# 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

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

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**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