Evaluation of Harmonic Contributions for Multi Harmonic Sources System Based on Mixed Entropy Screening and an Improved Independent Component Analysis Method

Evaluating the harmonic contributions of each nonlinear customer is important for harmonic mitigation in a power system with diverse and complex harmonic sources. The existing evaluation methods have two shortcomings: (1) the calculation accuracy is easily affected by background harmonics fluctuation; and (2) they rely on Global Positioning System (GPS) measurements, which is not economic when widely applied. In this paper, based on the properties of asynchronous measurements, we propose a model for evaluating harmonic contributions without GPS technology. In addition, based on the Gaussianity of the measured harmonic data, a mixed entropy screening mechanism is proposed to assess the fluctuation degree of the background harmonics for each data segment. Only the segments with relatively stable background harmonics are chosen for calculation, which reduces the impacts of the background harmonics in a certain degree. Additionally, complex independent component analysis, as a potential method to this field, is improved in this paper. During the calculation process, the sparseness of the mixed matrix in this method is used to reduce the optimization dimension and enhance the evaluation accuracy. The validity and the effectiveness of the proposed methods are verified through simulations and field case studies.


Introduction
Harmonic sources in a power system become complex and diverse as more nonlinear customers connected to the power grid [1,2]. For a bus with serious harmonic distortion, evaluating the contribution of each customer is essential to identify the dominant harmonic source and to design the effective scheme for harmonic pollution mitigation [3].
In general, the evaluation models are classified into two categories ( Figure 1): single point and multipoint models. In the single point model, the power grid is divided into the utility and the customer sides at the point of common coupling (PCC), indicating that there is only one suspicious harmonic source. The typical methods to solve this model include the fluctuation method [4], a serious of regression methods [5] and covariance method [6]. Yet, one common limitation of these methods is that they are only suitable when the background harmonics are stable. To reduce the impacts of the background harmonics, from 2015, independent component analysis method (ICA) [7,8] was applied to this researching field [9][10][11][12]. Compared with the classical methods, ICA has a higher calculation accuracy even when the background harmonics fluctuate. Thus, it becomes a popular technology to solve the single point model. sources system, this assumption is hard to hold. Moreover, during the execution of the multiple linear regression method, the background harmonics are still required to be stable. Thus, it has the same limitations as the least squares method. To ensure evaluation accuracy even when background harmonics fluctuate, Wang and co-workers [21] expanded the application of the complex independent component analysis method (CICA) from the single point model to the decentralized multipoint model. Compared with the former two methods, CICA has a superior ability to resist the impacts from the background harmonics. Yet, the calculation results are still unsatisfactory when the background harmonics fluctuate greatly. In addition, the essence of CICA is to reconstruct the source signals based on a negative entropy maximization algorithm. When many suspicious harmonic sources are involved, the calculation burden may increase, and the evaluation accuracy is thus decreased. Furthermore, the CICA method and the evaluation referred to by Wang and co-workers [21] rely on measurements with a Global Positioning System (GPS)-synchronized function that can synchronize the measuring time [18][19][20][21][22][23][24]. However, in engineering practice, it is not realistic to install GPS-based measurements at all related buses because it is uneconomical [6]. Owing to the above three limitations, the existing CICA-based evaluation method still has room for improvements. An overview of the developments and the limitations of the existing methods are shown in Figure 2. To overcome the shortcomings of the existing CICA method and further accurately evaluate the harmonic contribution for each harmonic source, an improved CICA method is proposed in this paper. First, the properties of asynchronous measurements are explored, and a model independent of GPS is proposed for evaluating harmonic contributions. Then, according to the Gaussianity of the measured harmonic data, a mixed entropy screening mechanism is proposed to select the data segments where the background harmonics are relatively stable. Finally, based on the sparseness of the mixed matrix in the blind source separation model, the optimization dimensions for maximizing the negative entropy Entropy 2020, 22, 323 4 of 24 in CICA are remarkably decreased. Thus, the calculation burden is significantly reduced, which enhances the evaluation accuracy of the harmonic contributions. Compared with the least squares and multiple linear regression methods, the corresponding assumptions are no longer needed in the proposed method. Meanwhile, as shown in Figure 2, the improved method has three advantages compared with the traditional CICA: (1) it does not rely on GPS-based measurements; (2) it is able to select data segments with stable background harmonics; and (3) it ensures a high calculation accuracy even when the background harmonics fluctuate and/or the number of suspicious harmonics sources is large. The performance of the proposed method is validated by simulations on the IEEE 14-bus system and field case studies with actual multi-infeed high voltage direct current (HVDC) system. For the convenience of reading and understanding, symbols involved in this paper are described in Table 1. Table 1. Description of the symbols involved in this paper. In the decentralized multipoint model of Figure 3, the h-th harmonic voltages at the concerned bus ( . U X ) are generated by all the harmonic sources in the power grid as shown in Figure 4 and Equation (1).
where Z X,i is the harmonic transfer impedance between customer i and bus X, and N − 1 is the number of suspicious harmonic sources. U X , we obtain the harmonic contribution for customer i as: Thus, accurately calculating the transfer impedance Z X,i is the key to evaluating the harmonic contribution for each customer. To calculate Z X,i , the first step is to estimate the harmonic currents . I c,i generated from each suspicious harmonic source [21]. This can be done by solving the single point model for each customer.
The equivalent harmonic circuit of customer i is shown in Figure 5, where the harmonic voltages . U PCC i and currents . I i can be measured directly at the PCC. . I u,i and Z u,i are respectively the harmonic current and impedance of the utility side. According to Figure 5, we have Generally, the fast-varying components of . I c,i and . I u,i separated from an average filter are approximately independent [9][10][11]21]. Thus, the CICA method can be adopted to solve Equation (3), and then . I c,i is obtained (the solving processes were introduced in [9,10] in detail.; to avoid repetition, the specific calculation steps are omitted here). After the harmonic currents . I c,i are generated from each harmonic source estimated, we establish the corresponding blind source separation model as [21]: Thus, accurately calculating the transfer impedance ZX,i is the key to evaluating the harmonic contribution for each customer. To calculate ZX,i, the first step is to estimate the harmonic currents c,i I  generated from each suspicious harmonic source [21]. This can be done by solving the single point model for each customer. The equivalent harmonic circuit of customer i is shown in Figure 5  According to Figure 5, we have Generally, the fast-varying components of c,i I  and u,i I  separated from an average filter are approximately independent [9][10][11]21]. Thus, the CICA method can be adopted to solve Eq. (3), and then c,i I  is obtained (the solving processes were introduced in [9-10] in detail.; to avoid repetition, the specific calculation steps are omitted here). After the harmonic currents c,i I  are generated from each harmonic source estimated, we establish the corresponding blind source separation model as [21]: where the superscript " fast " represents the fast-varying components of each signal; the separated signals T fast fast fast fast c,1 c,2 c, -1 0ˆˆN to the source signals with scaling indeterminacy [9][10][11]21]. This indeterminacy is indicated by the unknown complex coefficients ki. The transfer impedances ZX,i can be obtained as: T with scaling indeterminacy [9][10][11]21]. This indeterminacy is indicated by the unknown complex coefficients k i . The transfer impedances Z X,i can be obtained as: whereÂ i+1,i denotes the element that corresponds to the (i + 1)-st row and i-th column of matrixÂ. Based on the calculatedẐ X,i and Equation (4), we can recover the matrixÂ that does not contain k i , and thus, the scaling indeterminacy of the separated signals Y is solved. Finally, on the basis of the obtainedẐ X,i andˆ. U 0 , the harmonic contribution for each customer can be assessed by using Equations (1) and (2).

Model with Asynchronized Phasor Measurements
The above model is largely based on GPS-based synchronized phasor measurements, which is not practical as previously stated [6]. As a result, in many cases it is hard to obtain the relative phases between . U X and each . I i in Equation (1). To solve this problem, we propose a novel evaluation model in this section.

Properties of Asynchronous Measurements
Under conditions without GPS technology, a tiny and unknown time difference ∆t exists between the starting times of two measurements. For instance, the voltage waves measured at bus X satisfy where U x0 is the magnitude of the direct current (DC) component; H xh and φ Xh are the magnitude and initial angle of the h-th harmonic, respectively; and ω 1 = 2π f 1 where f 1 is the fundamental frequency. Assuming the starting time of measurement at bus A is ∆t behind that of measurement at bus X, we have Equation (8) can also be transformed into Adopting the discrete Fourier transform (DFT) for U A (t), we obtain the amplitude and initial angle for the h-th harmonic as H Ah and (φ Ah + α A ). Additionally, if the starting time of measurement at bus A is advanced by ∆t, the measurements at buses A and X will be synchronized, and thus, the corresponding DFT results become H Ah and φ Ah . Therefore, two conclusions can be drawn theoretically for the h-th harmonic obtained from the DFT:

•
The amplitudes are the same for the synchronous and asynchronous measurements.

•
The initial angular difference between the synchronous and asynchronous measurements is hω 1 ∆t (∆t is unknown).
In practice, all of the measuring data are usually divided into several segments to detect the harmonics. Since ∆t is the same for each segment, the angular difference between the synchronous and asynchronous measurements is constant for each data segment.
We further verify the properties of asynchronous measurements for three practice cases: (1) a wind farm; (2) a photovoltaic station; and (3) nonlinear loads with computers and light-emitting diode (LED) lights. The harmonic voltages (5th, 7th, and 11th) obtained from the DFT transformation for each time segment (0.02 s) are defined as U 1 . After manually creating a measurement delay, we define the corresponding DFT results as U 2 . Figure 6 presents the difference between U 1 and U 2 . For each case, |U 1 |−|U 2 |=0 and the differences between ∠U 1 and ∠U 2 are constant; thus, the aforementioned two properties are valid. Consequently, the amplitudes of harmonics detected from synchronous and asynchronous data are the same, while their phase differences are constant.

Proposed Model for Evaluating Harmonic Contributions Without GPS Technology
On the basis of the analysis above, we can transform Equation (4) by setting the starting time of measurement at bus X as the reference: where α i (i = 1, 2, · · · , N − 1) represents the relative phase between the synchronous and asynchronous measurements for customers i. Notably, the source signal . I c,i is still synchronous with . U X . Yet, because α i is unknown, we only have the asynchronous signals e jα i . I c,i . u 0 are independent from each other [9][10][11]21], Equation (10) can be solved via CICA as: Since the fluctuations of c,i I  and 0 U  are independent from each other [9][10][11]21], Equation (10) can be solved via CICA as: where the unknown complex coefficients ki indicate the scaling indeterminacy [9][10][11]21]. The transfer impedances ZX,i combined with αi can be obtained as: However, we still cannot separate the transfer impedances from αi because these angles are unknown. Yet, according to Equations (1) and (12), we can evaluate the harmonic contributions via  (1) and (12), we have Thus, c, X,Ẑ

A Mixed Entropy Screening Mechanism for Data Segments Selection
During the calculation process of CICA, the harmonic impedances should be constant to keep the matrix A invariant. To ensure this, first, the whole measured data are usually divided into several short segments. Then, we calculate the transfer impedances using each segment separately and where the unknown complex coefficients k i indicate the scaling indeterminacy [9][10][11]21].
The transfer impedances Z X,i combined with α i can be obtained as: However, we still cannot separate the transfer impedances from α i because these angles are unknown. Yet, according to Equations (1) and (12), we can evaluate the harmonic contributions via e −jα iẐ X,i without separating the transfer impedances from these unknown angles. When signal e jα i. I c,i multiplies e −jα i , the new signalˆ. I c,i is synchronized with . U X . Although α i is unknown, based on Equations (1) and (12), we have Thus,ˆ. I c,iẐX,i is obtained directly without calculating α i because the angle α i in e jα i. I c,i and e −jα iẐ X,i is just offset. Consequently, the harmonic contribution of each source can be evaluated without the synchronized phasor measurements.

A Mixed Entropy Screening Mechanism for Data Segments Selection
During the calculation process of CICA, the harmonic impedances should be constant to keep the matrix A invariant. To ensure this, first, the whole measured data are usually divided into several short segments. Then, we calculate the transfer impedances using each segment separately and average the results. It can be considered that the harmonic impedances are approximately constant during each single data segment because the corresponding time is quite short. Additionally, the calculation accuracy of CICA relies on stable background harmonics. A large fluctuation of the background harmonics may increase the calculation errors [9][10][11]21]. Therefore, if the fluctuation degree of the background harmonics is assessable, we can improve the calculation accuracy by choosing the data segments with relatively stable background harmonics for calculation, while eliminating the data segments with heavily fluctuating background harmonics. The problem is that the background harmonics cannot be measured directly. To overcome this difficulty, a mixed entropy screening mechanism is proposed in this section.
Based on the central limit theorem [25], the distribution of a signal x linear combined by many random signals tends to be Gaussian. In addition, if signal x is only dominated by a few of these signals, its Gaussian degree will be decreased. For instance, signal x is combined by four random real signals s i (i = 1, 2, 3, 4) with unit amplitude as: X = s 1 + s 2 + s 3 + s 4 . The sample size for each signal is 3000. Their distributions and Gaussian degrees are shown in Figure 7, where the Gaussian degree is assessed by the entropy of a signal as Equation (14) [26]. A high entropy corresponds to a strong Gaussian degree.
where M is the sample size of signal s, p(s m ) is the probability mass function of s. Since the standard Gaussian signal has the largest entropy among all random signals of equal variance [27], to make the entropy of different signals comparable, transformations in Equations (15) and (16) are performed for each signal.
where E{s} and std{s} are the mean value and standard deviation of signal s, respectively.
where s G is the standard Gaussian signal with zero-mean and unit variance. As shown in Figure 7, compared with the signals s i (i = 1, 2, 3, 4), the mixed signal x is closer to Gaussian distribution. Meanwhile, the entropy of signal x is also the biggest one among these signals, revealing that entropy can correctly reflect the Gaussian degree of each signal. After the amplitude of signal s 4 increases k times, the distribution and the entropy of signal x is presented in Figure 8. With the increasing of k, signal x is gradually dominated by s 4 , which causes the decline of the Gaussian degree of x, and so, the corresponding entropy declines. The above analyses can be applied into Equation (10) to assess the fluctuation degree of the background harmonic voltages . U 0 . Since the Gaussian degree of each complex signal in Equation (10) can be reflected by the Gaussian degree of their real and imaginary parts [11], we use the mixed entropy defined in Equation (17) to assess the Gaussian degree of a complex signal. For a data segment, two possible situations are discussed below: where the subscripts " x " and " y " respectively represent the real and imaginary parts of signal s.
For all the given data segments, the mixed entropy for their corresponding signal . U fast X can be obtained by using Equation (17). Furthermore, the average value of these mixed entropies can be calculated (defined as H m ). If H . U fast X > H m holds for a data segment, the corresponding background harmonics are stable. Otherwise, Equation (18) is required to further assess the fluctuation degree of the background harmonics.
A schematic diagram of the mixed entropy screening mechanism is shown in Figure 9.

Improved CICA Method
Although the mixed entropy screening mechanism can release the impacts of background harmonics, to further improve calculation accuracy, an improved CICA is proposed in this section. Before applying the CICA method, the observed signals X are preprocessed by centering and whitening to simplify the calculation [9][10][11]. The centering process transforms X into zero-mean signals, while the whitening process can be done as where Q is the whitening matrix. Λ and Γ are the diagonal and orthogonal matrices of eigenvalues of E{XX T }, respectively (the symbol E{.} denotes a mean value) [10].
The key step of CICA is to find a separating matrix W satisfying where the symbol T represents the Hermitian transpose. CICA is essentially an optimization algorithm based on the central limit theorem. Its goal is to maximize the non-Gaussianities of each signal in matrix Y, while the elements in matrix W are the variables to be optimized [7,8,28]. The non-Gaussianities of signal W T X w (signal Y) can be estimated via the negative entropy [7,11,29,30]: where s Gauss is a signal with zero mean and unit variance and obeys a Gaussian distribution, and G{.} is a nonlinear function. The non-Gaussianity of signal W T X w strengthens as the negative entropy J G (W T X w ) increases. Of note, the maxima of J G (W T X w ) are obtained when E G W T X W is optimal [7]. Furthermore, in another study [8], by defining g{.} as the derivative of G{.}, index σ c is deduced to solve this optimization problem: When σ c < 0, we should calculate the matrix W that maximizes E G W T X W 2 . Conversely, when σ c > 0, the matrix W that minimizes E G W T X W 2 should be solved [8]. The problem of maximizing the negative entropy J G (W T X w ) is now converted into optimizing E G W T X W 2 .
Despite the ordering and scaling indeterminacies of CICA [9][10][11]21], the optimal W satisfies Additionally, matrix A in Equation (10) is sparse to a certain degree. For an N × N matrix A, there are (N − 1) × (N − 1) elements naturally equal to 0 and one element equal to 1. However, in traditional CICA, these known elements are still treated as variables to be solved, which may cause two problems: (1) wasting the chance to reduce the optimization dimensions; and (2) increasing errors when the calculation results of these elements do not equal to their theoretical values (0 or 1). This paper uses the sparseness of matrix A to improve CICA and enhance the calculation accuracy. By assuming there are N − 1 suspicious harmonic sources in a system, we calculate the inverse of matrix A as Since matrix Q can be obtained from X, we can calculate its inverse Q −1 (defined as Θ). Thus, Equation (24) gives where W N,N and Θ N,N represent the element in the N th row and N th column of matrix W and Θ, respectively. Equation (26) can be rewritten as Once, W 1,1 , W 1,2 , · · · , W 1,N and W 2,N , W 3,N , · · · , W N,N are solved, the other elements in matrix W are obtained.
Furthermore, because we have Thus, W N,N can be obtained from W 2,N , W 3,N , · · · , W N−1,N , and Θ as Consequently, the optimization dimensions are greatly reduced from N × N to 2(N − 1), which significantly reduces the optimization burden. Moreover, the elements that are already known (0 or 1) in the solved matrix Awill be exactly equal to their proper values without the risk of error. The calculation accuracy is thus further enhanced. The flow chart of the improved CICA is shown in Figure 10.

Simulation Cases and Analysis
The IEEE-14 bus system [31] is used to validate the proposed method. The simulations are processed by the Matlab 8.5 program under the 5th harmonic. Bus 6 is set as the concerned bus, and suspicious harmonic sources are set at buses 13 and 14 ( Figure 11). The injected harmonic currents and background harmonic voltages are shown in Figure 12. Please note, all these harmonic data are per unit values, and the corresponding actual values can be converted by their base values in another study [31]. The harmonic data are averaged in 1 second, so we have 1 sample per second.   According to the structure and parameters of the IEEE-14 bus system [31], the harmonic impedances of the utility side are Z u,A = 0.1747 + 0.7313jp.u. and Z u,B = 0.2708 + 1.0004 jp.u.. Meanwhile, the harmonic impedances of the customer side are set as Z c,A = 0.5 + 2jp.u. and Z c,B = 2 + 11jp.u.. Throughout this simulation, Z u,A , Z u,B , Z c,A , and Z c,B are all unknowns and need to be solved. The above impedance values are just used as the references for analyzing the calculation errors.   Figure 9, the corresponding background harmonics are stable, while the harmonic currents generated by customer A or B fluctuate. The above conclusions are just consistent with the curves in Figure 12, and the effect of the mixed entropy screening mechanism is thus verified.

Multipoint Calculation
The key step of multipoint calculation is to solve the transfer impedance between each harmonic source and the concerned bus X (bus 6 in this case). As analyzed before, the calculated transfer impedances without synchronous measurements are Z α A X,A = e − jα A Z X,A and Z α B X,B = e − jα B Z X,B . According to Equation (13), the contributions from each harmonic source can be calculated even though α A and α B are unknown. Four methods are used for calculation: (1) the least squares method [18]; (2) the multiple linear regression method [20]; (3) traditional CICA; and (4) the proposed method. The calculation results are shown in Tables 3 and 4. For the least squares method, the calculation errors are huge because its necessary assumptions that the background harmonics are stable and that only one harmonic source fluctuates at a time are hard to hold for most of the segments. In addition, in engineering practice, with the increasing number of complex nonlinear customers, several suspicious harmonic sources usually exist for a concerned bus; thus, the basic assumptions of the least squares method are more difficult to satisfy. Meanwhile, the multiple linear regression method also requires that the background harmonics should not fluctuate; thus, the calculation errors are still high. Additionally, the calculation accuracy of traditional CICA is improved but still unsatisfactory because its calculation results are still impacted by the fluctuation of the background harmonics to a certain degree. In comparison, the improved CICA can accurately calculate the transfer impedances and evaluate the harmonic contribution correctly for each harmonic source. We further analyze the effects of the background harmonics on the calculation accuracy by setting . U 0 = k . U 0 . As the coefficient k increases, the background harmonics become more unstable. The calculation results of these four methods are shown in Figure 13. The calculation errors of the lease squares and multiple linear regression methods become terrible with the increasing of k. Although the results of the traditional CICA are modified compared with the former two methods, the errors still increase rapidly as the background harmonics increase. By contrast, the results of the improved CICA always have high accuracy, satisfying engineering requirements. Therefore, it is further verified that the improved CICA has a stronger ability to resist impacts from the background harmonics. To further explore the differences between the traditional and improved CICA, Figure 14 shows the amplitudes of the elements in the solved mixed matrix A. In the improved CICA, the elements A 2,2 , A 2,3 , A 3,1 , and A 3,3 are exactly equal to 0. However, in the traditional CICA, these elements do not exactly equal their theoretical values, which decrease the calculation accuracy. In addition, real-time implementation is important for an algorithm. Since the traditional CICA is derived from a fast fixed-point algorithm [8], it already has a high execution speed. Further, in the improved CICA, the optimization dimension is decreased based on the sparseness of the matrix A. Thus, the execution time is further reduced. For the whole 10 minutes data, the execution time of the improved CICA is just 0.43 seconds. Thus, with such a high execution speed, the proposed method can be used to evaluate the harmonic contributions in real time.

Field Case Verifications
The power grid for an actual multi-infeed HVDC system shown in Figure 15 is used to further verify the validity of the proposed method under the situation of asynchronous measurements. In this power system, converter stations, as the high-power harmonic sources in a high-voltage level, inject a lot of characteristic harmonics into the power system, which worsen the harmonic distortion in some areas of the grid [32,33]. Bus B23 with high 11th harmonic voltage content exceeding the Chinese standard limits is set as the concerned bus, while the four HVDC systems are the main suspicious harmonic sources. After applying the DFT analysis for the measured data, the 11th harmonic voltages at bus B23 and the currents at each HVDC system are shown in Figure 16. The sampling frequency is 10k Hz and the data resolution is 1 sample per second. Of note, the measured harmonic currents in Figure 16b are asynchronized with the harmonic voltages in Figure 16a. Meanwhile, the corresponding synchronized currents are also measured, and so, the results calculated from the synchronous case can be used as the reference. The harmonic contributions of the four HVDCs calculated from each method with the asynchronous data are shown in Figure 17 and Table 5. Since there are four suspicious harmonic sources, it is hard to satisfy the basic assumption of the least squares method that only one harmonic source fluctuates at a time. Thus, the calculation errors are large. The multiple linear regression method is based on the condition that the relative phase between . U X and each . U i is approximately constant. However, in engineering practice, this phase angle usually varies especially when . U X is generated by multiple harmonic sources. Therefore, the calculation errors of the multiple linear regression method are still large. In addition, for traditional CICA, the large number of suspicious harmonic sources increases the difficulty of signal separation. Thus, the calculation accuracy is low. In comparation, the optimization dimension of the improved CICA is decreased by adopting the sparseness of the matrix A. Hence, the calculation burden is greatly released, and the evaluation accuracy is satisfactory. Therefore, it can be concluded that HVDC2 is the dominant harmonic source for the concerned bus.   To further validate the above conclusion, the contents of the 11th harmonic voltage at the concerned bus are analyzed under different switching modes of harmonic filters at each HVDC. Although harmonic filters are expensive, they are still widely applied in the HVDC converter stations in China. Negative effects on technology and economy will occur without these filters. On the one hand, the harmonic currents injected into the power system far exceed the Chinese standard limits, and the voltages in most buses of the power gird will be seriously distorted. On the other hand, the potential economic loss caused by the characteristic harmonics of HVDCs can be far beyond the cost of the harmonic filters. As a result, filters are necessary for HVDC systems [34]. Figure 18 presents the harmonic impedance property of the double-tuned filters installed at the converter station. The impedance amplitudes corresponding to the 11th, 13th, 23rd, and 25th harmonics are low; thus, the relative characteristic harmonics can be filtered out.  Figure 19 indicates that when more filters are put into HVDC2, the contents of the 11th harmonic voltage at the concerned bus decrease obviously. In contrast, the harmonic mitigation effects are weak after putting more filters into other HVDCs. Consequently, HVDC2 is surely the dominant harmonic source for the concerned bus and the evaluation results from the improved CICA are further validated.

Conclusions
To evaluate the harmonic contributions accurately and economically, a novel method is proposed in this paper. First, an evaluation model independent of expensive GPS technology is developed. Then, a mixed entropy screening mechanism is designed to select the data segments with stable background harmonics. Last, the optimization dimensions of CICA are greatly reduced by using the sparseness of the mixed matrix, and so the evaluation accuracy is enhanced. The results of simulations and field case studies are summarized as: Thus, the fluctuation degree of background harmonics is successfully assessed by the proposed mixed entropy screening mechanism.
(2) In simulations and filed cases studies, the results of the proposed asynchronous evaluation model are consistent with the harmonic contributions evaluated from the synchronous measurement data, verifying the validity of the proposed GPS-free evaluation model. (3) Calculation precision of the traditional methods is low especially when the background harmonics are unstable and there are many suspicious harmonic sources. In comparison, owing to the effect of optimization dimensions reduction, the improved CICA always has high evaluation accuracy in simulations and field case studies.
Consequently, compared with the existing studies, the proposed method can evaluate the harmonic contributions accurately even when the background harmonics fluctuate and the number of suspicious harmonic sources is large. Further, the relative evaluation processes no longer rely on expensive GPS. By applying the proposed method, the cost of the harmonic contribution evaluation will greatly decrease. Meanwhile, the accurate evaluation results can be the basis of designing the punishment mechanism for nonlinear customers and identifying the dominated harmonic sources via the proposed method is significant to guide harmonic mitigation.
In future works, we will further apply the evaluation results to release the harmonic pollutions in the power grid.