Heat Price Field Calculation Based on the Extreme Problem of Searching the Optimal Load Flow in Heat-Supply Systems

The article proposes a method for calculating differentiated prices for heat at the nodes of a heat-supply system (HSS) based on the extreme economic formulation of the problem of searching for the optimal load flow in heat networks containing several heat sources. The problem boils down to finding the minimal total costs associated with heat generation and transportation, maintaining of material balances at heat network nodes (Kirchhoff’s first law), and constraints imposed on the heat source capacities. The problem is solved on the basis of Lagrange’s method of undetermined multipliers. An analysis of the initial optimization problem duality has shown that the dual variables (Lagrange’s multipliers) in balance constraints are nothing but nodal prices for heat, and the optimal load flow in a heat network obtained in the calculation process makes it possible to form the price field for the system as a whole. It is also shown that the prices for heat from the heat sources to end consumers increase in the direction of the steady-state optimal load flow in the heat network. With such an approach to solving the problem, it is also possible to determine the optimal action zones and levels of loading the heat sources taking into account their cost characteristics and the specified heat network’s physical, technical, and economic indicators. The nodal prices obtained from the calculations are in their economic essence marginal ones, i.e., prices based on calculating the limit low and limit high costs for generation and transportation of an additional unit of heat.