Abstract
The performance of electrical energy meters in non-sinusoidal conditions has been discussed since the early twentieth century and as of yet has not reached a fully comprehensive standardization. Within this context, this paper aims to update the present understanding on the subject through a closer look at the power definitions established by the IEEE Std. 1459-2010. The paper concentrates its goals across two different approaches. The first deals with the analytical development in the time domain, aiming at the decomposition of the instantaneous power in its different elementary components. The second, in turn, deals with the development of several calibration tests in different active electrical energy meters considering different voltage and current waveforms. The results show that the measurement deviations in non-sinusoidal conditions may be greater than 30% in some practical cases, which reinforces the need for more specific standards concerning the subject.
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Acknowledgements
The authors thank the Energisa Group for the financial support as part of the Program for Research and Technological Development of the Electricity Energy Sector, established by the Brazilian Electricity Regulatory Agency (ANEEL). Additionally, this study was financed in part by the Coordination for the Improvement of Higher Education Personnel—Brazil (CAPES)—Finance Code 001 and by the Brazilian National Council for Scientific and Technological Development (CNPq).
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Appendices
Appendix A: Data associated with the tests performed
Test number | Harmonic order | rms Voltage (V) | Angle (°) | rms Current (Amp) | Angle (°) |
|---|---|---|---|---|---|
#1 | 1 | 120.0 | 0.0 | 10.0 | 0.0 |
#2 | 1 | 120.0 | 0.0 | 10.0 | 0.0 |
5 | 12.0 | 0.0 | – | – | |
#3 | 1 | 120.0 | 0.0 | 10.0 | 0.0 |
5 | – | – | 5.0 | 0.0 | |
#4 | 1 | 120.0 | 0.0 | 10.0 | 0.0 |
5 | 10.0 | 0.0 | – | – | |
7 | – | – | 5.0 | 0.0 | |
#5 | 1 | 120.0 | 0.0 | 10.0 | 0.0 |
5 | 12.0 | 0.0 | 5.0 | 0.0 | |
#6 | 1 | 120.0 | 0.0 | 10.0 | 0.0 |
5 | 12.0 | 0.0 | 5.0 | 180.0 | |
#7 | 1 | 120.0 | 0.0 | 10.0 | 0.0 |
7 | 12.0 | 0.0 | 5.0 | 0.0 | |
#8 | 1 | 120.0 | 0.0 | 10.0 | 0.0 |
7 | 12.0 | 0.0 | 5.0 | 180.0 | |
#9 | 1 | 120.0 | 0.0 | 10.0 | -32.0 |
3 | 12.0 | 10.0 | 1.0 | 145.0 | |
5 | 24.0 | 45.0 | 2.0 | 170.0 | |
#10 | 1 | 120.0 | 0.0 | 10.0 | 329.4 |
3 | 1.380 | 166.8 | 8.416 | 292.8 | |
5 | 4.656 | 217.8 | 6.995 | 29.4 | |
7 | 0.780 | 272.3 | 5.313 | 36.5 | |
9 | 0.648 | 235.5 | 3.694 | 92.0 | |
11 | 0.804 | 50.0 | 2.618 | 141.1 | |
13 | 0.573 | 66.8 | 2.566 | 199.1 | |
15 | 0.030 | 325.8 | 0.792 | 310.7 | |
17 | 0.134 | 217.1 | 2.587 | 315.8 | |
19 | 0.076 | 189.9 | 1.031 | 63.9 | |
21 | 0.031 | 11.8 | 2.249 | 71.6 | |
23 | 0.028 | 241.3 | 0.726 | 147.9 | |
25 | 0.045 | 178.1 | 1.468 | 185.2 | |
27 | 0.003 | 180.3 | 0.864 | 260.6 |
Appendix B: Numerical results of the tests performed
Test | Meter | Reference | PMeas (W) | Error (%) | Test | Meter | Reference | PMeas (W) | Error (%) | ||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
P1 (W) | P (W) | Pref = P1 | Pref = P | P1 (W) | P (W) | Pref = P1 | Pref = P | ||||||
#1 | A.1 | 1200.0 | 1200.0 | 1209.89 | 0.82 | 0.82 | #6 | A.1 | 1200.0 | 1140.0 | 1171.87 | − 2.34 | 2.80 |
A.2 | 1200.0 | 1200.0 | 1187.71 | − 1.02 | − 1.02 | A.2 | 1200.0 | 1140.0 | 1168.07 | − 2.66 | 2.46 | ||
A.3 | 1200.0 | 1200.0 | 1198.15 | − 0.15 | − 0.15 | A.3 | 1200.0 | 1140.0 | 1142.39 | − 4.80 | 0.21 | ||
B.1 | 1200.0 | 1200.0 | 1199.16 | − 0.07 | − 0.07 | B.1 | 1200.0 | 1140.0 | 1138.48 | − 5.13 | − 0.13 | ||
B.2 | 1200.0 | 1200.0 | 1199.38 | − 0.05 | − 0.05 | B.2 | 1200.0 | 1140.0 | 1140.48 | − 4.96 | 0.04 | ||
B.3 | 1200.0 | 1200.0 | 1197.67 | − 0.19 | − 0.19 | B.3 | 1200.0 | 1140.0 | 1137.32 | − 5.22 | − 0.24 | ||
C.1 | 1200.0 | 1200.0 | 1199.07 | − 0.08 | − 0.08 | C.1 | 1200.0 | 1140.0 | 1151.18 | − 4.07 | 0.98 | ||
D.1 | 1200.0 | 1200.0 | 1211.58 | 0.96 | 0.96 | D.1 | 1200.0 | 1140.0 | 1150.65 | − 4.11 | 0.93 | ||
E.1 | 1200.0 | 1200.0 | 1192.62 | − 0.62 | − 0.62 | E.1 | 1200.0 | 1140.0 | 1145.25 | − 4.56 | 0.46 | ||
#2 | A.1 | 1200.0 | 1200.0 | 1190.40 | 0.80 | 0.80 | #7 | A.1 | 1200.0 | 1260.0 | 1238.46 | 3.21 | − 1.71 |
A.2 | 1200.0 | 1200.0 | 1210.82 | − 0.90 | − 0.90 | A.2 | 1200.0 | 1260.0 | 1222.26 | 1.86 | − 3.00 | ||
A.3 | 1200.0 | 1200.0 | 1202.40 | − 0.20 | − 0.20 | A.3 | 1200.0 | 1260.0 | 1253.79 | 4.48 | − 0.49 | ||
B.1 | 1200.0 | 1200.0 | 1201.08 | − 0.09 | − 0.09 | B.1 | 1200.0 | 1260.0 | 1256.93 | 4.74 | − 0.24 | ||
B.2 | 1200.0 | 1200.0 | 1200.62 | − 0.05 | − 0.05 | B.2 | 1200.0 | 1260.0 | 1257.64 | 4.80 | − 0.19% | ||
B.3 | 1200.0 | 1200.0 | 1202.52 | − 0.21 | − 0.21 | B.3 | 1200.0 | 1260.0 | 1256.08 | 4.67 | − 0.31 | ||
C.1 | 1200.0 | 1200.0 | 1200.93 | − 0.08 | − 0.08 | C.1 | 1200.0 | 1260.0 | 1231.68 | 2.64 | − 2.25 | ||
D.1 | 1200.0 | 1200.0 | 1189.32 | 0.89 | 0.89 | D.1 | 1200.0 | 1260.0 | 1261.63 | 5.14 | 0.13 | ||
E.1 | 1200.0 | 1200.0 | 1207.08 | − 0.59 | − 0.59 | E.1 | 1200.0 | 1260.0 | 1248.96 | 4.08 | − 0.88 | ||
#3 | A.1 | 1200.0 | 1200.0 | 1189.08 | 0.91 | 0.91 | #8 | A.1 | 1200.0 | 1140.0 | 1184.55 | − 1.29 | 3.91 |
A.2 | 1200.0 | 1200.0 | 1213.20 | − 1.10 | − 1.10 | A.2 | 1200.0 | 1140.0 | 1177.63 | − 1.86 | 3.30 | ||
A.3 | 1200.0 | 1200.0 | 1202.40 | − 0.20 | − 0.20 | A.3 | 1200.0 | 1140.0 | 1145.23 | − 4.56 | 0.46 | ||
B.1 | 1200.0 | 1200.0 | 1200.84 | − 0.07 | − 0.07 | B.1 | 1200.0 | 1140.0 | 1138.72 | − 5.11 | − 0.11 | ||
B.2 | 1200.0 | 1200.0 | 1200.84 | − 0.07 | − 0.07 | B.2 | 1200.0 | 1140.0 | 1141.41 | − 4.88 | 0.12 | ||
B.3 | 1200.0 | 1200.0 | 1201.92 | − 0.16 | − 0.16 | B.3 | 1200.0 | 1140.0 | 1138.94 | − 5.09 | − 0.09 | ||
C.1 | 1200.0 | 1200.0 | 1200.93 | − 0.08 | − 0.08 | C.1 | 1200.0 | 1140.0 | 1167.48 | − 2.71 | 2.41 | ||
D.1 | 1200.0 | 1200.0 | 1190.40 | 0.80 | 0.80 | D.1 | 1200.0 | 1140.0 | 1153.93 | − 3.84 | 1.22 | ||
E.1 | 1200.0 | 1200.0 | 1207.80 | − 0.65 | − 0.65 | E.1 | 1200.0 | 1140.0 | 1143.54 | − 4.71 | 0.31 | ||
#4 | A.1 | 1200.0 | 1200.0 | 1190.76 | 0.77 | 0.77 | #9 | A.1 | 1017.7 | 981.6 | 996.23 | − 2.11 | 1.49 |
A.2 | 1200.0 | 1200.0 | 1213.20 | − 1.10 | − 1.10 | A.2 | 1017.7 | 981.6 | 983.46 | − 3.36 | 0.19 | ||
A.3 | 1200.0 | 1200.0 | 1202.40 | − 0.20 | − 0.20 | A.3 | 1017.7 | 981.6 | 981.77 | − 3.53 | 0.01 | ||
B.1 | 1200.0 | 1200.0 | 1200.84 | − 0.07 | − 0.07 | B.1 | 1017.7 | 981.6 | 981.49 | − 3.55 | − 0.02 | ||
B.2 | 1200.0 | 1200.0 | 1200.60 | − 0.05 | − 0.05 | B.2 | 1017.7 | 981.6 | 981.71 | − 3.53 | 0.01 | ||
B.3 | 1200.0 | 1200.0 | 1202.64 | − 0.22 | − 0.22 | B.3 | 1017.7 | 981.6 | 980.18 | − 3.68 | − 0.15 | ||
C.1 | 1200.0 | 1200.0 | 1200.93 | − 0.08 | − 0.08 | C.1 | 1017.7 | 981.6 | 989.13 | − 2.80 | 0.76 | ||
D.1 | 1200.0 | 1200.0 | 1187.88 | 1.01 | 1.01 | D.1 | 1017.7 | 981.6 | 995.91 | − 2.14 | 1.45 | ||
E.1 | 1200.0 | 1200.0 | 1206.84 | − 0.57 | − 0.57 | E.1 | 1017.7 | 981.6 | 991.25 | − 2.60 | 0.98 | ||
#5 | A.1 | 1200.0 | 1260.0 | 1249.02 | 4.09 | − 0.87 | #10 | A.1 | 988.9 | 1033.2 | 1052.32 | 6.41 | 1.85 |
A.2 | 1200.0 | 1260.0 | 1229.47 | 2.46 | − 2.42 | A.2 | 988.9 | 1033.2 | 1051.58 | 6.33 | 1.78 | ||
A.3 | 1200.0 | 1260.0 | 1256.36 | 4.70 | − 0.29 | A.3 | 988.9 | 1033.2 | 1070.72 | 8.27 | 3.63 | ||
B.1 | 1200.0 | 1260.0 | 1257.32 | 4.78 | − 0.21 | B.1 | 988.9 | 1033.2 | 1071.82 | 8.38 | 3.73 | ||
B.2 | 1200.0 | 1260.0 | 1259.48 | 4.96 | − 0.04 | B.2 | 988.9 | 1033.2 | 1075.40 | 8.74 | 4.08 | ||
B.3 | 1200.0 | 1260.0 | 1256.57 | 4.71 | − 0.27 | B.3 | 988.9 | 1033.2 | 1285.68 | 30.01 | 24.43 | ||
C.1 | 1200.0 | 1260.0 | 1247.35 | 3.95 | − 1.00 | C.1 | 988.9 | 1033.2 | 1065.48 | 7.74 | 3.12 | ||
D.1 | 1200.0 | 1260.0 | 1264.34 | 5.36 | 0.34 | D.1 | 988.9 | 1033.2 | 1082.78 | 9.49 | 4.80 | ||
E.1 | 1200.0 | 1260.0 | 1230.64 | 2.55 | − 2.33 | E.1 | 988.9 | 1033.2 | 1071.25 | 8.32 | 3.68 | ||
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Macedo, J.R., Xavier, G.L., Gondin, I.N. et al. An update on the performance of active energy meters under non-sinusoidal conditions. Electr Eng 102, 1785–1794 (2020). https://doi.org/10.1007/s00202-020-00991-y
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DOI: https://doi.org/10.1007/s00202-020-00991-y