# Determining Optimal Power Flow Solutions Using New Adaptive Gaussian TLBO Method
## 4 Numerical Results
### Cases Definition
#### 4.2.2 Case 2: Minimizing the Total Fuel Cost Considering Multi-Fuel Sources  
A few generators in existing power systems are driven by several fuels, resulting in a cost function with piecewise quadratic characteristics. As a result, the OPF problem's objective function for these structures can be defined as follows [16]:  
\[F(P_{Gi})= \begin{cases}\alpha_{i1}+b_{i1}P_{Gi}+c_{i1}P_{Gi}^{2},p_{Gi}^{\min} \leq P_{Gi}\leq P_{Gi1}\\ \alpha_{i2}+b_{i2}P_{Gi}+c_{i2}P_{Gi}^{2};P_{Gi1}\leq P_{Gi}\leq P_{Gi2}\\ \ldots\\ \alpha_{ik}+b_{ik}P_{Gi}+c_{ik}P_{Gi}^{2};P_{GiR-1}\leq P_{Gi}\leq P_{Gi}^{ \max}\end{cases}  \tag{27}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Algorithm** & **Emission (t/h)** &
\begin{tabular}{c} **Fuel Cost** \\ **(USD/h)** \\ \end{tabular} & **V.D. (p.u.)** & **Power Losses (MW)** \\ \hline AGTLBO & 0.3662 & **800.481** & 0.9109 & 9.0222 \\ TLBO & 0.3668 & 800.674 & 0.9120 & 9.0198 \\ TS (Tabu Search) [5] & - & 802.290 & - & - \\ ABC (Artificial Bee Colony) [12] & 0.3651 & 800.660 & 0.9209 & 9.0328 \\ AGSO (Adaptive Group Search Optimization) [14] & 0.3703 & 801.750 & - & - \\ MHBMO (Modified Honey Bee Mating Optimization) [15] & - & 801.985 & - & 9.4900 \\ MSA (Moth Swarm Algorithm) [16] & 0.3665 & 800.510 & 0.9036 & 9.0345 \\ FPA (Flower Pollination Algorithm) [16] & 0.3596 & 802.798 & 0.3679 & 9.5406 \\ MFO (Moth-Flame Optimization) [16] & 0.3685 & 800.686 & 0.7577 & 9.1492 \\ MPSO-SFLA (Modified Particle Swarm Optimization and Shuffle Frog Leaping Algorithm) [21] & - & 801.750 & - & 9.5400 \\ GWO (Grey Wolf Optimizer) [25] & - & 801.410 & - & 9.3000 \\ MICA-TLA (Modified Imperialist Competitive Algorithm and Teaching Learning Algorithm) [45] & - & 801.049 & - & 9.1895 \\ DE (Differential Evolution) [49] & - & 802.390 & - & 9.4660 \\ SKH (Stud Krill Herd Algorithm) [52] & 0.3662 & 800.514 & - & 9.0282 \\ SFLA-SA (Shuffle Frog Leaping Algorithm and Simulated Annealing) [53] & - & 801.790 & - & - \\ ARCBBO (Adaptive Real Coded Biogeography-Based Optimization) [51] & 0.3663 & 800.516 & 0.8867 & 9.0255 \\ MGBICA (Modified Gaussian Bare-Bones Imperialist Competitive Algorithm) [54] & 0.3296 & 801.141 & - & - \\ IEP (Improved Evolutionary Programming) [50] & - & 802.460 & - & - \\ PSOGSA (Particle Swarm Optimization and Gravitational Search Algorithm) [55] & - & 800.499 & 0.1267 & 9.0339 \\ EP (Evolutionary Programming) [56] & - & 803.570 & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Optimal results by different algorithms for Case 1.  
where cost coefficients for the \(i\)th generator are represented by \(\alpha_{ik}\), \(b_{ik}\) and \(c_{ik}\) for fuel type \(k\). These circumstances are commonly considered just for the two first generators in several earlier papers, as described below [16]:  
\[OF_{2}=\left(\sum_{i=1}^{2}\alpha_{ik}+b_{ik}P_{Gi}+c_{ik}P_{Gi}^{2}\right)+ \left(\sum_{i=3}^{NG}\alpha_{i}+b_{i}P_{Gi}+c_{i}P_{Gi}^{2}\right)+P_{Cost} \tag{28}\]  
Reference [49] provided the data for the units with several fuel types. Table 3 summarizes the optimization results of the TLBO and AGTLBO algorithms, and these results are compared to those published in the previous works such as LTLBO (Levy Teaching-Learning-Based Optimization) [28], MDE (Modified Differential Evolution) [49], MFO (Moth-Flame Optimization) [16], IEP (Improved Evolutionary Programming) [50], MICATLA (Modified Imperialist Competitive Algorithm and Teaching Learning Algorithm) [45], FPA (Flower Pollination Algorithm) [16], SSO (Social Spider Optimization) [37], GABC (Gbest Guided Artificial Bee Colony Algorithm) [57], MSA (Moth Swarm Algorithm) [16], and MPSO-SFLA (Modified Particle Swarm Optimization and Shuffle Frog Leaping Algorithm) [21]. The table shows that AGTLBO has adequate optimization ability compared to other approaches. According to the obtained simulation data, the minimum cost obtained from AGTLBO is 646.4511 USD/h, which is less comparing to result of other algorithms.