# Closed-Form Expressions to Estimate the Mean and Variance of the Total Vector Error

^{*}

## Abstract

**:**

## 1. Introduction

## 2. PMU and Total Vector Error

## 3. Mean and Variance Estimation

#### 3.1. Problem Definition

#### 3.2. Mean and Variance Expression

#### 3.2.1. Real Part $\mathrm{Re}{\{\mathsf{\Delta}\widehat{X}\}}^{2}$

#### 3.2.2. Imaginary Part $\mathrm{Im}{\{\mathsf{\Delta}\widehat{X}\}}^{2}$

#### 3.2.3. Radicand

#### 3.2.4. Final Expressions

## 4. Numerical Validation

#### 4.1. Definition of the Tests

#### 4.2. Results

- The obtained expressions to estimate the mean value and variance of the TVE have been confirmed to be applicable and effective in all the realistic conditions tested.
- The phase delay and the gain error are the most critical sources of error that increase the TVE. Therefore, when the ADC of a PMU must be selected, those two values should be as low as possible.
- Due to its expression, the TVE always has a mean value different from zero and positive in all nonideal conditions (hence, in all practical applications). Furthermore, the TVE distribution is not normal (and, in particular, it lays in-between a normal and a chi-square distribution), hence the information of the variance should be completed with the desired confidence interval. To this purpose, the cumulative distribution function of the Nakagami distribution must be used:$$F\left(x\right)=\frac{\gamma \left(m,\frac{m}{\mathsf{\Omega}}{x}^{2}\right)}{\mathsf{\Gamma}\left(m\right)}$$

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Bertrand, P.; Mendik, M.; Hazel, T.; Tantin, P. Ct Saturation Calculations—Are They Applicable in the Modern World?—Part IV: Ct Sizing as per IEC Standards and the Benefits of Non-Conventional Instrument Transformers. In Proceedings of the Industry Applications Society 56th Annual Petroleum and Chemical Industry Conference, Anaheim, CA, USA, 14–16 September 2009. [Google Scholar]
- Heine, H.; Guenther, P.; Becker, F. New non-conventional instrument transformer (NCIT)—A future technology in gas insulated switchgear. In Proceedings of the IEEE Power Engineering Society Transmission and Distribution Conference, Dallas, TX, USA, 3–5 May 2016. [Google Scholar]
- Thomas, R.; Vujanic, A.; Xu, D.Z.; Sjödin, J.E.; Salazar, H.R.M.; Yang, M.; Powers, N. Non-conventional instrument transformers enabling digital substations for future grid. In Proceedings of the IEEE Power Engineering Society Transmission and Distribution Conference, Dallas, TX, USA, 3–5 May 2016. [Google Scholar]
- IEC 61869-6. Part 6: Additional General Requirements for Low-Power Instrument Transformers. In Instrument Transformers; International Standardization Organization: Geneva, Switzerland, 2016. [Google Scholar]
- IEC 61869-10. Part 10: Additional Requirements for Low-Power Passive Current Transformers. In Instrument Transformers; International Standardization Organization: Geneva, Switzerland, 2018. [Google Scholar]
- IEC 61869-11. Part 11: Additional Requirements for Low-Power Passive Voltage Transformers. In Instrument Transformers; International Standardization Organization: Geneva, Switzerland, 2018. [Google Scholar]
- Chakrabarti, S.; Kyriakides, E.; Bi, T.; Cai, D.; Terzija, V. Measurements get together. IEEE Power Energy Mag.
**2009**, 7, 41–49. [Google Scholar] [CrossRef] - Available online: https://www.ge.com/news/press-releases/ge-power-deliver-worlds-largest-wide-area-monitoring-system-wams (accessed on 14 May 2021).
- Hojabri, M.; Dersch, U.; Papaemmanouil, A.; Bosshart, P. A Comprehensive Survey on Phasor Measurement Unit Applications in Distribution Systems. Energies
**2019**, 12, 4552. [Google Scholar] [CrossRef][Green Version] - Mingotti, A.; Peretto, L.; Tinarelli, R. Accuracy evaluation of an equivalent synchronization method for assessing the time reference in power networks. IEEE Trans. Instrum. Meas.
**2018**, 67, 600–606. [Google Scholar] [CrossRef] - Mingotti, A.; Peretto, L.; Tinarelli, R. An equivalent synchronization for phasor measurements in power networks. In Proceedings of the AMPS 2017—IEEE International Workshop on Applied Measurements for Power Systems, Liverpool, UK, 20–22 September 2017. [Google Scholar]
- Almas, M.S.; Vanfretti, L. Impact of time-synchronization signal loss on PMU-based WAMPAC applications. In Proceedings of the IEEE Power and Energy Society General Meeting, Boston, MA, USA, 17–21 July 2016. [Google Scholar]
- Castello, P.; Muscas, C.; Pegoraro, P.A.; Sulis, S. Trustworthiness of PMU data in the presence of synchronization issues. In Proceedings of the 2018 IEEE International Instrumentation and Measurement Technology Conference: Discovering New Horizons in Instrumentation and Measurement, Houston, TX, USA, 14–17 May 2018; pp. 1–5. [Google Scholar]
- Shereen, E.; Delcourt, M.; Barreto, S.; Dan, G.; Le Boudec, J.; Paolone, M. Feasibility of time-synchronization attacks against PMU-based state estimation. IEEE Trans. Instrum. Meas.
**2020**, 69, 3412–3427. [Google Scholar] [CrossRef] - Abdolkhalig, A.; Zivanovic, R. Evaluation of IEC 61850-9-2 samples loss on total vector error of an estimated phasor. In Proceedings of the 2013 IEEE Student Conference on Research and Development, Putrajaya, Malaysia, 16–17 December 2013; pp. 269–274. [Google Scholar]
- Dickerson, W. Effect of PMU analog input section performance on frequency and ROCOF estimation error. In Proceedings of the 2015 IEEE International Workshop on Applied Measurements for Power Systems, Aachen, Germany, 23–25 September 2015. [Google Scholar]
- Zhang, J.; Tang, L.; Mingotti, A.; Peretto, L.; Wen, H. Analysis of white noise on power frequency estimation by DFT-based frequency shifting and filtering algorithm. IEEE Trans. Instrum. Meas.
**2020**, 69, 4125–4133. [Google Scholar] [CrossRef] - Sira, M.; Maslan, S.; Zachovalova, V.N.; Crotti, G.; Giordano, D. Modelling of PMU uncertainty by means of Monte Carlo method. In Proceedings of the Conference on Precision Electromagnetic Measurements, Ottawa, ON, Canada, 10–15 July 2016. [Google Scholar]
- Lixia, M.; Muscas, C.; Sulis, S. On the accuracy specifications of phasor measurement units. In Proceedings of the 2010 IEEE International Instrumentation and Measurement Technology Conference, Austin, TX, USA, 3–6 May 2010; pp. 1435–1440. [Google Scholar]
- Yang, G.Y.; Martin, K.E.; Østergaard, J. Investigation of PMU performance under TVE criterion. In Proceedings of the 2010 5th International Conference on Critical Infrastructure, Beijing, China, 20–22 September 2010. [Google Scholar]
- Singh, R.S.; Hooshyar, H.; Vanfretti, L. Assessment of time synchronization requirements for phasor measurement units. In Proceedings of the 2015 IEEE Eindhoven PowerTech, Eindhoven, The Netherlands, 29 June–2 July 2015. [Google Scholar]
- Chakrabarti, S.; Kyriakides, E. PMU measurement uncertainty considerations in WLS state estimation. IEEE Trans. Power Syst.
**2009**, 24, 1062–1071. [Google Scholar] [CrossRef][Green Version] - Mingotti, A.; Peretto, L.; Tinarelli, R. Uncertainty analysis of an equivalent synchronization method for phasor measurements. IEEE Trans. Instrum. Meas.
**2018**, 67, 2444–2452. [Google Scholar] [CrossRef] - Mingotti, A.; Peretto, L.; Tinarelli, R. A Closed-form Expression to Estimate the Uncertainty of THD Starting from the LPIT Accuracy Class. Sensors
**2020**, 20, 1804. [Google Scholar] [CrossRef] [PubMed][Green Version] - Mingotti, A.; Baldi, A.; Peretto, L.; Tinarelli, R. A general easy-to-use expression for uncertainty evaluation in residual voltage measurement. IEEE Trans. Instrum. Meas.
**2020**, 69, 1576–1584. [Google Scholar] [CrossRef] - Steinmetz, C.P. Theory and Calculation of Alternating Current Phenomena Vol. 4; McGraw-Hill Book Company, Inc.: New York, NY, USA, 1916. [Google Scholar]
- IEC 60255-118-1:2018. Measuring Relays and Protection Equipment, Part 118-1: Synchrophasor for Power Systems in Measurements; International Standardization Organization: Geneva, Switzerland, 2018. [Google Scholar]
- ISO/IEC Guide 98-3:2008. Uncertainty of Measurement—Part 3: Guide to the Expression of Uncertainty in Measurement (GUM:1995); International Standardization Organization: Geneva, Switzerland, 2008. [Google Scholar]
- Evans, M.; Hastings, N.; Peacock, B. “Chi Distribution”. §8.3 in Statistical Distributions, 3rd ed.; Wiley: New York, NY, USA, 2000; p. 57. [Google Scholar]
- Shang, Y. A central limit theorem for randomly indexed 𝑚-dependent random variables. Filomat
**2012**, 26, 713–717. [Google Scholar] [CrossRef][Green Version] - Paris, J.F. Nakagami-q (Hoyt) distribution function with applications. IEEE Electron. Lett.
**2009**, 45, 210–211. [Google Scholar] [CrossRef] - ISO/IEC Guide 98-3/Suppl. 1:2008. “Evaluation of Measurement Data—Supplement 1 to the “Guide to the Expression of Uncertainty in Measurement”—Propagation of Distributions Using a Monte Carlo Method”; International Standardization Organization: Geneva, Switzerland, 2008. [Google Scholar]

RV | Distribution | $\mathit{\mu}$ | ${\mathit{\sigma}}^{2}$ |
---|---|---|---|

$\mathit{g}$ | Uniform | 0 | ${G}^{2}/3$ |

$\mathit{o}$ | Uniform | 0 | ${O}^{2}/3$ |

$\mathit{l}$ | Uniform | 0 | ${L}^{2}/3$ |

$\mathit{r}$ | Uniform | 0 | ${R}^{2}/3$ |

$\mathsf{\Delta}\mathit{\psi}$ | Uniform | $\mathsf{\Delta}\psi /2$ | $\mathsf{\Delta}{\psi}^{2}/12$ |

Test | $\mathsf{\Delta}\mathit{\psi}\text{}\left(\mathbf{Rad}\right)$ | $\mathit{G}\text{}(\u2013)$ | $\mathit{L}\text{}\left(\mathbf{V}\right)$ | $\mathit{R}\text{}\left(\mathbf{V}\right)$ | $\mathit{X}\text{}\left(\mathbf{V}\right)$ |
---|---|---|---|---|---|

# 1 | $6\times {10}^{-4}$ | $2\times {10}^{-4}$ | $1.22\times {10}^{-3}$ | $3.66\times {10}^{-4}$ | 7 |

# 2 | $6\times {10}^{-4}$ | $2\times {10}^{-4}$ | $1.22\times {10}^{-3}$ | $3.66\times {10}^{-4}$ | 3 |

# 3 | $6\times {10}^{-4}$ | $2\times {10}^{-4}$ | $1.22\times {10}^{-3}$ | $3.66\times {10}^{-4}$ | 1 |

# 4 | $6\times {10}^{-4}$ | $2\times {10}^{-4}$ | $1.22\times {10}^{-3}$ | $3.66\times {10}^{-4}$ | 0.1 |

# 5 | $6\times {10}^{-3}$ | $2\times {10}^{-4}$ | $1.22\times {10}^{-3}$ | $3.66\times {10}^{-4}$ | 7 |

# 6 | $6\times {10}^{-4}$ | $2\times {10}^{-3}$ | $1.22\times {10}^{-3}$ | $3.66\times {10}^{-4}$ | 7 |

# 7 | $6\times {10}^{-4}$ | $2\times {10}^{-4}$ | $1.22\times {10}^{-2}$ | $3.66\times {10}^{-4}$ | 7 |

# 8 | $6\times {10}^{-4}$ | $2\times {10}^{-4}$ | $1.22\times {10}^{-3}$ | $3.66\times {10}^{-3}$ | 7 |

# 9 | $6\times {10}^{-3}$ | $2\times {10}^{-3}$ | $1.22\times {10}^{-2}$ | $3.66\times {10}^{-3}$ | 7 |

# 10 | $1.2\times {10}^{-2}$ | $2\times {10}^{-4}$ | $1.22\times {10}^{-3}$ | $3.66\times {10}^{-4}$ | 7 |

# 11 | $6\times {10}^{-4}$ | $4\times {10}^{-3}$ | $1.22\times {10}^{-3}$ | $3.66\times {10}^{-4}$ | 7 |

# 12 | $6\times {10}^{-4}$ | $2\times {10}^{-4}$ | $2.44\times {10}^{-2}$ | $3.66\times {10}^{-4}$ | 7 |

# 13 | $6\times {10}^{-4}$ | $2\times {10}^{-4}$ | $1.22\times {10}^{-3}$ | $7.32\times {10}^{-3}$ | 7 |

# 14 | $1.2\times {10}^{-2}$ | $4\times {10}^{-3}$ | $2.44\times {10}^{-2}$ | $7.32\times {10}^{-3}$ | 7 |

Test | ${\mathit{\mu}}_{TVE}$(−) | ${\mathit{\sigma}}_{TVE}$(−) | ${\widehat{\mathit{\mu}}}_{TVE}$(−) | ${\widehat{\mathit{\sigma}}}_{TVE}$(−) |

# 1 | $3.3\times {10}^{-4}$ | $1.6\times {10}^{-4}$ | $3.4\times {10}^{-4}$ | $1.4\times {10}^{-4}$ |

# 2 | $3.4\times {10}^{-4}$ | $1.6\times {10}^{-4}$ | $3.5\times {10}^{-4}$ | $1.5\times {10}^{-4}$ |

# 3 | $4.3\times {10}^{-4}$ | $2.2\times {10}^{-4}$ | $4.3\times {10}^{-4}$ | $2.1\times {10}^{-4}$ |

# 4 | $2.8\times {10}^{-3}$ | $1.5\times {10}^{-3}$ | $2.8\times {10}^{-3}$ | $1.5\times {10}^{-3}$ |

# 5 | $3.0\times {10}^{-3}$ | $1.7\times {10}^{-3}$ | $3.1\times {10}^{-3}$ | $1.5\times {10}^{-3}$ |

# 6 | $1.1\times {10}^{-3}$ | $5.3\times {10}^{-4}$ | $1.1\times {10}^{-3}$ | $4.7\times {10}^{-4}$ |

# 7 | $5.1\times {10}^{-4}$ | $2.6\times {10}^{-4}$ | $5.1\times {10}^{-4}$ | $2.6\times {10}^{-4}$ |

# 8 | $3.3\times {10}^{-4}$ | $1.6\times {10}^{-4}$ | $3.4\times {10}^{-4}$ | $1.4\times {10}^{-4}$ |

# 9 | $3.3\times {10}^{-3}$ | $1.6\times {10}^{-3}$ | $3.4\times {10}^{-3}$ | $1.4\times {10}^{-3}$ |

# 10 | $6.0\times {10}^{-3}$ | $3.4\times {10}^{-3}$ | $6.3\times {10}^{-3}$ | $3.0\times {10}^{-3}$ |

# 11 | $2.1\times {10}^{-3}$ | $1.1\times {10}^{-3}$ | $2.1\times {10}^{-3}$ | $1.0\times {10}^{-3}$ |

# 12 | $8.6\times {10}^{-4}$ | $4.5\times {10}^{-4}$ | $8.6\times {10}^{-4}$ | $4.5\times {10}^{-4}$ |

# 13 | $3.3\times {10}^{-4}$ | $1.6\times {10}^{-4}$ | $3.4\times {10}^{-4}$ | $1.4\times {10}^{-4}$ |

# 14 | $6.6\times {10}^{-3}$ | $3.1\times {10}^{-3}$ | $6.8\times {10}^{-3}$ | $2.9\times {10}^{-3}$ |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Mingotti, A.; Costa, F.; Peretto, L.; Tinarelli, R. Closed-Form Expressions to Estimate the Mean and Variance of the Total Vector Error. *Energies* **2021**, *14*, 4641.
https://doi.org/10.3390/en14154641

**AMA Style**

Mingotti A, Costa F, Peretto L, Tinarelli R. Closed-Form Expressions to Estimate the Mean and Variance of the Total Vector Error. *Energies*. 2021; 14(15):4641.
https://doi.org/10.3390/en14154641

**Chicago/Turabian Style**

Mingotti, Alessandro, Federica Costa, Lorenzo Peretto, and Roberto Tinarelli. 2021. "Closed-Form Expressions to Estimate the Mean and Variance of the Total Vector Error" *Energies* 14, no. 15: 4641.
https://doi.org/10.3390/en14154641