# Laplace-Domain Hybrid Distribution Model Based FDIA Attack Sample Generation in Smart Grids

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## Abstract

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## 1. Introduction

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- We propose an efficient FDIA attack sample generation method. Our method can quickly construct large-scale attack training samples for FDIA detection models, thereby solving the problem of sparse attack samples.
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- By analyzing the measured data changes of each node of the power system, individual Laplacian distribution can be established sequentially according to the change of the sensing measurement data of each node. A Laplace-domain hybrid distribution can be constructed to generate FDIA attack samples by combining multiple symmetric Laplace distribution models, which can better improve the concealment of the attack sample.
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- We conduct a large number of experiments by different detection models to verify that our attack samples are more deceptive than other attack samples. The experimental results demonstrate that our method outperforms traditional FDIA attack sample construction schemes in terms of attack strength and covert capability, while guaranteeing a low computational complexity.

## 2. False Data Injection Attack

## 3. Proposed Method

#### 3.1. The Framework of Proposed Scheme

#### 3.2. Laplace Distribution and Observation

#### 3.3. Hybrid Laplace Distribution Model

- (1)
- Firstly, determine the initial values of the model parameters according to the characteristics of the sensing measurement data, then the model parameter vector is ${\vartheta}_{k}^{\left(0\right)}=\left({b}_{1}^{\left(0\right)},{b}_{2}^{\left(0\right)},\cdots ,{b}_{k}^{\left(0\right)},{s}_{1}^{\left(0\right)},{s}_{2}^{\left(0\right)},\cdots ,{s}_{k}^{\left(0\right)},{\mu}_{1}^{\left(0\right)},{\mu}_{2}^{\left(0\right)},\cdots ,{\mu}_{k}^{\left(0\right)}\right)$. Correspondingly, the posterior probability of sample ${X}_{1},{X}_{2},\cdots {X}_{n}$ (where ${X}_{i}\in f({s}_{i}^{\left(0\right)},{\mu}_{i}^{\left(0\right)})$ ) under this initial condition can be expressed as:$${p}_{tj}^{\left(0\right)}=\frac{g\left({x}_{t},{b}_{j}^{\left(0\right)},{\mu}_{j}^{\left(0\right)},{s}_{j}^{\left(0\right)}\right)}{{\displaystyle {\displaystyle \sum _{i=1}^{k}}}g\left({x}_{t},{b}_{t}^{\left(0\right)},{\mu}_{t}^{\left(0\right)},{s}_{t}^{\left(0\right)}\right)}.$$Considering that the posterior probability of each component of the linear component is calculated circularly: For each component, the probability density of the sample data ${X}_{i}$ belonging to the component is calculated by using the initialized parameters ${s}_{i}$ and ${\mu}_{i}$, where the posterior probability can be calculated by matrix multiplication with the prior probability ${b}_{i}$, and the posterior probability meets the normalization condition, that is, $\sum _{j=1}^{k}}{p}_{tj}^{\left(0\right)}=1$. For any group of $\sum _{j=1}^{k}}{b}_{j}=1$, the assignment of samples to k components with ${p}_{tj}^{\left(0\right)}$ can be completed sequentially under the initial value ${\vartheta}_{k}^{\left(0\right)}$. Subsequently, the parameters of each component distribution can be obtained by the expectation algorithm.$$\left\{\begin{array}{c}{b}_{j}^{\left(1\right)}=\frac{1}{n}{\displaystyle \sum _{t=1}^{n}}{p}_{tj}^{\left(0\right)}\hfill \\ {\mu}_{j}^{\left(1\right)}=\frac{{\displaystyle \sum _{t=1}^{n}}{p}_{tj}^{\left(0\right)}{x}_{t}}{{\displaystyle \sum _{t=1}^{n}}{p}_{tj}^{\left(0\right)}}\hfill \\ {s}_{j}^{\left(1\right)}=\frac{\sqrt{{\displaystyle \sum _{t=1}^{n}}{p}_{tj}^{\left(0\right)}{({x}_{t}-{\mu}_{j}^{\left(1\right)})}^{2}}}{{\displaystyle \sum _{t=1}^{n}}{p}_{tj}^{\left(0\right)}}\hfill \end{array}\right.$$
- (2)
- In order to find the best model parameters, we need to introduce the maximum likelihood estimation (MLE) on sample ${X}_{1},{X}_{2},\cdots {X}_{n}$.$$L\left({\vartheta}_{k}\right)={\displaystyle \prod _{i=1}^{n}}{\displaystyle \sum _{t=1}^{k}}g\left({x}_{i},{b}_{t},{\mu}_{t},{s}_{t}\right)$$In order to maximize the likelihood function $L\left({\vartheta}_{k}\right)$ under the conditions of formula ${p}_{tj}^{}=\frac{g\left({x}_{t},{b}_{j}^{},{\mu}_{j}^{},{s}_{j}^{}\right)}{{\displaystyle \sum _{i=1}^{k}}g\left({x}_{t},{b}_{t}^{},{\mu}_{t}^{},{s}_{t}^{}\right)}$, we use the derivative method and make the derivative zero to solve the corresponding ${s}_{j}$ and ${\mu}_{j}$.
- (3)
- Finally, after m rounds of iteration, the result of $m+1$ rounds can be obtained, that is, the solution of ${b}_{j}^{(m+1)}$, ${\mu}_{j}^{(m+1)}$, ${s}_{j}^{(m+1)}$.$$\left\{\begin{array}{c}{b}_{j}^{(m+1)}=\frac{1}{n}{\displaystyle \sum _{t=1}^{n}}{p}_{tj}^{\left(m\right)}\hfill \\ {\mu}_{j}^{(m+1)}=\frac{{\displaystyle \sum _{t=1}^{n}}{p}_{tj}^{\left(m\right)}{x}_{t}}{{\displaystyle \sum _{t=1}^{n}}{p}_{tj}^{\left(m\right)}}\hfill \\ {s}_{j}^{(m+1)}=\frac{\sqrt{{\displaystyle \sum _{t=1}^{n}}{p}_{tj}^{\left(m\right)}{({x}_{t}-{\mu}_{j}^{(m+1)})}^{2}}}{{\displaystyle \sum _{t=1}^{n}}{p}_{tj}^{\left(m\right)}}\hfill \end{array}\right.$$In the iterative algorithm for maximum likelihood estimation of hybrid model parameters, the likelihood function is monotonically increasing, i.e., $L\left({\vartheta}_{k}^{\left(m+1\right)}\right)\u2a7eL\left({\vartheta}_{k}^{\left(m\right)}\right)$. This means that an $L\left({\vartheta}_{k}\right)$ maximum point can always be found in the iteration process, and the corresponding threshold $\epsilon $ can be given generally. When $\left|L\left({\vartheta}_{k}^{\left(m+1\right)}\right)-L\left({\vartheta}_{k}^{\left(m\right)}\right)\right|\u2a7d\epsilon $, the Likelihood function is the largest, and accordingly the iteration should stop to obtain the maximum likelihood estimation parameters.

## 4. Experimental Results and Discussions

#### 4.1. Attack Data Generation

#### 4.2. Experimental Setup

#### 4.3. Experimental Results and Discussions

#### 4.4. Impact of Noise Error

#### 4.5. Time Complexity Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Actual data distribution and standard Laplace distribution for different node data (node 1, node 2, node 3, node 4) in the IEEE 14-node system.

**Figure 3.**Variation distribution comparison for normal samples and generated attack samples. (

**a**) Normal measurement data variation. (

**b**) Generated measurement data variation.

**Figure 4.**Accuracy comparison of two attack sample generation schemes on the IEEE 14- and IEEE 118-node systems. In this experiment, CNN-based FDIA detection model and LSTM-based FDIA detection model are used to provide the testing results. (

**a**) CNN-based detection over the IEEE 14 system. (

**b**) LSTM-based detection over the IEEE 14 system. (

**c**) CNN-based detection over the IEEE 118 system. (

**d**) LSTM-based detection over the IEEE 118 system.

**Figure 6.**Comparison of two attack sample generation schemes under different noise levels. (

**a**) CNN-based detection model. (

**b**) LSTM-based detection model.

**Table 1.**Performance comparison of two attack sample generation schemes, the traditional FDIA sample generation and FDIA sample generation based on hybrid Laplace model (LMM-FDIA). In this test, CNN-based FDIA attack detection model is used over the IEEE 14-node system.

Attack Level | Attack Strength | Traditional Scheme | LMM-FDIA Based Scheme | ||||
---|---|---|---|---|---|---|---|

Precision | Recall | ${\mathit{F}}_{\mathbf{1}}$-Score | Precision | Recall | ${\mathit{F}}_{\mathbf{1}}$-Score | ||

Weak Attacks | 2% | 0.5485 | 0.6137 | 0.5613 | 0.3330 | 0.4520 | 0.3689 |

5% | 0.6243 | 0.6869 | 0.6408 | 0.4205 | 0.6950 | 0.5179 | |

10% | 0.7021 | 0.7132 | 0.7004 | 0.6556 | 0.6148 | 0.5499 | |

Moderate Attacks | 15% | 0.7813 | 0.7954 | 0.7826 | 0.6636 | 0.5541 | 0.5574 |

20% | 0.8372 | 0.8424 | 0.8358 | 0.7351 | 0.6776 | 0.6764 | |

25% | 0.8478 | 0.8732 | 0.8565 | 0.7794 | 0.7605 | 0.7595 | |

Strong Attacks | 30% | 0.8995 | 0.9094 | 0.8998 | 0.8280 | 0.8143 | 0.7981 |

40% | 0.9494 | 0.9452 | 0.9462 | 0.8541 | 0.8021 | 0.8090 | |

50% | 0.9754 | 0.9708 | 0.9725 | 0.9281 | 0.9186 | 0.9213 |

**Table 2.**Performance comparison of two attack sample generation schemes, the traditional FDIA sample generation and FDIA sample generation based on hybrid Laplace model (LMM-FDIA). In this test, LSTM-based FDIA attack detection model is used over the IEEE 14-node system.

Attack Level | Attack Strength | Traditional Scheme | LMM-FDIA Based Scheme | ||||
---|---|---|---|---|---|---|---|

Precision | Recall | ${\mathit{F}}_{\mathbf{1}}$-Score | Precision | Recall | ${\mathit{F}}_{\mathbf{1}}$-Score | ||

Weak Attacks | 2% | 0.5399 | 0.5417 | 0.5408 | 0.4943 | 0.4686 | 0.4811 |

5% | 0.6290 | 0.6000 | 0.6141 | 0.5000 | 0.4078 | 0.4492 | |

10% | 0.6704 | 0.6842 | 0.6959 | 0.5044 | 0.4799 | 0.4854 | |

Moderate Attacks | 15% | 0.7481 | 0.8063 | 0.7761 | 0.5244 | 0.4859 | 0.5045 |

20% | 0.8208 | 0.8307 | 0.8257 | 0.5249 | 0.6261 | 0.5054 | |

25% | 0.8409 | 0.9023 | 0.8705 | 0.5662 | 0.4706 | 0.5140 | |

Strong Attacks | 30% | 0.8586 | 0.9189 | 0.8877 | 0.6058 | 0.5787 | 0.6041 |

40% | 0.9509 | 0.9421 | 0.9465 | 0.6742 | 0.6495 | 0.6616 | |

50% | 0.9831 | 0.9569 | 0.9698 | 0.7843 | 0.5867 | 0.5960 |

**Table 3.**Performance comparison of two attack sample generation schemes, the traditional FDIA sample generation and FDIA sample generation based on hybrid Laplace model (LMM-FDIA). In this test, CNN-based FDIA attack detection model is used over the IEEE 118-node system.

Attack Level | Attack Strength | Traditional Scheme | LMM-FDIA Based Scheme | ||||
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Precision | Recall | ${\mathit{F}}_{\mathbf{1}}$-Score | Precision | Recall | ${\mathit{F}}_{\mathbf{1}}$-Score | ||

Weak Attacks | 2% | 0.5176 | 0.5485 | 0.4952 | 0.4372 | 0.5854 | 0.4360 |

5% | 0.5818 | 0.5714 | 0.5650 | 0.5933 | 0.4132 | 0.3958 | |

10% | 0.7241 | 0.6903 | 0.7005 | 0.7976 | 0.4121 | 0.5355 | |

Moderate Attacks | 15% | 0.8023 | 0.7874 | 0.7912 | 0.8586 | 0.5338 | 0.6525 |

20% | 0.8364 | 0.8263 | 0.8281 | 0.8619 | 0.6291 | 0.7165 | |

25% | 0.8746 | 0.8481 | 0.8583 | 0.9072 | 0.6810 | 0.7730 | |

Strong Attacks | 30% | 0.8926 | 0.8952 | 0.8917 | 0.9464 | 0.7250 | 0.8162 |

40% | 0.9374 | 0.9376 | 0.9364 | 0.9189 | 0.7970 | 0.8504 | |

50% | 0.9621 | 0.9693 | 0.9651 | 0.9272 | 0.9047 | 0.9157 |

**Table 4.**Performance comparison of two attack sample generation schemes, the traditional FDIA sample generation and FDIA sample generation based on hybrid Laplace model (LMM-FDIA). In this test, LSTM-based FDIA attack detection model is used over the IEEE 118-node system.

Attack Level | Attack Strength | Traditional Scheme | LMM-FDIA Based Scheme | ||||
---|---|---|---|---|---|---|---|

Precision | Recall | ${\mathit{F}}_{\mathbf{1}}$-Score | Precision | Recall | ${\mathit{F}}_{\mathbf{1}}$-Score | ||

Weak Attacks | 2% | 0.5328 | 0.1870 | 0.2769 | 0.4966 | 0.6444 | 0.5610 |

5% | 0.5718 | 0.6027 | 0.5869 | 0.5062 | 0.9808 | 0.6678 | |

10% | 0.7518 | 0.6126 | 0.6751 | 0.6737 | 0.2288 | 0.3417 | |

Moderate Attacks | 15% | 0.8008 | 0.7654 | 0.7827 | 0.8398 | 0.4703 | 0.6029 |

20% | 0.8728 | 0.7549 | 0.8096 | 0.9037 | 0.6748 | 0.7726 | |

25% | 0.8672 | 0.8735 | 0.8703 | 0.8817 | 0.7130 | 0.7884 | |

Strong Attacks | 30% | 0.9129 | 0.8708 | 0.8914 | 0.8338 | 0.9102 | 0.8703 |

40% | 0.8995 | 0.9670 | 0.9320 | 0.9499 | 0.8509 | 0.8977 | |

50% | 0.9555 | 0.9769 | 0.9661 | 0.9800 | 0.9373 | 0.9581 |

Attack Method | Characteristics | Challenges |
---|---|---|

FDIA [20] | Effectively avoiding BDD detection Low model complexity | Easy to detect by DL model Simple construction of attack vector |

AFDIA [26] | Effectively avoiding BDD detection High success rate | Poor robustness Easy to detect by DL model Large amount of model parameters |

D-FDIA [27] | Effectively avoiding BDD detection Low computational burden | Easy to detect by DL model Poor concealment of attack vector |

SG-FDIA [28] | Effectively avoiding BDD detection High time efficiency | Easy to detect by DL model Poor robustness |

M-AFDIA [29] | Effectively avoiding DL model detection Strong concealment of attack vectors | Easy to detect by BDD Long running time |

S-AFDIA [29] | Effectively avoiding BDD and DL model detection | High model complexity High operation cost Obtain comprehensive system information |

LMM-FDIA | Effectively avoiding BDD and DL model detection Low model complexity and running time Strong concealment of attack vector | Poor performance on small samples |

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## Share and Cite

**MDPI and ACS Style**

Wu, Y.; Zu, T.; Guo, N.; Zhu, Z.; Li, F.
Laplace-Domain Hybrid Distribution Model Based FDIA Attack Sample Generation in Smart Grids. *Symmetry* **2023**, *15*, 1669.
https://doi.org/10.3390/sym15091669

**AMA Style**

Wu Y, Zu T, Guo N, Zhu Z, Li F.
Laplace-Domain Hybrid Distribution Model Based FDIA Attack Sample Generation in Smart Grids. *Symmetry*. 2023; 15(9):1669.
https://doi.org/10.3390/sym15091669

**Chicago/Turabian Style**

Wu, Yi, Tong Zu, Naiwang Guo, Zheng Zhu, and Fengyong Li.
2023. "Laplace-Domain Hybrid Distribution Model Based FDIA Attack Sample Generation in Smart Grids" *Symmetry* 15, no. 9: 1669.
https://doi.org/10.3390/sym15091669