## 7 Conclusions  
This paper presented a convex VU optimization method that is paired with Monte Carlo simulation. The model showcases the benefit of convex model with relaxed VU definition, that was used for stochastic assessment of distribution network HC. The bus-injection model of the problem formulation enabled to estimate VU without load-flow calculations. Measured weekly voltage data were utilized for background VU modelling and total VU was estimated on 5% weekly tolerance condition. Similarly to [20], the convex relaxation of VU definition shows a credible result.  
As in [6], the results indicated higher VU levels at rural networks. Problem formulation with fixed phase selection showed maximum allowable PV injection to be limited by the background VU. Problem formulation with variable phase, however, demonstrated the ability of single-phase PVs to nearly double PV power injection. Higher background VU grants higher HC. As seen in other papers, consideration of other constraints, such as voltage limits, start to be more important. Optimal phase selection of single-phase PVs eliminates issue and other limits start to be the bottleneck for HC [18]. Furthermore, both fixed and variable phase models showed ability to attenuate VU considerably. PVs with power of the single-phase circuit capacity can attenuate VU in range of 70-99% of the simulated cases.  
Analysis of background negative sequence voltage variability, that is quantified by PR, revealed lower HC in case of predictable phase selection strategies, e.g., sequential phase selection. On the  
Figure 13: Average computation time.  
other hand, random phase selection of PVs will lead to almost no correlation between HC and PR. Moreover, the paper summarizes that PV HC is higher, if PVs are located closer to feeding substation. Lastly, the computation times of the presented algorithms are within the same range as the similarly formulated problem in [18].  
Conceptualization, V.P. and M.L.; methodology, V.P.; software, V.P.; validation, V.P. and M.L.; formal analysis, V.P.; investigation, V.P.; resources, M.L.; data curation, M.L.; writing--original draft preparation, V.P.; writing--review and editing, V.P.; visualization, V.P.; supervision, M.L.; project administration, M.L.; funding acquisition, M.L. All authors have read and agreed to the published version of the manuscript.  
This research was funded by Business Finland as a part of \"SolarX\" project.  
The authors of the paper would like to extend their gratitude towards Helen Sahkoverkko Oy and Caruna Networks Oy for benevolent help in conducting measurements.  
The authors declare no conflict of interest.  
The following abbreviations and nomenclature are used in this manuscript:  
HC Hosting capacity MIQP Mixed-integer quadratic programming PR Prevailing ratio PV Photovoltaic QP Quadratic programming VU Voltage unbalance Sets  
\(E_{b}\) Set of network buses \(E_{PV}\) Set of buses with PVs \(T_{\text{L-h}}\) Set of two time steps with the lowest and highest background VU \(T_{\text{w}}\) Set of 1008 steps of a week on 10-minute time resolution \(\Phi\) Set of phases \(a\), \(b\) and \(c\) Indices \(i,j,k\) Indices of buses \(t\) Index of time step \(\phi\) Index of phases Variables \(D_{i,\phi,t}^{\text{var}}\) Variable phase selection binary matrix \(l_{i,\phi,t}\) Negative sequence current injected by PV \(l_{i,t}^{R}\), \(l_{i,t}^{I}\) Real and imaginary parts of the injected PV current \(U_{j,t}^{I}\) Negative sequence voltage \(U_{j,t}^{I}\) and \(U_{j,t}^{I}\) Real and imaginary parts of the negative sequence voltage \(\theta_{\phi,t}\) Phase angle