 35 10.08 0VR-NLMS 28 9.12 0FRP-NLMS 31 3.95 0VRP-NLMS 22 3.34 0DSC PLL 82 4.53 0
Figure 10.
NMSD curves forType D voltage sagswith different NLMSalgorithms andDSC-PLL 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.5
–40
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DSC- PLL
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error. The accuracy and the speed of the estimation have a very satisfactory behavior withVRP-NLMS in the presence of harmonics.
Sub-harmonics are defined as non-integer harmonics with the frequency below the mainfundamental under consideration. The corresponding waveform of the grid voltage isshown Figure 14 (left) and polluted with a sub-harmonics component of 10 per cent ofamplitude and 2.5 Hz frequency. Also, it is polluted with 10 per cent of 3th inter-harmonicsand 10 per cent of 5th inter-harmonics, simultaneously.
The performance of the proposed and conventional-frequency estimation under sub- andinter-harmonics is shown in Figure 15. The VRP-NLMS strategy has steady state errors lessthan 0.1 Hz. As it can be observed, the proposed technique is less sensitive to the presence ofharmonics as compared to the other methods. The proposed estimator has robust again sub-and inter-harmonics.
6.3 Frequency estimation under frequency deviationsFigure 16a illustrates the superior performance of the VRP-NLMS estimator for a powersystem in the presence of a grid frequency step. The reference frequency is decreased to 49.8Hz at t = 1 sec, and it is recovered to the previous value at t = 1.1 sec. The estimation speedand accuracy by VRP-NLMS algorithm is more than other methods.
Figure 11.
Steady-statetrajectory in thealpha-beta plane
(right) and phasor(left) views of voltage
harmonics–1.2–0.9–0.6–0.3 0 0.3 0.6 0.9 1.2
–1.2
–0.9
–0.6
–0.3
0
0.3
0.6
0.9
1.2
Vα
Vβ
Harmonic Voltage
DSC outputVa
Vc
Vb
Harmonic
Voltage
Figure 12.
Frequency estimationfor voltage harmonicswith different NLMS
algorithms andDSC-PLL0.95 1 1.05 1.1 1.15 1.2 1.25 1.3
40
42
44
46
48
50
52
54
56
58
60
Time (sec)
Fre
qu
en
cy (H
z)
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VR- NLMS
FRP- NLMS
VRP- NLMS
DSC- PLL
Zone 1 Zone 2
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7. ConclusionIn this paper, a new NLMS-based DSC is proposed to estimate and track the frequency of thepower system. The simulation results validated the analytical findings under differentconditions such as voltage sags, unbalance and harmonic distortions and frequencyvariations. The proposed algorithm exhibits a considerably improved convergence rate and
Table III.
Measurements fromFigure 12
Settling time (ms)Maximum error (Hz)
Transient Steady state
Zone 1FR-NLMS 1 8.48 7.95VR-NLMS 1 7.36 4.87FRP-NLMS 25 3.43 0VRP-NLMS 21 3.75 0DSC PLL 73 5.62 0
Zone 2FR-NLMS 34 7.15 0VR-NLMS 31 5.22 0FRP-NLMS 29 1.17 0VRP-NLMS 26 0.93 0DSC PLL 89 2.05 0
Figure 13.
NMSD curves forvoltage harmonicswith different NLMSalgorithms andDSC-PLL 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.5
–40
–35
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–5
0
5
Time (sec)
NM
SD
(d
B)
FR- NLMS
VR- NLMS
FRP- NLMS
VRP- NLMS
DSC- PLL
Figure 14.
The Steady-statetrajectory in thealpha-beta plane(right) and the phasor(left) views of sub-andinter-harmonics intime domain –1.2 –0.9 –0.6 –0.3 0 0.3 0.6 0.9 1.2
–1.2
–0.9
–0.6
–0.3
0
0.3
0.6
0.9
1.2
Vα
Vβ
Harmonics
DSC output
1 1.1 1.2 1.3 1.4 1.5
–1
–0.5
0
0.5
1
Time (sec)
V (
p.u
.)
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tracking properties relative to the other existing NLMS methods. A DSC operator is placedbefore the NLMS algorithm in the proposed estimator. The positive sequence of voltage isseparated by the DSC operator in FRP and VRP-NLMS algorithms. The DSC operator in theFRP and VRP-NLMS algorithms operates as pre-filter, and it extracts the fundamentalcomponent of the voltage, and it is caused to lowmisalignment with the best performance.
Figure 15.
(a) Frequencyestimation; and (b)NMSD curves forvoltage sub- and
inter-harmonics withdifferent NLMSalgorithms and
DSC-PLL
1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.540
42
44
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56
Time (sec)
Fre
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Hz)
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FRP-NLMS
VRP-NLMS
DSC PLL
1.1 1.11 1.12 1.13 1.14 1.1549.5
50
50.5
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2–15
–10
–5
0
Time (sec)
NM
SD
(d
B)
FR NLMS VR NLMS FRP NLMS VRP NLMS DSC PLL
1.2
–12
–11
-10
–9
–8
–7
–6
(a)
(b)
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The VRP-NLMS algorithm is smarter than the FRP-NLMS algorithm because of VR. TheVR varies the step size of the estimation. The proposed step size has adaptive value andreduces the tradeoff between maladjustment and tracking ability of the fixed step sizeNLMS algorithm. The variable step size algorithm also reduces the sensitivity of themaladjustment to the level of non-stationarity. Also, the proposed algorithm has beencompared with DSC-PLL; the proposed algorithm is stable and immune to the disturbance.The VRP-NLMS has the best performance and fast convergence rate.
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Figure 16.
(a) Frequencyestimation; and (b)NMSD curves forfrequency step withdifferent NLMSalgorithms and DSC-PLL
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49.75
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Xia, Y. and Mandic, D.P. (2012), “Widely linear adaptive frequency estimation of unbalanced Three-Phase power systems”, IEEE Transactions on Instrumentation and Measurement, Vol. 61 No. 1,pp. 74-83.
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Further reading
de Souza, H.E., Bradaschia, F., Neves, F.A., Cavalcanti, M.C., Azevedo, G.M. and de Arruda, J.P. (2009),“A method for extracting the Fundamental-Frequency Positive-Sequence voltage vector basedon simple mathematical transformations”, IEEE Transactions on Industrial Electronics, Vol. 56No. 5, pp. 1539-1547.
Golestan, S., Freijedo, F.D., Vidal, A., Yepes, A.G., Guerrero, J.M. and Doval-Gandoy, J. (2016), “Anefficient implementation of generalized delayed signal cancellation pll”, IEEE Transactions onPower Electronics, Vol. 31 No. 2, pp. 1085-1094.
Corresponding authorMohammad Reza Besmi can be contacted at: [email protected]
For instructions on how to order reprints of this article, please visit our website:www.emeraldgrouppublishing.com/licensing/reprints.htmOr contact us for further details: [email protected]
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