## 1 Introduction
Moreover, a group of residential houses were exploited as an interconnected EH system with the aim of reducing the daily costs and in terms of the cost function of the battery life with the particle swarm optimization (PSO) method. The study in [18] provided a two-level optimization model of optimal planning. An active distribution system that consisted of DGs and several EHs could supply its surplus electricity to the market and was used daily. In [19], the planning of an integrated energy system with the participation of electricity, gas, and several smart EHs along with the consumer was presented using the hierarchical game method. In [20], an optimal planning model based on reliability was presented in order to connect EHs employing multi-carrier energy infrastructures. In [21], the planning of energy production in a smart energy network consisting of five examples of EHs to minimize the cost of energy supply and greenhouse gas emissions was investigated, and in [22], a hierarchical energy management system for minimizing the cost and peak shaving of the upstream network was presented in the local network consisting of different residential EHs. In [23], an EH optimization method was presented in the demand response energy market. The planning model for the performance of energy sources and energy storage by satisfying the constraints of the electricity and natural gas network considering the responsive load was presented using the water wave optimization (WWO) algorithm. In [24], EH system planning using wind and photovoltaic sources with optimal interaction between different sources to supply different system loads was investigated using quantum particle swarm optimization (QPSO) to minimize the total system cost. In [25], energy planning in a storage-based residential system was presented based on a multi-criteria optimization method with the participation of the demand side for minimizing production costs and maximizing the level of consumer satisfaction using the shuffled frog leaping algorithm (SFLA). In [26], the optimization of the EH to minimize operating costs, carbon emission, and energy efficiency based on a multi-objective optimization model was presented using VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR). In [27], EH management was developed with the participation of electricity, gas, and heating networks with the aim of minimizing the cost of operation in the presence of energy and storage resources using an ant-lion optimizer and krill herd optimization (ALO-KHO) algorithm. The review of the literature makes the necessity of an economic approach to the planning of interconnected EHs even more clear. In addition, economic analysis and evaluation based on the market model for interconnected EHs can be very effective in maximizing their profitability in the day-ahead electricity, gas, and heating markets, which have not been well addressed in previous studies. Furthermore, considering that optimal and coordinated planning of EH equipment improves network performance and system flexibility in cooperation with various types of electric, gas, and heating networks, the importance of using a powerful solver is felt. The key model for optimal modeling of multi-carrier EH systems is very time-consuming and challenging due to its nonlinear and multi-dimensional functions. To solve the problem mentioned in the literature, methods to approximate one-dimensional and multi-dimensional linear singularities to linearize the non-convex functions of natural gas transfer, generator cost, and compressor performance have been adopted, while using meta-heuristic algorithms, the linearization is not needed. The use of intelligent meta-heuristic algorithms based on iteration with high computing and optimization power can provide the conditions for achieving the maximum benefit of the EH by accurately determining the optimal capacity of EH equipment and, as a result, optimal and coordinated planning.