## 1 Introduction  
Industrial, commercial, and residential consumers are connected to energy networks, such as electricity, natural gas, and district heating or cooling [1]. Therefore, integrating these networks with each other and using the merits of each will increase the efficiency and reliability of the system, as well as optimal performance [2; 3]. In recent years, a general framework has been proposed that combines various energy carriers and performs conversion and storage in them in order to provide the required load on the consumer side; it is called an EH [4; 5]. The input to an EH is various types of energy, and the output is another form of energy demanded by different local or remote end-users [6]. An EH model can include CHP equipment, distributed generation, renewable energy sources, electrical and heating energy storage and boilers, and active loads. On the other hand, simultaneous and coordinated planning of all energy equipment at the point of consumption can improve network performance and system flexibility in cooperation with various types of electric, gas, and heating networks [7; 8]. Various studies have been conducted in the field of EH. In [9], the EH had various sources including cogeneration units, steam boilers, renewable resources, electric chillers, and absorption chillers, as well as electric, heating, and cooling energy storage devices in order to improve the flexibility of EHs. The results showed that, unlike the balance-based methods that do not guarantee the optimality of the response, in [10], the operation of a multi-carrier local distribution system in the case of islands that were separated from the main grid due to faults or important incidents was achieved. This network consisted of three EHs, each of which, while planning to optimally meet their demand, could minimize operating costs by exchanging with other EHs. In [11], a framework for the optimal operation of interconnected EHs to reduce costs, provide the energy needed by consumers, reduce greenhouse gas emissions, and strengthen the interaction of electricity and gas networks was presented using electricity to gas converters. In [12], a decentralized energy management framework was developed based on the interaction that enables coordination between EHs. In this research, to improve the economic performance of the integrated EH system, a trading platform was created to facilitate the self-organized trade of the integrated EH system. In [13], a two-level optimization framework was created to determine the optimal strategy of interaction between EHs and the distribution company, in which the total cost of operating the distribution network was minimized according to the limitations of the network at the upper level, and the total cost of each EH connected to the distribution network was minimized at the lower level. In [14], it was also investigated that the grid-connected EH system is a key model for the optimal modeling of multi-carrier energy systems. However, the direct calculation of the operating mode of this model is very time-consuming and challenging due to its nonlinear and multidimensional functions. To solve the above problem in the proposed method of this research, methods were adopted to approximate one-dimensional and multi-dimensional linear singularities for linearizing the non-convex functions of natural gas transmission, generator costs, and compressor performance. In [15], a new framework for the optimal management of EHs was presented. On this basis, each EH manages its production resources to plan the supply of demand to reduce the cost and emission of pollutants. In [16], it was reported that residential and commercial buildings with different consumption patterns could be controlled together and the distributed generation (DG) resources available in each of these hubs could be used to cover the lack of capacity in another hub. In [17], the linear approximation method was used to simplify the model of an interconnected system, including three EHs, to mitigate the computational costs.