**3. Proposed Model**

The methods used in the literature for the detection of attacks are successful up to a certain limit. If the attacker knows the whole network of the smart grid and makes an attack, it becomes difficult to detect those attacks. Therefore, we proposed a new power system model for an AC power flow network that is safe against stealth FDI attacks and the control room is able to detect these attacks in an efficient manner. The introduced model was based on the concept of dummy value. The smart grid meters will transmit both values, i.e., the actual value and the dummy value. No additional transmission lines and no extra buses will be used. There is no need for any extra meters in the proposed model. The vulnerabilities of the communication networks in supervisory control and data acquisition (SCADA) systems in the smart grid, such as unsophisticated bugs or communication failures, were not considered in this work. The application of the measured value and the dummy value in this article did not consider the error caused by the measurement equipment itself or any other reason. In this work, the error due to parametric variation of the meter or any other unknown reason was not taken into consideration. However, it may be incorporated into our future work. Moreover, this work focused on false data injection attacks in which the intruder hacks the measurement vector and injects the attack vector

into the measurement vector before it is received by the control room. Therefore, this study only considered targeted attacks.

The measurement vector for the AC power flow network contains the active and reactive powers injected into all the buses, active and reactive powers flowing through transmission lines in the forward direction, and active and reactive powers flowing in the backward direction. If a system has *b* number of buses and *t* number of transmission lines, then the measurement vector for the AC power flow network is given by

$$\mathbf{z}\_{\mathbf{y}} = \begin{bmatrix} \mathbf{p}\_{\mathbf{v}(\mathbf{y})} & \mathbf{q}\_{\mathbf{v}(\mathbf{y})} & \mathbf{p}\_{\mathbf{v}\mathbf{w}(\mathbf{y})} & \mathbf{q}\_{\mathbf{v}\mathbf{w}(\mathbf{y})} & \mathbf{p}\_{\mathbf{w}\mathbf{v}(\mathbf{y})} & \mathbf{q}\_{\mathbf{w}\mathbf{v}(\mathbf{y})} \end{bmatrix}^{\mathrm{T}} \tag{15}$$

where **z***<sup>y</sup>* is the measurement vector at the *y*th instant and **y** = 1, 2, 3, ... , *mt*. Here, *mt* represents the total number of instances. **pv**(**y**) and **qv**(**y**) are the vectors containing the active and reactive powers injected to all the buses at the *y*th instant. Both vectors will have a dimension of 1 × *b*. Similarly, **pvw**(**y**) and **qvw**(**y**) denote vectors having the active and reactive powers flowing through all the transmission in the forward direction at the *y*th instant. Both vectors have dimensions of 1 × *t*. Moreover, **pwv**(**y**) and **qwv**(**y**) represent the vectors of the active and reactive powers flowing through all the transmission lines in the backward direction at the *y*th 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:

$$\mathbf{Z} = \begin{bmatrix} \mathbf{z}\_1 & \mathbf{z}\_2 & \mathbf{z}\_3 & \dots & \dots & \dots & \dots & \mathbf{z}\_{\mathbf{mt}} \end{bmatrix}^\mathrm{T} \tag{16}$$

The dimensions of the measurement matrix are *mt* × *m*. The measurement vector after implementing the proposed system will become like this:

$$\mathbf{z\_{dy}} = \begin{bmatrix} p\_{v(y)}(1); p'\_{v(y)}(1); \dots \dots ; q\_{v(y)}(b); q'\_{v(y)}(b); \\ p\_{vw(y)}(1); p'\_{vw(y)}(1); \dots \dots ; q\_{vw(y)}(t); q'\_{vw(y)}(t); \\ p\_{uv(y)}(1); p'\_{uv(y)}(1); \dots \dots ; q\_{uv(y)}(t); q'\_{uv(y)}(t) \end{bmatrix}$$

The measurement vector containing the actual and dummy values is represented by **zdy**. Here, **pv**(**y**)(1) represents the first entry of the vector **pv**(**y**) and **qv**(**y**)(**b**) is the *b*th entry of the vector **q v**(**y**) . 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 *y*th instant are **p <sup>v</sup>**(**y**) and **q v**(**y**) . Similarly, other vectors containing dummy values of the active and reactive powers for transmission lines at the *y*th instant are denoted by **p vw**(**y**) , **q vw**(**y**) , **p wv**(**y**) , and **q wv**(**y**) . **zdy** will have dimensions of 2*m* × 1. The measurement matrix after including the dummy values will be

$$\mathbf{Z\_{d}} = \begin{bmatrix} \mathbf{z\_{d1}} & \mathbf{z\_{d2}} & \mathbf{z\_{d3}} & \dots & \dots & \dots & \dots & \mathbf{z\_{dmt}} \end{bmatrix}^{\mathrm{T}} \tag{17}$$

This measurement vector will have dimensions of *mt* × 2*m*. The Jacobian matrix of the proposed system at the *y*th instant is represented by **Hdy** and its dimensions are 2*m* × *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.

Realistic data of the AC power flow network was used for implementing and evaluating the proposed model. For this purpose, the load curves of a transmission organization known as PJM, which serves 13 states of the United States and the District of Columbia, were taken as a reference to generate the data of the power flow network. These load curves were based on realistic data. Therefore, our generated data were very close to the realistic

data of an AC power flow network. The data were generated for four different seasons, namely, summer, fall, winter, and spring, based on the standard realistic load curves given for each season.

The overall proposed model was divided into two scenarios. In the first scenario, a fixed dummy value was sent to the control room. However, in the second case, a variable dummy value was sent and it changed with the change of the actual value of power.
