Distribution Network Mixed-harmonic Separation Method Based on Characteristic Vector

The threat of harmonic pollution has become increasingly serious with the rapid increase in the scale and complexity of distribution networks. However, the harmonic source variety of distribution networks and the phenomena about harmonic characteristics become vague after harmonic mixture because harmonic governance and responsibility division cause much inconvenience. To address this point, a distribution network mixed-harmonic separation method based on characteristic vector is developed in this study. This method realizes the division of the current working state, harmonic emission levels, and harmonic governance responsibility of each harmonic source by equivalent analysis and reverse thinking. The results are verified through an experiment.


Introduction
Harmonic pollution is one of the most common power quality problems of power grids.Having a large amount of harmonics in a power grid negatively affects line facilities, electrical equipment, the power grid communication system, and the monitoring instrument; it also influences the safety or reliability of power grid facilities [1] .The harmonics of a power grid must be managed.Harmonic detection and analysis are the prerequisites to realizing harmonic governance, and only accurate harmonic detection can provide a good basis for harmonic governance [2] .
However, the scale and complexity of distribution networks have significantly improved with the continuous development of power systems and progress of the social economy.The gravity and complexity of distribution network harmonic pollution present an increasing trend with the parallel operation of distributed generation and new generation.Such a trend is mainly manifested in the technical and management levels [3][4] .In the technical level, a mixed harmonic is produced by mixing multiple harmonic sources, and their harmonic characteristics become blurred compared with those of the harmonic sources.This condition reduces the governance effect of a passive filter and other harmonic suppression measures to a certain extent.In the management level, the mixed harmonic governance responsibilities of the harmonic source become difficult under the influence of the harmonic source; this condition is not conducive for a power grid to control harmonic pollution effectively [5] .
To address these problems, this study analyzes the harmonic mixing process from the perspective of reverse thinking.The results of harmonic analysis are regarded as the harmonic characteristic parameters of each harmonic amplitude and phase angle.Harmonic amplitude and phase angle are employed to evaluate harmonic features and used as the basis for harmonic governance.To separate mixed harmonics, the harmonic mixing process is made equivalent to linear space vector synthesis [6] .
The harmonic characteristic vectors extracted from harmonic characteristic parameters are established, and the harmonic characteristic vector is utilized to realize vector decomposition.Based on the characteristic vector of mixed-harmonic separation method, real-time monitoring and calculation of harmonics, harmonic source workload rate current injection, and the mixing ratio of harmonic distribution networks are achieved [7][8] .

Forward analysis of harmonic mixing processing
The national standard recommendation algorithm (GB/T 14549-1993) assumes that harmonics of the same sequence will stack with each other when these two harmonics are mixed.The amplitudes before and after superposition meet the following relationship.( 2 cos ) If the harmonic phase angle is unknown or varies randomly, we can calculate it according to the following equation.( ) 2) The value of h k is presented in Table 1.The national standard recommendation algorithm mainly aims to calculate the superposition of two harmonic sources and estimates the harmonic phase angle in a random change of situation.The phase angle is not included in the harmonic parameters after the superposition of two harmonic sources.Therefore, only an approximate calculation of h k can be utilized to continue to calculate.A small difference in the harmonic phase angle of electric power causes a certain error.
The International Electrotechnical Commission (IEC) has recommended a calculation method of harmonic mixing.Each harmonic amplitude obtained after multi-source harmonic mixing is expressed as follows: 1 ( ) Where is the superposition principle index.The selection principle is indicated in Table 2. 2.0 2.0 2.0 In a distribution network, when the differences in the public joint phase angle of multiple harmonic sources are small, the amplitude can be calculated according to the manner of superposition, namely 1   , by setting 1   .In consideration of the characteristics of each harmonic source in power network harmonic and the load characteristic of power lines, each harmonic amplitude under normal circumstances can be simply calculated according to such a situation.

Formulation of harmonic characteristic vector
Considering the linear relationship between mixed harmonics and each harmonic source during harmonic mixing calculation based on superposition and owing to the different mechanisms of harmonics and the harmonic transmission channel among various harmonic sources, harmonics vary and are linearly independent of one another.Therefore, this characteristic can be used as a basis to determine the similarity between the harmonic mixing process and the linear space vector synthesis process and implement an equivalent analysis.Consequently, the harmonic amplitude sequence of each harmonic source is equivalent to the base 1 2 , , , m A A A  of vector space V , and the mixed harmonic amplitude sequence is equivalent to vector  in vector spaceV .The problem of separating mixed harmonics is then equivalently converted to solving the representing coefficient of vector  in vector spaceV .
The equivalent diagram between the harmonic separation process and vector decomposition is shown in Figure .1.In Figure 1, the horizontal two-way arrows represent the corresponding relationship between the arrows pointing to the content on both ends.In the diagram, the base and synthetic vectors of the linear space are known conditions.The vectors of basal projections of the synthetic vector are reversely derived in the directions by vector decomposition.The vectors are regarded as the solution vectors.The mixed vector coordinates and the module values of solution vectors are solved, and the relations among the module value sizes of solution vectors are analyzed.Accordingly, the calculation of such parameters as the work load rate, harmonic injection, and harmonic contribution of various harmonic sources and the calculation of mixed-harmonic separation in a distribution network can also be equivalent to gathering the harmonic characteristic vectors of the harmonic source and the distribution network's mixed harmonic in the aspect of harmonic analysis and separation [9][10][11] .
In view of the harmonic analysis results, the harmonic sequence number can be arranged from small to large, and a unified vector form called the harmonic characteristic vector can be obtained.The transformation between the algebraic problem of harmonic separation and the geometric problem of the calculation of a particular base coefficient in vector space is realized [12][13] .

Establishment and solution of the mixed-harmonic separation equation
The mixed-harmonic separation equation is based on general methods of vector decomposition., , , m A A A  .The amplitude matrixes of all harmonic sources are written as follows: The number 1 m workload rates of m different harmonic sources, namely, the coefficient vector of the amplitude matrixes of the harmonic sources, are written as follows: The amplitude characteristic vector of mixed harmonics with Y is expressed, and the harmonic positive superposition process can be written as follows: Hence, the mixed-harmonic separation problem is converted into a problem about reverse analysis of the harmonic positive superposition process, i.e., vector decomposition calculation in the linear space.Each row vector of matrix A can be regarded as a base in the linear space, and vector Y can be regarded as a synthetic vector formed by a series of bases according to a certain proportion.The mixed-harmonic separation equation shown below provides the corresponding synthesis coefficient of each base.
In the physical sense, the solution of this equation means that the harmonic levels of the harmonic source in the current working state are injected into the distribution network, and the ratio of the harmonic levels with their standard working state is injected into the distribution network, namely, the workload rate of various harmonic sources at present.On the basis of the workload rate, the injected harmonic level of each harmonic source of the maximum working state can be acquired according to the relevant data in the statistical analysis.The level is used to estimate injection i K of each current harmonic source in the system mixed harmonic, i.e.
  Where represent the harmonic current contents of various harmonic sources in a standard working condition.
The harmonic contribution rate c K of the various harmonic sources of the distribution network mixed harmonic is obtained after normalizing the injection.
Therefore, the separation calculation of the distribution network mix ed harmonic can be completed, and dynamic monitoring of each harmonic source in the current worki ng state, the harmonic emission status, and parameters (e.g., total harmonic contribution rate of the cur rent distribution network) is realized.

Distribution network mixed-harmonic separation experiment
According to the actual measurement data, the main harmonic sources of a distribution network in a given area include converter harmonics, industrial harmonics, and town residents' live load harmonics.Industrial harmonics include electric furnace steel harmonics in the form of AC arc furnace and singlecrystal silicon growth furnace, which is mainly com Table 3. Distribution characteristic parameters of each harmonic wave source in an area posed of DC electric arc furnace.The harmonic superposition of harmonic sources occurs in the public join point.According to related data records, the harmonic distribution characteristics of all the harmonic sources in a full-load working condition can be identified, as shown in Table 3.According to the amplitude and phase angle of each harmonic, we set the element parameters of a simulated harmonic source and performed fast Fourier transform (FFT) harmonic analysis based on windowed interpolation to calculate the harmonic characteristic vector of each harmonic.The characteristic vector of residents' live load harmonic is 1 [60.1, 0.2,9.6,0.1, 7.7, 0.2, 4.2, 0.1, 2.1, 0, 1.5, 0, 0.8, 0.3, 0, 0.1] A  (10)   The characteristic vector of converter harmonic is 2 [41.9, 0, 2.8, 0, 4.1, 0,3.9, 0,1.8, 0, 2.5, 0,1.6, 0, 0.2, 0, 0.1] A  (11)   The characteristic vector of electrical arc furnace steel harmonic is 3 [115.4,4,9.9, 22.3, 7.1,10.6,3.5,5.1, 0.9, 2 .1,0.5,1.0,0.2, 0.6, 0, 0.2, 0, 0.2] A  (12)   The characteristic vector of single-crystal silicon growth furnace harmonic is 4 [72.5, 0, 6.8, 0, 4.9, 0,3.9, 0,1.7, 0,16.4,0,1 4.4, 0, 0.3, 0, 0.2] A  (13)  We obtained harmonic source amplitude matrix Aas follows: 60.1 0.2 9.6 0.1 7.7 0.2 4.2 0.1 2.1 0 1.5 0 0.8 0 0.3 0 0.1 41.9 0 2.8 0 4.1 0 3.9 0 1.8 0 2.5 0 1.6 0 0.2 0 0.1 115.4 9.9 22.3 7.1 10.6 3.5 5.1 0.9 2.1 0.5 1.0 0.2 0.6 0 0.2 0 0.2 72.5 0 6.8 0 4.9 0 3.9 0 1.7 0 16.4 0 14.4 0 0.3 0 0.2 We acquired the calculation results after using the monitoring device to measure mixed-harmonic wave, introduced the results into a Matlab environment through data interface, and performed distribution network harmonic separation calculation.By comparing the differences between the calculation results and the recording data, the error rate of the calculations result of the distribution network mixed-harmonic separation method is shown in Figure 2. Error rate of the calculation result by the separation method of hybrid harmonic wave Comparison of the data reveals that the results of workload rate, harmonic injection, and harmonic contribution rate calculated by the distribution network mixed-harmonic separation calculation method based on characteristic vector are close to actual data.The calculation error rates of workload rate, harmonic injection, and harmonic contribution rate are within 2%.Considering the recording data, only two significant figures are retained, and the accuracy is low.Certain differences may exist between the actual and analyzed error rates, but they do not influence the calculation accuracy of the mixed-harmonic separation method.The experimental result proves that the distribution network mixed-harmonic separation method based on characteristic vector developed in this study has high accuracy and reliability.

Conclusions
The distribution network mixed-harmonic separation method based on characteristic vector developed in this study identifies the similarity between harmonic separation and vector decomposition problems by forward analyzing the harmonic mixing process in a distribution network.The harmonic was analyzed with FFT harmonic analysis method based on windowed interpolation.The harmonic characteristic vector was extracted.The mixed-harmonic separation equation and complete distribution network harmonic separation were solved from the analysis results.The work conducted in this investigation can be summarized as follows.
1) The proposed harmonic characteristic vector can effectively express the amplitude and phase angle of harmonics and provides a good characterization of the original harmonic.
2) Compared with the reserved standard working condition, the distribution network mixedharmonic separation method based on characteristic vector can solve the mixed-harmonic separation equation and provide subsequent calculations.The current working conditions of each harmonic were

Figure 1 .
Figure 1.Equivalent diagram between vector decomposition in linear space and harmonic separation.In Figure1, the horizontal two-way arrows represent the corresponding relationship between the arrows pointing to the content on both ends.In the diagram, the base and synthetic vectors of the linear space are known conditions.The vectors of basal projections of the synthetic vector are reversely derived in the directions by vector decomposition.The vectors are regarded as the solution vectors.The mixed vector coordinates and the module values of solution vectors are solved, and the relations among the module value sizes of solution vectors are analyzed.Accordingly, the calculation of such


amplitude sequences of m different harmonic sources.The characteristic vectors of these m different harmonic sources are then formed on the basis of 1 2

Figure 2 .
Figure 2.Error rate of the calculation result by the separation method of hybrid harmonic wave Comparison of the data reveals that the results of workload rate, harmonic injection, and harmonic contribution rate calculated by the distribution network mixed-harmonic separation calculation method based on characteristic vector are close to actual data.The calculation error rates of workload rate, harmonic injection, and harmonic contribution rate are within 2%.Considering the recording data, only two significant figures are retained, and the accuracy is low.Certain differences may exist between the actual and analyzed error rates, but they do not influence the calculation accuracy of the mixed-harmonic separation method.The experimental result proves that the distribution network mixed-harmonic separation method based on characteristic vector developed in this study has high accuracy and reliability.

Table 1 .
Evaluation reference table of h

Table 2 .
Evaluation reference table of  recommended by IEC

Table 3 .
Distribution characteristic parameters of each harmonic wave source in an area

Table 4 .
The results are shown below.The local electric power is according to experience and the data saved in the data recording device to verify the calculation results.The results are shown in the following table.Working load and contribution rates of each harmonic wave source in an area