# On the Importance of Characterizing Virtual PMUs for Hardware-in-the-Loop and Digital Twin Applications

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Characterization of the Calibrator

#### 2.1. The Calibrator Hardware Architecture

#### 2.2. Characterization and Testing of the Calibrator

#### 2.2.1. Signal Magnitude Test Case

- -
- A voltage signal varying from 80 to 120% of the rated value;
- -
- A current signal varying from 10 to 200% of the rated value.

#### 2.2.2. Harmonic Distortion Test Case

#### 2.2.3. Synchronization Test Case

#### 2.2.4. Frequency and ROCOF Test

#### 2.2.5. Phase Displacement Test

#### 2.3. Results of the Characterization Tests

#### 2.3.1. Signal Magnitude Test Results

#### 2.3.2. Harmonic Distortion Test Results

#### 2.3.3. Synchronization Test Results

#### 2.3.4. Frequency and ROCOF Test Results

#### 2.3.5. Phase Displacement Test Results

#### 2.3.6. Characterization Conclusions

## 3. RTS Environment

#### 3.1. Description of the RTS

- Oregano syn1588 PCIe NIC. It is a PCI Express Ethernet network interface that provides highly accurate clock synchronization via the IEEE 1588 Standard (accuracy of its oscillator higher than 0.05 ppm). The Oregano card can be synchronized with either a PPS signal or an IRIG-B signal from a GPS source (3.3 V signal).
- External Clock Adapter OP5964. It is used to receive and transmit the synchronization signal from the outside to the interfaces.
- RTSI (Real-Time System Interface) Synchronization Board. This board directly communicated with the FPGA, as can be seen from Figure 7 (black line).
- XILINX TE0741 KINTEX-7 FPGA. It accepts either OPAL-RT boards or RS422 signals. The types of synchronization allowed are LVDS and fiber optic.
- Analog input (AI) card OP5340. It features 16 synchronous differential analog input channels with a maximum voltage range of ±20 V, sampled at 400 kSa/s. The analog to digital converter (ADC) has a 16-bit resolution, and the minimum acquisition time is 2.5 µs per channel. The declared maximum noise of the analog card is 20 mV peak-to-peak. The ADC already includes anti-aliasing filters to remove frequencies higher than 600 kHz.
- Ethernet port. It is used to interface a laptop to the OP4510 RTS.

#### 3.2. Description of the PMU

## 4. Tests and Results

#### 4.1. PMU Testing

#### 4.1.1. Amplitude Tests

#### 4.1.2. Frequency Tests

#### 4.1.3. Harmonic Tests

#### 4.1.4. Phase Tests

#### 4.2. Tests Results

^{−8}, 10

^{−7}, 10

^{−7}, and 10

^{−5}for the amplitude, phase, frequency, and ROCOF, respectively.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**Schematics of the calibrator phase displacement test: (

**a**) “actual zero” test setup; (

**b**) 0, 30, 45, 90° phase-shifted sinusoids test setup.

**Figure 5.**Oscilloscope waveform acquisitions for the board synchronization evaluation. The PPS signal (

**blue**), the CTR signal (

**red**), the OUT signal (

**yellow**).

**Figure 6.**Distribution histograms of the delay measurement between the (

**a**) PPS and the CTR rising fronts and (

**b**) the PPS and the OUT rising fronts.

**Figure 9.**RMS vs. frequency of results in Table 16.

**Figure 10.**RMS for each harmonic test (Table 17).

DAC Properties | Value |
---|---|

Resolution | 16 bit |

Full range | ±10 V |

Max sampling rate | 900 kSa/s |

GPS Properties | Value |
---|---|

PPS accuracy | 15 ns (one sigma) |

10 MHz clock accuracy | $1.16\times {10}^{-12}$ Hz (one day average) |

10 MHz stability | See Allan deviation graph in [63] |

Set Point (% of Nominal Value) | Peak Set Point (V) | RMS Set Point V X_{1} |
---|---|---|

10 | 0.5 | ≈0.35 |

20 | 1 | ≈0.71 |

50 | 2.5 | ≈1.77 |

100 | 5 | ≈3.54 |

120 | 6 | ≈4.24 |

150 | 7.5 | ≈5.30 |

200 | 10 | ≈7.07 |

50 Hz Component (% of Nominal Value) | Harmonic Component Order h | Harmonic Component Magnitude (% of 50 Hz Signal Nominal Value) | |
---|---|---|---|

Single harmonic component test signal ${\mathit{X}}_{\mathit{h}}$ | 0 | From 2 to 50 | 10 |

Standard harmonic distortion test signal ${\mathit{X}}_{1+\mathit{h}}$ | 100 | 2 | 10 |

3 | 10 | ||

5 | 10 | ||

7 | 10 | ||

11 | 10 | ||

20 | 10 | ||

30 | 10 | ||

50 | 10 |

${\mathit{X}}_{1}{\mathit{RMS}}^{*}.$ (V) | ${\mu}_{{\mathit{X}}_{1}\mathit{RMS}}$ (V) | ${\sigma}_{{\mathit{X}}_{1}\mathit{RMS}}$ (V) | ${\mathit{u}}_{{\mathit{A},\mathit{X}}_{1}\mathit{RMS}}$ (V) | ${\mathit{u}}_{{\mathit{B},\mathit{X}}_{1}\mathit{RMS}}$ (V) | ${\mathsf{\Delta}}_{{\mathit{X}}_{1}\mathit{RMS}}$ (V) |
---|---|---|---|---|---|

0.3535534 | 0.3535314 | $2\times {10}^{-6}$ | $3\times {10}^{-7}$ | $3\times {10}^{-5}$ | $2\times {10}^{-5}$ |

0.7071068 | 0.7070302 | $2\times {10}^{-6}$ | $3\times {10}^{-7}$ | $4\times {10}^{-5}$ | $8\times {10}^{-5}$ |

1.767767 | 1.767642 | $3\times {10}^{-5}$ | $4\times {10}^{-6}$ | $2\times {10}^{-4}$ | $1\times {10}^{-4}$ |

3.535534 | 3.535152 | $2\times {10}^{-5}$ | $3\times {10}^{-6}$ | $3\times {10}^{-4}$ | $4\times {10}^{-4}$ |

4.242641 | 4.242152 | $2\times {10}^{-5}$ | $3\times {10}^{-6}$ | $3\times {10}^{-4}$ | $5\times {10}^{-4}$ |

5.303301 | 5.302879 | $2\times {10}^{-5}$ | $3\times {10}^{-6}$ | $3\times {10}^{-4}$ | $4\times {10}^{-4}$ |

7.071068 | 7.070681 | $2\times {10}^{-5}$ | $3\times {10}^{-6}$ | $4\times {10}^{-4}$ | $4\times {10}^{-4}$ |

$\mathit{h}.$ | ${\mathit{X}}_{\mathit{h}}{\mathit{RMS}}^{*}$ (V) | ${\mu}_{{\mathit{X}}_{\mathit{h}}\mathit{RMS}}$ (V) | ${\mathsf{\sigma}}_{{\mathit{X}}_{\mathit{h}}\mathit{RMS}}$ (V) | ${\mathit{u}}_{{\mathit{A},\mathit{X}}_{\mathit{h}}\mathit{RMS}}$ (V) | ${\mathit{u}}_{{\mathit{B},\mathit{X}}_{\mathit{h}}\mathit{RMS}}$ (V) | ${\mathsf{\Delta}}_{{\mathit{X}}_{\mathit{h}}\mathit{RMS}}$ (V) |
---|---|---|---|---|---|---|

5 | 0.3535534 | 0.3535502 | $1\times {10}^{-6}$ | $2\times {10}^{-7}$ | $3\times {10}^{-5}$ | $3\times {10}^{-6}$ |

50 | 0.3535131 | $1\times {10}^{-6}$ | $1\times {10}^{-7}$ | $4\times {10}^{-5}$ | $4\times {10}^{-5}$ |

$\mathit{h}.$ | ${\mathit{X}}_{1+\mathit{h}}{\mathit{RMS}}^{*}$ (V) | ${\mu}_{{\mathit{X}}_{1+\mathit{h}}\mathit{RMS}}$ (V) | ${\mathsf{\sigma}}_{{\mathit{X}}_{1+\mathit{h}}\mathit{RMS}}$ (V) | ${\mathit{u}}_{{\mathit{A},\mathit{X}}_{1+\mathit{h}}\mathit{RMS}}$ (V) | ${\mathit{u}}_{{\mathit{B},\mathit{X}}_{1+\mathit{h}}\mathit{RMS}}$ (V) | ${\mathsf{\Delta}}_{{\mathit{X}}_{1+\mathit{h}}\mathit{RMS}}$ (V) |
---|---|---|---|---|---|---|

2 | 3.5531676 | 3.552858 | $2\times {10}^{-5}$ | $3\times {10}^{-6}$ | $3\times {10}^{-4}$ | $2.8\times {10}^{-4}$ |

3 | 3.552888 | $2\times {10}^{-5}$ | $3\times {10}^{-6}$ | $3\times {10}^{-4}$ | $2.8\times {10}^{-4}$ | |

5 | 3.552913 | $2\times {10}^{-5}$ | $2\times {10}^{-6}$ | $3\times {10}^{-4}$ | $2.5\times {10}^{-4}$ | |

7 | 3.552886 | $2\times {10}^{-5}$ | $3\times {10}^{-6}$ | $3\times {10}^{-4}$ | $2.8\times {10}^{-4}$ | |

11 | 3.552876 | $2\times {10}^{-5}$ | $3\times {10}^{-6}$ | $4\times {10}^{-4}$ | $2.9\times {10}^{-4}$ | |

20 | 3.552886 | $2\times {10}^{-5}$ | $3\times {10}^{-6}$ | $4\times {10}^{-4}$ | $2.8\times {10}^{-4}$ | |

30 | 3.552893 | $2\times {10}^{-5}$ | $3\times {10}^{-6}$ | $4\times {10}^{-4}$ | $2.7\times {10}^{-4}$ | |

50 | 3.552720 | $7\times {10}^{-5}$ | $9\times {10}^{-6}$ | $4\times {10}^{-4}$ | $4.5\times {10}^{-4}$ |

${\mathit{f}}^{*}.$ (Hz) | ${\mu}_{\mathit{f}}$ (Hz) | ${\mathit{u}}_{\mathit{A},\mathit{f}}$ (Hz) | ${\mathit{u}}_{\mathit{B},\mathit{f}}$ (Hz) | ${\mathsf{\Delta}}_{\mathit{f}}$ (Hz) | ${\mu}_{\mathit{ROCOF}}$ (Hz) | ${\mathit{u}}_{\mathit{A},\mathit{ROCOF}}$ (Hz) | ${\mathit{u}}_{\mathit{B},\mathit{ROCOF}}$ (Hz) | ${\delta}_{\mathit{max}-\mathit{min}}$ (Hz) |
---|---|---|---|---|---|---|---|---|

45.0 | 45.000000 | $1\times {10}^{-6}$ | $3\times {10}^{-5}$ | $1.5\times {10}^{-7}$ | 0 | $2\times {10}^{-6}$ | $4\times {10}^{-5}$ | $8\times {10}^{-5}$ |

45.5 | 45.500002 | $1\times {10}^{-6}$ | $3\times {10}^{-5}$ | $1.5\times {10}^{-6}$ | − $3\times {10}^{-7}$ | $1\times {10}^{-6}$ | $4\times {10}^{-5}$ | $5\times {10}^{-5}$ |

46.0 | 46.000001 | $1\times {10}^{-6}$ | $3\times {10}^{-5}$ | $1.1\times {10}^{-6}$ | $2\times {10}^{-8}$ | $1\times {10}^{-6}$ | $4\times {10}^{-5}$ | $6\times {10}^{-5}$ |

46.5 | 46.500001 | $1\times {10}^{-6}$ | $3\times {10}^{-5}$ | $1.4\times {10}^{-6}$ | $9\times {10}^{-8}$ | $1\times {10}^{-6}$ | $4\times {10}^{-5}$ | $6\times {10}^{-5}$ |

47.0 | 47.000001 | $1\times {10}^{-6}$ | $3\times {10}^{-5}$ | $1.1\times {10}^{-6}$ | $2\times {10}^{-7}$ | $1\times {10}^{-6}$ | $4\times {10}^{-5}$ | $5\times {10}^{-5}$ |

47.5 | 47.500002 | $1\times {10}^{-6}$ | $3\times {10}^{-5}$ | $1.5\times {10}^{-6}$ | $4\times {10}^{-7}$ | $2\times {10}^{-6}$ | $4\times {10}^{-5}$ | $6\times {10}^{-5}$ |

48.0 | 48.000001 | $1\times {10}^{-6}$ | $3\times {10}^{-5}$ | $1.4\times {10}^{-6}$ | $2\times {10}^{-7}$ | $1\times {10}^{-6}$ | $4\times {10}^{-5}$ | $5\times {10}^{-5}$ |

48.5 | 48.500006 | $1\times {10}^{-6}$ | $3\times {10}^{-5}$ | $5.6\times {10}^{-6}$ | − $1\times {10}^{-7}$ | $1\times {10}^{-6}$ | $4\times {10}^{-5}$ | $6\times {10}^{-5}$ |

49.0 | 49.000003 | $1\times {10}^{-6}$ | $3\times {10}^{-5}$ | $2.9\times {10}^{-6}$ | − $2\times {10}^{-7}$ | $1\times {10}^{-6}$ | $4\times {10}^{-5}$ | $5\times {10}^{-5}$ |

49.5 | 49.500000 | $1\times {10}^{-6}$ | $3\times {10}^{-5}$ | $4.6\times {10}^{-7}$ | − $4\times {10}^{-8}$ | $2\times {10}^{-6}$ | $4\times {10}^{-5}$ | $7\times {10}^{-5}$ |

50.0 | 50.000002 | $8\times {10}^{-7}$ | $3\times {10}^{-5}$ | $2.3\times {10}^{-6}$ | 0 | $1\times {10}^{-6}$ | $4\times {10}^{-5}$ | $4\times {10}^{-5}$ |

50.5 | 50.500000 | $1\times {10}^{-6}$ | $3\times {10}^{-5}$ | − $4.1\times {10}^{-8}$ | − $5\times {10}^{-8}$ | $1\times {10}^{-6}$ | $4\times {10}^{-5}$ | $6\times {10}^{-5}$ |

51.0 | 51.000001 | $1\times {10}^{-6}$ | $3\times {10}^{-5}$ | $1.3\times {10}^{-6}$ | $1\times {10}^{-7}$ | $2\times {10}^{-6}$ | $4\times {10}^{-5}$ | $8\times {10}^{-5}$ |

51.5 | 51.500007 | $1\times {10}^{-6}$ | $3\times {10}^{-5}$ | $6.6\times {10}^{-6}$ | $2\times {10}^{-7}$ | $2\times {10}^{-6}$ | $4\times {10}^{-5}$ | $6\times {10}^{-5}$ |

52.0 | 52.000003 | $1\times {10}^{-6}$ | $3\times {10}^{-5}$ | $3.1\times {10}^{-6}$ | − $2\times {10}^{-8}$ | $2\times {10}^{-6}$ | $4\times {10}^{-5}$ | $7\times {10}^{-5}$ |

52.5 | 52.500004 | $1\times {10}^{-6}$ | $3\times {10}^{-5}$ | $4.1\times {10}^{-6}$ | − $1\times {10}^{-7}$ | $2\times {10}^{-6}$ | $4\times {10}^{-5}$ | $7\times {10}^{-5}$ |

53.0 | 53.000006 | $1\times {10}^{-6}$ | $3\times {10}^{-5}$ | $5.6\times {10}^{-6}$ | $8\times {10}^{-8}$ | $1\times {10}^{-6}$ | $4\times {10}^{-5}$ | $6\times {10}^{-5}$ |

53.5 | 53.500004 | $1\times {10}^{-6}$ | $3\times {10}^{-5}$ | $4.1\times {10}^{-6}$ | − $8\times {10}^{-8}$ | $1\times {10}^{-6}$ | $4\times {10}^{-5}$ | $6\times {10}^{-5}$ |

54.0 | 54.000005 | $1\times {10}^{-6}$ | $3\times {10}^{-5}$ | $5.3\times {10}^{-6}$ | $5\times {10}^{-8}$ | $1\times {10}^{-6}$ | $4\times {10}^{-5}$ | $6\times {10}^{-5}$ |

54.5 | 54.500001 | $1\times {10}^{-6}$ | $3\times {10}^{-5}$ | $1.3\times {10}^{-6}$ | − $2\times {10}^{-8}$ | $1\times {10}^{-6}$ | $4\times {10}^{-5}$ | $5\times {10}^{-5}$ |

55.0 | 55.000004 | $1\times {10}^{-6}$ | $3\times {10}^{-5}$ | $4.0\times {10}^{-6}$ | $7\times {10}^{-8}$ | $2\times {10}^{-6}$ | $4\times {10}^{-5}$ | $6\times {10}^{-5}$ |

${\mu}_{\mathit{\varphi}0}\left(\mathrm{rad}\right)$ | ${\mathit{u}}_{\mathit{A},\mathbf{\varphi}0}\left(\mathrm{rad}\right)$ |
---|---|

1.27 $\times {10}^{-7}$ | 4 $\times {10}^{-9}$ |

${\mathit{\varphi}}_{\mathit{OUT}}^{*}\left(\mathrm{rad}\right)$ | ${\mathit{\varphi}}_{\mathit{OUT}2}^{*}\left(\mathrm{rad}\right)$ | ${\mu}_{\mathit{\varphi}}\left(\mathrm{rad}\right)$ | ${\mathit{u}}_{\mathit{A},\mathit{\varphi}}\left(\mathrm{rad}\right)$ | ${\mathsf{\Delta}}_{\mathit{\varphi}}\left(\mathrm{rad}\right)$ |
---|---|---|---|---|

0 | 0 | $2.690\times {10}^{-6}$ | $4\times {10}^{-9}$ | $-$$2.7\times {10}^{-6}$ |

0.523598776 | 0 | 0.5236015 | $2\times {10}^{-7}$ | $-$$2.8\times {10}^{-6}$ |

0.785398163 | 0 | 0.7854010 | $3\times {10}^{-7}$ | $-$$2.9\times {10}^{-6}$ |

1.570796327 | 0 | 1.5707995 | $4\times {10}^{-7}$ | $-$$3.2\times {10}^{-6}$ |

Test Name | Peak (V) | RMS (V) | % of Rated (%) | Phase (rad) | Frequency (Hz) |
---|---|---|---|---|---|

A1 | 10 | 7.0710 | 200 | 0 | 50 |

A2 | 5 | 3.5355 | 100 | 0 | 50 |

A3 | 2.5 | 1.7677 | 50 | 0 | 50 |

A4 | 1 | 0.7071 | 20 | 0 | 50 |

A5 | 0.1 | 0.0707 | 10 | 0 | 50 |

Test Name | Peak (V) | RMS (V) | % of Rated (%) | Phase (rad) | Frequency (Hz) |
---|---|---|---|---|---|

F1 | 5 | 3.5355 | 100 | 0 | 48 |

F2 | 5 | 3.5355 | 100 | 0 | 48.5 |

F3 | 5 | 3.5355 | 100 | 0 | 49 |

F4 | 5 | 3.5355 | 100 | 0 | 49.5 |

F5 | 5 | 3.5355 | 100 | 0 | 50 |

F6 | 5 | 3.5355 | 100 | 0 | 50.5 |

F7 | 5 | 3.5355 | 100 | 0 | 51 |

F8 | 5 | 3.5355 | 100 | 0 | 51.5 |

F9 | 5 | 3.5355 | 100 | 0 | 52 |

Test Name | Peak (V) | RMS (V) | % of Rated (%) | Order (-) | % of 50 Hz Comp (%) |
---|---|---|---|---|---|

H1 | 5 | 3.5355 | 100 | 3 | 10 |

H2 | 5 | 3.5355 | 100 | 5 | 10 |

H3 | 5 | 3.5355 | 100 | 7 | 10 |

H4 | 5 | 3.5355 | 100 | 9 | 10 |

H5 | 5 | 3.5355 | 100 | 11 | 10 |

H6 | 5 | 3.5355 | 100 | 15 | 10 |

H7 | 5 | 3.5355 | 100 | 19 | 10 |

H8 | 5 | 3.5355 | 100 | 21 | 10 |

H9 | 5 | 3.5355 | 100 | 25 | 10 |

H10 | 5 | 3.5355 | 100 | 29 | 10 |

H11 | 5 | 3.5355 | 100 | 31 | 10 |

H12 | 5 | 3.5355 | 100 | 35 | 10 |

H13 | 5 | 3.5355 | 100 | 39 | 10 |

H14 | 5 | 3.5355 | 100 | 41 | 10 |

H15 | 5 | 3.5355 | 100 | 45 | 10 |

H16 | 5 | 3.5355 | 100 | 49 | 10 |

Test Name | Peak (V) | RMS (V) | % of Rated (%) | Frequency (Hz) | Phase (°) | Phase (rad) |
---|---|---|---|---|---|---|

P1 | 5 | 3.5355 | 100 | 50 | 0 | 0 |

P2 | 5 | 3.5355 | 100 | 50 | 10 | 0.1745 |

P3 | 5 | 3.5355 | 100 | 50 | 20 | 0.3490 |

P4 | 5 | 3.5355 | 100 | 50 | 30 | 0.5235 |

P5 | 5 | 3.5355 | 100 | 50 | 40 | 0.6981 |

P6 | 5 | 3.5355 | 100 | 50 | 50 | 0.8726 |

P7 | 5 | 3.5355 | 100 | 50 | 60 | 1.0471 |

P8 | 5 | 3.5355 | 100 | 50 | 70 | 1.2217 |

P9 | 5 | 3.5355 | 100 | 50 | 80 | 1.3962 |

P10 | 5 | 3.5355 | 100 | 50 | 90 | 1.5707 |

P11 | 5 | 3.5355 | 100 | 50 | 100 | 1.7453 |

Test Name | RMS (V) | ${\mathit{\sigma}}_{\mathit{R}\mathit{M}\mathit{S}}\left(\mathrm{V}\right)$ | Phase (rad) | ${\mathit{\sigma}}_{\mathit{P}\mathit{h}}\left(\mathrm{rad}\right)$ | Frequency (Hz) | ${\mathit{\sigma}}_{\mathit{F}\mathit{r}}\left(\mathrm{Hz}\right)$ | ROCOF (Hz/s) | ${\sigma}_{\mathit{R}}(\mathrm{Hz}/\mathrm{s})$ |
---|---|---|---|---|---|---|---|---|

A1 | 7.0692976 | $1\times {10}^{-7}$ | 0.0054920 | $4\times {10}^{-7}$ | 50.0000000 | $3\times {10}^{-7}$ | $3.24\times {10}^{-7}$ | $2\times {10}^{-5}$ |

A2 | 3.53433398 | $9\times {10}^{-8}$ | 0.0088586 | $4\times {10}^{-7}$ | 50.0000002 | $3\times {10}^{-7}$ | $1.77\times {10}^{-6}$ | $3\times {10}^{-5}$ |

A3 | 1.76701027 | $8\times {10}^{-8}$ | 0.0093632 | $4\times {10}^{-7}$ | 50.0000000 | $3\times {10}^{-7}$ | $-4.19\times {10}^{-6}$ | $7\times {10}^{-5}$ |

A4 | 0.70625170 | $8\times {10}^{-8}$ | $-$0.0035345 | $3\times {10}^{-7}$ | 50.0000004 | $3\times {10}^{-7}$ | $1.25\times {10}^{-6}$ | $2\times {10}^{-4}$ |

A5 | 0.06985583 | $8\times {10}^{-8}$ | 0.007932 | $3\times {10}^{-6}$ | 49.999975 | $2\times {10}^{-6}$ | $-1.35\times {10}^{-5}$ | $2\times {10}^{-3}$ |

Test Name | RMS (V) | ${\sigma}_{RMS}$(V) | Phase (rad) | ${\sigma}_{Ph}$(rad) | Frequency (Hz) | ${\sigma}_{Fr}$(Hz) | ROCOF (Hz/s) | ${\sigma}_{R}$(Hz/s) |

F1 | 3.5343752 | $1\times {10}^{-7}$ | 0.0073137 | $4\times {10}^{-7}$ | 48.0000000 | $3\times {10}^{-7}$ | $-9.19\times {10}^{-7}$ | $5\times {10}^{-5}$ |

F2 | 3.5344621 | $1\times {10}^{-7}$ | 0.0119328 | $4\times {10}^{-7}$ | 48.4999925 | $2\times {10}^{-7}$ | $3.94\times {10}^{-5}$ | $5\times {10}^{-5}$ |

F3 | 3.53431802 | $9\times {10}^{-8}$ | 0.0092100 | $4\times {10}^{-7}$ | 48.9999999 | $3\times {10}^{-7}$ | $-1.17\times {10}^{-6}$ | $5\times {10}^{-5}$ |

F4 | 3.5344581 | $1\times {10}^{-7}$ | 0.008737 | $1\times {10}^{-6}$ | 49.5000027 | $3\times {10}^{-7}$ | $-2.72\times {10}^{-6}$ | $5\times {10}^{-5}$ |

F5 | 3.53448223 | $8\times {10}^{-8}$ | $-$0.0031022 | $4\times {10}^{-7}$ | 49.9999971 | $3\times {10}^{-7}$ | $4.90\times {10}^{-5}$ | $3\times {10}^{-5}$ |

F6 | 3.5344243 | $1\times {10}^{-7}$ | $-$0.0027930 | $6\times {10}^{-7}$ | 50.4999894 | $2\times {10}^{-7}$ | $-3.13\times {10}^{-7}$ | $5\times {10}^{-5}$ |

F7 | 3.5342794 | $1\times {10}^{-7}$ | 0.0034803 | $5\times {10}^{-7}$ | 50.9999988 | $3\times {10}^{-7}$ | $1.24\times {10}^{-5}$ | $5\times {10}^{-5}$ |

F8 | 3.5344108 | $1\times {10}^{-7}$ | 0.0072208 | $4\times {10}^{-7}$ | 51.4999955 | $3\times {10}^{-7}$ | $9.18\times {10}^{-7}$ | $6\times {10}^{-5}$ |

F9 | 3.5342751 | $1\times {10}^{-7}$ | $-$0.0047451 | $4\times {10}^{-7}$ | 52.0000001 | $3\times {10}^{-7}$ | $-2.46\times {10}^{-6}$ | $6\times {10}^{-5}$ |

Test Name | RMS (V) | ${\mathit{\sigma}}_{\mathit{R}\mathit{M}\mathit{S}}\left(\mathrm{V}\right)$ | Phase (rad) | ${\mathit{\sigma}}_{\mathit{P}\mathit{h}}\left(\mathrm{rad}\right)$ | Frequency (Hz) | ${\mathit{\sigma}}_{\mathit{F}\mathit{r}}\left(\mathrm{Hz}\right)$ | ROCOF (Hz/s) | ${\mathit{\sigma}}_{\mathit{R}}(\mathrm{Hz}/\mathrm{s})$ |
---|---|---|---|---|---|---|---|---|

H1 | 3.53444568 | $8\times {10}^{-8}$ | 0.0055126 | $4\times {10}^{-7}$ | 50.0000008 | $3\times {10}^{-7}$ | $-1.60\times {10}^{-6}$ | $3\times {10}^{-5}$ |

H2 | 3.53421753 | $9\times {10}^{-8}$ | 0.0076177 | $8\times {10}^{-7}$ | 50.0000070 | $3\times {10}^{-7}$ | $-1.51\times {10}^{-6}$ | $4\times {10}^{-5}$ |

H3 | 3.53443023 | $9\times {10}^{-8}$ | 0.0089869 | $7\times {10}^{-7}$ | 50.0000075 | $3\times {10}^{-7}$ | $-1.17\times {10}^{-6}$ | $4\times {10}^{-5}$ |

H4 | 3.5342145 | $1\times {10}^{-7}$ | 0.0082917 | $9\times {10}^{-7}$ | 50.0000136 | $3\times {10}^{-7}$ | $1.65\times {10}^{-6}$ | $4\times {10}^{-5}$ |

H5 | 3.53439991 | $8\times {10}^{-8}$ | 0.0091382 | $5\times {10}^{-7}$ | 49.9999942 | $2\times {10}^{-7}$ | $1.99\times {10}^{-6}$ | $4\times {10}^{-5}$ |

H6 | 3.5343785 | $1\times {10}^{-7}$ | 0.0054355 | $5\times {10}^{-7}$ | 50.0000072 | $3\times {10}^{-7}$ | $2.26\times {10}^{-6}$ | $4\times {10}^{-5}$ |

H7 | 3.5343746 | $1\times {10}^{-7}$ | 0.0072207 | $4\times {10}^{-7}$ | 50.0000003 | $3\times {10}^{-7}$ | $3.37\times {10}^{-8}$ | $4\times {10}^{-5}$ |

H8 | 3.5343741 | $2\times {10}^{-7}$ | 0.0076282 | $4\times {10}^{-7}$ | 50.0000000 | $3\times {10}^{-7}$ | $3.14\times {10}^{-7}$ | $4\times {10}^{-5}$ |

H9 | 3.5342377 | $2\times {10}^{-7}$ | 0.0085004 | $9\times {10}^{-7}$ | 49.9999873 | $2\times {10}^{-7}$ | $-2.13\times {10}^{-7}$ | $5\times {10}^{-5}$ |

H10 | 3.5356639 | $1\times {10}^{-7}$ | 0.005984 | $6\times {10}^{-6}$ | 50.0000197 | $7\times {10}^{-7}$ | $-4.50\times {10}^{-6}$ | $7\times {10}^{-5}$ |

H11 | 3.5350865 | $1\times {10}^{-7}$ | 0.0081300 | $5\times {10}^{-7}$ | 50.0000089 | $3\times {10}^{-7}$ | $2.50\times {10}^{-6}$ | $5\times {10}^{-5}$ |

H12 | 3.5349436 | $1\times {10}^{-7}$ | 0.0070349 | $3\times {10}^{-7}$ | 49.9999980 | $2\times {10}^{-7}$ | $-2.99\times {10}^{-6}$ | $5\times {10}^{-5}$ |

H13 | 3.5348042 | $2\times {10}^{-7}$ | 0.0097929 | $3\times {10}^{-7}$ | 49.9999977 | $2\times {10}^{-7}$ | $-1.06\times {10}^{-6}$ | $5\times {10}^{-5}$ |

H14 | 3.5347333 | $1\times {10}^{-7}$ | 0.008859 | $2\times {10}^{-6}$ | 50.0000258 | $3\times {10}^{-7}$ | $7.53\times {10}^{-7}$ | $6\times {10}^{-5}$ |

H15 | 3.53484464 | $9\times {10}^{-8}$ | 0.0043363 | $4\times {10}^{-7}$ | 49.9999995 | $3\times {10}^{-7}$ | $9.23\times {10}^{-7}$ | $5\times {10}^{-5}$ |

H16 | 3.5346480 | $2\times {10}^{-7}$ | 0.008433 | $1\times {10}^{-6}$ | 50.0000127 | $3\times {10}^{-7}$ | $-1.27\times {10}^{-6}$ | $6\times {10}^{-5}$ |

Test Name | RMS (V) | ${\mathit{\sigma}}_{\mathit{R}\mathit{M}\mathit{S}}\left(\mathrm{V}\right)$ | Phase (rad) | ${\mathit{\sigma}}_{\mathit{P}\mathit{h}}\left(\mathrm{rad}\right)$ | Frequency (Hz) | ${\mathit{\sigma}}_{\mathit{F}\mathit{r}}\left(\mathrm{Hz}\right)$ | ROCOF (Hz/s) | ${\mathit{\sigma}}_{\mathit{R}}(\mathrm{Hz}/\mathrm{s})$ |
---|---|---|---|---|---|---|---|---|

P1 | 3.53407799 | $9\times {10}^{-8}$ | 0.0113214 | $8\times {10}^{-7}$ | 50.0000045 | $3\times {10}^{-7}$ | $2.46\times {10}^{-7}$ | $3\times {10}^{-5}$ |

P2 | 3.53414587 | $8\times {10}^{-8}$ | 0.1757891 | $4\times {10}^{-7}$ | 49.9999982 | $3\times {10}^{-7}$ | $1.93\times {10}^{-6}$ | $3\times {10}^{-5}$ |

P3 | 3.53423390 | $8\times {10}^{-8}$ | 0.3574863 | $4\times {10}^{-7}$ | 49.9999988 | $3\times {10}^{-7}$ | $-4.54\times {10}^{-6}$ | $3\times {10}^{-5}$ |

P4 | 3.53424212 | $8\times {10}^{-8}$ | 0.5307076 | $4\times {10}^{-7}$ | 49.9999991 | $3\times {10}^{-7}$ | $-1.11\times {10}^{-6}$ | $3\times {10}^{-5}$ |

P5 | 3.5341987 | $1\times {10}^{-7}$ | 0.7064876 | $7\times {10}^{-7}$ | 50.0000064 | $3\times {10}^{-7}$ | $-1.29\times {10}^{-5}$ | $3\times {10}^{-5}$ |

P6 | 3.53416886 | $9\times {10}^{-8}$ | 0.8733776 | $4\times {10}^{-7}$ | 49.9999999 | $3\times {10}^{-7}$ | $-9.72\times {10}^{-6}$ | $3\times {10}^{-5}$ |

P7 | 3.53417573 | $9\times {10}^{-8}$ | 1.0548074 | $4\times {10}^{-7}$ | 49.9999991 | $3\times {10}^{-7}$ | $-3.91\times {10}^{-6}$ | $3\times {10}^{-5}$ |

P8 | 3.5340813 | $1\times {10}^{-7}$ | 1.224595 | $2\times {10}^{-6}$ | 50.0000105 | $4\times {10}^{-7}$ | $-5.17\times {10}^{-6}$ | $4\times {10}^{-5}$ |

P9 | 3.5340916 | $1\times {10}^{-7}$ | 1.4034948 | $4\times {10}^{-7}$ | 49.9999980 | $3\times {10}^{-7}$ | $-7.36\times {10}^{-6}$ | $3\times {10}^{-5}$ |

P10 | 3.5340525 | $1\times {10}^{-7}$ | 1.5760936 | $8\times {10}^{-7}$ | 49.9999852 | $3\times {10}^{-7}$ | $-5.18\times {10}^{-6}$ | $3\times {10}^{-5}$ |

P11 | 3.53407339 | $8\times {10}^{-8}$ | 1.7542494 | $4\times {10}^{-7}$ | 49.9999982 | $3\times {10}^{-7}$ | $-9.10\times {10}^{-6}$ | $3\times {10}^{-5}$ |

Test Name | TVE (%) | ${\mathit{u}}_{\mathit{T}\mathit{V}\mathit{E}}(-)$ | FE (Hz) | ${\mathit{\sigma}}_{\mathit{F}\mathit{E}}\left(\mathrm{Hz}\right)$ | RFE (Hz/s) | ${\mathit{\sigma}}_{\mathit{R}\mathit{F}\mathit{E}}(\mathrm{Hz}/\mathrm{s})$ |
---|---|---|---|---|---|---|

A1 | 0.54970 | $3\times {10}^{-7}$ | $-2.89\times {10}^{-8}$ | $1\times {10}^{-6}$ | $3.42\times {10}^{-7}$ | $2\times {10}^{-5}$ |

A2 | 0.31161 | $3\times {10}^{-7}$ | $-3\times {10}^{-6}$ | $1\times {10}^{-6}$ | $5\times {10}^{-5}$ | $4\times {10}^{-5}$ |

A3 | 0.93710 | $3\times {10}^{-7}$ | $-3.41\times {10}^{-8}$ | $1\times {10}^{-6}$ | $-4.19\times {10}^{-6}$ | $7\times {10}^{-5}$ |

A4 | 0.37348 | $2\times {10}^{-7}$ | $3.69\times {10}^{-7}$ | $1\times {10}^{-6}$ | $1.25\times {10}^{-6}$ | $2\times {10}^{-4}$ |

A5 | 1.4512 | $9\times {10}^{-7}$ | $-2.5\times {10}^{-5}$ | $2\times {10}^{-6}$ | $-1.35\times {10}^{-5}$ | $3\times {10}^{-4}$ |

Test Name | TVE (%) | ${\mathit{u}}_{\mathit{T}\mathit{V}\mathit{E}}(-)$ | FE (Hz) | ${\mathit{\sigma}}_{\mathit{F}\mathit{E}}\left(\mathrm{Hz}\right)$ | RFE (Hz/s) | ${\mathit{\sigma}}_{\mathit{R}\mathit{F}\mathit{E}}(\mathrm{Hz}/\mathrm{s})$ |
---|---|---|---|---|---|---|

F1 | 0.73198 | $3\times {10}^{-7}$ | $-4.45\times {10}^{-8}$ | $1\times {10}^{-6}$ | $-9.19\times {10}^{-7}$ | $5\times {10}^{-5}$ |

F2 | 1.19348 | $3\times {10}^{-7}$ | $-7\times {10}^{-6}$ | $1\times {10}^{-6}$ | $4\times {10}^{-5}$ | $5\times {10}^{-5}$ |

F3 | 0.92148 | $3\times {10}^{-7}$ | $-5.35\times {10}^{-8}$ | $1\times {10}^{-6}$ | $-1.17\times {10}^{-6}$ | $5\times {10}^{-5}$ |

F4 | 0.87410 | $6\times {10}^{-7}$ | $3\times {10}^{-6}$ | $2\times {10}^{-6}$ | $-2.72\times {10}^{-6}$ | $5\times {10}^{-5}$ |

F5 | 0.31161 | $3\times {10}^{-7}$ | $-3\times {10}^{-6}$ | $1\times {10}^{-6}$ | $5\times {10}^{-5}$ | $4\times {10}^{-5}$ |

F6 | 0.28105 | $4\times {10}^{-7}$ | $-1.1\times {10}^{-5}$ | $1\times {10}^{-6}$ | $-3.13\times {10}^{-7}$ | $5\times {10}^{-5}$ |

F7 | 0.34979 | $4\times {10}^{-7}$ | $-1\times {10}^{-6}$ | $2\times {10}^{-6}$ | $1\times {10}^{-5}$ | $5\times {10}^{-5}$ |

F8 | 0.72267 | $3\times {10}^{-7}$ | $-4\times {10}^{-6}$ | $1\times {10}^{-6}$ | $9.18\times {10}^{-7}$ | $6\times {10}^{-5}$ |

F9 | 0.47577 | $3\times {10}^{-7}$ | $1.30\times {10}^{-7}$ | $1\times {10}^{-6}$ | $-2.46\times {10}^{-6}$ | $6\times {10}^{-5}$ |

Test Name | TVE (%) | ${\mathit{u}}_{\mathit{T}\mathit{V}\mathit{E}}(-)$ | FE (Hz) | ${\mathit{\sigma}}_{\mathit{F}\mathit{E}}\left(\mathrm{Hz}\right)$ | RFE (Hz/s) | ${\mathit{\sigma}}_{\mathit{R}\mathit{F}\mathit{E}}(\mathrm{Hz}/\mathrm{s})$ |
---|---|---|---|---|---|---|

H1 | 0.55204 | $3\times {10}^{-7}$ | $8.13\times {10}^{-7}$ | $1\times {10}^{-6}$ | $-1.60\times {10}^{-6}$ | $4\times {10}^{-5}$ |

H2 | 0.76254 | $5\times {10}^{-7}$ | $7\times {10}^{-6}$ | $1\times {10}^{-6}$ | $-1.51\times {10}^{-6}$ | $4\times {10}^{-5}$ |

H3 | 0.89909 | $4\times {10}^{-7}$ | $7\times {10}^{-6}$ | $1\times {10}^{-6}$ | $-1.17\times {10}^{-6}$ | $4\times {10}^{-5}$ |

H4 | 0.82986 | $5\times {10}^{-7}$ | $1.4\times {10}^{-5}$ | $1\times {10}^{-6}$ | $1.65\times {10}^{-6}$ | $4\times {10}^{-5}$ |

H5 | 0.91423 | $3\times {10}^{-7}$ | $-6\times {10}^{-6}$ | $1\times {10}^{-6}$ | $1.99\times {10}^{-6}$ | $4\times {10}^{-5}$ |

H6 | 0.54446 | $3\times {10}^{-7}$ | $7\times {10}^{-6}$ | $1\times {10}^{-6}$ | $2.26\times {10}^{-6}$ | $4\times {10}^{-5}$ |

H7 | 0.72270 | $3\times {10}^{-7}$ | $2.64\times {10}^{-7}$ | $1\times {10}^{-6}$ | $3.37\times {10}^{-8}$ | $4\times {10}^{-5}$ |

H8 | 0.76340 | $3\times {10}^{-7}$ | $2.86\times {10}^{-9}$ | $1\times {10}^{-6}$ | $3.14\times {10}^{-7}$ | $4\times {10}^{-5}$ |

H9 | 0.85068 | $5\times {10}^{-7}$ | $-1.3\times {10}^{-5}$ | $1\times {10}^{-6}$ | $-2.13\times {10}^{-7}$ | $5\times {10}^{-5}$ |

H10 | 0.6461 | $4\times {10}^{-6}$ | $2.0\times {10}^{-5}$ | $1\times {10}^{-6}$ | $-4.50\times {10}^{-6}$ | $7\times {10}^{-5}$ |

H11 | 0.81305 | $3\times {10}^{-7}$ | $9\times {10}^{-6}$ | $1\times {10}^{-6}$ | $2.50\times {10}^{-6}$ | $5\times {10}^{-5}$ |

H12 | 0.70363 | $2\times {10}^{-7}$ | $-2\times {10}^{-6}$ | $1\times {10}^{-6}$ | $-2.99\times {10}^{-6}$ | $5\times {10}^{-5}$ |

H13 | 0.97941 | $2\times {10}^{-7}$ | $-2\times {10}^{-6}$ | $1\times {10}^{-6}$ | $-1.06\times {10}^{-6}$ | $5\times {10}^{-5}$ |

H14 | 0.8861 | $9\times {10}^{-7}$ | $2.6\times {10}^{-5}$ | $1\times {10}^{-6}$ | $7.53\times {10}^{-7}$ | $6\times {10}^{-5}$ |

H15 | 0.43403 | $3\times {10}^{-7}$ | $-5.48\times {10}^{-7}$ | $1\times {10}^{-6}$ | $9.23\times {10}^{-7}$ | $5\times {10}^{-5}$ |

H16 | 0.8436 | $6\times {10}^{-7}$ | $1.3\times {10}^{-5}$ | $1\times {10}^{-6}$ | $-1.27\times {10}^{-6}$ | $6\times {10}^{-5}$ |

Test Name | TVE (%) | ${\mathit{u}}_{\mathit{T}\mathit{V}\mathit{E}}(-)$ | FE (Hz) | ${\mathit{\sigma}}_{\mathit{F}\mathit{E}}\left(\mathrm{Hz}\right)$ | RFE (Hz/s) | ${\mathit{\sigma}}_{\mathit{R}\mathit{F}\mathit{E}}(\mathrm{Hz}/\mathrm{s})$ |
---|---|---|---|---|---|---|

P1 | 0.31161 | $3\times {10}^{-7}$ | $-3\times {10}^{-6}$ | $1\times {10}^{-6}$ | $5\times {10}^{-5}$ | $4\times {10}^{-5}$ |

P2 | 0.13178 | $6\times {10}^{-7}$ | $-2\times {10}^{-6}$ | $1\times {10}^{-6}$ | $1.93\times {10}^{-6}$ | $4\times {10}^{-5}$ |

P3 | 0.84269 | $3\times {10}^{-7}$ | $-1\times {10}^{-6}$ | $1\times {10}^{-6}$ | $-4.54\times {10}^{-6}$ | $4\times {10}^{-5}$ |

P4 | 0.71170 | $3\times {10}^{-7}$ | $-9.34\times {10}^{-7}$ | $1\times {10}^{-6}$ | $-1.11\times {10}^{-6}$ | $4\times {10}^{-5}$ |

P5 | 0.83629 | $4\times {10}^{-7}$ | $6\times {10}^{-6}$ | $1\times {10}^{-6}$ | $-1\times {10}^{-5}$ | $4\times {10}^{-5}$ |

P6 | 0.08218 | $9\times {10}^{-7}$ | $-8.40\times {10}^{-8}$ | $1\times {10}^{-6}$ | $-9.72\times {10}^{-6}$ | $4\times {10}^{-5}$ |

P7 | 0.76181 | $3\times {10}^{-7}$ | $-8.71\times {10}^{-7}$ | $1\times {10}^{-6}$ | $-3.91\times {10}^{-6}$ | $4\times {10}^{-5}$ |

P8 | 0.2903 | $1\times {10}^{-6}$ | $1.1\times {10}^{-5}$ | $1\times {10}^{-6}$ | $-5.17\times {10}^{-6}$ | $4\times {10}^{-5}$ |

P9 | 0.72415 | $3\times {10}^{-7}$ | $-2\times {10}^{-6}$ | $1\times {10}^{-6}$ | $-7.36\times {10}^{-6}$ | $4\times {10}^{-5}$ |

P10 | 0.53129 | $5\times {10}^{-7}$ | $-1.5\times {10}^{-5}$ | $1\times {10}^{-6}$ | $-5.18\times {10}^{-6}$ | $4\times {10}^{-5}$ |

P11 | 0.89278 | $3\times {10}^{-7}$ | $-2\times {10}^{-6}$ | $1\times {10}^{-6}$ | $-9.10\times {10}^{-6}$ | $4\times {10}^{-5}$ |

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## Share and Cite

**MDPI and ACS Style**

Mingotti, A.; Costa, F.; Cavaliere, D.; Peretto, L.; Tinarelli, R.
On the Importance of Characterizing Virtual PMUs for Hardware-in-the-Loop and Digital Twin Applications. *Sensors* **2021**, *21*, 6133.
https://doi.org/10.3390/s21186133

**AMA Style**

Mingotti A, Costa F, Cavaliere D, Peretto L, Tinarelli R.
On the Importance of Characterizing Virtual PMUs for Hardware-in-the-Loop and Digital Twin Applications. *Sensors*. 2021; 21(18):6133.
https://doi.org/10.3390/s21186133

**Chicago/Turabian Style**

Mingotti, Alessandro, Federica Costa, Diego Cavaliere, Lorenzo Peretto, and Roberto Tinarelli.
2021. "On the Importance of Characterizing Virtual PMUs for Hardware-in-the-Loop and Digital Twin Applications" *Sensors* 21, no. 18: 6133.
https://doi.org/10.3390/s21186133