## 2 Problem Formulation
### The Linear CNEP Model  
As can be observed from the aforementioned problem model, constraints (4), (5), (7) and (8) are nonlinear. In addition, Equations (4) and (5) are non-convex due to the terms of \(\cos(\theta_{i,t,y,\omega}-\theta_{j,t,y,\omega})\) and \(\sin(\theta_{i,t,y,\omega}-\theta_{j,t,y,\omega})\)[33; 34; 35], and they include a binary variable. Therefore, the model of (1)-(18) is a non-convex mixed-integer nonlinear problem (MINLP). This will slow down the solving procedure and may result in infeasible solutions in actual large-scale networks [36; 37; 38]. In this section, an equivalent linear model is proposed to speed up the solving procedure and guarantee the global optimality of the solution. (A) The Linear approximation of AC power flow constraints: Nonlinear constraints in this section are (4) and (5). For linearizing these equations, the following assumptions are considered [38; 39]:  
* The difference in the voltage phases of the two ends of a transmission line is under 6 degrees or 0.105 radians. * Since the magnitude of bus voltages in the transmission network should be between 0.95 and 1.05 per unit, the magnitude of bus voltages is very close to 1 per unit. According to the mentioned assumptions, the bus voltages can be expressed as \(1+\Delta V\) where the value of \(\Delta V\) is much smaller than 1. Also, based on the first assumption, \(\cos(\theta_{i,t,y,\omega}-\theta_{  
where \(M\) is a large enough number which is assumed to be \(10^{6}\). Also, the linearization procedure of Equation (23) is described in the next section. (B) The linear approximation of system operation limitations: Equations (7), (8) and (23) are nonlinear. These relations represent circular inequalities, which denote the variation interval of active and reactive power inside a circle with a radius equal to the maximum apparent power. Nonetheless, determination of the linear equations corresponding to the circular plane with Equation (24) can be performed as illustrated in Figure 2[36; 37; 38]. Hence, the circular plane is obtained by unifying several square planes. For example, based on Figure 2, the linear description of the circular plane is as in Equation (25). \[y^{2}+x^{2}\leq r^{2} \tag{24}\]  
\[y\leq r,\,y\geq-r,\,x\leq r\&\,x\geq-r \tag{25}\]  
According to Figure 2, Equation (24) has a large linearization error. Thereby, to reduce the linearization error, the number of square planes with different angles from the horizontal axis should be increased. Thus, at first 360 degrees of circle perimeter are divided into equal parts \(\Delta\alpha\). Then, the line equation is linearized for each \(k\,\times\,\Delta\alpha\), where \(k\) is the counter of the linearized part. Finally, the calculated line equation is integrated with a circle with a radius less than or equal to \(r\). Therefore, the linear approximation of the circular plane is expressed as below:  
\[x\cos(k\Delta\alpha)+y\sin(k\Delta\alpha)\leq r\,k\in\varphi_{k}=\{0,1,\dots,n_{k}-1\} \tag{26}\]  
where \(n_{k}\) denotes the number of linearized parts. For example, if 180 square planes are used for the linearization of a circular plane, \(n_{k}\) and \(\Delta\alpha\) will be equal to 180 and 2, respectively. Consequently, the planes \(x\leq r\) and \(y\leq r\) correspond to \(k=0\) and \(k=45\). Therefore, the linear approximation of (23) and (8) is as follows. \[\cos(k\times\Delta\alpha)\times PL_{i,j,t,y,\omega}+\sin(k\times\Delta\alpha) \times QL_{i,j,t,y,\omega}\leq SL_{i,j}^{\max}\times x_{i,j}\quad\forall i,j, t,y,\omega,k \tag{27}\]  
\[\cos(k\times\Delta\alpha)\times PG_{i,t,y,\omega}+\sin(k\times\Delta\alpha) \times QG_{i,t,y,\omega}\leq SG_{i}^{\max}\quad\forall i,t,y,\omega,k \tag{28}\]  
Figure 2: The linearization of the circular plane. Therefore, the model of the mixed-integer linear problem (MILP) is proposed as:  
\[