**Aim.** Experimental study of main technological parameters of a ball mill during coal milling at a thermal power plant and development of a mathematical model for the mill on the basis of the obtained data. **Method.** Based on the analysis of the energy flows in the ball mill the relation between the vibration energy of the mill body and the amount of the material in the mill is defined. Experimental study, i.e. measurement of the main technological parameters (mill loading with the material, air mixture temperature at the mill output and differential pressure across the mill), was carried out at an acting ball mill. The heat and mass conservation laws as well as flow rate and heat exchange equations were applied to develop the mathematical model of a ball mill in the form of the system of nonlinear differential equations. The transient processes were simulated in Simulink (Matlab) and compared to the experimental data. The relative reduced errors for each parameter were calculated to estimate the adequacy of the developed mathematical model. **Results.** It was found that the relative productivity of the ball mill can be defined using the root mean square level of the vertical component of the vibration acceleration for the front bearing of the coal mill in the range from 2 to 6 kHz. For the maximum productivity of the mill this level is minimal and does not vary any more at further loading of the mill. The mathematical model of the ball mill was developed. The transient processes of the main parameters were simulated and compared to the obtained experimental data. The adequacy of the developed model was estimated. **Scientific novelty.** The mathematical model of the ball mill was developed in the form of system of nonlinear differential equations which provides the possibility for modeling the transient processes with sufficient accuracy. The relative reduced errors of the simulated transient processes with respect to the experimental processes were calculated. For the air mixture temperature variation this error is 5.0 %, for the mill loading signal it is 7.4 %, and for the differential pressure across the mill drum the error equals 11.2 %. **Practical significance.** The developed mathematical model can be applied at practice for studying the ball mill operation in various modes as well as for development of automatic control algorithms for coal milling process at thermal power plants.

1. Pistun, Y., Zagraj, V. & Skobalo, A. (2002). Automatic control and optimization of ball mills, Proc. of VIII Forum of Power Engineers, Techn. Univ. of Opole, May 29-31, 2002, ISBN 83-88492-04-7, Kabza, Z. (Ed.), pp. 575-581, Publ.House of Tech. Univ. of Opole, Opole, Poland.

2. Fedoryshyn, R.; Nykolyn, H.; Zagraj, V. & Pistun, Y. (2012). The improved system for automation and optimization of solid material grinding by means of ball mills. Annals of DAAAM for 2012 & Proceedings of the 23rd International DAAAM Symposium, ISBN 978-3-901509-91-9, ISSN 2304-1382, CDROM version, pp.053-056, Editor B. Katalinic, Published by DAAAM International, Vienna, Austria, EU, 2012.

3. T. Chai, L. Zhai, and H. Yue, (2011). "Multiple models and neural networks based decoupling control of ball mill coal-pulverizing systems," Journal of Process Control, vol. 21, no. 3, pp. 351-366.

https://doi.org/10.1016/j.jprocont.2010.11.007

4. Feng, L., Yang, F., Zhang, W. & Tian, H. (2019). "Model Predictive Control of Duplex Inlet and Outlet Ball Mill System Based on Parameter Adaptive Particle Swarm Optimization", Mathematical Problems in Engineering, vol. 2019.

https://doi.org/10.1155/2019/6812754

5. Lingfang, S., Jingmiao, S., Yinde, M., Congwei, F., Jibing, R. & Wei, Y. (2015), "Application research of PID-GPC algorithm in the ball mill system", Open Automation and Control Systems Journal, vol. 7, no. 1, pp. 157-166.

https://doi.org/10.2174/1874444301507010157

6. Formusatin V. P. (2007) Improvement of dust systems productivity at thermal power stations. - Power stations, No. 6, pp. 1-4 [in Russian].

7. Levit, G. (1991). Production of dust at thermal power stations, Energoatomizdat, ISBN 5-283-00151-2, Moscow, 384 p [in Russian].

8. Bai, Y. & He, F. (2015), "Modeling on the effect of coal loads on kinetic energy of balls for ball mills", Energies, vol. 8, no. 7, pp. 6859-6880.

https://doi.org/10.3390/en8076859

9. Druzhbliak O. M., Pistun Y. P., Trus A. I. (1984) Systems for feeding the ball mills. - Power engineering and electrification, No. 8, pp. 29-32 [in Russian].

10. Macku, L[ubomir] & Novosad, D[avid] (2017). Influence of Online Identification Methods on the Nonlinear Process Control, Proceedings of the 28th DAAAM International Symposium, pp.0216-0223, B. Katalinic (Ed.), Published by DAAAM International, ISBN 978-3-902734-11-2, ISSN 1726-9679, Vienna, Austria.

https://doi.org/10.2507/28th.daaam.proceedings.029

11. Opalka, J. & Hubka, L. (2015), "Nonlinear state and unmeasured disturbance estimation for use in power plant superheaters control", Procedia Engineering, pp. 1539.

https://doi.org/10.1016/j.proeng.2015.01.526

12. Filaretov, V[ladimir]; Zhirabok, A[lexey]; Zuev, A[lexander] & Protcenko, A[leksandr] (2016). Identification of Faults in Nonlinear Dynamic Systems, Proceedings of the 26th DAAAM International Symposium, pp.0470-0477, B. Katalinic (Ed.), Published by DAAAM International, ISBN 978-3-902734- 07-5, ISSN 1726-9679, Vienna, Austria.

https://doi.org/10.2507/26th.daaam.proceedings.062