# Optimal Harmonic Mitigation in Distribution Systems with Inverter Based Distributed Generation
## 6 Simulation Results and Discussion
### Simulation Results of Case 1  
The results of the filter planning in the first case are shown in Tables 3 and 4. The results show that the proposed method reduced effectively the maximum THD from 7.87% to 2% to meet the IEEE-519 standard. As obtained in Table 3, the 7 th order harmonic filter with a large size (\(Q_{c}\)) was chosen for solving the system harmonic distortion problem. Based on the concept shown in Figure 5, the filter passband increases with increasing the filter capacitive reactive power and increasing the order of tuned frequency. Therefore, the 7 th order harmonic filter has more flatten impedance characteristics curve rather than the 5 th order filter. Furthermore, the capacity of the selected filter is 2.33 MVAR, which indicates a  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Harmonic** & \multirow{2}{*}{1} & \multirow{2}{*}{5} & \multirow{2}{*}{7} & \multirow{2}{*}{11} & \multirow{2}{*}{13} & \multirow{2}{*}{17} & \multirow{2}{*}{19} & \multirow{2}{*}{23} & \multirow{2}{*}{25} \\
**Order** & & & & & & & & & \\
**Value (\%)** & 100 & 42 & 14.3 & 7.9 & 3.2 & 3.7 & 2.3 & 2.3 & 1.4 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Harmonic spectrum of six-pulse nonlinear loads.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Harmonic Order** & **Magnitude \%** & **Phase (deg.)** & **Harmonic Order** & **Magnitude \%** & **Phase (deg.)** \\ \hline
1 & 100 & \(-\)2.34 & 9 & 8 & 140.36 \\
3 & 20 & \(-\)15.29 & 11 & 5 & 65.54 \\
5 & 15 & \(-\)20.74 & 13 & 3 & 42.62 \\
7 & 10 & \(-\)30.85 & 15 & 2 & 153.28 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Harmonic spectrum of the inverter-based DGs.  
Figure 7: IEEE 69-bus distribution test system.  
very flat characteristics curve. Therefore, only one filter tuned at the 7 th order can mitigate both 5 th, 7 thv and other harmonic orders and keep the THD within the permissible limits.  
The objective functions' values, which are obtained in Table 4, show that the system minimum voltage has improved from 0.93 PU to 0.98 PU. The system power loss has also reduced from 51.5 to 29.8 kW. Moreover, the three considered objective functions have minimized using the proposed algorithm, which approves the ability of the proposed method to achieve multi-dimensional goals (technical and economic).  
The convergence curve of the WCA is shown in Figure 8. It shows the convergence speed of the WCA, and how it can reach the personal best within a few iterations. The THD of the system buses before and after using the filters are shown in Figure 9. Where the maximum THD decreased significantly from about 8% to 2% to not only meet the standard limit but also, considered as a preventive action to keep the system harmonic within limits during any future increase in the system harmonics.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Without Filter** & **With Filter** \\ \hline V\({}_{\text{max}}\) (p.u) & 1 & 1.0139 \\ V\({}_{\text{min}}\) (p.u) & 0.9329 & 0.9824 \\ P\({}_{\text{loss}}\) (kW) & 51.561 & 29.83 \\ THD\({}_{\text{max}}\) (\%) & 7.87 & 2 \\ \hline \multicolumn{4}{c}{**Fitness Functions**} \\ \hline \(F_{1}\) & \(F_{2}\) & \(F_{3}\) & \(F_{total}\) \\
1.9679 & 0.4545 & \(2.8657\times 10^{4}\) & 0.2284 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Results of objective functions of case 1.  
Figure 8: Convergence curve of the Water Cycle Algorithm (WCA).  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Filter No.** & **Placement** & \(f_{t}\) & **Q\({}_{\epsilon}\) (MVar)** & **R (\Omega)** & **L (mH)** & **C (\%F)** \\ \hline
1 & 39 & 7 & 2.3303 & 0.0655 & 0.0015 & 138.84 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Results of control variables for filter planning in case 1.  
Figure 10 shows that the installed filters can improve the system voltage profile and keep all the buses' voltages within the permissible limits. As it can be seen that many bus voltages (bus 40-bus 47) were out of limits before installing the filter.