## 6 Results
### Optimization Fixed Phase  
In the previous section, over the course of current simulations, VU levels always stay below the 2% limit in case of suburban and urban networks. Consequently, further optimization is focused on the rural network. The results of the QP problem formulation are presented. PV phase connection was fixed as entailed by the strategies. The histogram of maximum PV injected current can be seen in Figure 7. As concluded by visual inspection, the histograms follow lognormal distribution. As expected, strategy 1 performs worse than strategy 2 and 3. Phase selection that on average causes high VU will result in lower average HC. Strategies 2 and 3 are once again quite similar, except to the bump around  
Figure 5: Voltage unbalance levels in rural network. 20-25 kW for strategy 2. Starting the phase sequence at phase \(a\) at all the times will guarantee to show good result in part of randomly generated scenarios. Strategy 4, however, demonstrates the worst performance. The reason behind bad results is the changes of the negative sequence voltage vector. The phases of strategy 4 are fixed in the case where PV injected power is equal to the single-phase circuit capacity at particular background VU. However, if the injected power of PV plants is increased, the relative contribution of background unbalance to the resultant unbalance vector is decreasing. By increasing injected power even further, PV plants must rearrange connection phases to mitigate increasing unbalance caused by themselves. Phase selection must be reconsidered every time after the power is altered. The adaptive phase selection is better demonstrated in optimization with variable phases in Section 6.3. Recapping the VU histogram in Figure 5, a validation of the two figures can be made. In VU assessment, the PV injection was fixed and VU values were recorded. On the contrary, the VU limit is fixed in the maximization, and the PV injection is a variable. Both of the estimations are essentially focused around one point. To justify the percentile values of 95 of VU and 5 of PV power, the comparison is shown in Table 2.  
Figure 6: Voltage unbalance in suburban (**a**) and urban (**b**) networks. Figure 7: Maximum PV injection at fixed phases. The number of instances VU exceeding 2% limit is shown in the first row, remembering that the PV plant power in Monte Carlo simulation is equal to single-phase circuit capacity. Likewise, the number of cases PV plant power of Figure 7 is below the single-phase circuit capacity shown in the bottom row of the table. The values have a close match, except for strategy 4. As discussed above, connection phases of strategy 4 are selected, while the PV injected power is fixed to single-phase circuit capacity. During the optimization, however, the power is altered and the phase selection must be reconsidered for every unique power value. The optimal selection of the phases is presented in forthcoming section. Other strategies, however, suggest the validity of the 5 th percentile PV power estimation as an equivalent reference value for tolerance condition set by tthe EN 50160 standard. The small difference though can be explained by the equal power condition for PV plants, while the VU obtains a unique value for every bus. The results of minimizing the VU at fixed phase connection are shown in Figure 8. The histogram of VU attenuation is shown on (a), where it can be seen, that attenuation is hindered by the phase selection of the strategy 3 and strongly depends on the background VU phasor. Due to the fact that the phasor angle is fixed to the feeding voltage phasor, the attenuating vectors of single-phase PVs are not able to follow the vector of the background VU, and the attenuation is not always possible, as indicates the average attenuation value of 52%. As compared to other results of current paper, voltage unbalance mitigation subject to fixed phases is the only case where some iterations yield no improving result. Voltage unbalance was attenuated in 69.1% of the cases generated in Monte Carlo simulation.