# Adaptive Extended Kalman Filter with Correntropy Loss for Robust Power System State Estimation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Maximum Correntropy Criteria

#### 2.2. Review of Extended Kalman Filter

## 3. Adaptive Extended Kalman Filter With Correntropy Loss

#### 3.1. Extended Kalman Filter with Correntropy Loss

#### 3.2. Adaptive Extended Kalman Filter with Correntropy Loss

## 4. Adaptive Extended Kalman Filter With Correntropy Loss for PSDSE

#### 4.1. Power System Dynamic Model

#### 4.2. Adaptive Extended Kalman Filter with Correntropy Loss for Power System Forecasting-Aided State Estimation

- 1)
- Select the appropriate initial parameters: a proper kernel bandwidth $\sigma $ and a small positive $\epsilon $; Set an initial state value ${\widehat{x}}_{0|0}$ and corresponding covariance matrix ${\mathbf{P}}_{0|0}$; Let k = 1;
- 2)
- Use Equations (8) and (9) to calculate the ${\widehat{x}}_{k|k-1}$ and ${\mathbf{P}}_{k|k-1}$, and obtain the ${\mathbf{B}}_{p,k|k-1}$ by Cholesky decomposition;
- 3)
- Let k = 1 and ${\widehat{x}}_{k|k,0}={\widehat{x}}_{k|k-1}$, where ${\widehat{x}}_{k|k,t}$ stands for the estimated state at the fixed-point iteration k;
- 4)
- Calculate the state transition function using (45)–(47) and the Jacobian matrix ${H}_{k}$ using (51)–(60);
- 5)
- Get the estimates state ${\widehat{x}}_{k|k,t}$ by Equations (61)–(69);$${\widehat{x}}_{k|k,t}={\widehat{x}}_{k|k-1}+{\tilde{K}}_{k}\left({y}_{k}-{H}_{k}{\widehat{x}}_{k|k-1}\right)$$$${\tilde{K}}_{k}={\tilde{P}}_{k|k-1}{H}_{k}^{T}{\left({H}_{k}{\tilde{P}}_{k|k-1}{H}_{k}^{T}+{\tilde{R}}_{k}\right)}^{-1}$$$${\tilde{P}}_{k|k-1}={B}_{p,k|k-1}^{}{\tilde{C}}_{x,k}^{-1}{B}_{p,k|k-1}^{T}$$$${\tilde{R}}_{k}={B}_{r,k}{\tilde{C}}_{y,k}^{-1}{B}_{r,k}^{T}$$$${\tilde{C}}_{x,k}=diag\left({G}_{\sigma}\left({\tilde{e}}_{1,k}\right),\dots ,{G}_{\sigma}\left({\tilde{e}}_{n,k}\right)\right)$$$${\tilde{C}}_{y,k}=diag\left({G}_{\sigma}\left({\tilde{e}}_{n+1,k}\right),\dots ,{G}_{\sigma}\left({\tilde{e}}_{n+m,k}\right)\right)$$$${\tilde{e}}_{i,k}={d}_{i,k}-{w}_{i,k}{\widehat{x}}_{k|k,t-1}$$$${\tilde{R}}_{k}={R}_{k}{\left({H}_{k}{P}_{k|k-1}{H}_{k}^{T}+{R}_{k}\right)}^{-1}{R}_{k}+{H}_{k}{P}_{k}{H}_{k}^{T}$$$${\tilde{Q}}_{k}={K}_{k}\left({H}_{k}{P}_{k|k-1}{H}_{k}^{T}+{R}_{k}\right){K}_{k}^{T}$$
- 6)
- Compare the estimation of the current step and the estimation of the last step. If (70) holds, let ${\widehat{x}}_{k|k}={\widehat{x}}_{k|k,t}$ and continue to 7). Otherwise, $t+1\to t$, and go back to 5);$$\frac{\left|\right|{\widehat{x}}_{k|k,t}-{\widehat{x}}_{k|k,t-1}\left|\right|}{\left|\right|{\widehat{x}}_{k|k,t-1}\left|\right|}\le \epsilon $$
- 7)
- Moreover, the posterior matrix is updated as (71), $k+1\to k$ and go back to 2).$${P}_{k|k}=\left(I-{\tilde{K}}_{k}{H}_{k}\right){P}_{k|k-1}{\left(I-{\tilde{K}}_{k}{H}_{k}\right)}^{T}+{\tilde{K}}_{k}{R}_{k}{\tilde{K}}_{k}^{T}$$

## 5. Results

#### 5.1. Case 1: Gaussian Measurement Noise Environment

#### 5.2. Case 2: Gaussian Mixture Measurement Noise Environment

#### 5.3. Case 3: Laplace and Gaussian Mixture Measurement Noise Environment

#### 5.4. Case 4: the Nonlinear Variation of Loads

#### 5.5. Case 5: in Presence of Outliers

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**The overall performance of all algorithms in standard IEEE 14-bus under Gaussian mixture measurement noise environment.

**Figure 3.**The overall performance of all algorithms in standard IEEE 30-bus under Gaussian mixture measurement noise environment.

**Figure 6.**The overall performance of all algorithms in standard IEEE 14-bus under Laplace and Gaussian mixture measurement noise environment.

**Figure 7.**The overall performance of all algorithms in standard IEEE 30-bus under Laplace and Gaussian mixture measurement noise environment.

**Figure 8.**The mean absolute error of voltage angle of no.3 bus in IEEE 30-bus in different variation trend of load.

**Figure 9.**The root mean square error of voltage angle of no.3 bus in IEEE 30-bus in different variation trend of load.

**Figure 10.**The true value of voltage amplitude of no.3 bus in IEEE 30-bus test system and estimated value of other algorithms when the loads change follows a random fluctuation.

**Figure 11.**The true value of voltage angle of no.3 bus in IEEE 30-bus test system and estimated value of other algorithms when the loads change follows a random fluctuation.

**Figure 12.**The overall performance of all algorithms in standard IEEE 30-bus in presence of outliers.

EKF | UKF | A-EKF | MCC-EKF | AMCC-EKF | |
---|---|---|---|---|---|

Index J (p.u.) | 0.39 | 0.29 | 0.31 | 0.25 | 0.16 |

**Table 2.**The average overall performance of all algorithms in standard IEEE 14-bus under Gaussian mixture measurement noise environment.

EKF | UKF | A-EKF | MCC-EKF | AMCC-EKF | |
---|---|---|---|---|---|

Index J (p.u.) | 0.53 | 0.41 | 0.49 | 0.33 | 0.23 |

**Table 3.**The average overall performance of all algorithms in standard IEEE 30-bus under Gaussian mixture measurement noise environment.

EKF | UKF | A-EKF | MCC-EKF | AMCC-EKF | |
---|---|---|---|---|---|

Index J (p.u.) | 0.55 | 0.41 | 0.48 | 0.32 | 0.24 |

**Table 4.**The average overall performance of all algorithms in standard IEEE 30-bus under Laplace and Gaussian mixture measurement noise environment.

EKF | UKF | A-EKF | MCC-EKF | AMCC-EKF | |
---|---|---|---|---|---|

Index J (p.u.) | 0.65 | 0.48 | 0.53 | 0.36 | 0.28 |

**Table 5.**The average overall performance of all algorithms in standard IEEE 30-bus under Laplace and Gaussian mixture measurement noise environment.

EKF | UKF | A-EKF | MCC-EKF | AMCC-EKF | |
---|---|---|---|---|---|

Index J (p.u.) | 0.59 | 0.52 | 0.46 | 0.38 | 0.23 |

EKF | UKF | A-EKF | MCC-EKF | AMCC-EKF | |
---|---|---|---|---|---|

MAE | 0.06 | 0.05 | 0.05 | 0.03 | 0.01 |

RMSE | 0.25 | 0.21 | 0.23 | 0.18 | 0.12 |

**Table 7.**The average overall performance of all algorithms in standard IEEE 30-bus in presence of outliers.

EKF | UKF | A-EKF | MCC-EKF | AMCC-EKF | |
---|---|---|---|---|---|

Index J (p.u.) | 0.56 | 0.46 | 0.48 | 0.36 | 0.24 |

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**MDPI and ACS Style**

Zhang, Z.; Qiu, J.; Ma, W.
Adaptive Extended Kalman Filter with Correntropy Loss for Robust Power System State Estimation. *Entropy* **2019**, *21*, 293.
https://doi.org/10.3390/e21030293

**AMA Style**

Zhang Z, Qiu J, Ma W.
Adaptive Extended Kalman Filter with Correntropy Loss for Robust Power System State Estimation. *Entropy*. 2019; 21(3):293.
https://doi.org/10.3390/e21030293

**Chicago/Turabian Style**

Zhang, Zhiyu, Jinzhe Qiu, and Wentao Ma.
2019. "Adaptive Extended Kalman Filter with Correntropy Loss for Robust Power System State Estimation" *Entropy* 21, no. 3: 293.
https://doi.org/10.3390/e21030293