• Andrey Batukhtin Transbaikal state university
  • Irina Batukhtina Transbaikal state university
  • Maxim Bass Transbaikal state university
  • Sergey Batukhtin Transbaikal state university
  • Mihail Kobylkin Transbaikal state university
  • Marina Baranovskaya Transbaikal state university
  • Alena Baranovskaya Transbaikal state university
Keywords: district heating systems, first-order differential equations, regression, generalized regression neural network, radial basis neural network, linear neural network, scheduling, mathematical models, inertia, accuracy,


The article describes a new method for making management decisions on heat supply in district heating systems, based on solving a sequence of recurrence relations of first-order differential equations, enabling to synthesize daily schedules of heat supply in such systems. Using first order differential equations, we implement real-time daily heat supply scheduling, predict the time-temperature dependence for heating water in the supply line, and we form a decision on the thermal energy delivery on the basis of this information. The effectiveness of our method is confirmed by numerical modeling and comparative analysis of daily heat supply scheduling with the help of advanced intelligent decision making tools. For comparative analysis, we considered daily scheduling using a nonlinear regression model, a generalized regression neural network, a radial basis neural network, and a linear neural network. The effectiveness of our method was estimated on the basis of MAPE (mean absolute percentage error) and Accuracy coefficients. The model was recognized as most effective for which the MAPE value was maximum, and the Accuracy value tended to one hundred percent. Experimental studies showed that our proposed model has an advantage over the regression model by 1.68 times and over the neural models by more than 10.2 times when modeling for a hundred heating network sections. Thus, the main purpose of our study was to increase the accuracy of the method of making a managerial heat supply decision based on the experimental verification of a mathematical model of thermal inertia of a branched district heating system.


Sokolov, E.Ya. (1963) District heating introduction and heating networks. Moscow-Leningrad: State Energy Publishing House.

Baykasoğlu, A., & Ozsoydan, F. B. (2018). Dynamic scheduling of parallel heat treatment furnaces: A case study at a manufacturing system. Journal of Manufacturing Systems, vol. 46, 152–162, DOI:10.1016/j.jmsy.2017.12.005

Yang, Ch., Chen, W., Zhao, R., & Xu, T. (2018). Stochastic analysis of effective moment of inertia of cracked in-service reinforced concrete beams. Journal of Southwest Jiaotong University, vol. 53, no. 3, 492–499, DOI: 10.3969/j.issn.0258-2724.2018.03.009.

Magli, S., Lodi, C., Contini, F. M., Muscio, A., & Tartarini, P. (2016). Dynamic analysis of the heat released by tertiary buildings and the effects of urban heat island mitigation strategies. Energy and Buildings, vol. 114, 164–172, DOI:10.1016/j.enbuild.2015.05.037

Sheng, Shiqi, and Tu, Z.C. (2013). Universality of energy conversion efficiency for optimal tight-coupling heat engines and refrigerators. Journal of Physics A: Mathematical and Theoretical vol. 46, no. 40, DOI:10.1088/1751-8113/46/40/402001.

Bolatturk A. (2006). Determination of optimum insulation thickness for building walls with respect to various fuels and climate zones in Turkey. Appl Therm Eng, vol. 26, no.11, 12, 1301–1309. DOI: 10.1016/j.applthermaleng.2005.10.019.

Kalema, T., Jóhannesson, G., Pylsy, P., Hagengran, P. (2008). Accuracy of Energy Analysis of Buildings: A Comparison of a Monthly Energy Balance Method and Simulation Methods in Calculating the Energy Consumption and the Effect of Thermal Mass, vol. 32, no 2, 101-130, DOI: 10.1177/1744259108093920.

Stepanov V., Starikova, N. and Stepanova, T. (2000). Indices for estimation of energy conservation in space heating. Energy and buildings, vol. 30, no.3, 189-193, DOI: 10.1016/S0378-7788(99)00013-4.

Valero, A. (2006). Exergy accounting: capabilities and drawbacks. Energy, vol. 31, no. 1, 164-180, DOI: 10.1016/

Torío, H., Angelotti, A., Schmidt, D. (2009). Exergy analysis of renewable energy-based climatisation systems for buildings: a critical view. Energy and Buildings, vol. 41, no. 3, 248-271, DOI: 10.1016/j.enbuild.2008.10.006.

Romero, J.C., Linares, P. (2014). Exergy as a global energy sustainability indicator. A review of the state of the art. Renewable and Sustainable Energy Reviews, vol. 33, 427-442, DOI: 10.1016/j.rser.2014.02.012.

Bass, M.S., Batukhtin, A.G. (2011). An Integrated Approach for Optimizing the Operation of Modern Heat Supply Systems. Thermal Engineering, vol. 58, no. 8, 678–681, DOI:10.1134/S0040601511080052

Goryachikh N.V., Batukhtin A.G., Ivanov 2010. S.A. Some Methods for Making Cogeneration Stations More Maneuverable. Thermal Engineering, vol. 57, no. 10, 892–896, DOI:10.1134/S0040601510100125

Kicsiny, R. (2014). New delay differential equation models for heating systems with pipes, Int. J. Heat Mass Transfer, vol. 79, 807–815, DOI: 10.1016/j.ijheatmasstransfer.2014.08.058.

Bau, U., Braatz, AL., Lanzerath, F., Herty, M., Bardow, A. (2015). Control of adsorption chillers by a gradient descent method for optimal cycle time allocation International Journal of Refrigeration - Revue Internationale Du Froid, vol. 56, 52-64, DOI: 10.1016/j.ijrefrig.2015.03.026.

Kicsiny, R. (2017) Grey-box model for pipe temperature based on linear regression International Journal of Heat and Mass Transfer, vol. 107, 13-20, DOI: 10.1016/j.ijheatmasstransfer.2016.11.033/

Messerle V.E., Karpenko E.I., Ustimenko A.B., Lavrichshev O.A. (2013). Plasma preparation of coal to combustion in power boilers. Fuel Processing Technology, vol. 107, 93–98, DOI: 10.1016/j.fuproc.2012.07.001.

Messerle V.E., Karpenko E.I., Ustimenko A.B. (2014). Plasma Assisted Power Coal Combustion in the Furnace of Utility Boiler: Numerical Modelling and Full-Scale Test. Fuel, vol. 126, 294-300, DOI: 10.1016/j.fuel.2014.02.047.

Messerle V.E., Mosse A.L., Ustimenko A.B. (2016). Municipal Solid Waste Plasma Processing: Thermodynamic Computation and Experiment. IEEE Transactions on Plasma Science, vol. 99, 1-6, DOI: 10.1109/TPS.2016.2601107

Messerle V.E., Ustimenko A.B., Lavrichshev O.A. (2016). Comparative study of coal plasma gasification: Simulation and experiment. Fuel, vol. 164, 172-179, DOI:10.1016/j.fuel.2015.09.095

Gao H., Chui E., Runstedler A. (2010). Numerical investigation of plasma ignition process in a utility boiler. Proceedings of 6th International Workshop and Exhibition on Plasma Assisted Combustion (IWEPAC). Heilbronn, Germany.

Karpenko E. I., Rinchinov A. P., Karpenko Y. E., Bass, M. S., Batukhtin S. G. (2016). The results of the tests of experimental-industrial plasma-cyclone installation. Industrial Energetics, vol. 4, 24-27, DOI: 10.12973/ejmste/79043

Kaminski, K., Krzyzynski T. (2015). Modeling and Simulation of the Solar Collector Using Different Approaches. Mechatronics: Ideas, Challenges, Solutions and Applications, 131-151, DOI: 10.1007/978-3-319-26886-6_9

Hussain, MI., Ali, A., Lee, GH. (2016). Multi-module concentrated photovoltaic thermal system feasibility for greenhouse heating: Model validation and techno-economic analysis. Solar Energy, vol. 135, 719-730, DOI: 10.1016/j.solener.2016.06.053.

Beg, O. Anwar; Ali, Nasir; Zaman, Akbar. (2016). Computational modeling of heat transfer in an annular porous medium solar energy absorber with the P1-radiative differential approximation. Journal of the Taiwan Institute of Chemical Engineers, vol. 66, 258-268, DOI: 10.1016/j.jtice.2016.06.034

Li, H. and Chen, Zh. (2008). Overview of different wind generator systems and their comparisons. IET Renewable Power Generation, vol. 2, no. 2, 123-138, DOI: 10.1049/iet-rpg:20070044

Smith, E., Koolnapadol, N., Promvonge, P. (2012). Heat transfer behavior in a square duct with tandem wire coil element insert. ChinJChem Eng, vol. 20, 863–869, WOS:000311472200008

Thianpong, C., Yongsiri, K., Nanan, K. and Eiamsa-ard, S. (2012). Thermal performance evaluation of heat exchangers fitted with twisted-ring turbulators. IntCommun Heat Mass Transf, vol. 39, 861–868, DOI: 10.1016/j.icheatmasstransfer.2012.04.004

Kicsiny, R. (2016). Improved multiple linear regression based models for solar collectors. Renewable Energy, vol. 91, 224–232, DOI: 10.1016/j.renene.2016.01.056.

Stevanovic, V.D., B. Zivkovic, S. Prica, B. Maslovaric, V. Karamarkovic, V. Trkulja, (2009). Prediction of thermal transients in district heating systems. Energy Convers. Manage, vol. 50, 2167–2173, DOI: 10.1016/j.enconman.2009.04.034

Polyanin, A.D., Zaitsev, V.F. (2003). Handbook of Exact Solutions for Ordinary Differential Equations, 2nd Edition, Chapman & Hall/CRC, Boca Raton.

Karpenko E.I., Trusov B.G. (1995). А Comparative Analysis of Plasma and Fire Technologies of Pulverized Coal Ignition, Combustion and Gasification Using a Mathematical Model of chemically nonequilibrum system. Thermophysics and Aeromechanics, vol. 2, no. 23, p. 245-250.

Saleh, A.M., Mueller, D.W., Abu-Mulaweh, H.I. (2015). Flat-Plate Solar Collector in Transient Operation: Modeling and Measurements. Journal of Thermal Science and Engineering Applications, vol. 7, DOI: 10.1115/1.4028569

Etter, D.M., Kuncicky, D., Moore, H. (2004). Introduction to MATLAB 7, Springer.

Khelifa, A., Touafek, K., Ben Moussa, H., Tabet, I. (2016). Modeling and detailed study of hybrid photovoltaic thermal (PV/T) solar collector. Solar Energy, vol. 135, 169-176, DOI: 10.1016/j.solener.2016.05.048.

Gholampour, M., Ameri, M. (2015). Design Considerations of Photovoltaic/Thermal Air Systems: Energetic and Exergetic Approaches. Journal of Solar Energy Engineering-Transactions of the ASME, vol. 137 (031005), DOI: 10.1115/1.4029107

Khelifa, A., Touafek, K., Ben M. (2015). Approach for the modelling of hybrid photovoltaic-thermal solar collector. IET Renewable Power Generation, vol. 9, 207-217, DOI: 10.1049/iet-rpg.2014.0076.

Batukhtin A. (2017). Solving a Sequence of Recurrence Relations for First-Order Differential Equations. Eurasia Journal of Mathematics, Science and Technology Education, vol. 13, no 11, 7179-7191, DOI: 10.12973/ejmste/79043

Wang, D., Zhi, Y., Jia, H., Hou, K., Zhang. (2019). Optimal scheduling strategy of district integrated heat and power system with wind power and multiple energy stations considering thermal inertia of buildings under different heating regulation modes. Applied Energy, 341–358, DOI: 10.1016/j.apenergy.2019.01.199

Kim, Z., Shin, Y., Yu, J., Kim, G., & Hwang, S. (2019). Development of NOx removal process for LNG evaporation system: Comparative assessment between response surface methodology (RSM) and artificial neural network (ANN). Journal of Industrial and Engineering Chemistry, vol. 74, 136–147, DOI10.1016/j.jiec.2019.02.020

Gonzalez, P.A., Zamarreno, J.A. (2005). Prediction of hourly energy consumption in buildings based on a feedback artificial neural network. Energ. Build., vol. 37, no 6, p. 595-601.

Original Scientific Paper