identification of multiple harmonic sources in power systems using independent component analysis...

Eng Int Syst (2010) 1: 51–63© 2010 CRL Publishing Ltd Engineering

Intelligent Systems

Identification of multiple harmonicsources in power systems usingindependent component analysisand mutual information

Masoud Farhoodnea∗, Azah Mohamed, Hussain Shareef

Department of Electrical, Electronic and Systems Engineering, University Kebangsaan, MalaysiaE-mail: [email protected], [email protected]

This paper presents a novel technique for accurate determination of the probable locations of multiple harmonic sources in powersystems. In the proposed methodology, independent component analysis (ICA), which is one of the blind source separation techniques,is applied for estimating the profiles of injected harmonic currents produced by nonlinear loads. By using the ICA algorithm, thereconstruction of the injected harmonic currents can be implemented easily without prior knowledge about the system components.From the reconstructed harmonic current profiles, the mutual information theory is then applied for determining the exact location ofharmonic sources by computing the pair-wise mutual information between the extracted current trends and bus voltages. Numericalsimulations were made to prove the accuracy of the proposed method in locating multiple harmonic sources in the IEEE 30 and 34bus test distribution systems. Results proved that the proposed method can accurately determine the probable location of harmonicsources in radial and non radial power systems with mean square error less than 0.00035.

Keywords: Harmonic distortion; harmonic source location; power quality; multiple harmonic sources.

1. INTRODUCTION

Due to the penetration of various distributed nonlinear and har-monic producing loads, the diffusion of harmonics in powersystems is escalating and has become an important powerquality problem. In addition harmonics are the main rea-son for causing resonance problems, overheating in capacitorbanks, transformers and conductors, and telecommunicationconflictions. These problems may decrease the reliability andincrease the maintenance costs of power systems. From theabove-mentioned issues, it is important for power providersto identify the source of harmonic distortions so as to solve

∗E-mail: [email protected]

or lessen the problems related to harmonic distortion. Meth-ods for identifying the location of harmonic sources in powersystems are generally categorized as single point and multiplepoint strategies. The real power direction method [1] is one ofthe earliest single point method proposed in locating harmonicsources. However, the real power direction method has beenfound to be only 50 percent reliable and therefore the accuracyof the method is questionable [2]. Other single point meth-ods for harmonic source localization include incentive-basedmethod [3–4], the critical impedance method [5] and voltagemagnitude comparison method [6]. These methods requirecomplete knowledge of system parameters in harmonic fre-quency or implementation of switching tests for obtaining theharmonic impedances of the system. However, such data are

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IDENTIFICATION OF MULTIPLE HARMONIC SOURCES IN POWER SYSTEMS

not available in most of the time, and implementing switchingtests is not possible in practical power systems. More recentsingle point based methods such as total harmonic distortion(THD) method [7], harmonic vector method (HVM) [8], andIEEE1459–2000 standard based method [9] tried to solve theabove mentioned problems in the previous methods. Nonethe-less, these current methods still suffer from some problemssuch as the use of unsuitable index harmonic modeling andinability to determine the location of harmonic sources forindividual harmonic orders. In general, implementing singlepoint based methods are very economical in comparison withother methods, but such methods are unable to work in practi-cal systems because harmonic sources exist at different pointsin a power network. To tackle this problem, several methodshave been developed for locating multiple harmonic sourcesin power systems. Harmonic state estimation (HSE) [10–11]is one of the most popular multiple point based methods. But,the HSE based methods require complete knowledge about thesystem parameters for each individual harmonic order, whichare usually unknown or difficult to calculate in practice. In ad-dition, the measurement placement technique [12–13] whichis widely applied in HSE for optimizing the number of mea-surements in the system needs to have prior knowledge aboutharmonic sources and their locations. Another concern withHSE is that this approach needs various types of harmonicmeasurements such as voltage, real and reactive powers whichare costly for large systems. In [14], the weighted least squareestimation technique and Euclidean norm have been appliedto recognize the location of multiple harmonic sources. Inthis method, the weighted least square estimation techniqueand HSE are used to locate the possible buses with harmonicsources while the Euclidean norm determines the exact buseswith harmonic sources. However, the disadvantage of thismethod is the use of all types of harmonic measurements atall buses which is not economical. Another multiple pointbased method is the Independent component analysis (ICA)method [15]. In this method, measured voltages at the selectedbuses, determined by the measurement placement techniqueare used for estimating the harmonic impedance matrix ofthe system, and harmonic current profiles of the harmonicsources at each harmonic frequency. The minimum electri-cal distances between the estimated impedance matrix and theactual impedance matrix are then obtained by using the ex-haustive search technique. The buses with minimal distancesare considered as the location of harmonic sources. However,this method is not quite practical because it needs to determinethe actual impedance matrix of a system at each harmonic fre-quency. In addition, this method requires prior informationabout system parameters and historical records of the loadsfor solving indeterminacies related to ICA algorithm.

This paper presents a new method based on ICA and mutualinformation theory (MI) to determine the accurate location ofmultiple harmonic sources in a power system. In this method,ICA is used to rebuild or estimate the harmonic currents pro-duced by the harmonic sources while the MI theory is usedfor determining the locations of harmonic sources, respec-tively. The proposed method for identifying the location ofharmonic sources is different from the previous ICA method[15], in the fact that it uses the mutual information theory forfinal decision making stage in locating the harmonic sources.

In addition, unlike the above mentioned techniques, the pro-posed method only requires voltage measurements of all busesand eliminates other kinds of measurements. In addition, it isassumed that there are no prior information about the systemparameters, harmonic impedance matrix and characteristicsof harmonic sources. Hence estimation will be done com-pletely in blind situation. This assumption is significant inthe deregulated power systems where complete power systemparameters, load behavior and utility response is not knownbecause of economic and security restrictions in electricitymarket.

2. INDEPENDENT COMPONENTANALYSIS

Independent component analysis (ICA) is one of the blindsource separation techniques (BSS) based on the statisticalindependence between the source signals. Due to the gen-erality of the ICA model, it has found many applications indifferent areas such as audio separation, econometrics, brainimaging, and telecommunication data separation. Basically,ICA is a technique that transforms the observed signals intoa linear transform of source signals that are statistically andmutually independent from each other. To reconstruct thesource signals from the observed signals, certain assumptionsare considered such that the source signals are statistically in-dependent, all but one source signals must have non gaussiandistributions and the number of observations should be greateror equal to the number of sources [16].

Assuming that there are N sources and M measurements,the linear mixing model of ICA can be written as

xj = aj1s1 + aj2s2 + ... + ajnsn for all

j = 1, 2, ..., M (1)

The matrix representation of Eq. (1) is

X = AS (2)

where,

S(t) = [s1(t), s2(t), ..., sn(t)]T : N -dimensional vec-tor of unknown source signals

X(t) = [x1(t), x2(t), ..., xm(t)]T : M−dimensionalvector of observed signals

A : M ×N full column coefficient matrix aij calledas the mixing matrix

In ICA, the objective is to find estimates of the S and A

from the available observation vector X. The unknown sourcematrix S can be estimated as

Sest = WX (3)

where,

Sest: estimate of the sources S with N × T dimension

W : N ×Mseparating matrix which is the pseudo inverseof the mixing matrix A

52 Engineering Intelligent Systems

M. FARHOODNEA ET AL

For estimating the model of ICA given by Eq. (3), sev-eral optimization techniques can be used such as maximiza-tion of nongaussianity, minimization of mutual information,and maximum likelihood estimation [17]. Here, the ICA al-gorithm with maximization of nongaussianity of the sourcesignal has been applied. Based on this approach, the entropyof a random variable which is related to the information thatthe observation of the variable gives is considered. The morerandom the variable is, the larger is its entropy [17]. The en-tropy, H of a random vector, y with density function, py(η)

is defined as:

H(y) = −∫

py(η) log py(η)dη (4)

where y is a random variable with mean µ and standard devi-ation δ.

It should be noted that a gaussian random variable has thelargest entropy among all random variables with equal vari-ance. Therefore, entropy can be used as a measure of non-gaussianity of random variables, which is always nonnega-tive. An important measure of nongaussianity is given by thenormalized version of differential entropy which is called asnegentropy J and is defined as:

J (y) = H(ygauss) − H(y) (5)

where ygauss is a gaussian random vector with the same co-variance matrix as y.

Negentropy is always nonnegative, and it is zero if y hasa gaussian distribution. Estimating negentropy of Eq. (5)is very difficult because it needs to estimate the probabilityfunctions of the source signals. Hence, by using only onequadratic function, the approximation of J becomes [18]:

J (y) ∝ [E {G(y)} − E {G(v)}]2 (6)

where G is practically any nonquadratic function with slowvariation such as tanh(u), v is a gaussian variable with zeromean and unit variance and y is a random variable with zeromean and unit variance.

To estimate the independent components, maximization ofthe estimated negentropy (6) is necessary. By maximizing thesum of N one unit contrast functions and taking into accountthe constraint of decorrelation, an optimization problem isobtained as follows:

Maximizen∑

i=1

J (wTi x)

under constraint E{(wT

k x)(wTj x)

}= δjk (7)

where wi for i = 1, 2, ..., N are the rows of the separatingmatrix, W which is inverse of coefficient matrix A.

Before implementing ICA, the observed vector X is prepro-cessed by centering and whitening. Preprocessing in ICA isknown to be a suitable technique for simplifying the ICA esti-mation .[16]. By centering, the mean of vector X is subtractedand transformed to zero-mean variable, while by whitening;the observed vector X is transformed linearly to a new vectorX̃ where its components are uncorrelated and their variancesare unity. In whitening, Eq. (8) is considered,

X̃ = ED−1/2ET X (8)

where D and E are the diagonal matrix of eigenvalues andthe orthogonal matrix of eigenvectors, respectively. They areobtained from the covariance matrix of the observed vectorX.

For maximizing the contrast functions in Eq. (7), the Fas-tICA algorithm [18] is applied and it is described in terms ofa flowchart as shown in Figure 1.

Note that for convergence of the algorithm, the old and newvalues of w should be in the same direction.

3. MUTUAL INFORMATION THEORY

Basically, mutual information (MI) is a measure of the infor-mation that members of a set of random variables have onthe other random variables in the set. Using the concept ofentropy defined for continuous random variables in Eq. (4),the discrete form of entropy is derived as,

H(y) = −N∑

y=1

p(y) log p(y) (9)

The entropy in Eq. (9) can be interpreted as a measure ofuncertainty of the events with probability function p. In otherwords, entropy is small for events with probability close to0 or 1, and large for probabilities between 0 and 1. In fact,the entropy of a random variable can be defined as the degreeof information that the observation of the variable gives. Todetermine uncertainty about variable A in subsequence of tri-als in which B occurs, the conditional entropy is used [19].Assuming random variables A and B consisting of the N ele-ments of ai and bi for i = 1, 2, ..., N , the conditional entropycan be defined as:

H(A |bi ) = −N∑

i=1

p(ai |bi ) log p(ai |bi ) (10)

Assuming that B is the average of H(A |bi ), the conditionalentropy of variables A and B can be written as:

H(A |B ) =N∑

i=1

p(bi)H(A |bi ) (11)

By using Eq. (10) and (11), the mutual information I ofvariable A and B can be obtained as:

I (A,B) = H(A) − H(A |B ), I (A,B) ≥ 0 (12)

Equation (12) is symmetric and can be interpreted as theinformation about A contained in B and vice versa. If Aand B are independent, their mutual information should bezero. In addition, mutual information is a better function formeasuring dependency between two random variables ratherthan the correlation function [20]. The reason is that mutualinformation is able to measure general dependency betweentwo variables, while the correlation function can only measurelinear dependency between them.



End

Repeat the algorithm for each independent

component

No

Normalizing separating matrix w to a new

matrix with unite norm

If converged

Choose the number of

independent components

Centering of the observed

vector X

Whitening of the observed

vector X using Eq. (8)

Initializing weight vector w randomly with unit norm

Update separating matrix w using gradient

optimization algorithm to maximize

negentropy by taking the gradient of Eq.

(7)

Start

Yes

Figure 1 Implementation procedure of the FastICA.

4. HARMONIC SOURCELOCALIZATION USING ICA AND MI

For estimating the probable location of multiple harmonicsources in power systems firstly, consider the system equa-

tion under non-sinusoidal condition which can be written inmatrix form as:

V h = ZhIh (13)

where V , Z and I are the bus voltage, impedance and businjected current vectors, respectively and h is the harmonicorder of the frequency.

If voltage V and impedance Z are identified, then the har-monic current I can be computed using Eq. (13). However,it is difficult to form the impedance matrix because of com-plexity of obtaining the values of harmonic impedance of thesystem for individual harmonic orders. Assuming that in theabsence of any information of the system parameters and char-acteristics of non linear loads, the only available informationis the bus voltages which are obtained by distributed harmonicvoltage measurement devices for each harmonic order.

The linear system of equations for the harmonic load flow(13) under non – sinusoidal condition is in close similarity withthe model in Eq. (2) which makes ICA a useful instrument formultiple harmonic source estimation. In addition, fast fluctu-ations of the loads consumption during specific time that arerelated to second to second or minute to minute variations,have non gaussian characteristics and act independently [21–23], which satisfy the preliminary conditions for applying ICAalgorithm. Applying ICA to Eq. (13), X represents the har-monic voltage measurement vectors V, the mixing matrix Arepresents the harmonic impedance matrix Z in the harmonicdomain, and the estimated signal S represents the bus injectedcurrent vector. In Eq. (13), the current matrix I is zero forlinear loads and non zero for harmonic or non-linear loads.

From (13), the injected harmonic currents generated by thenon-linear loads at the respective buses have a relationshipwith the bus voltages. By using the mutual information the-ory, the produced harmonic current at each bus has maximummutual information with its own bus voltage and the mutualinformation between this harmonic current and other bus volt-ages is reduced due to the current division between branches.

The procedure in implementing the proposed method usingICA and mutual information theory for estimating the locationof multiple harmonic sources is summarized as follows:

Measure harmonic voltages at all buses.

I. Estimate current traces of the harmonic sources using thefast ICA algorithm (Figure 1).

II. Calculate MI between the reconstructed harmonic cur-rent traces and harmonic bus voltages.

III. Estimate the location of harmonic sources by using com-puted mutual information.

5. SIMULATION RESULTS

To validate the performance and accuracy of the proposedmethod in radial and non radial distribution systems, two testsystems include the non radial IEEE 30 bus and radial IEEE 34bus test systems are used. In both test systems, three harmonicsources are considered at the selected buses. In addition, forboth cases it is assumed that all loads are with constant power


M. FARHOODNEA ET AL

Figure 2 IEEE 30 bus non radial test system.

Figure 3 Actual and reconstructed harmonic currents at bus 7, 16, and 30 for the 5th harmonic order for the 30 bus test system.

factor and the harmonic loads are modeled as harmonic cur-rent injection sources with harmonic spectrums given in [24].The procedure of estimating the location of harmonic sourcesusing ICA begins with harmonic voltage measurements. Har-monic power flow simulations are performed to compute busharmonic voltages and these voltages are considered as mea-surement vector, X in the ICA algorithm. The injected har-monic currents are then reconstructed as illustrated in Figure1. As we use random number for initialization of the sepa-rating matrix w, an average of ten runs of the algorithm isconsidered. The locations of harmonic sources are then de-termined by using the mutual information theory. The testresults for the two test systems are presented in the followingsubsections.

5.1 Results of the Non Radial 30 Bus TestSystem

In this subsection, to verify the accuracy of the proposedmethod, a non radial IEEE 30 bus test system shown in Figure2 is used [25]. In this test system, three harmonic sourcescontaining the 5th, 7th, 11th, 13th, and 17th harmonic ordersare placed at bus 7, 16, and 30. The PCFLO software [26] isemployed to perform harmonic power flow simulations so asto determine the harmonic voltages at all the buses. To createvariations in the harmonic measurement vectors, harmonicpower flows are simulated for different loading conditions.The loads are then multiplied by the Laplace distributed ran-dom variables to create different operating conditions for each





load at each sampling time. Harmonic bus voltages are com-puted by solving the linear system equation (13) for each har-monic frequency of interest. In this case, 350 harmonic volt-age samples have been generated to represent the harmonicmeasurement vector, X. Since the number of harmonic mea-surements is greater than the number of harmonic sources,preprocessing procedures are taken into account to reduce thedimension of the harmonic measurement vectors, X [17]. Forreconstructing variations in the harmonic currents producedby the harmonic sources, the Fast ICA algorithm programmedin Matlab is used. Figures 3 to 7 show the actual and the ex-tracted current traces at bus 7, 16, and 30 for the 5th, 7th, 11th,13th and 17th harmonic orders, respectively. In the figures,the red lines represent the actual or original harmonic currentswhile the blue lines represent the reconstructed harmonic cur-rents over the 350 samples. It should be noted that for ease ofcomparison, both signals in each figure are normalized withtheir largest values in per-unit (p.u).

To evaluate the accuracy of the ICA algorithm, correlationcoefficients of the actual and the reconstructed signals arecomputed as shown in Table 1. From the table, it is clear thatall the correlation coefficients are within the acceptable range,that is, close to 1. This implies high accuracy between theestimated and the actual harmonic currents. Table 2 shows themean square error of the signals so as to quantify the differencebetween the estimated and the actual harmonic currents. Theresults of Table 1 and 2 show that the Fast ICA algorithm canrebuild signals with high accuracy in which the mean squareerror of the estimated and the actual harmonic currents is lessthan 0.0015.

To identify the location of harmonic sources, the mutualinformation theory has been applied using Eq. (15) to calcu-late the pair-wise mutual information between the extractedcurrent trends and bus voltage for each harmonic frequency.The results of the mutual information are shown in Table 3,in which each column and row represents the estimated cur-


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Table 1 Correlation coefficients between the actual and the reconstructedsignals for the 30 bus test system.

Harmonic order Bus 7 Bus 16 Bus 305 0.9811 0.7789 0.98787 0.9829 0.9316 0.993011 0.9847 0.8851 0.988713 0.9846 0.8422 0.988517 0.9696 0.9389 0.9842

rent and the bus harmonic voltage for individual harmonicorder, respectively. The intersection between each columnand row represents the computed mutual information betweenthe three estimated currents at each harmonic frequency, andthe specified bus harmonic voltage. As shown in Table 3,the underlined-bold mutual information values imply that therelated bus has the greatest mutual information between the

Table 2 Mean square error between the actual and reconstructed signals forthe 30 bus test system.

Harmonic order Bus 7 Bus 16 Bus 305 4.61e-4 1.52e-3 5.69e-47 9.76e-5 3.90e-4 4.00e-5

11 8.76e-5 6.56e-4 6.45e-513 8.81e-5 9.01e-4 6.54e-517 1.73e-4 3.48e-4 9.05e-5

estimated harmonic currents and voltages and therefore thisbus is identified as the location of harmonic source. For ex-ample, in Table 3 the mutual information of 2.387 between thefirst estimated current of the 5th harmonic order (first column)and the harmonic voltage at bus 7 (7th row) has the greatestvalue in the related column. Therefore, the location of one ofthe possible harmonic sources (for 5th harmonic order) is at



Figure 8 IEEE 34 bus radial test system.

Table 3 Mutual information between estimated currents and bus voltages for the IEEE30 bus test system.

Harmonic order5 7 11 13 17

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3Bus 1 1.644 0.747 1.026 0.739 0.820 1.575 0.836 1.909 0.903 1.649 0.769 0.920 0.869 1.793 0.831Bus 2 1.990 0.700 0.946 0.754 0.793 1.732 0.830 2.169 0.911 1.763 0.773 0.850 0.861 1.844 0.854Bus 3 1.628 0.755 1.036 0.739 0.820 1.575 0.836 1.909 0.903 1.649 0.769 0.920 0.868 1.790 0.824Bus 4 1.635 0.754 1.031 0.733 0.835 1.581 0.833 1.918 0.905 1.643 0.766 0.918 0.853 1.790 0.822Bus 5 2.335 0.753 0.856 0.782 0.726 1.730 0.768 2.269 0.766 1.183 0.766 0.779 0.762 1.746 0.785Bus 6 2.338 0.776 0.864 0.798 0.775 1.918 0.785 2.327 0.861 1.952 0.729 0.812 0.850 1.767 0.847Bus 7 2.387 0.753 0.856 0.782 0.726 2.330 0.768 2.469 0.766 2.182 0.767 0.778 0.762 1.846 0.785Bus 8 2.385 0.757 0.872 0.786 0.756 1.788 0.838 2.417 0.841 1.846 0.770 0.809 0.856 1.770 0.860Bus 9 0.966 0.728 1.353 0.759 1.007 1.066 0.746 0.779 1.438 1.115 0.755 1.213 1.344 0.992 0.810

Bus 10 0.754 0.625 1.581 0.711 1.218 0.865 0.741 0.933 1.692 0.962 0.734 1.543 2.015 0.922 0.792Bus 11 0.966 0.728 1.353 0.759 1.007 1.066 0.746 0.779 1.438 1.115 0.755 1.213 1.346 0.994 0.807Bus 12 0.612 0.564 1.603 0.682 1.493 0.787 0.705 0.737 1.671 0.922 0.709 1.779 2.065 0.890 0.738Bus 13 0.612 0.564 1.603 0.682 1.493 0.787 0.705 0.909 1.767 0.922 0.709 1.779 2.065 0.890 0.738Bus 14 0.782 0.704 2.413 0.698 1.476 0.794 0.705 0.909 1.767 0.897 0.699 1.835 2.084 0.892 0.742Bus 15 0.907 0.836 1.499 0.684 1.449 0.800 0.705 0.943 1.708 0.726 0.638 1.684 2.050 0.898 0.734Bus 16 0.850 0.688 2.541 0.679 1.726 0.738 0.667 0.930 1.784 0.899 0.713 1.856 2.421 0.909 0.850Bus 17 0.875 0.791 1.360 0.718 1.464 0.821 0.696 0.860 1.721 0.900 0.689 1.738 1.529 0.959 0.882Bus 18 0.828 0.742 2.392 0.655 1.357 0.835 0.692 0.943 1.720 0.920 0.775 1.791 2.002 0.892 0.752Bus 19 0.776 0.683 2.205 0.710 1.338 0.846 0.704 0.951 1.727 0.911 0.740 1.786 1.999 0.896 0.770Bus 20 0.746 0.717 2.163 0.714 1.305 0.822 0.741 0.976 1.751 0.906 0.759 1.710 2.022 0.907 0.774Bus 21 0.760 0.651 1.799 0.715 1.216 0.886 0.766 0.918 1.684 0.946 0.748 1.610 1.917 0.921 0.757Bus 22 0.735 0.615 1.820 0.709 1.176 0.828 0.747 0.944 1.655 0.895 0.763 1.600 1.905 0.901 0.758Bus 23 0.860 0.720 1.867 0.586 1.172 0.638 0.686 0.838 1.298 0.900 0.740 1.344 2.208 0.896 0.680Bus 24 0.915 0.674 1.856 0.749 1.068 0.875 0.705 0.968 1.436 0.943 0.815 1.502 2.088 0.889 0.734Bus 25 0.818 1.350 0.929 1.463 0.728 0.728 1.162 1.033 1.028 0.832 1.196 0.966 1.826 0.914 0.854Bus 26 0.792 1.302 0.860 1.410 0.762 0.713 1.141 1.002 1.003 0.801 1.105 0.944 1.787 0.906 0.858Bus 27 0.895 1.791 0.876 2.330 0.729 0.700 1.474 0.951 0.911 0.748 1.635 0.778 1.557 0.913 0.892Bus 28 2.059 0.817 0.984 0.779 0.705 1.441 0.807 2.313 0.890 1.604 0.762 0.819 0.829 1.738 0.850Bus 29 0.782 2.335 0.811 2.767 0.662 0.773 2.069 0.867 0.873 0.775 2.538 0.702 0.774 0.927 1.967Bus 30 0.782 2.406 0.841 2.945 0.670 0.708 2.311 0.829 0.820 0.777 2.553 0.732 0.749 0.796 2.739


M. FARHOODNEA ET AL



bus 7. From the results shown in Table 3, it is evident thatharmonic source locations are located at bus 7, 16, and 30and this proves the accuracy of the proposed harmonic sourcelocation method.

5.2 Results of the Radial 34 bus Test System

To prove the accuracy and ability of proposed method forlocating multiple harmonic sources in radial distribution net-works, the radial IEEE 34 bus test system shown in Figure8 is used [27]. In the test system, three harmonic sourcescontaining the 5th, 7th, 11th, 13th, and 17th harmonic ordersare located at bus 15, 25, and 33. In order to use the ICAalgorithm for reconstructing the injected harmonic currents,harmonic voltage measurements are generated using the har-monic power flow algorithm for radial systems [28-29] whichhas been implemented using the Matlab codes.

Similar to the 30 bus test system, harmonic power flows aresimulated for different loading conditions by means of multi-plying the Laplace distributed random variables at all loads tocreate different operating conditions for each load and at eachsampling time. Again, harmonic bus voltages are computedby solving the linear system for each harmonic order. In orderto reconstruct injected harmonic currents caused by the har-monic sources, 500 harmonic voltage samples have been gen-erated at each bus so as to represent harmonic measurementvector, X. by using preprocessing procedures, the dimensionof the harmonic measurement vectors X can be decreased soas to omit the redundant information of the data. Figures 9to 13 show the extracted current traces by using the Fast ICAalgorithm in comparison with the actual signals at bus 15, 25,and 33 for the 5th, 7th, 11th, 13th and 17th harmonic orders,respectively, in which the red lines represent the actual har-monic currents and the blue lines represent the reconstructed





harmonic currents over 500 samples and all signals are nor-malized with their largest values in per-unit (p.u).

To prove high accuracy between the estimated and the actualharmonic currents, the obtained correlation coefficients andmean square error of the signals are shown in Tables 4 and 5.The results of Tables 4 and 5 imply that compatibility degreebetween the reconstructed and original signals is greater than98% and mean square error of estimation is less than 4.33e-5. It is shown that the 34 bus test system has correlationcoefficient values closer to 1, and lesser mean square errorvalue in comparison with the results of the 30 bus test system.Hence, the accuracy of the results is due to the difference inthe sample size used in the two systems.

To identify the location of harmonic sources, the pair-wisemutual information between the extracted current trends andbus voltage have been obtained for each harmonic frequencyas shown in Table 6. In the Table, the underlined-bold mutualinformation values imply that the related bus has the greatestmutual information between the estimated harmonic currents

Table 4 Correlation coefficients between the actual and the reconstructedsignals for the 34 bus test system.

Harmonic order Bus 15 Bus 25 Bus 335 0.9892 0.9906 0.99457 0.9931 0.9909 0.9978

11 0.9892 0.9907 0.994413 0.9936 0.9972 0.995117 0.9930 0.9909 0.9944

Table 5 Mean square error between the actual and reconstructed signals forthe 34 bus test system.

Harmonic order Bus 15 Bus 25 Bus 335 4.33e-5 3.74e-5 2.21e-57 2.24e-5 3.81e-5 8.90e-6

11 4.32e-5 3.73e-5 2.22e-513 2.57e-5 1.11e-5 1.94e-517 2.79e-5 3.62e-5 2.23e-5


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Table 6 Mutual information between estimated currents and bus voltages for the 34 bus test system.

Harmonic order5 7 11 13 17

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3Bus 1 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000Bus 2 0.814 0.911 0.758 0.747 1.046 0.703 0.712 1.173 0.711 0.670 1.407 0.699 0.608 0.707 1.321Bus 3 0.814 0.911 0.758 0.747 1.046 0.703 0.712 1.173 0.711 0.670 1.407 0.699 0.608 0.707 1.321Bus 4 0.954 0.770 0.866 0.904 0.900 0.718 0.822 0.916 0.722 0.700 1.037 0.781 0.719 0.779 0.990Bus 5 1.066 0.752 0.905 1.013 0.814 0.820 0.931 0.842 0.824 0.680 0.881 0.876 0.732 0.814 0.902Bus 6 1.108 0.753 0.931 1.057 0.747 0.880 0.967 0.789 0.887 0.782 0.860 0.973 0.788 0.895 0.838Bus 7 0.958 0.759 1.004 1.160 0.740 0.807 1.068 0.752 0.843 0.770 0.819 1.109 0.904 0.860 0.792Bus 8 0.956 0.679 1.042 1.249 0.717 0.744 1.130 0.775 0.835 0.820 0.831 1.160 0.977 0.869 0.804Bus 9 0.866 0.687 1.135 1.377 0.711 0.780 1.257 0.737 0.745 0.769 0.753 1.273 1.055 0.828 0.774

Bus 10 0.829 0.649 1.184 1.460 0.674 0.696 1.306 0.703 0.740 0.707 0.773 1.333 1.084 0.782 0.754Bus 11 0.829 0.649 1.184 1.460 0.674 0.696 1.306 0.703 0.740 0.707 0.773 1.333 1.084 0.782 0.754Bus 12 0.829 0.649 1.184 1.460 0.674 0.696 1.306 0.703 0.740 0.707 0.773 1.333 1.084 0.782 0.754Bus 13 0.815 1.075 0.723 0.685 1.321 0.653 0.623 1.462 0.671 0.632 1.681 0.657 0.603 0.672 1.585Bus 14 0.709 1.328 0.667 0.667 1.695 0.647 0.630 1.808 0.664 0.631 1.998 0.629 0.585 0.678 1.938Bus 15 0.701 1.483 0.625 0.652 1.846 0.659 0.635 1.988 0.671 0.629 2.144 0.627 0.585 0.664 2.094Bus 16 0.701 1.443 0.625 0.652 1.546 0.659 0.635 1.888 0.671 0.629 2.005 0.627 0.585 0.664 1.973Bus 17 1.201 0.739 0.875 0.977 0.744 0.928 0.919 0.834 0.974 0.840 0.813 0.952 0.831 0.972 0.794Bus 18 1.284 0.744 0.855 0.991 0.788 0.998 0.942 0.803 0.991 0.878 0.828 0.916 0.796 1.037 0.764Bus 19 1.338 0.718 0.815 0.943 0.767 1.043 0.896 0.767 1.084 0.917 0.817 0.890 0.815 1.124 0.802Bus 20 1.414 0.733 0.765 0.917 0.745 1.094 0.867 0.750 1.095 0.964 0.776 0.901 0.802 1.194 0.792Bus 21 1.476 0.714 0.743 0.870 0.734 1.152 0.856 0.760 1.129 1.025 0.784 0.884 0.785 1.272 0.757Bus 22 1.564 0.714 0.721 0.852 0.712 1.191 0.810 0.741 1.195 1.086 0.786 0.868 0.781 1.311 0.727Bus 23 1.607 0.669 0.730 0.850 0.753 1.233 0.805 0.702 1.232 1.091 0.753 0.845 0.750 1.344 0.717Bus 24 1.705 0.664 0.717 0.829 0.742 1.297 0.805 0.727 1.313 1.125 0.734 0.874 0.697 1.432 0.725Bus 25 1.741 0.680 0.729 0.793 0.706 1.322 0.780 0.721 1.324 1.196 0.767 0.839 0.710 1.480 0.731Bus 26 1.641 0.680 0.729 0.793 0.706 1.122 0.780 0.721 1.242 1.155 0.767 0.839 0.710 1.365 0.731Bus 27 1.541 0.680 0.729 0.793 0.706 1.022 0.780 0.721 1.127 1.124 0.767 0.839 0.710 1.312 0.731Bus 28 0.958 0.759 1.004 1.160 0.740 0.807 1.068 0.752 0.843 0.770 0.819 1.109 0.904 0.860 0.792Bus 29 0.958 0.759 1.004 1.160 0.740 0.807 1.068 0.752 0.843 0.770 0.819 1.109 0.904 0.860 0.792Bus 30 0.958 0.759 1.004 1.160 0.740 0.807 1.068 0.752 0.843 0.770 0.819 1.109 0.904 0.860 0.792Bus 31 0.818 0.617 1.257 1.524 0.689 0.711 1.357 0.733 0.755 0.671 0.739 1.364 1.128 0.782 0.768Bus 32 0.827 0.663 1.321 1.596 0.672 0.696 1.385 0.722 0.727 0.698 0.718 1.421 1.161 0.779 0.792Bus 33 0.795 0.695 1.335 1.632 0.665 0.677 1.443 0.723 0.704 0.724 0.723 1.500 1.190 0.747 0.741Bus 34 0.795 0.695 1.235 1.432 0.665 0.677 1.343 0.723 0.704 0.724 0.723 1.325 0.968 0.747 0.741

Table 7 Mean square errors of the mutual information values for IEEE 30 and 34 bus test system.

Harmonic orders 5th 7th 11th 13th 17thIEEE 30 bus test system 1.29e-4 2.24e-4 2.69e-4 3.45e-4 1.26e-4IEEE 34 bus test system 6.96e-5 8.58e-5 9.94e-5 1.21e-4 9.81e-5




and voltages and this bus should identify as one of the pos-sible location of harmonic source. For example, the mutualinformation of 1.741 between the first estimated current of the5th harmonic order (first column) and the harmonic voltage atbus 25 (25th row) has the greatest value in the related column,which implies that bus 25 is the location of one of the possibleharmonic sources in this test system.

From the results shown in Table 6, it is evident that harmonicsource locations are located at bus 15, 25, and 33 for the 34bus test system, because these buses have the highest mutualinformation values. Thus, the harmonic source locations havebeen accurately estimated by the proposed method using ICAand mutual information theory.

Table 7 shows the mean square errors of the mutual infor-mation values for both 30 and 34 bus test systems. The resultsshows that the mutual information algorithm can estimate ex-act locations of the harmonic sources with the mean squareerrors less than 0.00035 and 0.00012 for the 30 and 34 bustest systems, respectively.

6. CONCLUSION

In this paper, a novel technique has been presented for locatingmultiple harmonic sources in radial and non radial distributionsystems, without prior information about the network compo-nents and branch parameters. In the proposed method, ICAwhich is one of the blind source separation techniques is ap-plied for estimating the injected harmonic currents producedby harmonic sources and the mutual information theory is ap-plied to identify the location of possible harmonic sources.ICA is also used to estimate the load profiles of the harmonicsources with different sample sizes. The results from the casestudies confirm that the proposed method can accurately iden-tify the location of multiple harmonic sources in both radialand non radial distribution systems. The proposed method ispromising because it uses only bus voltage measurements andthere is no need for real and reactive power measurements andother restricted power system information.

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