## 3 Numerical Results and Discussion
### IEEE 6-Bus Test System
Figure 5: Structure of 6-bus network that resulted from applying the proposed problem in the last year by using (**a**) NACTEP, (**b**) LACTEP, and (**c**) DCTEP. Table 5 represents the obtained results for the operation and investment cost, constructed lines, network active and reactive losses, computation time, and the situation of problem-solving for three models. Based on this table, the construction cost is equal for the three models, due to their unique configuration. The exploitation cost and network's active and reactive losses during the planning are equal in DC and linear AC power flow (models 1 and 3). Based on the obtained results, network losses in these two models are zero, due to the equations that are used in them. Since modeling based on nonlinear AC power flow (model 2) considers network losses, the operation cost of this model is greater than the two other ones. However, it is worth noting that model 2 as an MINLP problem achieves more accurate results, but its computation time is significantly larger than the computation time of models 1 and 3. Based on Table 6, model 2 does not have a globally optimal solution. Noticeably, the proposed TEP problem using linear AC power flow equations (model 3) possesses a slight approximation in comparison with model 2. It has a very small computation time and achieves a globally optimal solution. Also, the computation time of model 1 is short and it obtains an optimal solution, but the obtained result of this model is not feasible because the reactive power is not considered in DC power flow equations. Herein, the results of the stochastic TEP method with the standard deviation of 10% for active and reactive load uncertainty are shown in Figure 6 and Table 6. By comparing Figures 5b and 6, it is observed that in the stochastic model, two transmission lines are proposed to be constructed which include 2-6 and 4-6. Also, the expected active and reactive power of generators has been increased in comparison with the deterministic model, and the expected load bus voltages have been reduced. Based on Table 6, the total cost of operation and construction is about 153.368 M$, which will be equal to 93.062 M$ in the deterministic problem. The cost difference is 60.306 M$, which can be considered the prediction cost or the uncertainty cost or the security cost due to considering the load uncertainty. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **LACPF** \\ \hline Expected operation cost ($) & 3,368,200 \\ \hline Investment cost (M$) & 150 \\ \hline Total cost (M$) & 153.3682 \\ \hline Constructed lines & 2–6, 4–6 \\ \hline Computation time (s) & 5 \\ \hline Solution situation & Optimal \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Results of different factors in 6-bus network for LACPF. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **DCTEP** & **NACTEP** & **LACTEP** \\ \hline Operation cost ($) & 3,062,000 & 3,224,000 & 3,062,000 \\ \hline Investment cost (M$) & 90 & 90 & 90 \\ \hline Total cost (M$) & 93.062 & 93.224 & 93.062 \\ \hline Constructed lines & 4–6 & 4–6 & 4–6 \\ \hline Daily active power Loss (MWh) & 0 & 3.23 & 0 \\ \hline Daily Reactive power Loss (MVARh) & 0 & 5.83 & 0 \\ \hline Computation time (s) & 4 & 432 & 4 \\ \hline Solution situation & Optimal & Local optimal & Optimal \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Results of different factors in 6-bus network for different power flow methods. (B) CNEP capability: In this section, three cases are implemented on a 6-bus network to study the proposed stochastic CNEP problem as follows:  
* Case I: The stochastic TEP method based on the linear AC power flow. * Case II: The stochastic linear AC power flow-based CNEP (LAC-CNEP) method, considering renewable sources and the planning of transmission lines. * Case III: The proposed stochastic LAC-CNEP method that is formulated in (29)-(31).