# Determining Optimal Power Flow Solutions Using New Adaptive Gaussian TLBO Method
## 4 Numerical Results
### Cases Definition
#### 4.2.1 Case 1: Minimizing the Fuel Cost  
In natural power systems, generation units use fossil fuel resources such as oil and natural gas, and their fuel costs can be formulated as follows [16]:  
\[OF_{1}=\sum_{i=1}^{NG}F_{i}(P_{Gi})=\sum_{i=1}^{NG}\Bigl{(}\alpha_{i}+b_{i}P_{Gi} +c_{i}P_{Gi}^{2}\Bigr{)}+P_{Cost} \tag{25}\]  
\[P_{Cost}=\lambda_{P}\Bigl{(}P_{G1}-P_{G1}^{lim}\Bigr{)}^{2}+\lambda_{Q}\sum_{i= 1}^{NG}\Bigl{(}Q_{Gi}-Q_{Gi}^{lim}\Bigr{)}^{2}+\lambda_{V}\sum_{i=1}^{NPQ} \Bigl{(}V_{Li}-V_{Li}^{lim}\Bigr{)}^{2}+\lambda_{S}\sum_{i=1}^{NTL}\Bigl{(}S_{ li}-S_{li}^{lim}\Bigr{)}^{2} \tag{26}\]  
A fuel cost is given by \(F_{i}\), and an output is given by \(P\bar{G}_{i}\), respectively, for the \(i\)th generator. There are three cost coefficients called \(\alpha_{i}\), \(b_{i}\) and \(c_{i}\), as well as \(NG\), which indicates the number of generators overall. Detailed information about the cost coefficients can be found in [51]. \(P_{Cost}\) is the penalty cost of constraints violations. In Table 2, TLBO and AGTLBO algorithms are compared in terms of fuel costs (USD/h), emissions (t/h), power losses (MW), and V.D. (p.u.). A comparison is made between these results and those reported in recent literature such as TS (Tabu Search) [5], ABC (Artificial Bee Colony) [12], AGSO (Adaptive Group Search Optimization) [14], MHBMO (Modified Honey Bee Mating Optimisation) [15], MSA (Moth Swarm Algorithm) [16], FPA (Flower Pollination Algorithm) [16], MFO (Moth-Flame Optimization) [16], MPSO-SFLA (Modified Particle Swarm Optimization and Shuffle Frog Leaping Algorithm) [21], GWO (Grey Wolf Optimizer) [25], MICATLA (Modified Imperialist Competitive Algorithm and Teaching Learning Algorithm) [45], DE (Differential Evolution) [49], SKH (Stud Krill Herd Algorithm) [52], SFLA-SA (Shuffle Frog Leaping Algorithm and Simulated Annealing) [53], ARCBBO (Adaptive Real Coded Biogeography-Based Optimization) [51], MGBICA (Modified Gaussian Bare-Bones Imperialist Competitive Algorithm) [54], IEP (Improved Evolutionary Programming) [50], PSOGSA (Particle Swarm Optimization and Gravitational Search Algorithm) [55], and EP  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Control Variables**} & \multicolumn{3}{c}{**Limits**} & & & & & & & & & & \\ \cline{2-10}
**Settings** & **Lower** & **Upper** & **Case 1** & **Case 2** & **Case 3** & **Case 4** & **Case 5** & **Case 6** & **Case 7**  
(Evolutionary Programming) [56]. As Table 2 shows, the proposed algorithm (Case 1) is suitable and effective for optimizing this objective function compared to other algorithms. According to the obtained simulation data, the minimum cost obtained from AGTLBO is 800.4811 USD/h, which is less comparing to result of other algorithms. By employing the AGTLBO, the objective function of TLBO, TS, ABC, AGSO, MHBMO, MSA, FPA, MFO, MPSO-SFLA, GWO, MICA-TLA, DE, SKH, SFLA-SA, ARCBBO, MGBICA, IEP, PSOGSA, and EP are decreased by 0.1930 USD/h, 1.8090 USD/h, 0.1790 USD/h, 1.2690 USD/h, 1.5040 USD/h, 0.0290 USD/h, 2.3170 USD/h, 0.2050 USD/h, 1.2690 USD/h, 0.9290 USD/h, 0.5680 USD/h, 1.9090 USD/h, 0.0330 USD/h, 1.3090 USD/h, 0.0350 USD/h, 0.0660 USD/h, 1.9790 USD/h, 0.0180 USD/h, and 3.0890 USD/h, respectively, which demonstrates the effectiveness of AGTLBO.