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Documents published in Scipedia

• Reducing the number of input variables through symbolic regression

  D. Brkić, P. Praks, M. Marek, U. Ilić, Z. Stajić

  Electronic Research Archive (2025). 2. Brkić, D., Praks, P., Marek, M., Ilić, U., Stajić, Z. (2025). Reducing the number of input variables through symbolic regression. Electronic Research Archive, Vol. 33, No. 9, paper No. 5158-5178.

  
  

  Abstract

  Symbolic regression, a type of machine learning technique, can efficiently disregard variables that are not significant to the final output, even if they were initially preselected as inputs. Various input parameters are tested in the three examples presented here, where the outputs are modeled using symbolic regression: estimating the middle plasma torch temperature used for waste gasification, the active energy of a solar power plant, and the diameter of a pipe with a known flow and pressure drop through it. Final highly accurate formulas are produced after numerous attempts with lower performances. The process for rejecting the parameters without or with limited influence is automatic and can be performed without human intervention and supervision. The results obtained using symbolic regression are easily interpretable by human experts. This approach shows how to use machine learning-based modeling as an additional tool for sensitivity analysis.

  Abstract
  Symbolic regression, a type of machine learning technique, can efficiently disregard variables that are not significant to the final output, even if they were initially preselected [...]

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• Maintenance optimization for unavailability enhancement of representative interconnected infrastructure based on minimum cost

  P. Praks, R. Fujdiak, D. Brkić

  Science Progress (2025). Briš, R., Praks, P., Fujdiak, R., Vrtal, M. and Brkić, D., 2025. Maintenance optimization for unavailability enhancement of representative interconnected infrastructure based on minimum cost. Science Progress, 108(3), p.00368504251366357.

  
  

  Abstract

  This paper explores the feasibility of solving a maintenance optimization problem in an interconnected smart grid system, comprising a power grid and a communication network, to reduce system unavailability. The unavailability, which must be in practice under the control of a system operator, is particularly sensitive to critical components in the power grid that must be under preventive maintenance (PM). The main goal is to find an optimal setup of PM within the specified mission time, minimizing system operation costs and reducing time-dependent unavailability. The method for unavailability quantification was remade to include different stochastic models for the unavailability calculation of system components working in different maintenance modes. A cost model is suggested to estimate the cost of various maintenance configurations. By applying these methodological tools designed to benefit users of any complex system, an optimal PM policy was developed for the selected smart grid. This policy reduces grid unavailability by approximately 20% and lowers costs by about 8.5% compared to a configuration without maintenance.

  Abstract
  This paper explores the feasibility of solving a maintenance optimization problem in an interconnected smart grid system, comprising a power grid and a communication network, [...]

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• Natural Gas Heating in Serbian and Czech Towns: The Role of Urban Topologies and Building Typologies

  D. Brkić, Z. Stajić

  Urban Science (2025). Brkić, D.; Stajić, Z.; Temeljkovski Novaković, D. Natural Gas Heating in Serbian and Czech Towns: The Role of Urban Topologies and Building Typologies. Urban Sci. 2025, 9, 284. https://doi.org/10.3390/urbansci9070284

  
  

  Abstract

  This article presents an analysis on natural gas heating in residential areas, focusing on two primary systems: (1) local heating, where piped gas is delivered directly to individual dwellings equipped with autonomous gas boilers, and (2) district heating, where gas or an alternative fuel powers a central heating plant, and the generated heat is distributed to buildings via a thermal network. The choice between these systems should first consider safety and environmental factors, followed by the urban characteristics of the settlement. In particular, building typology—such as size, function, and spatial configuration—and urban topology, referring to the relative positioning of buildings, play a crucial role. For example, very tall buildings often exclude the use of piped gas due to safety concerns, whereas in other cases, economic efficiency becomes the determining factor. To support decisionmaking, a comparative cost analysis is conducted, assessing the required infrastructure for both systems, including pipelines, boilers, and associated components. The study identifies representative residential building types in selected urban areas of Serbia and Czechia that are suitable for either heating approach. Additionally, the article examines the broader energy context in both countries, with emphasis on recent developments in the natural gas sector and their implications for urban heating strategies.

  Abstract
  This article presents an analysis on natural gas heating in residential areas, focusing on two primary systems: (1) local heating, where piped gas is delivered directly to [...]

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• European Efficiency Schemes for Domestic Gas Boilers: Estimation of Savings in Heating of Settlements

  D. Brkić

  Algorithms (2025). Brkić, D. European Efficiency Schemes for Domestic Gas Boilers: Estimation of Savings in Heating of Settlements. Algorithms 2025, 18, 416. https://doi.org/10.3390/a18070416

  
  

  Abstract

  This article aims to evaluate the seasonal efficiency of natural gas boilers used in European households, highlighting the cost effectiveness, environmental benefits, and user comfort associated with higher-efficiency models, particularly those based on condensing technology. The study applies a standardized algorithm used in European energy labeling schemes to calculate the seasonal efficiency of household gas boilers. It further includes a comparative analysis of selected boiler models available on the Serbian market and outlines a step-by-step method for estimating gas savings when replacing older, less efficient boilers with modern units. Condensing boilers demonstrate significantly higher seasonal efficiency than standard models by recovering additional heat from exhaust gases. These improved boilers produce lower greenhouse gas emissions and offer annual fuel savings of approximately 10% to 30%, depending on the boiler’s age, system design, and usage patterns. The results also confirm the direct correlation between seasonal efficiency and annual fuel consumption, validating the use of efficiency-based cost comparisons. The analysis focuses on residential gas boilers available in the Serbian market, although the models examined are commonly distributed across Europe. The findings highlight the important role of energy efficiency labels—based on a standardized algorithm—in guiding boiler selection, helping consumers and policymakers make informed decisions that promote energy savings and reduce environmental impact. This article contributes to the theoretical and practical understanding of gas boiler efficiency by integrating algorithm-based evaluation with market data and user-centered considerations. It offers actionable insights for consumers, energy advisors, and policymakers in the context of Europe’s energy transition. Verifying the efficiency calculations of gas boilers requires a careful combination of theoretical methods, measured data, and adherence to standards.

  Abstract
  This article aims to evaluate the seasonal efficiency of natural gas boilers used in European households, highlighting the cost effectiveness, environmental benefits, and [...]

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• Experimental Analysis of Magnetic Focusing of the Plasma Arc of a Cutting Torch

  M. Marek, D. Brkić, P. Praks, T. Kozubek, J. Frantík

  Materials (2025). Marek, M.; Brkić, D.; Praks, P.; Kozubek, T.; Frantík, J. Experimental Analysis of Magnetic Focusing of the Plasma Arc of a Cutting Torch. Materials 2025, 18, 1811. https://doi.org/10.3390/ma18081811

  
  

  Abstract

  This study aimed to verify the possibility of stabilizing and focusing a plasma column generated by a plasma cutter. The simulation performed by the COMSOL Multiphysics software is based on the actual configuration and geometry of the burner. This article presented a universal computational method based on FEM simulations, focusing on the deflection of the current of electrically charged particles in a magnetic field within the context of a plasma cutting torch. The simulations estimate the optimal shape and positioning of a focused electron beam for various magnetic lens positions and plasma stream energies, revealing that higher initial electron energies lead to a more even beam focus. Among the configurations tested, positioning the cathode 3 mm above the ring-shaped permanent magnet proved most effective, maintaining beam linearity and minimizing electron scattering, making it suitable for practical implementations.

  Abstract
  This study aimed to verify the possibility of stabilizing and focusing a plasma column generated by a plasma cutter. The simulation performed by the COMSOL Multiphysics software [...]

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• Fire Hazards Caused by Equipment Used in Offshore Oil and Gas Operations: Prescriptive vs. Goal-Oriented Legislation

  D. Brkić

  Fire (2025). https://www.mdpi.com/2571-6255/8/1/29

  
  

  Abstract

  Тhis article offers a concise overview of the best practices for safety in offshore oil and gas operations, focusing on the risks associated with various types of equipment, particularly on the risk of fire. It identifies specific machinery and systems that could pose hazards, assesses their potential impact on safety, and explores conditions that may lead to accidents. Some of the largest accidents were analyzed for their associations with fire hazards and specific equipment. Two primary regulatory approaches to offshore safety are examined: the prescriptive approach in the United States (US) and the goal-oriented approach in Europe. The prescriptive approach mandates strict compliance with specific regulations, while in the goal-oriented approach a failure to adhere to recognized best practices can result in legal accountability for negligence, especially concerning human life and environmental protection. This article also reviews achievements in safety through the efforts of regulatory authorities, industry collaborations, technical standards, and risk assessments, with particular attention given to the status of Mobile Offshore Drilling Units (MODUs). Contrary to common belief, the most frequent types of accidents are not those involving a fire/explosion caused by the failure of the Blowout Preventer (BOP) after a well problem has already started. Following analysis, it can be concluded that the most frequent type of accident typically occurs without fire and is due to material fatigue. This can result in the collapse of the facility, capsizing of the platform, and loss of buoyancy of mobile units, particularly in bad weather or during towing operations. It cannot be concluded that accidents can be more efficiently prevented under a specific type of safety regime, whether prescriptive or goal-oriented.

  Abstract
  Тhis article offers a concise overview of the best practices for safety in offshore oil and gas operations, focusing on the risks associated with various types of equipment, [...]

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• Serbian energy sector in the global political landscape amid the Russia‑Ukraine war: a focus on perspectives of integration into the European Union

  D. Brkić

  Discover Energy (2024). https://link.springer.com/article/10.1007/s43937-024-00056-2

  
  

  Abstract

  This article consists of two main parts, both in relation to Serbia’s accession to the European Union EU in relation to its energy sector: (1) Political and policy issues, and (2) Energy production, consumption and pricing. Each is heavily influenced by the Russia-Ukraine War. Political issues are primarily related to Kosovo, which unilaterally declared independence from Serbia. Regarding supply, Serbia’s energy sector is affected unevenly: while electricity production remains self-sufficient, shortages in oil and gas necessitate imports. Natural gas imports, primarily from Russia, now bypass Ukraine, aided by the new Turk Stream pipeline via Black Sea and also new EU-funded interconnector with Bulgaria. Newly introduced EU sanctions restrict Russian crude oil imports especially via maritime routes, but Russian influence still dominates domestic refineries and hydrocarbon extraction. Serbia’s heavy reliance on environmentally unfriendly lignite conflicts with EU renewable goals, but on the other hand protests against small hydropower projects for capturing rivers in pipes are frequent nowadays. Recent issues in the largest thermal power plant led to costly temporary imports. Despite challenges, Serbia navigates a complex energy landscape, balancing geopolitical realities with domestic and EU objectives while addressing environmental concerns, energy security and its national interest.

  Abstract
  This article consists of two main parts, both in relation to Serbia’s accession to the European Union EU in relation to its energy sector: (1) Political and policy issues, [...]

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• Two iterative methods for sizing pipe diameters in gas distribution networks with loops

  D. Brkić

  Computation (2024). Brkić D. Two Iterative Methods for Sizing Pipe Diameters in Gas Distribution Networks with Loops. Computation. 2024; 12(2):25.

  
  

  Abstract

  Closed-loop pipe systems allow the possibility of the flow of gas from both directions across each route, ensuring supply continuity in the event of a failure at one point, but their main shortcoming is in the necessity to model them using iterative methods. Two iterative methods of determining the optimal pipe diameter in a gas distribution network with closed loops are described in this paper, offering the advantage of maintaining the gas velocity within specified technical limits, even during peak demand. They are based on the following: (1) a modified Hardy Cross method with the correction of the diameter in each iteration and (2) the node-loop method, which provides a new diameter directly in each iteration. The calculation of the optimal pipe diameter in such gas distribution networks relies on ensuring mass continuity at nodes, following the first Kirchhoff law, and concluding when the pressure drops in all the closed paths are algebraically balanced, adhering to the second Kirchhoff law for energy equilibrium. The presented optimisation is based on principles developed by Hardy Cross in the 1930s for the moment distribution analysis of statically indeterminate structures. The results are for steady-state conditions and for the highest possible estimated demand of gas, while the distributed gas is treated as a noncompressible fluid due to the relatively small drop in pressure in a typical network of pipes. There is no unique solution; instead, an infinite number of potential outcomes exist, alongside infinite combinations of pipe diameters for a given fixed flow pattern that can satisfy the first and second Kirchhoff laws in the given topology of the particular network at hand.

  Abstract
  Closed-loop pipe systems allow the possibility of the flow of gas from both directions across each route, ensuring supply continuity in the event of a failure at one point, [...]

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• Serbian Energy Sector in a Gap Between East and West

  D. Brkić

  Serbian energy sector in a gap between east and west (2024). 1. Brkić, D. (2024). Serbian energy sector in a gap between east and west. Energy Exploration & Exploitation, Vol. 42, No. 1, 330-340., https://doi.org/10.1177/01445987231215445, eISSN 2048-4054 (online version) - ISSN 0144-5987 (printed version)

  
  

  Abstract

  Serbia’s energy sector is heavily reliant on Russian influence. On the other hand, Serbia’s status as a candidate country for joining the European Union (EU) membership requires active working toward diversifying energy sources of supply. In the past decade, Serbia has secured a reduced price for natural gas through a bilateral agreement with Russia, addressing the shortfall in its domestic production. The former agreement priced Russian gas at US$270 per thousand cubic meters and expired in 2021. The new deal links gas prices to crude oil and ranges between US $310 and US$408, maintaining its competitive position as one of Europe’s lowest import prices. Furthermore, alongside the new gas pipeline for Russian gas exports, the EU is funding the construction of a new interconnector, both with entry points from Bulgaria. Serbia also faces significant dependence on crude oil, and this reliance is compounded by the inability to import it from Russia any longer. Opposite, Serbia is usually self-sufficient in electricity production which still remains under state ownership. The domestic exploration and processing of oil and gas, as well as the sole underground gas storage facility in Serbia, have partial ownership by Russian Gazprom while the transportation of gas is under the full control of the Serbian government. This Communication about the energy situation in the Republic of Serbia put particular emphasis on the evolving political dynamics in the global energy market with a specific focus on the Russia– Ukraine war. The topic is also linked to the contentious status of the southern Serbian autonomous province, recognized as an independent state by the majority of Western nations but not by Serbia. It is feared that Serbia’s energy dependence on Russia could have significant ramifications for its EU candidacy.

  Abstract
  Serbia’s energy sector is heavily reliant on Russian influence. On the other hand, Serbia’s status as a candidate country for joining the European Union (EU) [...]

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• Symbolic Regression Approaches for the Direct Calculation of Pipe Diameter

  D. Brkić, P. Praks, R. Praksová, T. Kozubek

  Axioms (2023). Brkić, D.; Praks, P.; Praksová, R.; Kozubek, T. Symbolic Regression Approaches for the Direct Calculation of Pipe Diameter. Axioms 2023, 12, 850. https://doi.org/10.3390/axioms12090850

  
  

  Abstract

  This study provides novel and accurate symbolic regression-based solutions for the calculation of pipe diameter when flow rate and pressure drop (head loss) are known, together with the length of the pipe, absolute inner roughness of the pipe, and kinematic viscosity of the fluid. PySR and Eureqa, free and open-source symbolic regression tools, are used for discovering simple and accurate approximate formulas. Three approaches are used: (1) brute force of computing power, which provides results based on raw input data; (2) an improved method where input parameters are transformed through the Lambert W-function; (3) a method where the results are based on inputs and the Colebrook equation transformed through new suitable dimensionless groups. The discovered models were simplified by the WolframAlpha simplify tool and/or the equivalent Matlab Symbolic toolbox. Novel models make iterative calculus redundant; they are simple for computer coding while the relative error remains lower compared with the solution through nomograms. The symbolic-regression solutions discovered by brute force computing power discard the kinematic viscosity of the fluid as an input parameter, implying that it has the least influence.

  Abstract
  This study provides novel and accurate symbolic regression-based solutions for the calculation of pipe diameter when flow rate and pressure drop (head loss) are known, together [...]

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Articles (Documents not available in Scipedia)

• Discussion of “Friction modeling of flood flow simulations” by Vasilis Bellos, Ioannis Nalbantis, and George Tsakiris

  D Brkić, P Praks

  Journal of Hydraulic Engineering 146 (5), article 07020007, 2020

• Discussion of “Approximate analytical solutions for the Colebrook equation” by Ali R. Vatankhah

  D Brkić, P Praks

  Journal of Hydraulic Engineering 146 (2), article 07019011, 2020

• A possible publishing model profitable for authors

  D Brkić

  GRIN Verlag, 2019

• Correction: Brkić, D., and Praks, P. Accurate and efficient explicit approximations of the Colebrook flow friction equation based on the Wright ω-function. Mathematics 2019, 7, 34

  D Brkić, P Praks

  Mathematics 7 (10), article 951, 2019

• Discussion of “Economics and statistical evaluations of using Microsoft Excel Solver in pipe network analysis”

  D Brkić

  Journal of Pipeline Systems Engineering and Practice 9 (3), article 07018002, 2018

• Discussion of “Exact analytical solutions of the Colebrook-White equation”

  D Brkić

  Journal of Hydraulic Engineering ASCE 143 (9), article 07017007, 2017

• Discussion of “Gene expression programming analysis of implicit Colebrook–White equation in turbulent flow friction factor calculation”

  D Brkić

  Journal of Petroleum Science and Engineering 124, 399-401, 2014

• Discussion of “Method to cope with zero flows in Newton solvers for water distribution systems”

  D Brkić

  Journal of Hydraulic Engineering 140 (4), 456-459, 2014

• Very accurate explicit approximations for calculation of the Colebrook friction factor

  Ž Ćojbašić, D Brkić

  International Journal of Mechanical Sciences 67, 10-13, 2013

• Discussion of “Sustainable electric power system: Is it possible? Case study: Croatia”

  D Brkić

  Journal of Energy Engineering 138 (3), 163-164, 2012

• Discussion of “Jacobian matrix for solving water distribution system equations with the Darcy-Weisbach head-loss model”

  D Brkić

  Journal of Hydraulic Engineering 138 (11), 1000-1001, 2012

• Discussion of “Water distribution system analysis: Newton-Raphson method revisited”

  D Brkić

  Journal of Hydraulic Engineering 138 (9), 822-824, 2012

• Can pipes be actually really that smooth?

  D Brkić

  International Journal of Refrigeration 35 (1), 209-215, 2012

• Comparison of the Lambert W‐function based solutions to the Colebrook equation

  D Brkić

  Engineering Computations, 2012

• Determining friction factors in turbulent pipe flow

  D Brkić

  Chemical Engineering N.Y. 119 (3), 34-39, 2012

• Comments on "Experimental study of the pressure drop in the cathode side of air-forced open-cathode proton exchange membrane fuel cells"

  D Brkić

  International Journal of Hydrogen Energy 37 (14), 10963-10964, 2012

• Discussion of “Classroom activities to illustrate concepts of Darcy’s Law and hydraulic conductivity”

  D Brkić, V Mitrović

  Journal of Professional Issues in Engineering Education and Practice 137 (1 …, 2011

• Iterative methods for looped network pipeline calculation

  D Brkić

  Water Resources Management 25 (12), 2951-2987, 2011

• An explicit approximation of Colebrook's equation for fluid flow friction factor

  D Brkić

  Petroleum Science and Technology 29 (15), 1596-1602, 2011

• New explicit correlations for turbulent flow friction factor

  D Brkić

  Nuclear Engineering and Design 241 (9), 4055-4059, 2011

• Review of explicit approximations to the Colebrook relation for flow friction

  D Brkić

  Journal of Petroleum Science and Engineering 77 (1), 34-48, 2011

• Comments on “Settling velocities of particulate systems 15: Velocities in turbulent Newtonian flows”

  D Brkić

  International Journal of Mineral Processing 92 (3-4), 201-202, 2009

• Serbian gas sector in the spotlight of oil and gas agreement with Russia

  D Brkić

  Energy Policy 37 (5), 1925-1938, 2009

• An improvement of Hardy Cross method applied on looped spatial natural gas distribution networks

  D Brkić

  Applied Energy 86 (7-8), 1290-1300, 2009

• Serbian, Russian pipeline accord enhances European gas security

  D Brkić

  Oil & Gas Journal 106 (48), 52-54, 2008

• Systematic approach to natural gas usage for domestic heating in urban areas

  D Brkić, TI Tanasković

  Energy 33 (12), 1738-1753, 2008

• Prirodni gas kao gorivo za grejanje

  D Brkić

  Zadužbina Andrejević, 2006

• Određivanje graničnih parametara upotrebe prirodnog gasa u Beogradu

  D Brkić

  University of Belgrade, Faculty of Mining and Geological Engineering, 2005