AHarmonicImpedanceEstimationMethodBasedonthe … · 2020. 1. 3. · Generally,theexistingmethodsneedoneoronemore assumptions,whichincludesthatthebackgroundhar …

Research ArticleA Harmonic Impedance Estimation Method Based on theCauchy Mixed Model

Zhirong Tang 1 Huaqiang Li 2 Fangwei Xu2 Qin Shu2 and Yue Jiang2

1College of Nuclear Technology and Automation Engineering Chengdu University of Technology Chengdu 610059 China2College of Electrical Engineering Sichuan University Chengdu 610065 China

Correspondence should be addressed to Huaqiang Li lihuaqiangscueducn

Received 3 January 2020 Accepted 2 March 2020 Published 25 March 2020

Academic Editor Francesco Riganti-Fulginei

Copyright copy 2020 Zhirong Tang et al)is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this paper a new method without any tradition assumption to estimate the utility harmonic impedance of a point of commoncoupling (PCC) is proposed But the existing estimation methods usually are built on some assumptions such as the backgroundharmonic is stable and small the harmonic impedance of the customer side is much larger than that of utility side and theharmonic sources of both sides are independent However these assumptions are unpractical to modern power grid which causesvery wrong estimation)e proposed method first uses a CauchyMixedModel (CMM) to express the Norton equivalent circuit ofthe PCC because we find that the CMM can right fit the statistical distribution of the measured harmonic data for any PCC bytesting and verifying massive measured harmonic data Also the parameters of the CMM are determined by the expectationmaximization algorithm (EM) and then the utility harmonic impedance is estimated by means of the CMMrsquos parametersCompared to the existingmethods the main advantages of our method are as follows it can obtain the accurate estimation resultsbut it is no longer dependent of any assumption and is only related to the measured data distribution Finally the effectiveness ofthe proposed method is verified by simulation and field cases

1 Introduction

With the continual increase of the capacity of renewableenergy and power electronic equipments in power grid theutility harmonic problems are more and more serious andthe public requirements for power quality are more andmore raised and thus the harmonic control becomes moresignificant [1 2] )e harmonic suppression requires ac-curately acquiring the harmonic impedance of the utilityside [3ndash7]

At present the mainstream methods for estimating theutility harmonic impedance are noninvasive which mainlyinclude the linear regression method [8ndash13] fluctuationquantity (FQ) method [14 15] independent random vector(IRV) method [16] and independent component analysis(ICA) method [17ndash22] )e linear regression method useslinear regression to calculate Norton equivalent circuitequation coefficient and obtain the harmonic impedance)e FQ method uses the sign characteristics of the ratio of

harmonic voltage fluctuation to harmonic current fluctua-tion at a point of common coupling (PCC) to estimate theutility harmonic impedance )e linear regression methodand FQ method can effectively estimate the utility harmonicimpedance under the condition that the background har-monic is stable and small and the customer harmonicimpedance is much larger than that of utility side )e IRVmethod assumes the harmonic current of the PCC is weak-correlation with the utility harmonic voltage ie assumesimplicitly that the harmonic impedance of customer side ismuch larger than that of utility side which means that theharmonic current of the PCC is dominated by the customer)us the covariance of the harmonic current of the PCC andthe utility harmonic voltage can be approximated to zeroand then by solving the covariance equation the utilityharmonic impedance can be obtained )e main charac-teristic of ICA is that it can simultaneously figure out theharmonic impedances of both sides However ICA requiresthat the harmonic sources of both sides are independent

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 1580475 13 pageshttpsdoiorg10115520201580475

Generally the existing methods need one or one moreassumptions which includes that the background har-monic is stable and small the harmonic impedance of thecustomer side is much larger than that of utility side andthe harmonic sources of both sides are independentHowever for modern power grid these assumptions arehard to be held up For instance the harmonic sourcescaused by renewable energy and power electronic equip-ment in power grid widely distribute the harmonic sourcesamong power electronic equipments interact and har-monics amplify for transmission lines and cables )us it isobviously unreasonable that the background harmonic isassumed to be very small and stable [17] Besides filters orreactive compensation capacitors are usually installed onthe customer side which may cause the harmonic im-pedances of both sides to be approximately equivalent Asfor the independence hypothesis of the harmonic sourcesof both sides is much idealized and many researchers havepointed out that the assumption may not meet the practice[4 5 17 20 22]

In this paper a novel method without any assumption isproposed which employs a Cauchy Mixed Model (CMM) tofit the distribution characteristic of the measured data at thePCC because we have found that the statistical distributionof the harmonic measurements of the PCC can be depictedwith CMM through the analysis of the field harmonic dataSo a fitting CMM will be chosen to match the distributionrelationship between the harmonic voltage the harmoniccurrent and the utility harmonic impedance of the PCCAlso then the utility harmonic impedance can be obtainedby the expectation maximization algorithm (EM)

)is paper mainly studies from the following fouraspects

(1) )e principle of CMM is introduced and the Nortonequivalent circuit of the PCC is expressed as a CMM

(2) )e feasibility of CMM is verified by field measureddata

(3) )e CMM estimation processes are derived in detail(4) )e effectiveness and robustness of the proposed

method are verified by comparing the estimationresults of simulation and field data with existingalgorithms

)is paper is organized as follows Section 2 presents theanalysis for the validity of CMM Section 3 derives theprocess of estimating the utility impedance Sections 4 and 5verify the validity of the proposed method by simulation andfield case )e paper is concluded in Section 6

2 Basic Principles

21HarmonicEquivalentAnalysisModel Norton equivalentcircuit of the utility side and customer side is shown inFigure 1

In Figure 1 _Is is the utility harmonic current and Zs isthe harmonic impedance on the utility side _UPCC and _IPCCare the harmonic voltage and current measured at the PCCrespectively

According to the Norton equivalent circuit the equationcan be written as

Zs_Is _UPCC + Zs

_IPCC (1)

We have

Δ _Is(n) _Is(n + 1) minus _Is(n)

Δ _IPCC(n) _IPCC(n + 1) minus _IPCC(n)

Δ _UPCC(n) _UPCC(n + 1) minus _UPCC(n)

n 1 2 N minus 1

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(2)

where n is the location of the sampling point and N is thenumber of sampling points Δ _Is(n) is the fluctuation of theutility harmonic current source Δ _IPCC(n) is the harmoniccurrent fluctuation at the PCC and Δ _UPCC(n) is the har-monic voltage fluctuation at the PCC

)en equation (1) can be expressed as

ZsΔ _Is Δ _UPCC + ZsΔ _IPCC (3)

where

Δ _Is Δ _Is(1)Δ _Is(2) middot middot middot Δ _Is(N minus 1)1113966 1113967

Δ _UPCC Δ _UPCC(1)Δ _UPCC(2) middot middot middot Δ _UPCC(N minus 1)1113966 1113967

Δ _IPCC Δ _IPCC(1)Δ _IPCC(2) middot middot middot Δ _IPCC(N minus 1)1113966 1113967

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(4)

)us equation (3) can be rewritten asZsminusx minusZsminusy

Zsminusy Zsminusx

⎡⎣ ⎤⎦ΔIsminusx

ΔIsminusy

⎡⎣ ⎤⎦ ΔUPCCminusx

ΔUPCCminusy

⎡⎣ ⎤⎦ +Zsminusx minusZsminusy

Zsminusy Zsminusx

⎡⎣ ⎤⎦

middotΔIPCCminusx

ΔIPCCminusy

⎡⎣ ⎤⎦

(5)

where ΔIsminusx and ΔIsminusy are the real and imaginary parts of theutility harmonic current respectively ΔUPCCminusx andΔUPCCminusy are the real and imaginary parts of the harmonicvoltage measured at the PCC respectively ΔIPCCminusx andΔIPCCminusy are the real and imaginary parts of the harmoniccurrent measured at the PCC respectively Zsminusx and Zsminusy arethe real and imaginary of the harmonic impedance on theutility side respectively

)erefore

ΔUs ΔUPCC + 1113954ZsΔIPCC (6)

where

PCC

İs Zs Zc İc

İPCC

UPCC

Customer sideUtility side

Figure 1 )e Norton equivalent circuit

2 Mathematical Problems in Engineering

ΔUs ΔUsx ΔUsy1113960 1113961T

Zsminusx minusZsminusy

Zsminusy Zsminusx


ΔIsminusy

⎡⎣ ⎤⎦

ΔUPCC ΔUPCCminus x ΔUPCCminus y1113960 1113961T

1113954Zs Zsminusx minusZsminusy

Zsminusy Zsminusx

⎡⎣ ⎤⎦

ΔIPCC ΔIPCCminus x ΔIPCCminus y1113960 1113961T

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(7)

22 Cauchy Mixed Model A D-dimensional Cauchy prob-ability density function is

φ1113874x | μ 11139441113875 Γ((1 + D)2)

Γ(12)πD2| 1113936 |(12) 1 +(x minus μ)T1113936

minus1(x minus μ)1113960 1113961

((1+D)2)

(8)

where μ is the data center point 1113936 is the symmetric sem-idefinite matrix D is the data dimension and Γ(middot) is theGamma function

If given observation data X x1 x2 xN1113864 1113865 the CMMis

P(X | α μ 1113944) 1113944K

k1αkφ X μkΣk

11138681113868111386811138681113872 1113873 (9)

where K α α1 α2 middot middot middot αK1113864 1113865 μ μ1 μ2 μK1113864 1113865 and 1113936

11139361 11139362 1113936K1113864 1113865 are the number of models weightedvalues data center points and covariance matricesrespectively

23 Feasibility Analysis of theModel )is section introducesthe fitting methods of data distribution and the test of fittingresults by CMM First we introduce the Copula functionand Sklarrsquos theorem [23 24]

Definition 1 A copula is a function C of D variables on theunit D-cube [01]D with the following properties

(1) )e range of C is the unit interval [01](2) C(x) is zero for all x (x1 xD) isin [01]D for which

at least one coordinate equals zero(3) C(x) xd if all coordinates of x are 1 except the dth

one(4) C is D-increasing in the sense that for every ale b in

[01]D the measure ΔCab assigned by C to the D-box

[a b] [a1 b1]times times [aD bD] is nonnegative ie

ΔCab 1113944

ε1 εD( )isin 01

(minus1)ε1++εD C ε1a1 + 1 minus ε1( 1113857b1 (

εDaD + 1 minus εD( 1113857bD1113857ge 0

(10)

)e key theorem due to Sklar clarifies the relations ofdependence and the copula of a distribution

Theorem 1 Let F denote a D-dimensional distributionfunction with margins F1 FD 4en there exists a D-copula C such that for all real x (x1 xD)

F(x) C F1 x1( 1113857 FD xD( 1113857( 1113857 (11)

Therefore the C is unique that isC(x) F F

minus11 x1( 1113857 F

minus1D xD( 11138571113872 1113873 (12)

where Fminus11 (middot) Fminus1

D (middot) are quasi-inverses of the marginaldistribution functions

To verify whether the CMM can fit the probabilitydistribution of the harmonic voltage _UPCC and current _IPCCdata we combine the two-dimensional t-copula function[25] with the Monte Carlo method to generate a set ofrandom data conforming to CMM )en the CumulativeDistribution Functions (CDF) of field data and generateddata are calculated and expressed as curves respectivelyFinally we use the KolmogorovndashSmirnov (KS) test [26] todetect the maximum distance between two CDF curves

The steps of generating the CMM data are as follows

(1) K models are selected and the CMM is adopted toPCC data )en the corresponding weight αk andcovariance matrix 1113936k are obtained

(2) Convert the obtained covariance matrix 1113936k into acorrelation matrix ρk

(3) )e t-Copula function is used to generate randomtwo-dimensional Cauchy number of K groups as-sociated with ρk

(4) Generate N random numbers ni from 0 to max αk Ifni lt αk select the random number set of the kth two-dimensional Cauchy model

According to the abovementioned method the CMM isused to fit the distribution of two groupsrsquo field data )e firstset of field data is conducted at the site where a 150 kV busbar feeds a 100MW Electric Arc Furnace (EAF) )e totalsample time is 10 hours and the calculated 3rd harmonicsampling values are averaged per minute then the numberof the all samples is 600 CMM random numbers aregenerated corresponding to the EAF data )e CDF of thedata generated by CMM and the EAF data are shown inFigure 2

The results of the KS test are shown in Table 1In Table 1H 0 indicates acceptance of the hypothesis at

the significance level of 5 P is the probability of accep-tance and KSSTAT is the value of the test statistic It can beseen from Figure 2 and Table 1 that the CMM can fit EAFdata

The second set of field data is taken from a city powergrid with multiple DC terminals and the measured data arecollected from 600 data points (20 sampled data per minute)at a 500 kV bus of the DC terminal for the 11th harmonicvoltage and current CMM random numbers are generatedcorresponding to the DC terminal data )e CDF of the datagenerated by CMM and DC terminal data are shown inFigure 3

The results of the KS test are shown in Table 2The meanings of the parameters in Table 2 are the same

as those in Table 1 According to Figure 3 and Table 2 it is

Mathematical Problems in Engineering 3

obvious that the CMM can fit DC terminal data even thoughthere is a slight deviation between the CDF curve of theCMM and that of DC terminal

3 Utility Harmonic Impedance Estimation

In the previous section the feasibility of the CMMmethod isverified In this section the impedance estimation methodbased on CMM is described

For the observation dataΔUPCC ΔUPCC(1)ΔUPCC(2) ΔUPCC(N minus 1)1113864 1113865

ΔIPCC ΔIPCC(1)ΔIPCC(2) ΔIPCC(N minus 1)1113864 11138651113896 the

joint probability function of the CMM under utility im-pedance 1113954Zs 1113954Zs(1) 1113954Zs(2) 1113954Zs(K)1113966 1113967 is

P(θ) P ΔUPCCΔIPCC | θ( 1113857

1113945Nminus1

n1P ΔUPCC(n)ΔIPCC(n) | θ( 1113857

(13)

where θ (θ1 θ2 θK) θk (μk 1113936k 1113954Z(k)) By takingthe logarithm likelihood function for P(θ) we know

l(θ) argmaxθ

logP(θ) argmaxθ

middot log 1113945Nminus1


argmaxθ

1113944

Nminus1

n1log P ΔUPCC(n)ΔIPCC(n) | θ( 1113857

argmaxθ

1113944

Nminus1

n1log 1113944

K

k1αkφ ΔUPCC(n)ΔIPCC(n) θk

11138681113868111386811138681113872 1113873

(14)

EAFCMM

0

02

04

06

08

1

CDF

ndash200 ndash100 0 200 300ndash300 100

(a)

EAFCMM

0

02

04

06

08

1

CDF

ndash300 ndash100 0 100 200ndash200

(b)

EAFCMM

0

02

04

06

08

1

CDF

ndash1000 0 1000 2000ndash2000

(c)

EAFCMM

ndash1000 0 1000 2000 3000ndash20000

02

04

06

08

1

CDF

(d)

Figure 2 )e CDF of the data generated by CMM and the EAF data (a) Real part of current (b) Imaginary part of current (c) Real part ofvoltage (d) Imaginary part of voltage

Table 1 KS test result of CDF of EAF

ParameterCurrent Voltage

Real Imaginary Real ImaginaryH 0 0 0 0P 07584 07416 07710 08479KSSTAT 00299 00303 00295 00272


It is very difficult to directly find the value of θ fromequation (14) Since the EM algorithm has stable estimationability for the mixture model the EM algorithm is appliedhere )e EM algorithm consists of two steps namelyE-step calculating the expectation of the log likelihoodfunction of the complete data and M-step obtaining a newparameter value by maximizing the value of the intermediateamount

E-step according to Bayesian formula we can find theposterior probability of the nth group of samples(ΔUPCC(n)ΔIPCC (n)) generated by the kth Cauchy model

cnk αkφ ΔUPCCΔIPCC θk

11138681113868111386811138681113872 1113873

1113936Kk1αkφ ΔUPCCΔIPCC θk

11138681113868111386811138681113872 1113873 (15)

For equation (15) cnk is a known probability value andsatisfies equation (16)

1113944

K

k1cnk 1 (16)

)en we obtain the following function l(θ) based on theprinciple of Jensenrsquos inequality

l(θ) argmaxθ

1113944

Nminus1

n1log 1113944

K

k1cnk

αkφ ΔUPCC(n)ΔIPCC(n) θk

11138681113868111386811138681113872 1113873

cnk

ge argmaxθ

1113944

Nminus1

n11113944

K

k1cnklog


11138681113868111386811138681113872 1113873

cnk

(17)

0

02

04

06

08

1CD

F

DC terminalCMM

0 50ndash50

(a)

DC terminalCMM

0

02

04

06

08

CDF

0 50ndash50

(b)

DC terminalCMM

ndash2000 0 2000 4000ndash40000

02

04

06

08

1

CDF

(c)

DC terminalCMM

ndash2000 0 2000 4000ndash40000

02

04

06

08

1

CDF

(d)

Figure 3)e CDF of the data generated by CMM and DC terminal data (a) Real part of current (b) Imaginary part of current (c) Real partof voltage (d) Imaginary part of voltage

Table 2 KS test result of CDF of DC terminal




By adjusting the lower bound function of equation (17)the maximum value of the lower bound function can beobtained so that equation (17) can gradually approach the

optimal solution or obtain the local optimal solution asfollows

l(θ) argmaxθ

1113944

Nminus1

n11113944

K

k1cnklog

αkφ ΔUPCC(n)ΔIPCC(n) | θ( 1113857

cnk

argmaxθ

1113944

Nminus1

n11113944

K

k1cnk log αkΓ

1 + D

21113874 1113875 minus log Γ

12

1113874 1113875πD2minus log cnk minus log Σk

1113868111386811138681113868111386811138681113868111386812

1113882

minus1 + D

2log 1 + ΔUPCC(n) + 1113954Zs(k)ΔIPCC(n) minus μk1113872 1113873

Tmiddot Σminus1k ΔUPCC(n) + 1113954Zs(k)ΔIPCC (n) minus μk1113872 11138731113876 11138771113883

(18)

M-Step is to calculate a new round of the iterative pa-rameter model We assume that

Ψnk 1 + ΔUPCC(n) + 1113954Zs(k)ΔIPCC(n) minus μk1113872 1113873T

middot Σminus1k ΔUPCC(n) + 1113954Zs(k)ΔIPCC(n) minus μk1113872 1113873(19)

So the partial derivative of equation (18) for μk is

zl(θ)

zμk

minus 1113944Nminus1

n1cnk middot

1 + D

2middot1Ψnk

middot Σminus1k + ΣminusTk1113872 1113873

middot ΔUPCC(n) + 1113954Zs(k)ΔIPCC(n) minus μk1113872 1113873

(20)

Let equation (20) be equal 0 we have

μk 1113944

Nminus1

n1cnk middotΔUPCC(n) + 1113954Zs(k)ΔIPCC(n)

Ψnk

⎛⎝ ⎞⎠ 1113944

Nminus1

n1

cnk

Ψnk

⎛⎝ ⎞⎠

minus 1

(21)

Since equation (21) is nonlinear it is impossible toseparate μk )erefore in this step equation (21) can beiterated until convergence

)e partial derivative of equation (18) for 1113936k is obtainedas follows

zl(θ)

z1113936 k

minus1 + D

21113944

Nminus1

n1

cnk

Ψnk

middot 1113944minus1

k

ΔUPCC(n)( +1113954Zs(k)ΔIPCC(n) minus μk1113873⎡⎣

middot ΔUPCC(n)( +1113954Zs(k)ΔIPCC(n) minus μk1113873TΣminus1k minus ΣminusT

k 1113877

(22)

Let (22) be equal to 0 and after simplification the updateequation of 1113936k is

1113944k

(1 + D) 1113936

Nminus1n1 cnkΨnk middot ΔUPCC(n) + 1113954Zs(k)ΔIPCC(n) minus μk1113872 11138731113960 ΔUPCC(n) + 1113954Zs(k)ΔIPCC(n) minus μk1113872 1113873

T1113877

1113936Nminus1n1 cnk

(23)

Similarly since (23) is also nonlinear we cannot separate1113936k )us in this step (23) can be iterated until convergence

Since 1113954Zs(k) is a negative diagonal matrix its elementscannot be directly updated as a whole and each elementneeds to be updated separately )e following transforma-tion can be performed

1113954Zs(k)ΔIPCC(n) Zsminusx(k) minusZsminusy(k)

Zsminusy(k) Zsminusx(k)⎡⎣ ⎤⎦

ΔIPCCminusx(n)

ΔIPCCminusy(n)⎡⎣ ⎤⎦

ΔIPCCminusx(n) minusΔIPCCminusx(n)

ΔIPCCminusx(n) ΔIPCCminusx(n)1113890 1113891

Zsminusx(k)

Zsminusy(k)⎡⎣ ⎤⎦

WnRk

(24)

where Wn ΔIPCCminusx(n) minusΔIPCCminusx(n)

ΔIPCCminusx(n) ΔIPCCminusx(n)1113890 1113891 and

Rk Zsminus x(k) Zsminus y(k)1113960 1113961T )e cost function for Rk is

l Rk( 1113857 1113944Nminus1

n11113944

K

k1cnk log 1 + ΔUPCC(n) + WnRk minus μk( 1113857

T1113960


k

ΔUPCC(n) + WnRk minus μk( 1113857⎤⎦

(25))e partial derivative of equation (25) for Rk is

zl Rk( 1113857

zRk

1113944Nminus1

n1cnk

WTn 1113936

minus1k ΔUPCC(n) minus μk( 1113857 + WT

n 1113936minus1k WnRk

Ψnk

(26)


Let equation (26) be equal to 0 then the update equationof Rk is

Rk minus 1113944Nminus1

n1cnk middot

WTn 1113936

minus 1k WnRk

Ψnk

⎡⎣ ⎤⎦

minus 1

1113944

Nminus1

n1cnk middot

WTn 1113936

minus1k UPCC(n) minus μk( 1113857

Ψnk

⎡⎣ ⎤⎦

(27)Furthermore

l(α) l(θ) minus η 1113944K

k1αk minus 1⎛⎝ ⎞⎠ η N minus 1 (28)

We can obtain

αk 1

N minus 11113944

Nminus1

n1cnk (29)

For equations (16) and (29)

1113944

K

k1αk 1 (30)

)e μk 1113936kRk and αk pairs of parameters are updated byiterating over equations (21) (23) (27) and (29) untilconvergence Since the harmonic data corresponding to eachmodel affects the utility harmonic impedance we use aweighted value αk for Rk as follows

Zs [1 j] middot 1113944

K

k1αkRk (31)

)e steps of this method are in Algorithm 1

4 Simulation Analysis

Simulations are conducted to verify the effectiveness of theproposed method in accordance of Figure 1 )ey areprogrammed by Matlab2017a Both the customer side andthe utility side adopt a parallel model of harmonic currentsource and harmonic impedance )e specific parameters ofthe simulation model are set as follows the amplitude of theutility harmonic impedance Zs is 10Ω and the phase angleis 80deg (radian value is 13963))e amplitude of the customerharmonic impedance Zc is 10m(m |ZcZs|)Ω | middot | is the

module and the phase angle is 33deg (radian value is 05760) _Is

and _Ic are_Is(n) 200q middot 1 + 05 sin(n) + λ1rs(n)( 1113857

middot expminus30 middot 1 + 01cs(n) minus 005( 1113857π

180j

_Ic(n) 200 middot 1 + 05 sin(n) + λ2rc(n)( 1113857

middot exp30 middot 1 + 01cc(n) minus 005( 1113857π

180j

λ1 05

λ2 05

n 1 2 N

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(32)

where λ1 and λ2 are the disturbance factors belonging to[01] within the value range of λ1 and λ2 we have q asymp IsIcsin(middot) is a sine function rs(n) and rc(n) are randomnumbers that obey the standard normal distribution andcs(n) and cc(n) are random numbers that follow a uniformdistribution Also the unit is Ampere

According to the abovementioned parameters 300harmonic samples are generated randomly and the utilityharmonic impedance is estimated by the CMM FluctuationQuantity (FQ) Binary Regression (BR) Independent Ran-dom Vector (IRV) and Independent Component Analysis(ICA) methods respectively

41 Influence of Background Harmonic on Utility ImpedanceEstimation To analyze the influence of background har-monic on the utility impedance estimation we set q from 05to 12 in steps of 01 and λ1 λ2 05 When m 3 theimpedance of both sides is approximately equal Here wedefine the correlation coefficient ρ of the current of bothsides as follows

ρ 1113936

Nh1

_Is(h) minus E _Is1113872 11138731113872 1113873 _Ic(h) minus E _Ic1113872 11138731113872 1113873lowast11138681113868111386811138681113868

11138681113868111386811138681113868

1113936Nh1

_Is(h) minus E _Is1113872 11138731113872 1113873 _Is(h) minus E _Is1113872 11138731113872 1113873lowast

1113969

1113936Nh1

_Ic(h) minus E _Ic1113872 11138731113872 1113873 _Ic(h) minus E _Ic1113872 11138731113872 1113873lowast

1113969 (33)

where E(middot) is the expected value of the data and lowast is theconjugate plural of a complex data In the simulation the ρ isabout 043 and the relative errors of the harmonic im-pedance amplitude and the phase relative errors are shownin Figure 4 respectively

It can be seen from Figure 4 when the impedance of bothsides is close the relative errors of amplitude and phase ofthe five methods gradually increase with the increase ofbackground harmonics However compared with the otherfour methods the CMM method is the most reliable esti-mate of the utility harmonic impedance When qge 07 the

FQ BR IRV and ICA methods fail to estimate the utilityimpedance

When m 10 the consumer impedance is much largerthan the utility impedance the value of impedance on eachside is approximately equal )e amplitude relative errorsand phase relative errors of the five methods are shown inFigure 5

According to Figure 5 and related analysis in the pre-vious section the five methodsrsquo estimation errors increasewith the increase of amplitude ratio q In addition with theincrease of current ratio q the estimation accuracy of the


CMM method is higher than that of other four methods)erefore the CMM method can effectively estimate theutility harmonic impedance when the background harmonicincreases

42 Influence of the Continuous Change of BackgroundHarmonic and Harmonic Impedance on Utility ImpedanceEstimation For further analyzing the influence of thecontinuous change of background harmonic and harmonic

CMM

FQ

BR

IRV

ICA

0

20

40

60

80

Am

plitu

de re

lativ

e err

or (

)

10 11 12090806 0705Current ratio q

(a)

CMM

FQ

BR

IRV

ICA

0

10

20

30

40

Phas

e rel

ativ

e err

or (

)

10 11 12090706 0805Current ratio q

(b)

Figure 4 Relative errors of amplitude and phase of Zs (m 3) (a) Amplitude relative errors (b) Phase relative errors

Input Harmonic voltage _UPCC and current _IPCC at PCCOutput Utility side harmonic impedance estimate ZsStep1 According to equations Equations (1)ndash(3) (5) convert _UPCC and _IPCC in vector form to ΔUPCC and ΔIPCC in matrix formStep2 Select the appropriate model parameter K(Kge 1) and updating E-step parameter cnk by iteration equation (15)Step3 Update M-step parameters μk 1113936k αk by equations (21) (23) and (29)Step4 Update parameter Rk by equation (27)Step5 Estimate Zs from equation (31)

ALGORITHM 1 CMM algorithm

0

10

20

30

40

Am

plitu

de re

lativ

e err

or (

)

06 07 08 09 10 11 1205Current ratio q

CMM

FQ

BR

IRV

ICA

(a)

0

5

10

15

20

Phas

e rel

ativ

e err

or (

)

06 07 08 09 10 11 1205Current ratio q

CMM

FQ

BR

IRV

ICA

(b)



impedance on utility impedance estimation set q rangesfrom 05 to 12 and the impedance ratio of both sides rangesfrom 15 to 150 and for every q (its step size is 005) and m(its step size is 025) And set the disturbance factorsλ1 λ2 04 the correlation coefficient ρ of current of bothsides is about 046 )e relative errors of amplitude andphase of utility impedance are shown in Figure 6

According to Figure 6 when the correlation coefficient ρof the current amplitudes on each side is about 063 therelative error of the amplitude and phase angle of the CMMmethod is less than 12 and 10 respectively even underthe condition that the impedances of both sides are ap-proximately equivalent and the background harmonic islarge which verifies the robustness and effectiveness of theCMM method

43 Influence of Correlation of Harmonic Currents of BothSides on Utility Harmonic Impedance Estimation To furtherverify the effectiveness of the CMM method when thecurrents of both sides are not independent or are stronglycorrelated we conducted another simulation Sincechanging the interference of both sides of the current canchange the independence of the current we first set thedisturbance factors λ1 λ2 λ and then set λ from 01 to 10with a step size of 01 With the change in value of λ thecorrelation coefficient ρ of the current of both sides is asshown in Table 3 (the current ratio q is fixed at 05 and theimpedance ratio m is fixed at 5)

As can be seen from Table 3 the correlation coefficient ρdecreases with the increase of disturbance parameter λ Alsothe amplitude relative errors and phase relative errors of thefive methods are shown in Figure 7

From Figure 7 when the correlation is strong betweenthe currents of the consumer side and the utility side theestimation error ratio of the ICA method on the utilityharmonic impedance is the highest followed by the FQ BRand IRV methods In addition it can be seen that theperformance of the CMM method is significantly morestable which does not depend on the correlation of currentson each side )erefore it proves that the CMM methodprovides reliable utility harmonic impedance regardless ofthe strong or weak correlation between the consumer sidecurrent and the utility side current

5 Practical Cases

)e feasibility and effectiveness of the proposed method areverified by simulation in Section 4 In this section thepracticability of the proposed method is verified by twopractical cases under different backgrounds

51 Case 1 To verify the practicability of the proposedmethod the field data of the EAF are used for verificationand the harmonic voltage and current amplitudes of the EAFare shown in Figure 8 )e estimation values of the fivemethods with 600 sample points are shown in Table 4

In Table 4 the estimation results of the CMM BR IRVand ICA methods are equivalent while the amplitude es-timation result of the FQ method is quite different fromthose of other four methods To further verify the reliabilityof the proposed method in estimating the utility harmonicimpedance the abovementioned 600 data are divided into 11subintervals )e first 10 subintervals each of which has 54sample points and the last subinterval have 60 sample

Am

plitu

de re

lativ

eer

rors

()

Impedance ratio mBackground harmonic

ratio q05 06 07 08 09 10 11 120

510

15

0

1210

8642

10

8

6

4

2

(a)

05 06 07 08 09 10 11 120

510

15Impedance ratio m

765

21

34

8

0123456789

Phas

e rel

ativ

e err

ors (

)

Background harmonic

ratio q

(b)

Figure 6 Relative errors of amplitude and phase of Zs under the continuous change of background harmonic and harmonic impedance(a) Amplitude relative errors (b) Phase relative errors

Table 3 )e value of the correlation coefficient (m 5 q 05)

Parameter λ ρ λ ρ

Value

01 09146 06 0369702 08078 07 0309303 06551 08 0232204 05345 09 0122805 04400 10 00936


points )e five methods are used to estimate the utilityimpedance in per hour )e effectiveness of impedanceestimation of the five methods is shown in Figure 9

As can be seen from Figure 9 compared with the otherfour methods the CMM method produces the most stableestimation results of the amplitude and phase radian of theutility harmonic impedance in each time period )e FQmethod is unstable Compared with the CMM method theBR method still shows great fluctuations on its

performances )e performance of the IRV method isroughly equivalent to the one of the CMM method but inthe 9th time period the fluctuation of the IRV method islarger than that of the CMM method )is result proves theaccuracy and robustness of the CMM method

52 Case 2 )e amplitudes of harmonic voltage and currentof the DC terminal data are shown in Figure 10 In theabsence of load the utility side is directly connected into

01 02 03 04 05 06Disturbance factor λ

080

20

40

80A

mpl

itude

rela

tive e

rror

()

09

60

1007

CMM

FQ

BR

IRV

ICA

(a)

Phas

e rel

ativ

e err

or (

)

Disturbance factor λ

0

5

15

20

10

01 02 03 04 05 06 08 09 1007

CMM

FQ

BR

IRV

ICA

(b)

Figure 7 Amplitude relative errors and phase relative errors of five algorithms with different λ values (a) Amplitude relative errors(b) Phase relative errors

0

500

1000

1500

2000

2500

Am

plitu

de (V

)

400 6000 200Voltage data

(a)

0

50

100

150

200

250

Am

plitu

de (A

)

0 400 600200Current data

(b)

Figure 8 Amplitudes of Voltage and Current at the PCC of EAF (a) Voltage amplitude (b) Current amplitude

Table 4 Impedance values estimated by five algorithms for the EAF data

Impedance estimationAlgorithm

CMM FQ BR IRV ICAAmplitude (Ω) 101131 110819 101796 101199 101351Phase (rad) 12220 12537 11847 12121 11697


a closed loop with wires )en the measured utility short-circuit capacity is 34580 MVA

)e estimation values of the five methods with 600sample points are shown in Table 5

And then the 11th harmonic data are divided into 10subintervals and each subinterval has 60 sample pointsFigure 11 shows the estimation values with per subinterval

According to the short-circuit capacity the approximatevalue of the utility harmonic impedance is 795Ω It can beseen from Table 5 that the impedance amplitude estimated

by the CMM method is closest to the rough estimation ofshort-circuit capacity As can be seen from Figure 11(a) theestimation results of the CMMmethod in each data segmentare more stable than those of the other four methods )eBR IRV and FQ methods have the same amplitude esti-mation fluctuation but the amplitude estimated by the FQmethod is the smallest )e fluctuation of the ICA method isrelatively large Figure 11(b) shows that the phase estimationvalues of the CMM BR and IRV methods are fluctuatingaround 10630 rad and the proposed method is the most

Am

plitu

de (Ω

)

CMM

FQ

BR

IRV

ICA

5

10

15

20

2 3 4 5 6 7 8 9 10 111Time (h)

(a)

Phas

e (ra

d)

CMM

FQ

BR

IRV

ICA

ndash2

ndash1

0

1

2

2 3 4 5 6 7 8 9 10 111Time (h)

(b)

Figure 9 Utility impedance estimation effect of five methods in per hour (a) Utility impedance amplitude (b) Utility impedance phase

Am

plitu

de (V

)

1200

1300

1400

1500

1600

100 200 300 400 500 6000Voltage data

(a)

Am

plitu

de (A

)

165

17

175

18

185

100 200 300 400 5000Current data

(b)

Figure 10 Amplitudes of voltage and current at the PCC of DC terminal (a) Voltage amplitude (b) Current amplitude

Table 5 Impedance values estimated by five algorithms for the DC terminal data




robust )e phase estimation of the FQ method is thesmallest )e ICA method is unstable for phase estimation

6 Conclusions

In this paper a novel method based on CMM for estimatingutility harmonic impedance is proposed In theory CMMcan approach any probability model and in practice it iseasy to verify that field harmonic data of PCC can effectivelybe matched with CMM )erefore we convert the Nortonequivalent circuit model at the PCC into a CMM andthrough adjusting the parameters of the CMM with the EMalgorithm the optimal estimation of utility harmonic im-pedance is calculated )e distinguishing feature of themethod can obtain accurate and stable estimation resultswithout any assumption conditions Simulations and fieldcases verify the effectiveness of the proposed method

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Acknowledgments

)is work was supported by National Natural ScienceFoundation of China (51877141)

References

[1] Q Liu Y Li L Luo Y Peng and Y Cao ldquoPower qualitymanagement of PV power plant with transformer integratedfiltering methodrdquo IEEE Transactions on Power Deliveryvol 34 no 3 pp 941ndash949 2019

[2] Q Liu Y Li S Hu and L Luo ldquoA transformer integratedfiltering system for power quality improvement of industrialDC supply systemrdquo IEEE Transactions on Industrial Elec-tronics vol 67 no 5 pp 3329ndash3339 2020

[3] G K Singh A K Singh and R Mitra ldquoA simple fuzzy logicbased robust active power filter for harmonics minimizationunder random load variationrdquo Electric Power Systems Re-search vol 77 no 8 pp 1101ndash1111 2007

[4] A Moradifar K G Firouzjah and A A Foroud ldquoCom-prehensive identification of multiple harmonic sources usingfuzzy logic and adjusted probabilistic neural networkrdquoNeuralComputing amp Applications vol 31 no 3-4 pp 1ndash14 2017

[5] T Zang Z He Y Wang L Fu Z Peng and Q Qian ldquoApiecewise bound constrained optimization for harmonic re-sponsibilities assessment under utility harmonic impedancechangesrdquo Energies vol 10 no 7 pp 936ndash956 2017

[6] R Nasim H Mohsen S Keyhan et al ldquoHigh-frequency oscil-lations and their leading causes in dc microgridsrdquo IEEE Trans-actions on Energy Conversion vol 32 no 4 pp 1479ndash1491 2017

[7] S Duan R Johan B Gerhard et al ldquoContinuous event-basedharmonic impedance assessment using online measure-mentsrdquo IEEE Transactions on Instrumentation and Mea-surement vol 65 no 10 pp 2214ndash2220 2016

[8] S Mehdi and H Mokhtari ldquoA method for determination ofharmonics responsibilities at the point of common couplingusing data correlation analysisrdquo IET Generation Transmissionamp Distribution vol 8 no 1 pp 142ndash150 2014

[9] Z Yunyan C Wei D Chun et al ldquoA harmonic measurementmode based on partial least-squares regression methodrdquo inProceedings of the Electric Power Conference pp 1ndash5 IEEEVancouver Canada October 2008

[10] Z Abolfazl and M Hossein ldquoNew method for assessing theutility harmonic impedance based on fuzzy logicrdquo IETGeneration Transmission amp Distribution vol 11 no 10pp 2448ndash2456 2017

[11] X Y Xiao and H G Yang ldquo)e method of estimatingcustomers harmonic emission level based on bilinear re-gressionrdquo in Proceedings of the IEEE International Conferenceon Electric Utility Deregulation Restructuring and PowerTechnologies vol 2 no 10 pp 662ndash665 IEEE Hong KongChina April 2004

Am

plitu

de (Ω

)

775

780

785

790

795

800

2 3 4 5 6 7 8 9 101Subinterval

CMM

FQ

BR

IRV

ICA

(a)

Phas

e (ra

d)

104

105

106

107

2 3 4 5 6 7 8 9 101Subinterval

CMM

FQ

BR

IRV

ICA

(b)

Figure 11 Impedance estimation effect of five methods on utility side in each data segment (a) Amplitude (b) Phase


[12] T Zang Z He L Fu Y Wang and Q Qian ldquoAdaptivemethod for harmonic contribution assessment based on hi-erarchical K-means clustering and Bayesian partial leastsquares regressionrdquo IET Generation Transmission amp Distri-bution vol 10 no 13 pp 3220ndash3227 2016

[13] Y Wang H E Mazin W Xu and B Huang ldquoEstimatingharmonic impact of individual loads using multiple linearregression analysisrdquo International Transactions on ElectricalEnergy Systems vol 26 no 4 pp 809ndash824 2016

[14] H Yang P Porotte and A Robert ldquoAssessing the harmonicemission level from one particular customerrdquo in Proceedingsof the 3rd International Conference on Power Quality (ICPQ)Amsterdam Netherlands 1994

[15] AM Dan ldquoIdentification of individual harmonic sources andevaluation their contribution in the harmonic distortionlevelrdquo in Proceedings of the IEEE Power amp Energy SocietyGeneral Meeting pp 1ndash6 IEEE Calgary Canada July 2009

[16] J Hui H Yang S Lin andM Ye ldquoAssessing utility harmonicimpedance based on the covariance characteristic of randomvectorsrdquo IEEE Transactions on Power Delivery vol 25 no 3pp 1778ndash1786 2010

[17] F Xu H Yang J Zhao et al ldquoStudy on constraints forharmonic source determination using active power direc-tionrdquo IEEE Transactions on Power Delivery vol 33 no 6pp 2683ndash2692 2018

[18] F Karimzadeh S Esmaeili and S H Hosseinian ldquoA novelmethod for noninvasive estimation of utility harmonic im-pedance based on complex independent component analysisrdquoIEEE Transactions on Power Delivery vol 30 no 4pp 1843ndash1852 2015

[19] M Novey and T Adali ldquoComplex ICA by negentropymaximizationrdquo IEEE Transactions on Neural Networksvol 19 no 4 pp 596ndash609 2008

[20] F Karimzadeh S Hossein Hosseinian and S EsmaeilildquoMethod for determining utility and consumer harmoniccontributions based on complex independent componentanalysisrdquo IET Generation Transmission amp Distributionvol 10 no 2 pp 526ndash534 2016

[21] X Zhao and H Yang ldquoA new method to calculate the utilityharmonic impedance based on FastICArdquo IEEE Transactionson Power Delivery vol 31 no 1 pp 381ndash388 2016

[22] Y Wang W Xu J Yong et al ldquoEstimating harmonic impactof individual loads using harmonic phasor datardquo Interna-tional Transactions on Electrical Energy Systems vol 27no 10 Article ID e2384 2017

[23] L Wang J Zeng and Y Hong ldquoEstimation of distributionalgorithm based on copula theoryrdquo in Proceedings of the IEEECEC 2009 pp 1057ndash1063 Trondheim Norway May 2009

[24] J Strelen and F Nassaj ldquoAnalysis and generation of randomvectors with copulasrdquo in Proceedings of the 2007 WinterSimulation Conference (WSC2007) pp 488ndash496WashingtonDC USA 2007

[25] C Villa and F J Rubio ldquoObjective priors for the number ofdegrees of freedom of a multivariate t distribution and thet-copulardquo Computational Statistics amp Data Analysis vol 124pp 197ndash219 2018

[26] F Antoneli F M Passos L R Lopes et al ldquoA Kolmogorov-Smirnov test for the molecular clock based on Bayesian en-sembles of phylogeniesrdquo PLoS One vol 13 no 1 Article IDe0190826 2018


Generally the existing methods need one or one moreassumptions which includes that the background har-monic is stable and small the harmonic impedance of thecustomer side is much larger than that of utility side andthe harmonic sources of both sides are independentHowever for modern power grid these assumptions arehard to be held up For instance the harmonic sourcescaused by renewable energy and power electronic equip-ment in power grid widely distribute the harmonic sourcesamong power electronic equipments interact and har-monics amplify for transmission lines and cables )us it isobviously unreasonable that the background harmonic isassumed to be very small and stable [17] Besides filters orreactive compensation capacitors are usually installed onthe customer side which may cause the harmonic im-pedances of both sides to be approximately equivalent Asfor the independence hypothesis of the harmonic sourcesof both sides is much idealized and many researchers havepointed out that the assumption may not meet the practice[4 5 17 20 22]

In this paper a novel method without any assumption isproposed which employs a Cauchy Mixed Model (CMM) tofit the distribution characteristic of the measured data at thePCC because we have found that the statistical distributionof the harmonic measurements of the PCC can be depictedwith CMM through the analysis of the field harmonic dataSo a fitting CMM will be chosen to match the distributionrelationship between the harmonic voltage the harmoniccurrent and the utility harmonic impedance of the PCCAlso then the utility harmonic impedance can be obtainedby the expectation maximization algorithm (EM)

)is paper mainly studies from the following fouraspects

(1) )e principle of CMM is introduced and the Nortonequivalent circuit of the PCC is expressed as a CMM

(2) )e feasibility of CMM is verified by field measureddata

(3) )e CMM estimation processes are derived in detail(4) )e effectiveness and robustness of the proposed

method are verified by comparing the estimationresults of simulation and field data with existingalgorithms

)is paper is organized as follows Section 2 presents theanalysis for the validity of CMM Section 3 derives theprocess of estimating the utility impedance Sections 4 and 5verify the validity of the proposed method by simulation andfield case )e paper is concluded in Section 6

2 Basic Principles

21HarmonicEquivalentAnalysisModel Norton equivalentcircuit of the utility side and customer side is shown inFigure 1

In Figure 1 _Is is the utility harmonic current and Zs isthe harmonic impedance on the utility side _UPCC and _IPCCare the harmonic voltage and current measured at the PCCrespectively

According to the Norton equivalent circuit the equationcan be written as

Zs_Is _UPCC + Zs

_IPCC (1)

We have

Δ _Is(n) _Is(n + 1) minus _Is(n)

Δ _IPCC(n) _IPCC(n + 1) minus _IPCC(n)

Δ _UPCC(n) _UPCC(n + 1) minus _UPCC(n)

n 1 2 N minus 1

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(2)

where n is the location of the sampling point and N is thenumber of sampling points Δ _Is(n) is the fluctuation of theutility harmonic current source Δ _IPCC(n) is the harmoniccurrent fluctuation at the PCC and Δ _UPCC(n) is the har-monic voltage fluctuation at the PCC

)en equation (1) can be expressed as

ZsΔ _Is Δ _UPCC + ZsΔ _IPCC (3)

where

Δ _Is Δ _Is(1)Δ _Is(2) middot middot middot Δ _Is(N minus 1)1113966 1113967

Δ _UPCC Δ _UPCC(1)Δ _UPCC(2) middot middot middot Δ _UPCC(N minus 1)1113966 1113967

Δ _IPCC Δ _IPCC(1)Δ _IPCC(2) middot middot middot Δ _IPCC(N minus 1)1113966 1113967

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(4)

)us equation (3) can be rewritten asZsminusx minusZsminusy

Zsminusy Zsminusx


ΔIsminusy

⎡⎣ ⎤⎦ ΔUPCCminusx

ΔUPCCminusy

⎡⎣ ⎤⎦ +Zsminusx minusZsminusy

Zsminusy Zsminusx

⎡⎣ ⎤⎦

middotΔIPCCminusx

ΔIPCCminusy

⎡⎣ ⎤⎦

(5)

where ΔIsminusx and ΔIsminusy are the real and imaginary parts of theutility harmonic current respectively ΔUPCCminusx andΔUPCCminusy are the real and imaginary parts of the harmonicvoltage measured at the PCC respectively ΔIPCCminusx andΔIPCCminusy are the real and imaginary parts of the harmoniccurrent measured at the PCC respectively Zsminusx and Zsminusy arethe real and imaginary of the harmonic impedance on theutility side respectively

)erefore

ΔUs ΔUPCC + 1113954ZsΔIPCC (6)

where

PCC

İs Zs Zc İc

İPCC

UPCC

Customer sideUtility side

Figure 1 )e Norton equivalent circuit




Zsminusy Zsminusx


ΔIsminusy

⎡⎣ ⎤⎦



Zsminusy Zsminusx

⎡⎣ ⎤⎦


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(7)


φ1113874x | μ 11139441113875 Γ((1 + D)2)

Γ(12)πD2| 1113936 |(12) 1 +(x minus μ)T1113936

minus1(x minus μ)1113960 1113961

((1+D)2)

(8)



P(X | α μ 1113944) 1113944K

k1αkφ X μkΣk

11138681113868111386811138681113872 1113873 (9)










ΔCab 1113944

ε1 εD( )isin 01



(10)



F(x) C F1 x1( 1113857 FD xD( 1113857( 1113857 (11)


minus11 x1( 1113857 F

minus1D xD( 11138571113872 1113873 (12)























1113945Nminus1


(13)


l(θ) argmaxθ

logP(θ) argmaxθ



argmaxθ

1113944

Nminus1


argmaxθ

1113944

Nminus1

n1log 1113944

K


11138681113868111386811138681113872 1113873

(14)

EAFCMM

0

02

04

06

08

1

CDF


(a)

EAFCMM

0

02

04

06

08

1

CDF


(b)

EAFCMM

0

02

04

06

08

1

CDF

ndash1000 0 1000 2000ndash2000

(c)

EAFCMM

ndash1000 0 1000 2000 3000ndash20000

02

04

06

08

1

CDF

(d)









11138681113868111386811138681113872 1113873


11138681113868111386811138681113872 1113873 (15)


1113944

K

k1cnk 1 (16)


l(θ) argmaxθ

1113944

Nminus1

n1log 1113944

K

k1cnk


11138681113868111386811138681113872 1113873

cnk

ge argmaxθ

1113944

Nminus1

n11113944

K

k1cnklog


11138681113868111386811138681113872 1113873

cnk

(17)

0

02

04

06

08

1CD

F

DC terminalCMM

0 50ndash50

(a)

DC terminalCMM

0

02

04

06

08

CDF

0 50ndash50

(b)

DC terminalCMM

ndash2000 0 2000 4000ndash40000

02

04

06

08

1

CDF

(c)

DC terminalCMM

ndash2000 0 2000 4000ndash40000

02

04

06

08

1

CDF

(d)








l(θ) argmaxθ

1113944

Nminus1

n11113944

K

k1cnklog


cnk

argmaxθ

1113944

Nminus1

n11113944

K

k1cnk log αkΓ

1 + D

21113874 1113875 minus log Γ

12


1113868111386811138681113868111386811138681113868111386812

1113882

minus1 + D



(18)





zl(θ)

zμk


n1cnk middot

1 + D

2middot1Ψnk



(20)


μk 1113944

Nminus1


Ψnk

⎛⎝ ⎞⎠ 1113944

Nminus1

n1

cnk

Ψnk

⎛⎝ ⎞⎠

minus 1

(21)



zl(θ)

z1113936 k

minus1 + D

21113944

Nminus1

n1

cnk

Ψnk


k



k 1113877

(22)


1113944k

(1 + D) 1113936


T1113877


(23)





ΔIPCCminusx(n)




Zsminusx(k)


WnRk

(24)




l Rk( 1113857 1113944Nminus1

n11113944

K


T1113960


k



zl Rk( 1113857

zRk

1113944Nminus1

n1cnk

WTn 1113936



Ψnk

(26)




n1cnk middot

WTn 1113936

minus 1k WnRk

Ψnk

⎡⎣ ⎤⎦

minus 1

1113944

Nminus1

n1cnk middot

WTn 1113936


Ψnk

⎡⎣ ⎤⎦

(27)Furthermore



We can obtain

αk 1

N minus 11113944

Nminus1

n1cnk (29)


1113944

K

k1αk 1 (30)


Zs [1 j] middot 1113944

K

k1αkRk (31)







180j



180j

λ1 05

λ2 05

n 1 2 N

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(32)




ρ 1113936

Nh1


11138681113868111386811138681113868

1113936Nh1


1113969

1113936Nh1


1113969 (33)









CMM

FQ

BR

IRV

ICA

0

20

40

60

80

Am

plitu

de re

lativ

e err

or (

)

10 11 12090806 0705Current ratio q

(a)

CMM

FQ

BR

IRV

ICA

0

10

20

30

40

Phas

e rel

ativ

e err

or (

)

10 11 12090706 0805Current ratio q

(b)




0

10

20

30

40

Am

plitu

de re

lativ

e err

or (

)

06 07 08 09 10 11 1205Current ratio q

CMM

FQ

BR

IRV

ICA

(a)

0

5

10

15

20

Phas

e rel

ativ

e err

or (

)

06 07 08 09 10 11 1205Current ratio q

CMM

FQ

BR

IRV

ICA

(b)








5 Practical Cases




Am

plitu

de re

lativ

eer

rors

()


ratio q05 06 07 08 09 10 11 120

510

15

0

1210

8642

10

8

6

4

2

(a)

05 06 07 08 09 10 11 120

510

15Impedance ratio m

765

21

34

8

0123456789

Phas

e rel

ativ

e err

ors (

)

Background harmonic

ratio q

(b)




Value

01 09146 06 0369702 08078 07 0309303 06551 08 0232204 05345 09 0122805 04400 10 00936







080

20

40

80A

mpl

itude

rela

tive e

rror

()

09

60

1007

CMM

FQ

BR

IRV

ICA

(a)

Phas

e rel

ativ

e err

or (

)


0

5

15

20

10

01 02 03 04 05 06 08 09 1007

CMM

FQ

BR

IRV

ICA

(b)


0

500

1000

1500

2000

2500

Am

plitu

de (V

)


(a)

0

50

100

150

200

250

Am

plitu

de (A

)


(b)











Am

plitu

de (Ω

)

CMM

FQ

BR

IRV

ICA

5

10

15

20

2 3 4 5 6 7 8 9 10 111Time (h)

(a)

Phas

e (ra

d)

CMM

FQ

BR

IRV

ICA

ndash2

ndash1

0

1

2

2 3 4 5 6 7 8 9 10 111Time (h)

(b)


Am

plitu

de (V

)

1200

1300

1400

1500

1600

100 200 300 400 500 6000Voltage data

(a)

Am

plitu

de (A

)

165

17

175

18

185

100 200 300 400 5000Current data

(b)







6 Conclusions


Data Availability




Acknowledgments


References












Am

plitu

de (Ω

)

775

780

785

790

795

800

2 3 4 5 6 7 8 9 101Subinterval

CMM

FQ

BR

IRV

ICA

(a)

Phas

e (ra

d)

104

105

106

107

2 3 4 5 6 7 8 9 101Subinterval

CMM

FQ

BR

IRV

ICA

(b)





















Zsminusy Zsminusx


ΔIsminusy

⎡⎣ ⎤⎦



Zsminusy Zsminusx

⎡⎣ ⎤⎦


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(7)


φ1113874x | μ 11139441113875 Γ((1 + D)2)

Γ(12)πD2| 1113936 |(12) 1 +(x minus μ)T1113936

minus1(x minus μ)1113960 1113961

((1+D)2)

(8)



P(X | α μ 1113944) 1113944K

k1αkφ X μkΣk

11138681113868111386811138681113872 1113873 (9)










ΔCab 1113944

ε1 εD( )isin 01



(10)



F(x) C F1 x1( 1113857 FD xD( 1113857( 1113857 (11)


minus11 x1( 1113857 F

minus1D xD( 11138571113872 1113873 (12)























1113945Nminus1


(13)


l(θ) argmaxθ

logP(θ) argmaxθ



argmaxθ

1113944

Nminus1


argmaxθ

1113944

Nminus1

n1log 1113944

K


11138681113868111386811138681113872 1113873

(14)

EAFCMM

0

02

04

06

08

1

CDF


(a)

EAFCMM

0

02

04

06

08

1

CDF


(b)

EAFCMM

0

02

04

06

08

1

CDF

ndash1000 0 1000 2000ndash2000

(c)

EAFCMM

ndash1000 0 1000 2000 3000ndash20000

02

04

06

08

1

CDF

(d)









11138681113868111386811138681113872 1113873


11138681113868111386811138681113872 1113873 (15)


1113944

K

k1cnk 1 (16)


l(θ) argmaxθ

1113944

Nminus1

n1log 1113944

K

k1cnk


11138681113868111386811138681113872 1113873

cnk

ge argmaxθ

1113944

Nminus1

n11113944

K

k1cnklog


11138681113868111386811138681113872 1113873

cnk

(17)

0

02

04

06

08

1CD

F

DC terminalCMM

0 50ndash50

(a)

DC terminalCMM

0

02

04

06

08

CDF

0 50ndash50

(b)

DC terminalCMM

ndash2000 0 2000 4000ndash40000

02

04

06

08

1

CDF

(c)

DC terminalCMM

ndash2000 0 2000 4000ndash40000

02

04

06

08

1

CDF

(d)








l(θ) argmaxθ

1113944

Nminus1

n11113944

K

k1cnklog


cnk

argmaxθ

1113944

Nminus1

n11113944

K

k1cnk log αkΓ

1 + D

21113874 1113875 minus log Γ

12


1113868111386811138681113868111386811138681113868111386812

1113882

minus1 + D



(18)





zl(θ)

zμk


n1cnk middot

1 + D

2middot1Ψnk



(20)


μk 1113944

Nminus1


Ψnk

⎛⎝ ⎞⎠ 1113944

Nminus1

n1

cnk

Ψnk

⎛⎝ ⎞⎠

minus 1

(21)



zl(θ)

z1113936 k

minus1 + D

21113944

Nminus1

n1

cnk

Ψnk


k



k 1113877

(22)


1113944k

(1 + D) 1113936


T1113877


(23)





ΔIPCCminusx(n)




Zsminusx(k)


WnRk

(24)




l Rk( 1113857 1113944Nminus1

n11113944

K


T1113960


k



zl Rk( 1113857

zRk

1113944Nminus1

n1cnk

WTn 1113936



Ψnk

(26)




n1cnk middot

WTn 1113936

minus 1k WnRk

Ψnk

⎡⎣ ⎤⎦

minus 1

1113944

Nminus1

n1cnk middot

WTn 1113936


Ψnk

⎡⎣ ⎤⎦

(27)Furthermore



We can obtain

αk 1

N minus 11113944

Nminus1

n1cnk (29)


1113944

K

k1αk 1 (30)


Zs [1 j] middot 1113944

K

k1αkRk (31)







180j



180j

λ1 05

λ2 05

n 1 2 N

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(32)




ρ 1113936

Nh1


11138681113868111386811138681113868

1113936Nh1


1113969

1113936Nh1


1113969 (33)









CMM

FQ

BR

IRV

ICA

0

20

40

60

80

Am

plitu

de re

lativ

e err

or (

)

10 11 12090806 0705Current ratio q

(a)

CMM

FQ

BR

IRV

ICA

0

10

20

30

40

Phas

e rel

ativ

e err

or (

)

10 11 12090706 0805Current ratio q

(b)




0

10

20

30

40

Am

plitu

de re

lativ

e err

or (

)

06 07 08 09 10 11 1205Current ratio q

CMM

FQ

BR

IRV

ICA

(a)

0

5

10

15

20

Phas

e rel

ativ

e err

or (

)

06 07 08 09 10 11 1205Current ratio q

CMM

FQ

BR

IRV

ICA

(b)








5 Practical Cases




Am

plitu

de re

lativ

eer

rors

()


ratio q05 06 07 08 09 10 11 120

510

15

0

1210

8642

10

8

6

4

2

(a)

05 06 07 08 09 10 11 120

510

15Impedance ratio m

765

21

34

8

0123456789

Phas

e rel

ativ

e err

ors (

)

Background harmonic

ratio q

(b)




Value

01 09146 06 0369702 08078 07 0309303 06551 08 0232204 05345 09 0122805 04400 10 00936







080

20

40

80A

mpl

itude

rela

tive e

rror

()

09

60

1007

CMM

FQ

BR

IRV

ICA

(a)

Phas

e rel

ativ

e err

or (

)


0

5

15

20

10

01 02 03 04 05 06 08 09 1007

CMM

FQ

BR

IRV

ICA

(b)


0

500

1000

1500

2000

2500

Am

plitu

de (V

)


(a)

0

50

100

150

200

250

Am

plitu

de (A

)


(b)











Am

plitu

de (Ω

)

CMM

FQ

BR

IRV

ICA

5

10

15

20

2 3 4 5 6 7 8 9 10 111Time (h)

(a)

Phas

e (ra

d)

CMM

FQ

BR

IRV

ICA

ndash2

ndash1

0

1

2

2 3 4 5 6 7 8 9 10 111Time (h)

(b)


Am

plitu

de (V

)

1200

1300

1400

1500

1600

100 200 300 400 500 6000Voltage data

(a)

Am

plitu

de (A

)

165

17

175

18

185

100 200 300 400 5000Current data

(b)







6 Conclusions


Data Availability




Acknowledgments


References












Am

plitu

de (Ω

)

775

780

785

790

795

800

2 3 4 5 6 7 8 9 101Subinterval

CMM

FQ

BR

IRV

ICA

(a)

Phas

e (ra

d)

104

105

106

107

2 3 4 5 6 7 8 9 101Subinterval

CMM

FQ

BR

IRV

ICA

(b)


























1113945Nminus1


(13)


l(θ) argmaxθ

logP(θ) argmaxθ



argmaxθ

1113944

Nminus1


argmaxθ

1113944

Nminus1

n1log 1113944

K


11138681113868111386811138681113872 1113873

(14)

EAFCMM

0

02

04

06

08

1

CDF


(a)

EAFCMM

0

02

04

06

08

1

CDF


(b)

EAFCMM

0

02

04

06

08

1

CDF

ndash1000 0 1000 2000ndash2000

(c)

EAFCMM

ndash1000 0 1000 2000 3000ndash20000

02

04

06

08

1

CDF

(d)









11138681113868111386811138681113872 1113873


11138681113868111386811138681113872 1113873 (15)


1113944

K

k1cnk 1 (16)


l(θ) argmaxθ

1113944

Nminus1

n1log 1113944

K

k1cnk


11138681113868111386811138681113872 1113873

cnk

ge argmaxθ

1113944

Nminus1

n11113944

K

k1cnklog


11138681113868111386811138681113872 1113873

cnk

(17)

0

02

04

06

08

1CD

F

DC terminalCMM

0 50ndash50

(a)

DC terminalCMM

0

02

04

06

08

CDF

0 50ndash50

(b)

DC terminalCMM

ndash2000 0 2000 4000ndash40000

02

04

06

08

1

CDF

(c)

DC terminalCMM

ndash2000 0 2000 4000ndash40000

02

04

06

08

1

CDF

(d)








l(θ) argmaxθ

1113944

Nminus1

n11113944

K

k1cnklog


cnk

argmaxθ

1113944

Nminus1

n11113944

K

k1cnk log αkΓ

1 + D

21113874 1113875 minus log Γ

12


1113868111386811138681113868111386811138681113868111386812

1113882

minus1 + D



(18)





zl(θ)

zμk


n1cnk middot

1 + D

2middot1Ψnk



(20)


μk 1113944

Nminus1


Ψnk

⎛⎝ ⎞⎠ 1113944

Nminus1

n1

cnk

Ψnk

⎛⎝ ⎞⎠

minus 1

(21)



zl(θ)

z1113936 k

minus1 + D

21113944

Nminus1

n1

cnk

Ψnk


k



k 1113877

(22)


1113944k

(1 + D) 1113936


T1113877


(23)





ΔIPCCminusx(n)




Zsminusx(k)


WnRk

(24)




l Rk( 1113857 1113944Nminus1

n11113944

K


T1113960


k



zl Rk( 1113857

zRk

1113944Nminus1

n1cnk

WTn 1113936



Ψnk

(26)




n1cnk middot

WTn 1113936

minus 1k WnRk

Ψnk

⎡⎣ ⎤⎦

minus 1

1113944

Nminus1

n1cnk middot

WTn 1113936


Ψnk

⎡⎣ ⎤⎦

(27)Furthermore



We can obtain

αk 1

N minus 11113944

Nminus1

n1cnk (29)


1113944

K

k1αk 1 (30)


Zs [1 j] middot 1113944

K

k1αkRk (31)







180j



180j

λ1 05

λ2 05

n 1 2 N

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(32)




ρ 1113936

Nh1


11138681113868111386811138681113868

1113936Nh1


1113969

1113936Nh1


1113969 (33)









CMM

FQ

BR

IRV

ICA

0

20

40

60

80

Am

plitu

de re

lativ

e err

or (

)

10 11 12090806 0705Current ratio q

(a)

CMM

FQ

BR

IRV

ICA

0

10

20

30

40

Phas

e rel

ativ

e err

or (

)

10 11 12090706 0805Current ratio q

(b)




0

10

20

30

40

Am

plitu

de re

lativ

e err

or (

)

06 07 08 09 10 11 1205Current ratio q

CMM

FQ

BR

IRV

ICA

(a)

0

5

10

15

20

Phas

e rel

ativ

e err

or (

)

06 07 08 09 10 11 1205Current ratio q

CMM

FQ

BR

IRV

ICA

(b)








5 Practical Cases




Am

plitu

de re

lativ

eer

rors

()


ratio q05 06 07 08 09 10 11 120

510

15

0

1210

8642

10

8

6

4

2

(a)

05 06 07 08 09 10 11 120

510

15Impedance ratio m

765

21

34

8

0123456789

Phas

e rel

ativ

e err

ors (

)

Background harmonic

ratio q

(b)




Value

01 09146 06 0369702 08078 07 0309303 06551 08 0232204 05345 09 0122805 04400 10 00936







080

20

40

80A

mpl

itude

rela

tive e

rror

()

09

60

1007

CMM

FQ

BR

IRV

ICA

(a)

Phas

e rel

ativ

e err

or (

)


0

5

15

20

10

01 02 03 04 05 06 08 09 1007

CMM

FQ

BR

IRV

ICA

(b)


0

500

1000

1500

2000

2500

Am

plitu

de (V

)


(a)

0

50

100

150

200

250

Am

plitu

de (A

)


(b)











Am

plitu

de (Ω

)

CMM

FQ

BR

IRV

ICA

5

10

15

20

2 3 4 5 6 7 8 9 10 111Time (h)

(a)

Phas

e (ra

d)

CMM

FQ

BR

IRV

ICA

ndash2

ndash1

0

1

2

2 3 4 5 6 7 8 9 10 111Time (h)

(b)


Am

plitu

de (V

)

1200

1300

1400

1500

1600

100 200 300 400 500 6000Voltage data

(a)

Am

plitu

de (A

)

165

17

175

18

185

100 200 300 400 5000Current data

(b)







6 Conclusions


Data Availability




Acknowledgments


References












Am

plitu

de (Ω

)

775

780

785

790

795

800

2 3 4 5 6 7 8 9 101Subinterval

CMM

FQ

BR

IRV

ICA

(a)

Phas

e (ra

d)

104

105

106

107

2 3 4 5 6 7 8 9 101Subinterval

CMM

FQ

BR

IRV

ICA

(b)






















11138681113868111386811138681113872 1113873


11138681113868111386811138681113872 1113873 (15)


1113944

K

k1cnk 1 (16)


l(θ) argmaxθ

1113944

Nminus1

n1log 1113944

K

k1cnk


11138681113868111386811138681113872 1113873

cnk

ge argmaxθ

1113944

Nminus1

n11113944

K

k1cnklog


11138681113868111386811138681113872 1113873

cnk

(17)

0

02

04

06

08

1CD

F

DC terminalCMM

0 50ndash50

(a)

DC terminalCMM

0

02

04

06

08

CDF

0 50ndash50

(b)

DC terminalCMM

ndash2000 0 2000 4000ndash40000

02

04

06

08

1

CDF

(c)

DC terminalCMM

ndash2000 0 2000 4000ndash40000

02

04

06

08

1

CDF

(d)








l(θ) argmaxθ

1113944

Nminus1

n11113944

K

k1cnklog


cnk

argmaxθ

1113944

Nminus1

n11113944

K

k1cnk log αkΓ

1 + D

21113874 1113875 minus log Γ

12


1113868111386811138681113868111386811138681113868111386812

1113882

minus1 + D



(18)





zl(θ)

zμk


n1cnk middot

1 + D

2middot1Ψnk



(20)


μk 1113944

Nminus1


Ψnk

⎛⎝ ⎞⎠ 1113944

Nminus1

n1

cnk

Ψnk

⎛⎝ ⎞⎠

minus 1

(21)



zl(θ)

z1113936 k

minus1 + D

21113944

Nminus1

n1

cnk

Ψnk


k



k 1113877

(22)


1113944k

(1 + D) 1113936


T1113877


(23)





ΔIPCCminusx(n)




Zsminusx(k)


WnRk

(24)




l Rk( 1113857 1113944Nminus1

n11113944

K


T1113960


k



zl Rk( 1113857

zRk

1113944Nminus1

n1cnk

WTn 1113936



Ψnk

(26)




n1cnk middot

WTn 1113936

minus 1k WnRk

Ψnk

⎡⎣ ⎤⎦

minus 1

1113944

Nminus1

n1cnk middot

WTn 1113936


Ψnk

⎡⎣ ⎤⎦

(27)Furthermore



We can obtain

αk 1

N minus 11113944

Nminus1

n1cnk (29)


1113944

K

k1αk 1 (30)


Zs [1 j] middot 1113944

K

k1αkRk (31)







180j



180j

λ1 05

λ2 05

n 1 2 N

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(32)




ρ 1113936

Nh1


11138681113868111386811138681113868

1113936Nh1


1113969

1113936Nh1


1113969 (33)









CMM

FQ

BR

IRV

ICA

0

20

40

60

80

Am

plitu

de re

lativ

e err

or (

)

10 11 12090806 0705Current ratio q

(a)

CMM

FQ

BR

IRV

ICA

0

10

20

30

40

Phas

e rel

ativ

e err

or (

)

10 11 12090706 0805Current ratio q

(b)




0

10

20

30

40

Am

plitu

de re

lativ

e err

or (

)

06 07 08 09 10 11 1205Current ratio q

CMM

FQ

BR

IRV

ICA

(a)

0

5

10

15

20

Phas

e rel

ativ

e err

or (

)

06 07 08 09 10 11 1205Current ratio q

CMM

FQ

BR

IRV

ICA

(b)








5 Practical Cases




Am

plitu

de re

lativ

eer

rors

()


ratio q05 06 07 08 09 10 11 120

510

15

0

1210

8642

10

8

6

4

2

(a)

05 06 07 08 09 10 11 120

510

15Impedance ratio m

765

21

34

8

0123456789

Phas

e rel

ativ

e err

ors (

)

Background harmonic

ratio q

(b)




Value

01 09146 06 0369702 08078 07 0309303 06551 08 0232204 05345 09 0122805 04400 10 00936







080

20

40

80A

mpl

itude

rela

tive e

rror

()

09

60

1007

CMM

FQ

BR

IRV

ICA

(a)

Phas

e rel

ativ

e err

or (

)


0

5

15

20

10

01 02 03 04 05 06 08 09 1007

CMM

FQ

BR

IRV

ICA

(b)


0

500

1000

1500

2000

2500

Am

plitu

de (V

)


(a)

0

50

100

150

200

250

Am

plitu

de (A

)


(b)











Am

plitu

de (Ω

)

CMM

FQ

BR

IRV

ICA

5

10

15

20

2 3 4 5 6 7 8 9 10 111Time (h)

(a)

Phas

e (ra

d)

CMM

FQ

BR

IRV

ICA

ndash2

ndash1

0

1

2

2 3 4 5 6 7 8 9 10 111Time (h)

(b)


Am

plitu

de (V

)

1200

1300

1400

1500

1600

100 200 300 400 500 6000Voltage data

(a)

Am

plitu

de (A

)

165

17

175

18

185

100 200 300 400 5000Current data

(b)







6 Conclusions


Data Availability




Acknowledgments


References












Am

plitu

de (Ω

)

775

780

785

790

795

800

2 3 4 5 6 7 8 9 101Subinterval

CMM

FQ

BR

IRV

ICA

(a)

Phas

e (ra

d)

104

105

106

107

2 3 4 5 6 7 8 9 101Subinterval

CMM

FQ

BR

IRV

ICA

(b)





















l(θ) argmaxθ

1113944

Nminus1

n11113944

K

k1cnklog


cnk

argmaxθ

1113944

Nminus1

n11113944

K

k1cnk log αkΓ

1 + D

21113874 1113875 minus log Γ

12


1113868111386811138681113868111386811138681113868111386812

1113882

minus1 + D



(18)





zl(θ)

zμk


n1cnk middot

1 + D

2middot1Ψnk



(20)


μk 1113944

Nminus1


Ψnk

⎛⎝ ⎞⎠ 1113944

Nminus1

n1

cnk

Ψnk

⎛⎝ ⎞⎠

minus 1

(21)



zl(θ)

z1113936 k

minus1 + D

21113944

Nminus1

n1

cnk

Ψnk


k



k 1113877

(22)


1113944k

(1 + D) 1113936


T1113877


(23)





ΔIPCCminusx(n)




Zsminusx(k)


WnRk

(24)




l Rk( 1113857 1113944Nminus1

n11113944

K


T1113960


k



zl Rk( 1113857

zRk

1113944Nminus1

n1cnk

WTn 1113936



Ψnk

(26)




n1cnk middot

WTn 1113936

minus 1k WnRk

Ψnk

⎡⎣ ⎤⎦

minus 1

1113944

Nminus1

n1cnk middot

WTn 1113936


Ψnk

⎡⎣ ⎤⎦

(27)Furthermore



We can obtain

αk 1

N minus 11113944

Nminus1

n1cnk (29)


1113944

K

k1αk 1 (30)


Zs [1 j] middot 1113944

K

k1αkRk (31)







180j



180j

λ1 05

λ2 05

n 1 2 N

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(32)




ρ 1113936

Nh1


11138681113868111386811138681113868

1113936Nh1


1113969

1113936Nh1


1113969 (33)









CMM

FQ

BR

IRV

ICA

0

20

40

60

80

Am

plitu

de re

lativ

e err

or (

)

10 11 12090806 0705Current ratio q

(a)

CMM

FQ

BR

IRV

ICA

0

10

20

30

40

Phas

e rel

ativ

e err

or (

)

10 11 12090706 0805Current ratio q

(b)




0

10

20

30

40

Am

plitu

de re

lativ

e err

or (

)

06 07 08 09 10 11 1205Current ratio q

CMM

FQ

BR

IRV

ICA

(a)

0

5

10

15

20

Phas

e rel

ativ

e err

or (

)

06 07 08 09 10 11 1205Current ratio q

CMM

FQ

BR

IRV

ICA

(b)








5 Practical Cases




Am

plitu

de re

lativ

eer

rors

()


ratio q05 06 07 08 09 10 11 120

510

15

0

1210

8642

10

8

6

4

2

(a)

05 06 07 08 09 10 11 120

510

15Impedance ratio m

765

21

34

8

0123456789

Phas

e rel

ativ

e err

ors (

)

Background harmonic

ratio q

(b)




Value

01 09146 06 0369702 08078 07 0309303 06551 08 0232204 05345 09 0122805 04400 10 00936







080

20

40

80A

mpl

itude

rela

tive e

rror

()

09

60

1007

CMM

FQ

BR

IRV

ICA

(a)

Phas

e rel

ativ

e err

or (

)


0

5

15

20

10

01 02 03 04 05 06 08 09 1007

CMM

FQ

BR

IRV

ICA

(b)


0

500

1000

1500

2000

2500

Am

plitu

de (V

)


(a)

0

50

100

150

200

250

Am

plitu

de (A

)


(b)











Am

plitu

de (Ω

)

CMM

FQ

BR

IRV

ICA

5

10

15

20

2 3 4 5 6 7 8 9 10 111Time (h)

(a)

Phas

e (ra

d)

CMM

FQ

BR

IRV

ICA

ndash2

ndash1

0

1

2

2 3 4 5 6 7 8 9 10 111Time (h)

(b)


Am

plitu

de (V

)

1200

1300

1400

1500

1600

100 200 300 400 500 6000Voltage data

(a)

Am

plitu

de (A

)

165

17

175

18

185

100 200 300 400 5000Current data

(b)







6 Conclusions


Data Availability




Acknowledgments


References












Am

plitu

de (Ω

)

775

780

785

790

795

800

2 3 4 5 6 7 8 9 101Subinterval

CMM

FQ

BR

IRV

ICA

(a)

Phas

e (ra

d)

104

105

106

107

2 3 4 5 6 7 8 9 101Subinterval

CMM

FQ

BR

IRV

ICA

(b)





















n1cnk middot

WTn 1113936

minus 1k WnRk

Ψnk

⎡⎣ ⎤⎦

minus 1

1113944

Nminus1

n1cnk middot

WTn 1113936


Ψnk

⎡⎣ ⎤⎦

(27)Furthermore



We can obtain

αk 1

N minus 11113944

Nminus1

n1cnk (29)


1113944

K

k1αk 1 (30)


Zs [1 j] middot 1113944

K

k1αkRk (31)







180j



180j

λ1 05

λ2 05

n 1 2 N

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(32)




ρ 1113936

Nh1


11138681113868111386811138681113868

1113936Nh1


1113969

1113936Nh1


1113969 (33)









CMM

FQ

BR

IRV

ICA

0

20

40

60

80

Am

plitu

de re

lativ

e err

or (

)

10 11 12090806 0705Current ratio q

(a)

CMM

FQ

BR

IRV

ICA

0

10

20

30

40

Phas

e rel

ativ

e err

or (

)

10 11 12090706 0805Current ratio q

(b)




0

10

20

30

40

Am

plitu

de re

lativ

e err

or (

)

06 07 08 09 10 11 1205Current ratio q

CMM

FQ

BR

IRV

ICA

(a)

0

5

10

15

20

Phas

e rel

ativ

e err

or (

)

06 07 08 09 10 11 1205Current ratio q

CMM

FQ

BR

IRV

ICA

(b)








5 Practical Cases




Am

plitu

de re

lativ

eer

rors

()


ratio q05 06 07 08 09 10 11 120

510

15

0

1210

8642

10

8

6

4

2

(a)

05 06 07 08 09 10 11 120

510

15Impedance ratio m

765

21

34

8

0123456789

Phas

e rel

ativ

e err

ors (

)

Background harmonic

ratio q

(b)




Value

01 09146 06 0369702 08078 07 0309303 06551 08 0232204 05345 09 0122805 04400 10 00936







080

20

40

80A

mpl

itude

rela

tive e

rror

()

09

60

1007

CMM

FQ

BR

IRV

ICA

(a)

Phas

e rel

ativ

e err

or (

)


0

5

15

20

10

01 02 03 04 05 06 08 09 1007

CMM

FQ

BR

IRV

ICA

(b)


0

500

1000

1500

2000

2500

Am

plitu

de (V

)


(a)

0

50

100

150

200

250

Am

plitu

de (A

)


(b)











Am

plitu

de (Ω

)

CMM

FQ

BR

IRV

ICA

5

10

15

20

2 3 4 5 6 7 8 9 10 111Time (h)

(a)

Phas

e (ra

d)

CMM

FQ

BR

IRV

ICA

ndash2

ndash1

0

1

2

2 3 4 5 6 7 8 9 10 111Time (h)

(b)


Am

plitu

de (V

)

1200

1300

1400

1500

1600

100 200 300 400 500 6000Voltage data

(a)

Am

plitu

de (A

)

165

17

175

18

185

100 200 300 400 5000Current data

(b)







6 Conclusions


Data Availability




Acknowledgments


References












Am

plitu

de (Ω

)

775

780

785

790

795

800

2 3 4 5 6 7 8 9 101Subinterval

CMM

FQ

BR

IRV

ICA

(a)

Phas

e (ra

d)

104

105

106

107

2 3 4 5 6 7 8 9 101Subinterval

CMM

FQ

BR

IRV

ICA

(b)





















CMM

FQ

BR

IRV

ICA

0

20

40

60

80

Am

plitu

de re

lativ

e err

or (

)

10 11 12090806 0705Current ratio q

(a)

CMM

FQ

BR

IRV

ICA

0

10

20

30

40

Phas

e rel

ativ

e err

or (

)

10 11 12090706 0805Current ratio q

(b)




0

10

20

30

40

Am

plitu

de re

lativ

e err

or (

)

06 07 08 09 10 11 1205Current ratio q

CMM

FQ

BR

IRV

ICA

(a)

0

5

10

15

20

Phas

e rel

ativ

e err

or (

)

06 07 08 09 10 11 1205Current ratio q

CMM

FQ

BR

IRV

ICA

(b)








5 Practical Cases




Am

plitu

de re

lativ

eer

rors

()


ratio q05 06 07 08 09 10 11 120

510

15

0

1210

8642

10

8

6

4

2

(a)

05 06 07 08 09 10 11 120

510

15Impedance ratio m

765

21

34

8

0123456789

Phas

e rel

ativ

e err

ors (

)

Background harmonic

ratio q

(b)




Value

01 09146 06 0369702 08078 07 0309303 06551 08 0232204 05345 09 0122805 04400 10 00936







080

20

40

80A

mpl

itude

rela

tive e

rror

()

09

60

1007

CMM

FQ

BR

IRV

ICA

(a)

Phas

e rel

ativ

e err

or (

)


0

5

15

20

10

01 02 03 04 05 06 08 09 1007

CMM

FQ

BR

IRV

ICA

(b)


0

500

1000

1500

2000

2500

Am

plitu

de (V

)


(a)

0

50

100

150

200

250

Am

plitu

de (A

)


(b)











Am

plitu

de (Ω

)

CMM

FQ

BR

IRV

ICA

5

10

15

20

2 3 4 5 6 7 8 9 10 111Time (h)

(a)

Phas

e (ra

d)

CMM

FQ

BR

IRV

ICA

ndash2

ndash1

0

1

2

2 3 4 5 6 7 8 9 10 111Time (h)

(b)


Am

plitu

de (V

)

1200

1300

1400

1500

1600

100 200 300 400 500 6000Voltage data

(a)

Am

plitu

de (A

)

165

17

175

18

185

100 200 300 400 5000Current data

(b)







6 Conclusions


Data Availability




Acknowledgments


References












Am

plitu

de (Ω

)

775

780

785

790

795

800

2 3 4 5 6 7 8 9 101Subinterval

CMM

FQ

BR

IRV

ICA

(a)

Phas

e (ra

d)

104

105

106

107

2 3 4 5 6 7 8 9 101Subinterval

CMM

FQ

BR

IRV

ICA

(b)
























5 Practical Cases




Am

plitu

de re

lativ

eer

rors

()


ratio q05 06 07 08 09 10 11 120

510

15

0

1210

8642

10

8

6

4

2

(a)

05 06 07 08 09 10 11 120

510

15Impedance ratio m

765

21

34

8

0123456789

Phas

e rel

ativ

e err

ors (

)

Background harmonic

ratio q

(b)




Value

01 09146 06 0369702 08078 07 0309303 06551 08 0232204 05345 09 0122805 04400 10 00936







080

20

40

80A

mpl

itude

rela

tive e

rror

()

09

60

1007

CMM

FQ

BR

IRV

ICA

(a)

Phas

e rel

ativ

e err

or (

)


0

5

15

20

10

01 02 03 04 05 06 08 09 1007

CMM

FQ

BR

IRV

ICA

(b)


0

500

1000

1500

2000

2500

Am

plitu

de (V

)


(a)

0

50

100

150

200

250

Am

plitu

de (A

)


(b)











Am

plitu

de (Ω

)

CMM

FQ

BR

IRV

ICA

5

10

15

20

2 3 4 5 6 7 8 9 10 111Time (h)

(a)

Phas

e (ra

d)

CMM

FQ

BR

IRV

ICA

ndash2

ndash1

0

1

2

2 3 4 5 6 7 8 9 10 111Time (h)

(b)


Am

plitu

de (V

)

1200

1300

1400

1500

1600

100 200 300 400 500 6000Voltage data

(a)

Am

plitu

de (A

)

165

17

175

18

185

100 200 300 400 5000Current data

(b)







6 Conclusions


Data Availability




Acknowledgments


References












Am

plitu

de (Ω

)

775

780

785

790

795

800

2 3 4 5 6 7 8 9 101Subinterval

CMM

FQ

BR

IRV

ICA

(a)

Phas

e (ra

d)

104

105

106

107

2 3 4 5 6 7 8 9 101Subinterval

CMM

FQ

BR

IRV

ICA

(b)
























080

20

40

80A

mpl

itude

rela

tive e

rror

()

09

60

1007

CMM

FQ

BR

IRV

ICA

(a)

Phas

e rel

ativ

e err

or (

)


0

5

15

20

10

01 02 03 04 05 06 08 09 1007

CMM

FQ

BR

IRV

ICA

(b)


0

500

1000

1500

2000

2500

Am

plitu

de (V

)


(a)

0

50

100

150

200

250

Am

plitu

de (A

)


(b)











Am

plitu

de (Ω

)

CMM

FQ

BR

IRV

ICA

5

10

15

20

2 3 4 5 6 7 8 9 10 111Time (h)

(a)

Phas

e (ra

d)

CMM

FQ

BR

IRV

ICA

ndash2

ndash1

0

1

2

2 3 4 5 6 7 8 9 10 111Time (h)

(b)


Am

plitu

de (V

)

1200

1300

1400

1500

1600

100 200 300 400 500 6000Voltage data

(a)

Am

plitu

de (A

)

165

17

175

18

185

100 200 300 400 5000Current data

(b)







6 Conclusions


Data Availability




Acknowledgments


References












Am

plitu

de (Ω

)

775

780

785

790

795

800

2 3 4 5 6 7 8 9 101Subinterval

CMM

FQ

BR

IRV

ICA

(a)

Phas

e (ra

d)

104

105

106

107

2 3 4 5 6 7 8 9 101Subinterval

CMM

FQ

BR

IRV

ICA

(b)
























Am

plitu

de (Ω

)

CMM

FQ

BR

IRV

ICA

5

10

15

20

2 3 4 5 6 7 8 9 10 111Time (h)

(a)

Phas

e (ra

d)

CMM

FQ

BR

IRV

ICA

ndash2

ndash1

0

1

2

2 3 4 5 6 7 8 9 10 111Time (h)

(b)


Am

plitu

de (V

)

1200

1300

1400

1500

1600

100 200 300 400 500 6000Voltage data

(a)

Am

plitu

de (A

)

165

17

175

18

185

100 200 300 400 5000Current data

(b)







6 Conclusions


Data Availability




Acknowledgments


References












Am

plitu

de (Ω

)

775

780

785

790

795

800

2 3 4 5 6 7 8 9 101Subinterval

CMM

FQ

BR

IRV

ICA

(a)

Phas

e (ra

d)

104

105

106

107

2 3 4 5 6 7 8 9 101Subinterval

CMM

FQ

BR

IRV

ICA

(b)




















6 Conclusions


Data Availability




Acknowledgments


References












Am

plitu

de (Ω

)

775

780

785

790

795

800

2 3 4 5 6 7 8 9 101Subinterval

CMM

FQ

BR

IRV

ICA

(a)

Phas

e (ra

d)

104

105

106

107

2 3 4 5 6 7 8 9 101Subinterval

CMM

FQ

BR

IRV

ICA

(b)



































Documents

AHarmonicImpedanceEstimationMethodBasedonthe … · 2020. 1. 3. · Generally,theexistingmethodsneedoneoronemore assumptions,whichincludesthatthebackgroundhar …