Calibration of On-Board Energy Measurement Systems Installed in Locomotives for AC Distorted Current and High Voltage Waveforms and Determination of Its Uncertainty Budget
Abstract
:1. Introduction
2. Calibration Set-Up
- Mode C “Current with harmonics”: fictive electrical power is generated by injecting a sinusoidal current (50 Hz or 16.7 Hz) or a phase-fired current waveform up to 500 A rms with harmonic content up to 5 kHz in the current loop (see Figure 5). Simultaneously, a 50 Hz or 16.7 Hz high voltage is applied.
- Mode V “Voltage with harmonics”: fictive electrical power is generated by applying a sinusoidal high voltage up to 25 kV, 50 Hz or 15 kV 16.7 Hz with harmonic content up to 5 kHz (see Figure 4) to the current loop. Simultaneously, a 50 Hz or 16.7 Hz current or a phase-fired current waveform up to 500 A is induced on the loop.
2.1. Reference Energy Measuring System
2.2. Characterization and Implementation
3. Uncertainty Analysis
3.1. Model Functions
3.1.1. Model Functions for Rms Voltage and Current, Active, Apparent and Non-Active Power Obtained by Digital Sampling at the Low Voltage Side
- T = 1/f: period of the power system.
- f: fundamental frequency of the power system.
- t0: time origin of the starting measuring times.
- v0: acquired voltage in the first sampling interval of the period under consideration.
- vk: acquired voltage in the kth sampling interval of the period under consideration.
- vn: acquired voltage in the last sampling interval of the period under consideration.
- n: Number of sample intervals per period.
- hs = 1/fs: Sampling interval.
- fs: Sampling rate.
- Nc: Number of periods included in the measuring time interval used to determine the energy.
- ik: acquired current in the kth sampling interval of the period under consideration.
- i0: acquired current in the first sampling interval of the period under consideration.
- in: acquired current in the last sampling interval of the period under consideration.
3.1.2. Model Functions of Voltage and Current Sampled Values at the High Voltage Side
- : sampled voltage value acquired at the low voltage side.
- : constant term of error of the DMM, for DCV measurements, typically a percentage of the voltage reading.
- : additional term of error of the DMM, affected by the ratio between the full-scale voltage of the DMM, , and the absolute value of the sampled voltage .
- : calibrated scale factor of the high voltage divider.
- : corrections of the voltage divider scale factor, such us drift (), temperature coefficient (, short term stability () and non-linearity with voltage ().
- : sensitivity coefficient of the voltage phase displacement.
- : corrections of the sampled voltage value due to phase displacements. The phase shift of the DMM (negligible), the high voltage divider phase calibration (), the high voltage divider phase drift (negligible), the high voltage divider residual phase correction ).
- : sampled voltage value measured at the shunt.
- : constant term of error of the DMM connected to the shunt, for DCV measurements.
- : additional term of error of the DMM connected to the shunt, affected by the ratio between the full-scale voltage of the DMM, , and the absolute value of the sampled voltage .
- : calibrated scale factor of the current transducer.
- : corrections of the current transducer scale factor, such us drift (), temperature coefficient () and non-linearity with current ) (see note below).
- : calibrated value of the shunt resistor.
- : corrections of the calibrated shunt value, such us: drift (), temperature coefficient (), and variation versus frequency ().
- : sensitivity coefficient of the voltage phase displacement in the current circuit.
- : corrections of the sampled voltage value due to phase displacements. The phase shift of the DMM (negligible), the current sensor phase calibration (), the current sensor phase drift (), the current sensor residual phase correction () and the shunt phase error ().
3.1.3. Model Functions for the Quantities at the High Voltage Side
- is the rms voltage value calculated by trapezoidal rule from a finite number of samples Vk (k = 1: n) taken from the recorded voltage signal.
- is the current rms value calculated by trapezoidal rule from a finite number of samples Ik (k = 1:n) taken from the recorded voltage signal.
- Sp3: correction of the active power value due to synchronization error between both DMM.
3.2. Uncertainty Analysis
3.2.1. Uncertainty of the Calculated Rms Voltage and Current Values at the High Voltage Side
- : represents the uncertainty in p.u. of the correction term δv1.
- : represents the uncertainty in p.u. of the correction term δv2.
- : represents the calibration uncertainty in p.u. of the high voltage divider scale factor,.
- : represents the uncertainty in p.u. factor δj,VD.
- : represents the uncertainty in p.u. of the correction terms related to the phase displacements due to the digitizer and the divider.
- : represents the sensitivity coefficient of each lth correction term.
- : represents the uncertainty in p.u. of the correction terms related to the signal processing restrictions related to a rms of the voltage in a period T starting at tj.
- : represents the calibration uncertainty in p.u. of the current transducer scale factor,.
- : represents the uncertainty of the correction term related to the current transducer δl,CT.
- : represents the calibration uncertainty in p.u. of the current shunt,.
- : represents the uncertainty of the correction terms related to the phase displacements due to current transducer, shunt resistance, digitizer.
- : represents the sensitivity coefficient of each lth correction term.
- : represents the uncertainty in p.u. of the correction terms related to the signal processing restrictions related to a rms of the current in a period T starting at tj.
3.2.2. Uncertainty of the Calculated Power Quantities at the High Voltage Side
Active Power
Apparent Power
Non-active power
3.2.3. Uncertainty of the Signal Processing Contributions
Uncertainty Contribution Due to Trapezoidal Integration Rule ()
Uncertainty Contribution Due to an Integration Time Different to the Complete Period ()
Uncertainty Due to the Synchronization Error between Multimeters ()
4. Application of the Uncertainty Estimation of the EMS Calibration Set Up
5. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Measuring System | Component | Type | Performance |
---|---|---|---|
Voltage | Resistive divider | LCOE | 1060/1 |
Measuring cable | RG-58 | 50 Ω, 4 m | |
Matching impedance | LCOE | 1.1 MΩ | |
Multimeter (Master) | Keithley DMM7510 | 700 V, 10 A, 1 MS/s | |
Current | Fluxgate current sensor | LEM ITN-900-S | 636 A, 1500/1 A |
DC source | LCOE | ±15 VDC | |
Shunt | LCOE | 10 Ω | |
Multimeter (Slave) | Keithley DMM7510 | 700 V, 10 A, 1 MS/s | |
Energy calculation | Software | LCOE |
Expanded Uncertainty | % |
---|---|
U[P] (%) | 0.23 |
U[VRMS] (%) | 0.15 |
U[IRMS] (%) | 0.10 |
U[S] (%) | 0.18 |
U[N] (%) | 0.43 |
Influence Parameters Xi | Symbol | Estimate | Standard Uncertainty u(xi) | Probability Distribution | Sensitivity Coefficient ci | Uncertainty Contribution ui(y) |
---|---|---|---|---|---|---|
Digitization of Voltage If VRMS(tj) = 10% VFS | 0 | 0.012/√3% (I) | Rectangular | 1 | 0.0069% | |
0 | 0.0075/√3% (I) | Rectangular | 0.039% | |||
Digitization of Current If IRMS(tj) = 10% IFS | 0 | 0.012/√3% (I) | Rectangular | 1 | 0.0069% | |
0 | 0.0075/√3% (I) | Rectangular | 0.055% | |||
VD SF calibrat. uncertainty | 1059 | 0.1/2% (II) | Normal | 1 | 0.05% | |
VD SF drift | 0 | 0.02/√3% (I) | Rectangular | 1 | 0.011% | |
VD temperature coefficient | 0 | 0.0075/√3% (I) | Rectangular | 1 | 0.0043% | |
VD short term stability | 0 | 0.06/√3% (I) | Rectangular | 1 | 0.035% | |
VD non-linearity | 0 | 0.03/√3% (I) | Rectangular | 1 | 0.017% | |
VD phase uncertainty | 0 | 58/2 μrad (II) | Normal | −6.26 × 10−5%/μrad | 0.0018% | |
VD residual phase error | 0 | 1452/2 μrad (II) | Normal | −6.26 × 10−5%/μrad | 0.045% | |
CT SF calibration uncertainty | 1500 | 0.0070/2% (II) | Normal | 1 | 0.0035% | |
CT SF drift | 0 | 0.005/√3% (I) | Rectangular | 1 | 0.0029% | |
CT temperature coefficient | 0 | 0.006/√3% (I) | Rectangular | 1 | 0.0034% | |
CT SF non-linearity | 0 | 0.0036/√3% (I) | Rectangular | 1 | 0.0021% | |
CT S. F. phase calibration | 0 | 58/2 μrad (II) | Normal | −6.26 × 10−5%/μrad | 0.0018% | |
CT S. F. phase error drift | 0 | 293/√3 μrad (I) | Rectangular | −6.26 × 10−5%/μrad | 0.0106% | |
CT residual phase error | 0 | 668/2 μrad (II) | Normal | −6.26 × 10−5%/μrad | 0.021% | |
Shunt calibration uncertainty | 9.979 Ω | 0.05/2% (II) | Normal | 1 | 0.025% | |
Shunt drift | 0 | 0.01/√3% (I) | Rectangular | 1 | 0.0057% | |
Shunt temperature coeffic. | 0 | 0.05/√3% (I) | Rectangular | 1 | 0.029% | |
Shunt resistance variation vs. frequency | 0 | 0.01/√3% (I) | Rectangular | 1 | 0.0057% | |
Shunt residual phase error | 0 | Negligible | Normal | 1 | 0% | |
V integration trapezoidal rule | 0 | 0.028/√3% (III) | Rectangular | 1 | 0.0161% | |
I integration trapezoidal rule | 0 | 0.028/√3% (III) | Rectangular | 1 | 0.0161% | |
V Samples taken for a period | 0 | 5·10−5/√3% (IV) | Rectangular | 1 | 0.00003% | |
I Samples taken for a period | 0 | 5·10−5/√3% (IV) | Rectangular | 1 | 0.00003% | |
Synchroniz. between DMM | 0 | 314μrad/√3 (V) | Rectangular | −6.26 × 10−5%/μrad | 0.011% | |
Combined variance | 0.0136% | |||||
Combined standard uncert. | 0.117% | |||||
Expanded uncertainty | 0.23% |
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Khamlichi, A.; Garnacho, F.; Simon, P.; Rovira, J.; Ramirez, A. Calibration of On-Board Energy Measurement Systems Installed in Locomotives for AC Distorted Current and High Voltage Waveforms and Determination of Its Uncertainty Budget. Sensors 2021, 21, 7967. https://doi.org/10.3390/s21237967
Khamlichi A, Garnacho F, Simon P, Rovira J, Ramirez A. Calibration of On-Board Energy Measurement Systems Installed in Locomotives for AC Distorted Current and High Voltage Waveforms and Determination of Its Uncertainty Budget. Sensors. 2021; 21(23):7967. https://doi.org/10.3390/s21237967
Chicago/Turabian StyleKhamlichi, Abderrahim, Fernando Garnacho, Pascual Simon, Jorge Rovira, and Angel Ramirez. 2021. "Calibration of On-Board Energy Measurement Systems Installed in Locomotives for AC Distorted Current and High Voltage Waveforms and Determination of Its Uncertainty Budget" Sensors 21, no. 23: 7967. https://doi.org/10.3390/s21237967