Correlation between harmonic magnitudes and phase angles

The possible correlations between harmonic magnitudes and phase angles should be identified in order to include the relationship between variables in the model. Moreover, the correlations between variables may lead to the simplification of the model, in which only some variables have to be modeled in detail, and the others are calculated based on the correlations.

Three different correlations are analyzed:

• Relation between magnitudes and phase angles of the same harmonic order.

• Relation between magnitudes of different harmonic orders.

• Relation between phase angles of different harmonic orders.

The Pearson correlation coefficient SMP(see definition in appendix B.4) is used to identify the correlation between the different variables in a systematic way. In this case, a good correlation between variables is accepted if |SMP| > 0.7. Lower values of |SMP| indicate that between

4.5. Correlation between harmonic magnitudes and phase angles

variables there is a non-linear relationship or that there is no relationship at all. The analysis is complemented using scatter plots, which allow a easy verification of conclusions drawn with the |SMP| values and the identification of the possible non-linear relationships between variables.

The correlation analysis is applied to the data of one complete week, and the data of only one workday during two different periods (1-3 a.m. and 7-9 p.m.). The data of one day during different time periods is used in order to verify that the correlation is independent of the time, and to reduce the possible influence of other factors, like the voltage distortion. Moreover, the analysis is applied to the phase currents and the corresponding symmetrical components.

Fig. 4.15 shows the percentage of sites with good correlation between magnitudes and phase angles of the same harmonic, for the data of line conductor A (I_A^(h)), the balanced component (I_b^(h)) and the first unbalanced component (I_u1^(h)). The results obtained for the line conductors B and C are similar to the results of line conductor A, while the results of the second unbalanced component are similar to the results of the first unbalanced component. The amount of sites with good correlations do not exceed 40% in any of the considered cases. The correlation seems to improve for the balanced component, but still most of the sites do not show a clear linear relationship between magnitudes and phase angles of the same harmonic order.

0%

10%

20%

30%

40%

50%

1 3 5 7 9 11 13 15 1 3 5 7 9 11 13 15 1 3 5 7 9 11 13 15

All data

Workday, 1-3 a.m.

Workday, 7-9 p.m.

Harmonic order (h)

Cases with good correlation

( ) A

I h I_b^{( )h} I_u1^{( )h}

Figure 4.15.: Amount of sites with good correlation between magnitudes and phase angles

Scatter plots confirm the results obtained with the |SMP| values. Fig. 4.16 shows exemplary the scatter plots of the magnitudes and phase angles of the third harmonic of three different sites (line conductor A). There are some sites with clear correlation between variables, like site C in the figure, but in most of the cases the correlation is not clear. In general, the relation between variables is different for each residential site, and no unique relation between variables for all residential networks can be easily defined.

The relation between magnitudes of the same harmonic order, and between phase angles of the same harmonic order were analyzed in the same way, but no clear relation between magni-tudes or phase angles could be identified, except for the relation between the first and second unbalanced component of the fundamental currents, which show a clear linear relationship for most than 70% of the sites.

It is possible that the relation between variables is linked to the electrical characteristics of the network, the voltage distortion and/or the impedance characteristic; therefore, the relation be-tween variables for each site is different. Detailed and controlled measurements in different networks should be perform in order to verify the correlations between variables. In this the-sis, each variable will be treated independently and the correlation between variables will be neglected.

49

4. Characteristics of harmonic currents in residential low-voltage networks

160 180 200 220 240

5 10 15 20

Site A

5 10 15 20 5 10 15 20

Site B Site C

(3) Ain°ϕ

(3)

A in A

I I_A⁽³⁾ in A I_A⁽³⁾ in A

P 0.43

SM = − SMP=0.27 SM_P= −0.77

Figure 4.16.: Relation between magnitude and phase angle of the third harmonic of three resi-dential sites

4.6. Chapter summary

Measurements of 37 residential low-voltage networks are available to develop a harmonic emission model of aggregate customers using a measurement-based approach. The measure-ments were made during the winter time to reduce the influence of seasonal variations. More-over, only networks with mainly residential customers and with none or few photovoltaic sys-tems were selected, in order to analyze and model the real harmonic current emission of pure residential customers. According to the initial characterization, the harmonic currents of ag-gregate residential customers have the following characteristics:

• The harmonic current magnitudes depend on the type and number of customers con-nected to the network. Harmonic current magnitudes increase with the number of cus-tomers. Moreover, networks with mainly single-family houses have higher harmonic current magnitudes than networks with apartments.

• The harmonic phase angles do not vary randomly in the complex plane, but they are con-centrated in a prevailing direction for most harmonic orders. Most of the residential sites show a similar direction of harmonic phase angles, especially for the fundamental, third, fifth, seventh, and ninth harmonic orders. For higher harmonic orders, the variation of phase angles is higher for each site, and there are more differences between sites. There is no clear relation between the type and number of customers with the harmonic phase angles.

• Harmonic current magnitudes show a daily pattern for most harmonic orders, which is linked to the daily activities of residential customers. There are also differences between workdays and weekends, which results in variations of the daily patterns. Harmonic phase angles do not show a clear daily pattern.

• The unbalance of the fundamental and the harmonic currents is significantly high, usu-ally higher than 10%. The unbalance increases with the harmonic order.

• Magnitudes and phase angles do not show a clear correlation. Magnitudes and phase angles may be treated as independent variables.

Based on the characteristics of the harmonic currents, a modeling methodology can be defined, which is explained in the next chapter.