**6. Results of the Variable Dummy Value Model**

The proposed model with the fixed dummy value can be bypassed, as the dummy value is constant for all the instances. The dummy value should change with the change in the actual value. In the variable dummy value model, the limitation of the fixed value dummy model is overcome by changing the dummy value at every instant. The variable dummy value model was implemented for the AC power flow model of the IEEE 14-bus system. A dummy value was selected in such a way that it should remain close to the actual value of that power and this feature depends upon the values of constants used in the linear function. For the selection of the constants, the MLR model is built for the calculation of all dummy values. Before using the MLR model, we must select the dummy values as outputs to find the relationship between the input and output. To select the dummy value of power at any instant for MLR, any value is picked from its actual values that occurred for the whole one year prior to that instant. The multivariate linear regression model was run for all the linear equations, and we obtained the best values of the constants for a particular equation that gave the minimum cost for that equation. Table 2 shows the values of constants at the first instant for the first five buses and the first five transmission lines.

Figure 8 shows the learning of the MLR model when finding the parameters of the linear equation used to find the dummy values of *P*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.

**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 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.


**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.

In the control room, the dummy values were again calculated by using the obtained actual values from the measurement vector and those recalculated dummy values were subtracted from the obtained dummy values to find the residue. The residue should be zero for a secured system. The results of the proposed model of the variable dummy value are shown in Figure 9, where the model was evaluated using simple and stealth attacks. Safe measurements are also shown in the figure. The residue is plotted along the vertical

axis. For safe measurements, the value of the residue was zero, as shown in the bar graph. However, for simple attacks and stealth attacks, the residue had some value greater than zero. Therefore, simple and stealth attacks were detected by our proposed variable dummy value model. As a result, the limitations of the fixed dummy value approach were handled by this variable dummy value model, and stealth FDI attacks were easily detected in the control room.

**Figure 9.** Results of the variable dummy value model for simple and stealth attacks.
