Methods and Models for Optimal Managing of District Heating System Accounting Reliability Based on Lagrange Multiplier Method and Markov Random Process

The main purpose of the study is to determine the optimal condition of the district heating system functioning while minimizing the operating costs for the production and distribution of thermal energy, accounting the expected economic damages from possible heating undersupplies in case of failures of system components. To identification the parameters corresponding to these conditions, a methodological approach is proposed, which consists in the comprehensive calculation of nodal prices for thermal energy and reliability indices for the studied system when different component failures. The determination of nodal prices for thermal energy is based on the Lagrange multiplier method, while the calculation is carried out accounting the different cost of production at heating sources and the distribution of thermal energy flows on the heating network. Within the framework of the studied methodological problem, heating undersupplies to consumers in emergency conditions of the system and the corresponding economic equivalents (damages) are used as reliability indices. Assessment of probabilities of emergency conditions (states) is carried out using models of the markov random process. The general methodological basis for modeling thermal-hydraulic conditions in the system, including emergency ones, are the models of the theory of hydraulic circuits. The developed methods and models, along with the main formulated problem of optimal managing, allow identifying “bottlenecks” in the studied system, corresponding to the maximum economic damages from heating undersupplies to consumers based on obtained reliability indices. A computational experiment was carried out using the proposed methodology. The results are analyzed, their graphical interpretations are presented, and directions for further research are formulated.