# Detection and Prevention of False Data Injection Attacks in the Measurement Infrastructure of Smart Grids

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## Abstract

**:**

## 1. Introduction

#### 1.1. Power System State Estimation

**z**. The measurement vector contains the real forward powers, reactive forward powers, real backward powers, reactive backward powers, real powers injected into all the buses, reactive powers injected into all buses, voltage magnitudes, and voltage angles [20,21,22,23]. The measurement vector

**z**and the state variable

**x**have the following relationship:

**h(x)**represents the non-linear function that gives the dependencies between measured values and the state variables, and it can be found using the power system topology.

**e**represents random noise of Gaussian form with a zero mean and some known covariance.

**W**is the weighting matrix, as given in [26]. This is an unconstrained optimization problem whose first-order optimality condition is given by:

**H**represents the Jacobian matrix and $\widehat{x}$ is taken as the vector of the estimated states. An iterative process can be used for solving this non-linear equation [27].

- The voltage magnitudes of all the buses are very close to each other and they are assumed to be “1 pu”.
- The active power transmission through the transmission lines is taken as lossless, i.e., there are no losses in the transmission lines.
- The value of reactive power injected into all the buses, as well as flowing through the transmission lines, is taken as zero.
- There is a small difference in the voltage angles of two buses such that “Sin(δϕ) ≈ δϕ”

**H**is known as the Jacobian matrix of the power system topology. If the measurement vector has m values and the number of states is n, then the Jacobian matrix

**H**will have an order of “m × n”. In (4),

**x**contains the bus voltage angles.

**z**contains the values of active powers flowing through the transmission lines and injected into all the buses.

**H**is constant during each iteration of the linearization process. In the DC power flow model (4), the Jacobian matrix

**H**is constant throughout. Equation (4) will be valid for each iteration of the linearization model (3). Therefore, the same notation is adopted for both the linearized model (3) and the DC power flow model (4).

**R**represents the covariance matrix of

**e**. The estimated states, as well as the measurement vector

**z**, are used for the calculation of the measurement residue.

_{2}-norm is calculated for

**r**.

**r**) is done with the threshold

**τ**for finding the presence of bad data. The X

_{2}—test is used for the determination of the threshold

**τ**.

#### 1.2. Stealth False Data Injection (FDI) Attack

**H**is fully known to the attacker.

**H**is used for the construction of an undetectable attack. Stealth false data injection (FDI) is given as follows [17,19,22,28,29,30]:

**a**represents the vector of false data that is added to the measurement vector

**z**. The attacker hacks the data from the communication line and injects the attack vector

**a**into it, where

**a**=

**Hc**.

**H**is determined with the help of those measurements of power. The whole power system topology can be understood with the help of

**H**. The dependence of one power value on the other powers can be found using

**H**. This leads the attacker to make an undetectable attack. In fact, it tells the attacker which specific values of power the attacker will have to change with one particular change in power. To understand the whole power network, the formation of the Jacobian matrix

**H**is the most important component. The vector

**c**is multiplied by matrix

**H**and the resultant is added to the actual measurements when undertaking a stealth attack.

**c**. It is assumed that

**c**~N(0,${\sigma}_{c}^{2}$), where the false state variance is represented by ${\sigma}_{c}^{2}$.

#### 1.3. Contributions

- It was shown that the bad data filter of the state estimation was only useful for detecting bad measurement data and could not efficiently detect a stealth FDI, making the system vulnerable to all such attacks.
- A fixed dummy value model was proposed and it was shown that the false data attacks that went undetected by the bad data filter could be successfully detected.
- Since the dummy value in the fixed dummy value model is kept fixed, the intruder may obtain a clue about it and may change the measurement, keeping the same dummy value, therefore causing this model to be vulnerable to FDI attacks. To address the vulnerability of the fixed dummy value model, another technique for the variable dummy value model was also proposed, which was shown to successfully counter such attacks.

## 2. Literature Review

_{1}norm and nuclear norm. This mixed norm optimization problem was solved using the augmented Lagrange method of multipliers in order to obtain a good convergence rate. In [22], the false data detection problem was considered a matrix separation problem. FDI attacks are sparse in nature. To separate the states of the power system from the anomalies, a mechanism was developed. The problem was solved using two methods, namely, low-rank matrix factorization and nuclear norm minimization.

## 3. Proposed Model

**y**= 1, 2, 3,…, mt. Here, mt represents the total number of instances. ${p}_{v\left(y\right)}$ and ${q}_{v\left(y\right)}$ are the vectors containing the active and reactive powers injected to all the buses at the yth instant. Both vectors will have a dimension of 1 × b. Similarly, ${p}_{vw\left(y\right)}$ and ${q}_{vw\left(y\right)}$ denote vectors having the active and reactive powers flowing through all the transmission in the forward direction at the yth instant. Both vectors have dimensions of 1 × t. Moreover, ${p}_{wv\left(y\right)}$ and ${q}_{wv\left(y\right)}$ represent the vectors of the active and reactive powers flowing through all the transmission lines in the backward direction at the yth instant. The complete measurement vector will have a dimension of m × 1. The state vector

**x**contains the voltage magnitudes and voltage angles of all the buses. However, the Jacobian matrix will have a dimension of m × n, where m is the total number of values in the measurement vector and n is the total number of values in the state vector. The measurement vectors at all the instances can be placed together to obtain the measurement matrix as follows:

**z**. Here, ${p}_{v\left(y\right)}\left(1\right)$ represents the first entry of the vector ${p}_{v\left(y\right)}$ and ${q}_{v\left(y\right)}\left(b\right)$ is the bth entry of the vector ${q}_{v\left(y\right)}^{\prime}$. The dummy values of the power are present on the even indexes of the new measurement vector. The vectors of the dummy values containing the active and reactive powers injected to all the buses at the yth instant are ${p}_{v\left(y\right)}^{\prime}$ and ${q}_{v\left(y\right)}^{\prime}$. Similarly, other vectors containing dummy values of the active and reactive powers for transmission lines at the yth instant are denoted by ${p}_{vw\left(y\right)}^{\prime}$,${q}_{vw\left(y\right)}^{\prime}$,${p}_{wv\left(y\right)}^{\prime}$, and ${q}_{wv\left(y\right)}^{\prime}$.

_{dy}**z**will have dimensions of 2m × 1. The measurement matrix after including the dummy values will be

_{dy}**H**and its dimensions are 2m × n. There are different methods to find the Jacobian matrix. To make a stealth attack, it is necessary for the attacker to determine the Jacobian matrix. The attacker hacks both the dummy and actual values and creates a Jacobian matrix to attack the system.

_{dy}#### Fixed Dummy Value Model

**,**and ${z}_{s}\left(lq\right)$ represents the lqth entry of the sth historical measurement vector. $mt$ gives the total number of historical measurement vectors. The lth entry of the dummy values vector ${q}_{v\left(y\right)}^{\prime}$ is found by calculating the mean of the lqth entry of $mt$ historical measurement vectors. By using this procedure, the dummy values of all the reactive powers injected into the buses can be calculated. In (20) and (21), ${p}_{vw\left(y\right)}^{\prime}\left(l\right)$ and ${q}_{vw\left(y\right)}^{\prime}\left(l\right)$ represent the lth entry of each of the dummy measurement vectors ${p}_{vw\left(y\right)}^{\prime}$ and ${q}_{vw\left(y\right)}^{\prime}$

**,**respectively. ${z}_{s}\left(lpv\right)$ and ${z}_{s}\left(lqv\right)$ denote the lpvth and lqvth entries of the sth historical measurement vector, respectively. The lth entry of each of the dummy measurement vectors ${p}_{vw\left(y\right)}^{\prime}$ and ${q}_{vw\left(y\right)}^{\prime}$ is calculated by finding the mean of the lpvth and lqvth entries of $mt$ historical measurement vectors, respectively. Therefore, the dummy values of the active and reactive powers flowing through all the transmission lines can be calculated by using Equations (20) and (21), respectively.

**d**present in the control room. When the system is hacked by the attacker, 2m power values will be obtained by the attacker in ${z}_{dy}$ instead of m values. In the next step, the Jacobian matrix will be constructed by the attacker and a stealthy attack will be done in this way:

## 4. Simulations and Results of the Fixed Dummy Value Model

#### 4.1. Data Generation

#### 4.2. DC State Estimation

**c**to make the attack vector. The values of vector c were selected randomly between −1 and 1 such that it had zero mean and a variance of 2. The attack vector was added to the measurement vector to make the attack. The results of the DC state estimation are shown in Figure 2 and Figure 3. Three types of measurements are shown in Figure 2, namely, safe measurements, simple attack measurements, and the measurements for a stealth attack. A safe zone based on the threshold is also shown in the figure. The measurements outside the safe zone are considered as attacked. The results show that the safe measurement points were present in the safe zone and points of simple attacks were outside the zone. However, measurements affected by stealth attacks are also found in the safe zone, i.e., they are declared as safe by the DC state estimation. They should have to appear outside the safe zone. Therefore, DC SE is not capable of detecting stealth FDI attacks. Similarly, Figure 3 also shows the results of DC state estimation in the form of a bar graph. The residue was calculated for every measurement and the difference of that residue from the threshold is plotted along the vertical axis. For a safe measurement, the value of the difference should be positive, as the residue of that measurement should be less than the threshold. In the graph, the safe measurements are labeled with 1, measurements of simple attacks are labeled with 0, and stealth-attacked measurements are labeled with −1. The results show that the safe measurements and stealth-attacked measurements had positive values of difference. However, the value of the difference was negative for simple attacks. This means that the simple-attacked measurements were termed as attacked by the DC state estimation but measurements having stealth attacks were considered safe. Therefore, simple attacks were detected by the DC state estimation, but stealth attacks bypassed detection.

#### 4.3. AC State Estimation

#### 4.4. Fixed Dummy Value Model (Results)

#### 4.5. Limitations of Fixed Dummy Value Model

## 5. Variable Dummy Value Model

**=**[1 ${p}_{vkz}$ ${p}_{v}\left(k\right)$]

^{T}. ${\beta}_{k}$ denotes the kth parameter vector and ${\beta}_{k}$

**=**[${\beta}_{3vpk}{\beta}_{2vpk}{\beta}_{1vpk}$]

^{T}. ${\beta}_{1vpk}$, ${\beta}_{2vpk},$ and ${\beta}_{3vpk}$ are the constants to be learned for each dummy value of the active power injected into the buses. Therefore, for each dummy value, a different vector of constants is used. Depending upon the hypothesis, the cost function for the multivariate linear regression can be written as

## 6. Results of the Variable Dummy Value Model

_{1}. The constants of the equation of the line that best fit the training data were found. Constants for all the linear equations were found in this way and those equations were embedded in the meters to calculate the dummy values. Table 3 shows the actual values and the dummy values for the variable dummy value model at a single instant for the first five buses and first five transmission lines.

## 7. Discussion

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Categorization of safe measurements, simple attacks, and stealth attacks based on the threshold in a DC SE.

**Figure 4.**Categorization of safe measurements, simple attacks, and stealth attacks based on the threshold in an AC SE.

**Figure 6.**Detection results of the fixed dummy value model for simple and stealth attacks based on dummies.

**Figure 8.**Training of the multivariate linear regression model to find the constants of the equation used to calculate the dummy values of P

_{1}.

**Table 1.**Active and reactive powers injected into the first 5 buses and the active and reactive powers flowing through the first 5 transmission lines in the forward and backward directions.

Active Powers Injected into the Buses | |||

Bus No. | Actual Value (MW) | Dummy Value (MW) | |

1 | 232.11 | 196.3 | |

2 | 18.41 | 21.06 | |

3 | −93.94 | −82.22 | |

4 | −47.88 | −41.72 | |

5 | −7.58 | −6.63 | |

Reactive Powers Injected into the Buses | |||

Bus No. | Actual Value (MVAR) | Dummy Value (MVAR) | |

1 | −16.49 | −10.3 | |

2 | 30.79 | 21.73 | |

3 | 5.98 | −0.27 | |

4 | 3.9 | 3.9 | |

5 | −1.6 | −1.6 | |

Active Powers Flowing through the Transmission Lines in Forward Direction | |||

From | To | Actual Value (MW) | Dummy Value (MW) |

1 | 2 | 156.65 | 131.57 |

1 | 5 | 75.46 | 64.73 |

2 | 3 | 73.11 | 63.66 |

2 | 4 | 56.14 | 49.21 |

2 | 5 | 41.53 | 36.6 |

Reactive Powers Flowing through the Transmission Lines in Forward Direction | |||

From | To | Actual Value (MVAR) | Dummy Value (MVAR) |

1 | 2 | −20.35 | −14.04 |

1 | 5 | 3.86 | 3.74 |

2 | 3 | 3.57 | 4.71 |

2 | 4 | −1.54 | −1.62 |

2 | 5 | 1.17 | 0.81 |

Active Powers Flowing through the Transmission Lines in Backward Direction | |||

From | To | Actual Value (MW) | Dummy Value (MW) |

1 | 2 | −152.37 | −128.41 |

1 | 5 | −72.7 | −62.63 |

2 | 3 | −70.79 | −61.85 |

2 | 4 | −54.46 | −47.89 |

2 | 5 | −40.62 | −35.88 |

Reactive Powers Flowing through the Transmission Lines in Backward Direction | |||

From | To | Actual Value (MVAR) | Dummy Value (MVAR) |

1 | 2 | 27.58 | 17.83 |

1 | 5 | 2.21 | −0.38 |

2 | 3 | 1.55 | −1.68 |

2 | 4 | 3.01 | 2 |

2 | 5 | −2.1 | −2.31 |

**Table 2.**Constants used for the calculation of dummy values of active and reactive powers injected into the first 5 buses and the active and reactive powers flowing through the first five transmission lines in the forward and backward directions.

Calculation of Dummy Values of Active Powers Injected into the Buses | ||||

Bus No. | ${\mathit{\beta}}_{1}$ | ${\mathit{\beta}}_{2}$ | ${\mathit{\beta}}_{3}$ | |

1 | 0 | 40.81 | 196.30 | |

2 | 0 | 3.1 | 21.06 | |

3 | 0 | 13.45 | −82.22 | |

4 | 0 | 6.83 | −41.72 | |

5 | 0 | 1.09 | −6.63 | |

Calculation of Dummy Values of Reactive Powers Injected into the Buses | ||||

Bus No. | ${\beta}_{1}$ | ${\beta}_{2}$ | ${\beta}_{3}$ | |

1 | 0 | 6.73 | −10.30 | |

2 | 0 | 10.78 | 21.73 | |

3 | 0 | 7.29 | −0.27 | |

4 | 0 | 1.13 | 5.13 | |

5 | 0 | 8.25 | −9.88 | |

Calculation of Dummy Values of Active Powers Flowing through the Transmission Lines in Forward Direction | ||||

From | To | ${\beta}_{1}$ | ${\beta}_{2}$ | ${\beta}_{3}$ |

1 | 2 | 0.91 | 29.57 | 131.57 |

1 | 5 | −1.22 | 10.93 | 64.73 |

2 | 3 | −9.36 | −1.44 | 63.66 |

2 | 4 | −5.18 | −2.62 | 49.21 |

2 | 5 | −3.07 | −2.48 | 36.6 |

Calculation of Dummy Values of Reactive Powers Flowing through the Transmission Lines in Forward Direction | ||||

From | To | ${\beta}_{1}$ | ${\beta}_{2}$ | ${\beta}_{3}$ |

1 | 2 | 5.97 | −1.03 | −14.04 |

1 | 5 | 4.91 | 5.21 | 3.74 |

2 | 3 | 1.08 | 0.13 | 4.71 |

2 | 4 | −0.13 | −0.03 | −1.62 |

2 | 5 | −0.22 | −0.27 | 0.81 |

Calculation of Dummy Values of Active Powers Flowing through the Transmission Lines in Backward Direction | ||||

From | To | ${\beta}_{1}$ | ${\beta}_{2}$ | ${\beta}_{3}$ |

1 | 2 | 2.8 | −24.45 | −128.41 |

1 | 5 | 2.8 | −8.55 | −62.63 |

2 | 3 | 8.85 | 1.35 | −61.85 |

2 | 4 | 4.9 | 2.48 | −47.89 |

2 | 5 | 2.95 | 2.38 | −35.88 |

Calculation of Dummy Values of Reactive Powers Flowing Through the Transmission Lines in Backward Direction | ||||

From | To | ${\beta}_{1}$ | ${\beta}_{2}$ | ${\beta}_{3}$ |

1 | 2 | 5.35 | 16.66 | 17.83 |

1 | 5 | 1.65 | 4.65 | −0.38 |

2 | 3 | −3.25 | −0.53 | −1.68 |

2 | 4 | −0.74 | −0.39 | 2 |

2 | 5 | −0.16 | −0.04 | −2.31 |

**Table 3.**Active and reactive powers injected into the first 5 buses and the active and reactive powers flowing through the first 5 transmission lines in the forward and backward directions for the variable dummy value model.

Active Powers Injected to the Buses | |||

Bus No. | Actual Value (MW) | Dummy Value (MW) | |

1 | 232.11 | 232.02 | |

2 | 18.41 | 14.74 | |

3 | −93.94 | −154.82 | |

4 | −47.88 | −87.64 | |

5 | −7.58 | −9.95 | |

Reactive Powers Injected into the Buses | |||

Bus No. | Actual Value (MVAR) | Dummy Value (MVAR) | |

1 | −16.49 | −45.96 | |

2 | 30.79 | −26.41 | |

3 | 5.98 | −32.63 | |

4 | 3.9 | 0.58 | |

5 | −1.6 | −44.16 | |

Active Powers Flowing through the Transmission Lines in Forward Direction | |||

From | To | Actual Value (MW) | Dummy Value (MW) |

1 | 2 | 156.65 | 131.57 |

1 | 5 | 75.46 | 64.73 |

2 | 3 | 73.11 | 63.66 |

2 | 4 | 56.14 | 49.21 |

2 | 5 | 41.53 | 36.6 |

Reactive Powers Flowing through the Transmission Lines in Forward Direction | |||

From | To | Actual Value (MVAR) | Dummy Value (MVAR) |

1 | 2 | −20.35 | −14.04 |

1 | 5 | 3.86 | 3.74 |

2 | 3 | 3.57 | 4.71 |

2 | 4 | −1.54 | −1.62 |

2 | 5 | 1.17 | 0.81 |

Active Powers Flowing through the Transmission Lines in Backward Direction | |||

From | To | Actual Value (MW) | Dummy Value (MW) |

1 | 2 | −152.37 | −128.41 |

1 | 5 | −72.7 | −62.63 |

2 | 3 | −70.79 | −61.85 |

2 | 4 | −54.46 | −47.89 |

2 | 5 | −40.62 | −35.88 |

Reactive Powers Flowing through the Transmission Lines in Backward Direction | |||

From | To | Actual Value (MVAR) | Dummy Value (MVAR) |

1 | 2 | 27.58 | 17.83 |

1 | 5 | 2.21 | −0.38 |

2 | 3 | 1.55 | −1.68 |

2 | 4 | 3.01 | 2 |

2 | 5 | −2.1 | −2.31 |

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## Share and Cite

**MDPI and ACS Style**

Shahid, M.A.; Ahmad, F.; Albogamy, F.R.; Hafeez, G.; Ullah, Z.
Detection and Prevention of False Data Injection Attacks in the Measurement Infrastructure of Smart Grids. *Sustainability* **2022**, *14*, 6407.
https://doi.org/10.3390/su14116407

**AMA Style**

Shahid MA, Ahmad F, Albogamy FR, Hafeez G, Ullah Z.
Detection and Prevention of False Data Injection Attacks in the Measurement Infrastructure of Smart Grids. *Sustainability*. 2022; 14(11):6407.
https://doi.org/10.3390/su14116407

**Chicago/Turabian Style**

Shahid, Muhammad Awais, Fiaz Ahmad, Fahad R. Albogamy, Ghulam Hafeez, and Zahid Ullah.
2022. "Detection and Prevention of False Data Injection Attacks in the Measurement Infrastructure of Smart Grids" *Sustainability* 14, no. 11: 6407.
https://doi.org/10.3390/su14116407