This paper presents a practical and effective novel approach to curve fit electromechanical (EM) overcurrent (OC) relay characteristics. Based on singular value decomposition (SVD), the curves are fitted with equation in state space under modal coordinates. The relationships between transfer function and Markov parameters are adopted in this research to represent the characteristic curves of EM OC relays. This study applies the proposed method to two EM OC relays: the GE IAC51 relay with moderately inverse-time characteristic and the ABB CO-8 relay with inverse-time characteristic. The maximum absolute values of errors of hundreds of sample points taken from four time dial settings (TDS) for each relay between the actual characteristic curves and the corresponding values from the curve-fitting equations are within the range of 10 milliseconds. Finally, this study compares the SVD with the adaptive network and fuzzy inference system (ANFIS) to demonstrate its accuracy and identification robustness.
Power generation systems generally have few large generators connected directly to their subtransmission networks and distribution networks. Thus, the fault currents of buses do not differ much from those of transmission lines. This makes low-cost, reliable, and easily coordinated electromechanical (EM) overcurrent (OC) relays suitable for protection coordination relay in the subtransmission networks and distribution networks. Although some older relays have been replaced by new digital ones, there are still many EM OC relays in service.
The operation principle of the EM OC relay is to introduce an electric current into the coil of an electromagnet to produce eddy currents with phase differences. This in turn generates induction torque on the rotation disc of the relay. The proper contact closing time can be set by adjusting the distance between the fixed and the movable contacts, achieving protection coordination between upstream and downstream. However, due to the mechanical nature of the relay, there are inertial and frictional effects. Therefore, unlike a digital relay [
Accurate representations of the inverse-time EM OC relay characteristics play an important role in the coordination of power network protection schemes. In the early days, researchers were interested in fitting EM OC relay characteristics curves [
This paper applies the Hankel matrices and the singular value decomposition (SVD) [
The paper proposed a new application algorithm to fit the characteristic curves of the EM OC relays, using one unified equation to represent their characteristics. Finally, this study uses the SVD method to fit the characteristic curves of EM OC relays. Comparing the results with those obtained by [
The content of this paper is as follows: Section
The characteristic of an EM OC relay is determined by its magnetic circuit design, and the manufacturers provide the characteristics in the relay manuals with curves in a two-dimensional plot with
Various exponential and polynomial forms of equations are summarized and recommended by the IEEE Committee [
A customized characteristic Equation (
Take the characteristic curves of the ABB’s EM OC relay CO-8 as an example [
The actual characteristic curves and the curves fitted by (
Consider the case [
The actual characteristic curves and the modified curves fitted by (
The values of
Researchers are applying more artificial neural network and fuzzy model techniques [
Based on the concept of transfer function, this paper proposes an algorithm to represent the characteristic curves of EM OC relays to calculate Markov parameters [
The SVD method is as follows [
Find the estimated operating time
Use
Apply SVD to the Hankel matrix
Determine the proper dimension
Calculate the matrices
Transform the state space Equation (
Obtain the unit impulse response sequence from (
Derive the equation of the fitted relay characteristic curve by transforming back to continuous
Equation (
The characteristic curves of an EM OC relay can be represented by a digital state space model, and the desired bound of the maximum absolute value of errors may be established by selecting an appropriate system model order
The following case study involves two EM OC relays: the GE moderately inverse-time relay IAC51 [
Four characteristic curves corresponding to TDS 1, 4, 7, and 10 were selected for curve fitting and 486 sample points of relay operating time are taken for
The 26 parameters in (
TDS | 1 | 4 | 7 | 10 |
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4 | 9 | 12 | 13 |
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TDS: time dial setting;
The number of the smooth waveform components
Table
The curve-fitting errors of 17 actual operating times for each TDS of the IAC51 relay in milliseconds.
TDS | 1 | 4 | 7 | 10 | ||||||||
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Top | SVD | Err | Top | SVD | Err | Top | SVD | Err | Top | SVD | Err |
| ||||||||||||
1.5 | 1115.0 | 1112.6 | 2.34 | 4738.9 | 4738.9 | 0.02 | 9067.1 | 9067.1 | 0.01 | 14322.0 | 14322.0 | 0.16 |
2 | 753.7 | 756.8 | 3.11 | 3009.2 | 3011.1 | 1.87 | 5700.4 | 5702.3 | 1.89 | 8960.3 | 8964.3 | 3.99 |
3 | 530.5 | 529.2 | 1.31 | 2009.6 | 2008.3 | 1.30 | 3772.1 | 3770.0 | 2.05 | 5900.0 | 5896.7 | 3.25 |
4 | 442.2 | 443.4 | 1.15 | 1616.0 | 1616.3 | 0.32 | 3013.9 | 3014.0 | 0.10 | 4698.1 | 4697.7 | 0.49 |
6 | 362.4 | 361.8 | 0.61 | 1266.8 | 1266.2 | 0.59 | 2342.1 | 2340.9 | 1.22 | 3632.3 | 3631.1 | 1.21 |
8 | 323.3 | 322.4 | 0.82 | 1094.7 | 1094.4 | 0.34 | 2010.9 | 2010.4 | 0.48 | 3107.9 | 3106.7 | 1.18 |
10 | 298.2 | 298.6 | 0.47 | 985.7 | 986.2 | 0.48 | 1801.4 | 1802.0 | 0.66 | 2776.2 | 2778.1 | 1.92 |
13 | 275.0 | 275.2 | 0.22 | 881.8 | 881.9 | 0.12 | 1601.5 | 1601.3 | 0.29 | 2457.8 | 2455.8 | 2.04 |
16 | 259.3 | 259.3 | 0.04 | 812.5 | 812.8 | 0.34 | 1468.8 | 1469.2 | 0.42 | 2249.4 | 2250.6 | 1.25 |
19 | 247.4 | 247.6 | 0.20 | 763.2 | 763.1 | 0.04 | 1373.9 | 1373.6 | 0.28 | 2099.8 | 2099.0 | 0.84 |
22 | 239.5 | 238.9 | 0.66 | 726.4 | 726.2 | 0.18 | 1302.1 | 1302.0 | 0.12 | 1985.9 | 1984.6 | 1.29 |
26 | 230.6 | 230.1 | 0.49 | 686.3 | 686.7 | 0.46 | 1224.8 | 1226.1 | 1.36 | 1864.8 | 1865.8 | 1.06 |
30 | 223.7 | 223.4 | 0.21 | 655.8 | 655.8 | 0.08 | 1166.7 | 1166.5 | 0.18 | 1772.3 | 1773.0 | 0.66 |
34 | 217.4 | 218.1 | 0.66 | 630.5 | 630.8 | 0.29 | 1118.8 | 1119.1 | 0.24 | 1695.6 | 1696.3 | 0.67 |
39 | 211.6 | 212.4 | 0.88 | 604.8 | 604.7 | 0.11 | 1070.0 | 1069.4 | 0.66 | 1619.2 | 1619.0 | 0.15 |
44 | 207.1 | 207.5 | 0.41 | 583.8 | 583.9 | 0.08 | 1029.2 | 1029.4 | 0.20 | 1554.7 | 1553.9 | 0.88 |
49 | 202.9 | 203.0 | 0.09 | 566.3 | 566.1 | 0.26 | 994.7 | 994.3 | 0.38 | 1500.4 | 1499.8 | 0.64 |
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TDS: time dial setting;
Table
Summary of the SVD results for the IAC51 relay.
TDS | Max_Err/ |
Max_Err%/ |
AV | AV% |
---|---|---|---|---|
1 | 4.05/1.8 | 0.48/41.4 | 0.53 | 0.19 |
4 | 3.62/2.2 | 0.16/3.9 | 0.28 | 0.03 |
7 | 5.08/2.8 | 0.15/3.9 | 0.50 | 0.03 |
10 | 8.73/2.2 | 0.20/12.1 | 0.89 | 0.04 |
TDS: time dial setting; Max_Err/
Finally, Figure
The actual characteristic curves and the curves fitted by (
Four characteristic curves corresponding to TDS 0.5, 3, 6, and 9 were selected for curve fitting, and 488 sample points of relay operating time were taken for
The 40 parameters in (
TDS | 0.5 | 3 | 6 | 9 |
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9 | 12 | 15 | 19 |
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Please refer to Table
This simple operating equation contains 40 parameters and includes
Table
The curve-fitting errors of 17 actual operating times for each TDS of the CO-8 relay in milliseconds.
TDS | 0.5 | 3 | 6 | 9 | ||||||||
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Top | SVD | Err | Top | SVD | Err | Top | SVD | Err | Top | SVD | Err |
| ||||||||||||
1.5 | 1682.2 | 1682.1 | 0.11 | 15254.0 | 15254.0 | 0.10 | 31907.0 | 31907.0 | 0.26 | 50681.0 | 50681.0 | 0.16 |
2 | 760.5 | 764.2 | 3.69 | 6182.1 | 6182.6 | 0.49 | 13232.0 | 13232.0 | 0.21 | 21256.0 | 21257.0 | 1.11 |
3 | 343.9 | 343.6 | 0.25 | 2577.6 | 2581.6 | 3.98 | 5334.0 | 5337.1 | 3.10 | 8449.4 | 8450.2 | 0.78 |
4 | 213.7 | 214.3 | 0.60 | 1578.5 | 1578.0 | 0.44 | 3341.9 | 3339.4 | 2.51 | 5222.6 | 5218.7 | 3.92 |
6 | 128.1 | 127.6 | 0.55 | 1004.0 | 1002.7 | 1.24 | 2062.6 | 2060.3 | 2.32 | 3221.3 | 3218.3 | 3.02 |
8 | 99.7 | 99.7 | 0.01 | 821.3 | 820.2 | 1.12 | 1618.9 | 1618.4 | 0.44 | 2542.7 | 2540.0 | 2.66 |
10 | 86.1 | 86.3 | 0.19 | 711.7 | 712.9 | 1.13 | 1407.3 | 1409.4 | 2.16 | 2210.1 | 2211.9 | 1.79 |
13 | 75.8 | 75.7 | 0.04 | 618.3 | 617.9 | 0.42 | 1244.0 | 1243.5 | 0.56 | 1944.8 | 1944.7 | 0.10 |
16 | 70.1 | 70.0 | 0.10 | 563.1 | 563.1 | 0.01 | 1145.0 | 1145.3 | 0.30 | 1783.3 | 1783.9 | 0.62 |
19 | 66.5 | 66.6 | 0.13 | 526.5 | 527.1 | 0.56 | 1078.3 | 1078.6 | 0.28 | 1672.8 | 1672.7 | 0.06 |
22 | 64.7 | 64.6 | 0.07 | 501.4 | 501.2 | 0.12 | 1030.0 | 1029.9 | 0.06 | 1589.8 | 1589.9 | 0.13 |
26 | 62.9 | 62.9 | 0.01 | 476.0 | 476.2 | 0.23 | 982.0 | 982.0 | 0.02 | 1506.4 | 1506.5 | 0.02 |
30 | 61.4 | 61.4 | 0.00 | 458.5 | 458.1 | 0.47 | 946.5 | 946.1 | 0.36 | 1442.9 | 1442.6 | 0.33 |
34 | 60.1 | 60.4 | 0.35 | 443.7 | 444.2 | 0.45 | 917.4 | 917.8 | 0.38 | 1390.5 | 1391.4 | 0.93 |
39 | 59.6 | 59.6 | 0.02 | 430.7 | 430.6 | 0.19 | 889.1 | 889.1 | 0.02 | 1339.7 | 1339.5 | 0.14 |
44 | 59.0 | 58.9 | 0.09 | 419.3 | 419.5 | 0.20 | 865.5 | 865.5 | 0.03 | 1297.0 | 1296.7 | 0.27 |
49 | 58.6 | 58.7 | 0.12 | 409.8 | 409.9 | 0.11 | 845.0 | 845.1 | 0.14 | 1260.1 | 1259.9 | 0.15 |
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Please refer to Table
Table
Summary of the SVD results for the CO-8 relay.
TDS | Max_Err/ |
Max_Err%/ |
AV | AV% |
---|---|---|---|---|
0.5 | 3.69/2.0 | 0.73/3.9 | 0.17 | 0.16 |
3 | 8.22/3.9 | 0.50/3.9 | 0.42 | 0.05 |
6 | 7.72/2.5 | 0.29/5.2 | 0.53 | 0.03 |
9 | 6.08/4.4 | 0.18/6.9 | 0.58 | 0.03 |
Please refer to Table
The absolute values of error of 488 sample points with TDS 3 CO8 relay are shown in Figure
Absolute values of error for
Finally, Figure
The actual characteristic curves and the curves fitted by (
The following paragraphs analyze and compare the SVD fitting results for the IAC51 and CO-8 EM OC relays.
The maximum absolute values of errors of the hundreds of sample points calculated by SVD are within the millisecond range, which is accurate enough for practical purposes. However, more fitted waveform components can be selected if greater precision is desired. Although the maximum absolute values of errors and maximum absolute values of percentage errors primarily occur at small
The SVD method can be applied to different types of inverse-time EM OC relay. As shown in Table
SVD can be applied to achieve excellent protection coordination between widely used digital relays and conventional EM OC relays. The procedures may be outlined below using CO-8 relay as an example, with a coordination time interval of 0.3 second.
Set
First, apply SVD to determine the settings of the CO-8 relay and then calculate the operating time for the digital relay by adding 0.3 s to, or subtracting 0.3 s from, that of the CO-8 relay if the digital relay is to function as a backup or primary relay, respectively.
In 2008, Geethanjali and Slochanal applied fuzzy logic and artificial neural network on six adaptive network and fuzzy inference system (ANFIS) algorithms (tri-mf5, gauss-mf5, gbell-mf5, tri-mf7, gauss-mf7, and gbell-mf7) for the characteristic curve fitting of one EM OC relay: the very inverse-time CRP9 relay with
The 26 parameters in (
TDS | 4 | 7 | 10 |
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9 | 9 | 9 |
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Please refer to Table
This simple operating equation includes
For each TDS, Table
Comparison of results by ANFIS and SVD for the CRP9 relay.
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Top | ANFIS | Err1 | Err1% | SVD | Err | Err% |
---|---|---|---|---|---|---|---|
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4 | 1352 | 1368 | 16 | 1.18 | 1352.00 | 0.00 | 0.00 |
8 | 605 | 609 | 4 | 0.66 | 605.00 | 0.00 | 0.00 |
12 | 459 | 456 | 3 | 0.65 | 459.00 | 0.00 | 0.00 |
16 | 375 | 374 | 1 | 0.27 | 375.00 | 0.00 | 0.00 |
20 | 358 | 356 | 2 | 0.56 | 358.00 | 0.00 | 0.00 |
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AV = 5.2 | AV% = 1.59★ | AV = 0.00 | AV% = 0.00 | ||||
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4 | 2459 | 2477 | 18 | 0.73 | 2459.00 | 0.00 | 0.00 |
8 | 1153 | 1132 | 21 | 1.82 | 1153.00 | 0.00 | 0.00 |
12 | 894 | 883 | 11 | 1.23 | 894.00 | 0.00 | 0.00 |
16 | 744 | 739 | 5 | 0.67 | 744.00 | 0.00 | 0.00 |
20 | 669 | 665 | 4 | 0.60 | 669.00 | 0.00 | 0.00 |
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AV = 11.8 | AV% = 0.61★ | AV = 0.00 | AV% = 0.00 | ||||
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4 | 3935 | 3963 | 28 | 0.71 | 3935.00 | 0.00 | 0.00 |
8 | 1754 | 1738 | 16 | 0.91 | 1754.00 | 0.00 | 0.00 |
12 | 1304 | 1305 | 1 | 0.08 | 1304.00 | 0.00 | 0.00 |
16 | 1123 | 1124 | 1 | 0.09 | 1123.00 | 0.00 | 0.00 |
20 | 1024 | 1027 | 3 | 0.29 | 1024.00 | 0.00 | 0.00 |
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AV = 9.8 | AV% = 0.64★ | AV = 0.00 | AV% = 0.00 |
Nine methods, which include the coefficient analytical model and fuzzy model in [
Average of absolute values of percentage errors of coefficient analytical model, fuzzy model, SVD method, and six ANFIS methods for CRP9 relay.
In summary, SVD outperforms ANFIS in curve fitting the operating time characteristics of the EM OC relay CRP9. Once again, this confirms the accuracy and identification robustness of the SVD method.
This study proposes an algorithm based on singular value decomposition (SVD) and fits the characteristic curves corresponding to eight TDS of two different types of inverse-time EM OC relays. The method decomposes the waveforms of the curves, according to their eigenvalues and corresponding eigenvectors, into various smooth and oscillating components without converting them to the frequency domain. Results show that the SVD performs exceedingly well in every respect regarding four different types of absolute values of errors. Although the maximum absolute values of errors and maximum absolute values of percentage errors primarily occur at small
The ability to accurately represent the characteristic curves of the EM OC relays by a simple equation can provide good protection coordination between conventional EM OC relays and digital relays for subtransmission systems and distribution systems. The formula is a unified equation by SVD algorithm, and it can fit the characteristics of any piecewise nonlinear continuous smooth descending curves by simply changing the values of its parameters. Such convenience or advantage can be exploited not only for OC protection equipments such as EM OC relays or power fuses in power systems for practical applications or future studies, but also for any subject in any field with such characteristic.