## 4 Conclusions  
We have developed a workflow for the digital modeling of unbalanced and non-sinusoidal power flows, as well as that of electromagnetic fields of power supply systems of railroads, the external network of which represents compact power transmission lines of different designs. Based on our simulation results, we can conclude that compact power transmission lines with three-segment and concentric arrangements of conductors scored best in terms of power quality and energy efficiency. However, the designs of such transmission lines are quite complex and entail higher construction costs.  
We have proposed a technique for determining the EMF strength of compact overhead lines that power traction substations. To that end, eight COHL designs have been considered.  
The models developed by the authors for load flow analysis and EMF of COHLs are versatile, and they can be extended to COHLs of any type. Their use in the practice of  
Figure 23: Coefficients of variation of amplitudes of electric field strengths: \(k_{v}=\frac{\sigma}{\tau_{\text{max}}};\sigma\) is the standard deviation; \(\overline{E}_{\text{max}}\) is the mean value of the EF amplitude—the callout numbers correspond to those of Figure 15.  
Figure 24: Coefficients of variation of magnetic field amplitudes—the callout numbers correspond to those of Figure 15.  
designing external power supply systems for railroads will be instrumental to making informed decisions on the choice of rational designs of high-voltage power transmission lines.  
The proposed models of power systems and power supply systems for railroads enable the evaluation of efficacy and application of COHLs for power supply of traction substations. The approach detailed in the article stands apart from other research in that it is systems-based and treats COHLs as inseparable from a complex power system. In contrast to traditional methods based on a single-line representation, our models were built using the three-phase (ABC) reference frame, which provides the most adequate modeling of power system load flows with single-phase traction loads that create significant unbalances in the electrical networks that feed them. The simulation took into account the dynamics of changes in traction loads by simulating the movement of trains along real-world rail track profiles.  
Conceptualization, K.S. and A.K.; methodology, A.K., I.F., E.V. and K.S.; software, A.K.; validation A.K., I.F., E.V. and K.S.; formal analysis, I.F. and E.V.; investigation, A.K., I.F., E.V. and K.S.; resources, K.S.; data curation, A.K., I.F., and E.V.; writing--original draft preparation, A.K., I.F., and E.V.; writing--review and editing, A.K. and K.S.; visualization, A.K., I.F., and E.V.; supervision, A.K.; project administration, A.K. and K.S.; funding acquisition, K.S. All authors have read and agreed to the published version of the manuscript.  
The research was carried out under State Assignment No. \"Applied Scientific Research\" for \"Development of methods, algorithms, and software for modeling load flows of DC traction power supply systems and electromagnetic fields in traction substations of AC railroads\".  
Data Availability Statement: Data sharing not applicable. No new data were created or analyzed in this study. Data sharing is not applicable to this article.  
The authors declare no conflict of interest.