# Determining Optimal Power Flow Solutions Using New Adaptive Gaussian TLBO Method
## 4 Numerical Results
### Cases Definition
#### 4.2.5 Case 5: Minimizing the Fuel Cost Considering Valve-Point Effects (VPE)  
The generators have typically functioned with fossil fuels in power systems. The fuel consumption can be controlled by adjusting the valves of fuel supply reservoirs. Therefore, by altering the fuel cost function of Case 1, VPE can be modeled [53]:  
\[OF_{5}=OF_{1}+\sum_{i=1}^{NG}|d_{i}\sin[e_{i}(P_{Gi}^{\text{min}}-P_{Gi})]| \tag{31}\]  
where the cost coefficients for the \(i\)th generating unit are \(d_{i}\) and \(e_{i}\).  
Table 6 shows the results attained by the AGTLBO algorithm and the solutions obtained in previous publications such as PSO (Particle Swarm Optimization) [11], and SP-DE (Differential Evolution Integrated Self-Adaptive Penalty) [61], demonstrating the efficacy of the proposed AGTLBO. According to the obtained simulation data, the minimum cost obtained from AGTLBO is 832.1624 USD/h, which is less comparing to result of other algorithms. By employing the AGTLBO, the objective function of TLBO, PSO, and SP-DE are decreased by 0.3701 USD/h, 0.5247 USD/h, and 0.3189 USD/h, respectively, which demonstrates the effectiveness of AGTLBO.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Algorithm** & **Emission (t/h)** & **Fuel Cost (USD/h)** & **V.D. (p.u.)** & **Power Losses (MW)** \\ \hline AGTLBO & 0.4379 & **832.1624** & 0.8592 & 10.6863 \\ TLBO & 0.4382 & 832.5325 & 0.8654 & 10.6870 \\ PSO (Particle Swarm Optimization) [11] & - & 832.6871 & - & - \\ SP-DE (Differential Evolution Integrated Self-Adaptive Penalty) [61] & 0.43651 & 832.4813 & 0.7504 & 10.6762 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Optimal results by different algorithms for Case 5.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Algorithm** & **Emission (t/h)** & **Fuel Cost (USD/h)** & **V.D. (p.u.)** & **Power Losses (MW)** \\ \hline AGTLBO & 0.2073 & 967.6336 & 0.9086 & **3.0906** \\ TLBO & 0.2072 & 967.7195 & 0.8821 & 3.1104 \\ ABC (Artificial Bee Colony) [12] & 0.2073 & 967.6810 & 0.9008 & 3.1078 \\ FPA (Flower Pollination Algorithm) [16] & 0.2076 & 967.1138 & 0.3893 & 3.5661 \\ MFO (Moth-Flame Optimization) [16] & 0.2073 & 967.6785 & 0.9156 & 3.1111 \\ MSA (Moth Swarm Algorithm) [16] & 0.2073 & 967.6636 & 0.8887 & 3.1005 \\ DSA (Differential Search Algorithm) [22] & 0.2083 & 967.6493 & - & 3.0945 \\ GWO (Grey Volt Optimization) [25] & - & 963.8800 & - & 3.4100 \\ Java [32] & - & 967.6827 & 0.1272 & 3.1035 \\ ARCBBO (Adaptive Real Coded biogeography-Based Optimization) [51] & 0.2073 & 967.66605 & 0.8913 & 3.1009 \\ EEA (Efficient Evolutionary Algorithm) [58] & - & 952.3785 & - & 3.2823 \\ EGA (Enhanced Genetic Algorithm) [58] & - & 967.9300 & - & 3.2440 \\ EGA-DQLF (Enhanced Genetic Algorithm to Decoupled Quadratic Load & - & 967.8600 & 0.1218 & 3.2008 \\ ALC-PSO (Particle Swarm Optimization with an Aging Leader and & - & 967.7683 & - & 3.1700 \\ Challenges) [60] & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 5: Optimal results by different algorithms for Case 4.  
To compare and better test the performance of the AGTLBO, the problem is considered a multiobjective optimization issue by solving more than one objective function at the same time for Cases 6, 7, and 8.