General Modeling of Interconnected Hubs in Series and Parallel Structures

. Energy decision-makers have considered solutions to meet the demand for a ﬀ ordable and highly reliable energy due to population growth and technological advancement. One of these possibilities is the utilization of several energy carriers in one system as an energy hub. Instead of optimizing a single energy carrier, the energy hub optimizes a system with many energy carriers, including electricity, natural gas, and local heat, by use of its converters and storages. In this paper, a novel and new general framework is proposed to evaluate and compare both series and parallel connections of n hub to analyze cost and reliability aspects by considering coupling matrices. In order to compare the outcomes of the connections of several comparable hubs in both series and parallel modes, new indices are proposed and evaluated from the aspect of evaluating the amount of energy not supplied and the amount of energy input to each hub. Additionally, simulations are run for a variety of scenarios in order to better assess the proposed model and investigate each type of connection by evaluating the proposed performance indices. The results show that in all the examined scenarios, the total cost of the energy carriers in the series mode (link in the output) is lower than in the parallel mode (link in the input).


Motivation and
Aim. An energy hub (EH), or multicarrier energy system, is a new way to provide energy in such a way that different energy carriers at the input of the hub can be converted into other types of energy carriers at the output by the converters used in the energy hub. EHs are equipped with storage devices for storing electrical energy, heat, and other energy carriers, which are stored during nonpeak times and used during peak times. The ever-increasing demand for energy, especially electrical energy, and the grid's inability to meet demand in a sustainable manner have led to new solutions such as energy hub systems. EH systems have the ability to work individually as well as the ability to connect to each other and exchange energy. Many studies have been done in the field of individual energy hub systems and a limited number of connected hubs. In the studies conducted around energy hub systems, more focus has been on providing the optimal value of hub input carriers, and some have focused on optimizing the internal components of the energy hub. Some studies have also investigated the effect of demand response programs in the energy hub system [1,2]. Modeling, investigating, and comparing connected energy hubs in two general modes of connecting hubs at the input (parallel connection) or connecting hubs on both the input and output sides (series connection) are subjects that have not been particularly considered so far. The purpose of this paper is to propose a new general framework to investigate and compare the benefits of connecting energy hub systems in series or parallel, considering the importance of energy hub systems and their increasing development.
1.2. Literature Review. Energy hub systems can be operated independently or integrated. Interconnected hubs are a model in which interaction takes place between several hubs. For this purpose, a multistage linearization method can be applied to the hubs. Accordingly, the whole system can be formulated as a mixed integer linear programming (MILP) problem. One-dimensional and multidimensional linear approximation methods are used to simplify the nonconvex functions of natural gas transmission, generator cost functions, and compressor operation [3]. In [4], a two-objective function (minimizing the cost of different types of energy and minimizing the difference between the cost of purchased energy and the income of energy sales) has been used to optimize the EH. In [5], the interaction between two EHs is modeled as a game. Each EH operator (as an actor) participates in the energy pricing market and tries to maximize its profit according to the satisfaction of energy consumption. In reference [6], strategies for planning and managing energy hub systems under uncertainty are developed using chance-based optimization. Convexification techniques are utilized to formulate the nonlinear flow of electricity and gas, which results in the formulation of the second-order cone problem (SOCP), which is useful for global optimization. Reference [7] has proposed an optimal operation model for managing several hubs with electrical and thermal energy needs, which aims to reduce the total operation cost of the hubs. Reference [8] addresses the interconnected electrical and natural gas system planning issue in terms of the demand-side flexibility of energy hubs. The integrated demand response (IDR) concept has been created for hubs in this regard. Reference [9] shows a reliability-based ideal organizing model for the linking of energy hubs with several energy facilities. From a set of candidate routes that meet probabilistic reliability criteria, the suggested organizing issue defines a low-cost network of transmission lines and natural gas pipelines to link EHs. In [10], a risk-based framework for controlling linked EHs in multiple-energy carrier (MEC) systems is provided to reduce the estimated system operating costs. In [11], three basic schemes for the organization of energy hubs on the demand side, i.e., individual, shared energy market, and demand information collection, are studied under a stochastic framework with respect to probabilistic load forecasts. In [12], a new model is presented to analyze the effects of demand response in an interconnected EHs system. In [13], an integrated demand response scheme is designed to coordinate the operation of electricity-to-gas converters, heat pumps, diverse storage devices, and flexible loads in the framework of advanced modeling of energy hubs. With the aim of exploiting the potential of complementary energy, maximizing the use of sources of clean energy, and jointly reducing operational expenses, an organized operation simulation of an incorporated multienergy carrier structure depending on the linearized coupling connection has been developed in this paper [14].
In [15], a two-level multifunction optimization framework for the interaction of energy hub and distribution network systems has been developed, which minimizes the total cost of distribution network operation according to network constraints in high-level (distribution network) and low-level (energy hub) problems. The proposed model is a nonlinear problem that is linearized using the KKT (Karush-Kuhn-Tucker) conditions and constraints. In [16], microgrids are considered multiple energy poles to meet different energy needs. Reference [17] proposes particle density optimization (PSO) combined with a numerical method, collectively referred to as the decomposition technique. This method decomposes the optimization problem into storage scheduling and other elements of the energy hub system and solves them separately with PSO and "interior point" numerical methods. In [18], a centralized passive energy management framework that enables coordination between multiple energy hubs (MEHs) has been developed. Aiming to improve the economic exploitation of the interconnected hub system, a peer-to-peer (P2P) transaction platform for self-organized trading of MEH has been established. Reference [19] provides an energy management and routing controlling technique for new kinds based on ER utilizing lower and higher levels. In the lower level, the multienergy converting equipment and the storage device in the local energy system are precisely modeled, and the multienergy optimization plan is accomplished with the lowest cost of multienergy electricity, gas, and cooling. In the upper layer, the energy flow of the electricity, heating, and natural gas networks is normalized by mathematically exact extraction in the extended energy network. The extended energy dissipation concept of a multienergy system is obtained with a unified theoretical solution. In addition, two types of ER transactions are proposed using market-based priority ranking methods to meet the desired incentive purpose by moving transaction energy waste from one to the other, whose priority is decided by the price and transaction volume. In this article, the structure of the series has been investigated, but no comparison has been made between the two structures. Reference [20] examines a completely distributed system for power trade in a hyperactive power industry. The proposed model provides a peer-to-peer trade mechanism among customers. This approach is intended for businesses, industries, and home energy centers to meet their relevant requirements in a model with the lowest cost. The alternate orientation of the multiplier technique has been used to depict distributed power flow in this research. The ideal use of EHs is depicted as a typical combined integers linear programming optimization problem. The related decision variables of the EH functioning are moved to the peer-to-peer marketplace, and the alternating orientation of multiplier technique is used to ensure limited information trade and deal with the issue of information security. Two alternative scenarios are explored in this work to demonstrate the usefulness of the peer-to-peer power trading paradigm, which are named unified and synchronized operating modes. In an integrated setting, there is not any P2P exchange of energy, but in the integrated structure, the electricity market is considered through P2P exchange. The obtained results confirm that the coordinated model can effectively manage energy trading through peer-to-peer for different energy centers and achieve the lowest cost of operation for energy centers in the system. In this article, the structure of the series has been investigated, but no comparison has been made between the two structures. In [21], a peer-topeer multisource trading framework is proposed for multienergy microgrids. In this framework, connected microgrids not only meet multienergy demands with local biogas, solar, and wind hybrid renewable energies but also actively combine their available multienergy resources and communications to provide safe and high-quality services. They trade multienergy trading and multi-microgrid communication, which is an unsolvable optimization issue due to their intrinsic strong 2 International Journal of Energy Research links to numerous sources and independent decision-making. Therefore, the primary problem is described as a Nash bargaining problem, and then it is broken into the subproblem of the distribution of many social resources and the subproblem of income allocation. The suggested approach has been implemented and compared on a three-microgrid system during a 24-hour planning period. In this article, the structure of the series has been investigated, but no comparison has been made between the two structures. In [22], it investigates the energy production of five energy hubs in a smart energy grid that has been set up to minimize energy costs and greenhouse gas (GHG) emissions under different scenarios in a region in Canada. In [23], the estimation of production, distribution, and conversion of multienergy sources through the optimal setup of the hub and the new operating model is presented. The impact of renewable energy sources (RES) and powerto-gas (P2G) on the overall cost of the system has been investigated. In [24], EHs are used in a grid-based distribution structure in two forms: primary and additional hubs. The goal of [25] is to establish rivalry in the commerce electricity market through the existence of an integrated demand response program to decrease the expenses of buyers and boost the profits of retailers. This gives customers more freedom to reduce their energy costs. Reference [26] presents a comprehensive methodology to identify the most effective synchronized functioning of grid-connected EHs and local power systems dependent on the widespread use of wind energy. Reference [27] describes the synchronized energy management of hubs in various networks based on the collaboration of hubs in day-ahead marketplaces. Reference [28] deals with the problem of optimal energy management in multicarrier energy networks in the presence of interconnected energy hubs.
Reference [29] focuses on complementing multiple energy sources in energy hubs to solve the possible congestion of the distribution network. Reference [30] presents a decentralized framework for optimal distribution of integrated power distribution and natural gas systems (IDGS) in grid network hubs. In reference [31], a complicated system of energy hubs has been simulated and optimized under various situations to investigate the economic feasibility and potential decrease in carbon dioxide pollution. Reference [32] discusses the oversight of EHs linked to gas, electricity, and heat systems, where the hub is incorporated as an integrated structure between dispersed carriers and systems for storing energy. Reference [33] introduces a new computational model using a modular approach to create an interconnected urban infrastructure, including energy, building, and transportation sectors. Reference [34] considers an intelligent multicarrier energy system where users are equipped with energy storage and conversion devices (for example, an energy hub). In reference [35], with the goal of decreasing the difficulty of centralized optimum dispatching in large-scale systems and safeguarding the data security of separate hubs, an ordered optimization technique for ER-based energy networks is described. Reference [20] examines a completely decentralized model for electricity exchange in a power market. The suggested system provides a P2P trade mechanism among users. This version is intended for commercial, advertisement, and domestic needs in a lowcost model. In this study, the alternate direction technique of multiplier technique is used to represent distributed power flow. In [36], a robust chance-bound model is developed to handle production and consumption uncertainty in multicarrier energy hubs. The broad formulation of the problem is established in the framework of mixed integer linear programming. The major purpose of the model that is suggested is to enhance the loading factor with uncertainty while resolving the decision maker's acceptable risk index. In addition, an operational strategy for energy hubs in the presence of electricity demand, heating, and cooling, as well as uncertainties about renewable electricity generation, is presented in [37]. A twostage probabilistic framework for the combined construction and management of an EH in the presence of renewable electric and heat energy-storing devices is given in [38]. Since severe uncertainties are linked with electric loads, heating, and cooling, as well as wind turbine power generation, their consequences are displayed in the suggested model. The model also takes demand response and integrated demand response programmers into account. Additionally, the issue is resolved using continuous and discrete approaches using real-code genetic algorithms and binary-code genetic algorithms. A microenergy network (MEN) that combines natural gas and electrical power with wind, solar, and energy storage devices is optimally modeled in reference [39]. The suggested strategy, which seeks to reduce operational expenses and emissions of greenhouse gases, is built on energy hubs. The growing utilization of sources of clean energy and energy storage technologies to deliver secure electricity while lowering operating expenses and environmental impacts is the driving force behind this research. The ideal operations issue with MEN is resolved using a broad algebraic modeling framework. In this article, the connection of interconnected hubs is not investigated, but we have investigated the connection of hubs in two structures in our research. Reference [40] adopts an integrated perspective on the optimal capacity design and operation of energy hubs. It also discusses the potential for increasing the reliability of energy hub systems while reducing operational costs by sharing energy between multiple energy hubs through networking. In this article, the connection of interconnected hubs is not investigated, but we have investigated the connection of hubs in two structures. Table 1 also includes a comprehensive comparison of previous works for clarity and the benefits of the proposed method.

Contribution.
As mentioned in the previous section, all the reviewed studies have examined only the series or parallel mode; a comparison between the two modes has not been investigated in any research. The goal of this paper is to provide a new general framework to evaluate and compare both series and parallel connections of n hub and to analyze these two modes in terms of cost and reliability. Therefore, the most important contribution of this paper is as follows: (i) Presenting a general framework and new formulation for making n interconnected energy hubs in series and parallel structures Finally, the conclusion of the paper is presented in the last section.

Problem Formulation
In general, hubs can be connected together in both series and parallel ways. As shown in Figure 1, which shows the series connection of hubs, the first hub is fed from the upstream energy network, and its output is the input of the   International Journal of Energy Research second hub. This procedure continues until the hub n-1. The hub n is only a consumer hub that receives its energy through the output section of the n -1th hub. In addition, Figure 2 represents the parallel connection of the hubs. The hubs are linked at the input so that each hub is fed by the common upstream network, and the output of the hubs is not related to each other. In this structure, the order of placement of the n hubs is not important and does not make a difference in the result. Each hub includes three carriers of electric energy, natural gas, and local heat at the entrance. It also has a transformer, CHP, a heat exchanger, and two types of electrical and thermal consumers. Hubs include electrical and heat storage (HS), and in series mode, all hubs except the nth hub have a power-to-gas (P2G) converter.
The main parts in the modeling of each hub are the expression of the relationship between the input carriers and the demand in the output section, which is done according to (1) and with the help of the coupling matrix [46]. : ð1Þ In this regard, L α , L β , ⋯, and L ω are the energy demands of different carriers in the output section; P α , P β , ⋯, and P ω are the energy carriers in the input section; and C αα to C ωω are the coupling matrix coefficients. For instance, the coupling matrices of a hub with a P2G converter, including electrical, gas, and heat loads, and a hub without a P2G converter including electric and heat loads are formulated in (2) and (3), respectively.

Proposed Method
In this section, as shown in Figure 3, first, the input data is given to the problem along with common constraints in both series and parallel modes. Then, the objective function related to parallel and series modes is executed, and the amount of generation of energy carriers and storages is determined. Finally, the proposed indices related to each mode (series, parallel, series, and parallel) are presented.

Linear Optimization Formulation.
In general, the mathematical formulation for a linear problem considering equality and inequality constraints is as follows [48]: min z = c 1 x 1 +c 2 x 2 +⋯ +c n x n , s:t: A 11 x 1 +A 12 x 2 +⋯ +A 1n x n ⊕ b 1 , where the symbol ⊕ specifies the type of constraint and is defined as follows: = is the equality constraint; ≥ is the greater than or equal to constraint; ≤ is the less than or equal to constraint.

3.2.
Modeling N Hubs in Parallel Mode. The proposed objective function for optimal integrated operation of several connected hubs in parallel mode is as follows: In the above equation, π i ðtÞ is the price of the ith energy carrier in hour t, P i,j is the ith type energy carrier entering the jth hub, VOLL k is the specified value for the unsupplied energy of the kth type load, and UP k,j is the amount of unsupplied energy of the kth type of load in the jth hub.
The main coupling formula for n hubs including the storages is defined in (5). In this equation, L j ðtÞ is the vector of energy carrier load of the jth hub, C j is the coefficients of the coupling matrix of the jth hub, and Q j ðtÞ is the rate of charge and discharge of the storages for the jth hub. In addition, the amount of energy input to hub 1 to N j per carrier is presented in (6).
3.3. Modeling N Hubs in Series Mode. The proposed objective function for optimal and integrated operation of several connected hubs in series mode is defined in (7). In addition, the main relationships related to the modeling of n hubs in series are proposed in (8) and (9).
5 International Journal of Energy Research where P i,j+1 ðtÞ is the energy carrier of the ith type entering the (j + 1) th hub, L j ðtÞ is the energy carrier load of the jth hub, C j is the coefficients of the coupling matrix, and Q j ðtÞ express the amount of charge and discharge of storage for the jth hub and UP j ðtÞ is the unsupplied power of energy hub j at hour t. In the series structure of multiple hubs, the total amount of input energy of each carrier is equal to the total amount of input energy to hub 1.     (10). In addition, according to (11), the last hour energy of the storage device must be equal to its initial energy.
The limits of the input energy carriers, the amount of charging and discharging of storage devices, the amount of energy stored in storage devices, and the energy not supplied are presented in the following equations.
In the following, the proposed indices will be introduced and explained.

Proposed Indices.
In this section, in order to provide a better and clearer comparison of the simulation results in various scenarios, new indices are proposed as follows. The index presented in (17) shows the ratio of the cost of electrical energy to the total cost (ROEC). Similarly, the ratio of gas and local heat cost to the total cost is proposed in (18) Furthermore, the amount of unsupplied electrical energy compared to the demanded electrical load is presented in (20). Similarly, the ratio of unsupplied thermal (ROUH) The ratio of the electrical input energy (ROEE) of the jth hub to the total electrical input energy in parallel and series modes is proposed in (22) and (23).
And finally, the ratio of the heat input energy (ROHE) of the jth hub to the total thermal input energy in parallel and series modes is presented in (24) and (25).
In the following, the simulation results of the proposed framework will be presented and analyzed.

Case Study
In general, the proposed method has the ability to implement n hubs in series and parallel modes. However, in this section, the results of the proposed formulation for both considered structures of two hubs (i.e., series and parallel), are analyzed separately. This linear optimization problem is implemented using "linprog" function in MATLAB software. The consumption load of the hubs as well as the price  Increasing the consumption load of the hubs Scenario 5 Increasing the price of energy carriers purchased from the upstream network Scenario 6 Effect of failures in the storage of the hub    Table 2 in the appendix. Also, the generation power of energy carriers in parallel connection in the basic state is shown in Table 3. In the whole 24 hours, the gas price is 7 $/pu, and the local heat price is 6.5 $/pu. The outage cost for electricity is equal to 280 $/pu, and the outage cost for heat is equal to 130 $/pu. In addition, the converter coefficients are as follows [1]: Series : Also, several scenarios for comparing hubs in two parallel and series modes are considered and examined according to Table 4. (Scenario 1). The information of electric energy, gas, and heat carriers, as well as the share of each hub of these carriers in the case of series connections of hubs, is presented in Table 5.

Basic Mode
The cost of different energy carriers in two series and parallel modes for the base mode is shown in Figure 4. As can be seen, all of the carriers except the gas carrier and also the total cost, which is more important, are more economical in series mode. According to Figure 5, the reason for the higher cost of purchasing gas in the base mode is the lack of access of hub no. 2 to the upstream network, and hub no. 1 must also meet the needs of hub no. 2. Since the P2G converter is not economical, the entire needs of hub no. 2 are purchased from upstream. Since there is no unsupplied

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International Journal of Energy Research energy in this state, the indices of ROUH and ROUE are equal to zero. Also, the ratio of different energy carriers in each hub to the total energy of the two hubs is shown in Figure 6. As it is known, in the series connection of the input energy carriers at hub no. 1, they have a higher percentage than at hub no. 2, which is a natural alternative to the type of connection. Also, in a parallel connection completely according to the type of load, it can be seen that the amount of electric and thermal carriers used for hub no. 1 is 54.37% and 72.52%, respectively, which is more than hub no. 2.

Difference in Type of Hub Customers (Scenario 2)
4.2.1. Commercial Customers for Two Hubs. In this scenario (Scenario 2.1), both hubs have similar commercial customers and are not connected to residential loads. According to Figure 7, in series mode, the costs of electric and thermal energy are lower than in parallel mode, but the cost of gas is higher than in parallel mode, which is due to the removal of the output link of two hubs in parallel mode. While in series mode, the gas needed by hub 2 is supplied by hub 1, but in parallel mode, it is purchased directly from the upstream network and is generally more economical. In this case, there is no unsupplied energy, and the total cost is $486.45 in series mode and $677.76 in parallel mode. It can be clearly seen that the serial mode is more economical in terms of costs.

Residential Customers for Two
Hubs. In this case (Scenario 2.2), both hubs have similar residential customers and are not connected to commercial loads. According to Figure 8, in the series mode, the cost of electric and thermal energy is lower than in the parallel mode, but the gas cost is higher than in the parallel mode, which is the same as the case of commercial customers. In this case, there is no unsupplied energy and the total cost is $952.67 in series mode and $1268 in parallel mode. (Scenario 3) 4.3.1. Disturbance at the Electric Entrance. In this scenario (Scenario 3.1), the disturbance occurred at 7:00 p.m., which is the peak hour of electric energy consumption for both hubs, and electric energy is not supplied. Therefore, a penalty must be paid for this unsupplied energy. As shown in Figure 9, in series mode this penalty is 166.4 $/h, and in parallel mode, it is equal to 109.2 $/h, but the total cost is still more economical in series mode.

4.3.2.
Disturbance at the Gas Entrance. In this case, during the peak hours of energy consumption, the gas has been cut off and the disturbance has been simulated, but as shown in Table 6, there is no blackout or unsupplied energy in any case, which is due to the cheapness of local heat compared to gas. And the local heat capacity responds to the heat load, so cutting gas has no effect on the unsupplied energy. The total cost is lower in series mode than in parallel mode.

4.4.
Increasing the Consumption Load of the Hubs (Scenario 4). In this scenario, we increased the load by 20% in the 4 hours when we had the highest energy consumption, and according to the results, there was no unsupplied energy because the increase in load was compensated by increasing the purchase from upstream and using the capacities in the hubs. As it is shown in Figure 10, with the increase of the electric load, the ROEC index, which represents the ratio of the cost of electric carriers to the total cost, has increased in the series mode compared to the basic mode. Also, with the increase in heat load, the ROHC index, which represents the ratio of local heat cost to total cost, has increased in the series mode compared to the basic mode.
In parallel with the increase in electrical load, the ROEC index has increased slightly compared to the basic mode. Also, with the increase in thermal load, the ROHC index has also increased in parallel mode compared to the basic mode. In addition, with the increase in electric load, the ROGC index has increased in parallel mode compared to the basic mode, which indicates the increase in gas purchase with the increase in load. The total cost of the series is more economical than the parallel mode, but there is no significant difference. In this scenario, the price of electricity has been increased by 50% in the most expensive hours, which did not cause a blackout, but the total cost of electricity has increased compared to the basic mode (Scenario 1), according to  Figure 9: The cost of energy carriers for the occurrence of a disturbance in the electrical input.

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International Journal of Energy Research storage, which cannot use the cheaply stored energy in expensive hours. According to Figure 11, the total cost in series mode is equal to $843.04, and in parallel mode, it is equal to $981.19, which is more economical in series mode. As shown in Figure 12, the ROEE 1 and ROEE 2 indices, which represent the ratio of electric input energy to each hub to the total electric energy, increased with the failure of the electric storage compared to the base state. Also, the electricity cost ratio has also increased in parallel mode due to the lack of storage and direct connection to the upstream network, resulting in an increase in purchases and an increase in costs.

4.7.
Comparison of the Proposed Scenarios. According to Figure 13, the total cost of a parallel connection is higher in all scenarios compared to the total cost of a series  13 International Journal of Energy Research connection. Also, by applying different scenarios, it can be seen that the costs have increased compared to the base case in both series and parallel connections, except for customers to businesses, which has reduced costs compared to the base case due to the low load. It can be seen that the effect of the type of residential customer has significantly increased the cost compared to the base case compared to other scenarios, which is due to high load hours and high consumption of energy carriers.

Conclusion
The energy hub plays a significant role in increasing flexibility, efficiency, and dependability as a crucial component of energy supply systems. Reliable and economical solutions are required to meet the energy demand due to the rise in demand for various energy carriers brought on by population growth, technological development, and network infrastructure expansion. In this regard, the interconnected energy hub system enhances a system with many energy carriers, such as electricity, natural gas, and local heat, which produces a more efficient outcome than the single carrier mode. One of the difficult problems in smart grids nowadays is the scheduling of interconnected energy hubs. In this paper, a new general framework for modeling many interconnected energy hubs in both series and parallel structures is proposed. In addition, several indices are proposed and analyzed in a variety of scenarios. Among the most important results, the following can be mentioned: (i) In all the examined scenarios, the total cost of energy carriers is lower in series mode (link in the output) than in parallel mode (link in the input) (ii) The total cost of energy carriers in the series mode is higher in all scenarios except scenario 2.1 compared to the base mode, where scenario 2.1 is 39.14% more economical than the base mode (iii) The highest total cost in series mode corresponds to scenario 3.1 with a value of $960.3, which is 20.14% more than the base mode. In addition, the total cost of the base case is almost equal to scenario 3.2 (iv) The total cost in the parallel mode for scenario 2.1 is equal to $677.8, which is 30.33% more economical than the base mode, and the highest total cost is for scenario 2.2, which is 30.33% more expensive than the base mode (v) Unsupplied energy is zero in all scenarios except scenario 3.1 (disturbance at the electric entrance), so that the shutdown penalty in series and parallel mode is equal to 166.4 $/h and 109.2 $/h, respectively. In other words, the shutdown cost in parallel mode is 34.37% more economical than in the series mode Therefore, it is concluded that the first hub needs more reliable elements than other hubs, especially in the series mode, because its failure causes a major disruption in the entire system. The advantage of a series structure compared to a parallel structure is lower operating costs, although it requires that elements with higher capacity and reliability be used in the initial hubs. This advantage of the series structure has been proven in the simulation of the base mode as well as in different scenarios, and in general, it can be said that the probability of unsupplied power in the parallel mode is lower than in the series mode due to the independence of the hubs from each other.   Input stored thermal energy in hour t (p.u.) Q e ðtÞ: Battery charging and discharging in hour t (p.u.) Q h ðtÞ: Charging and discharging thermal storage in hour t (p.u.) P i ðtÞ: Input energy of carrier i in hour t (p.u.) P i,j ðtÞ: The ith type energy carrier input to the jth hub in hour t (p.u.) P e,j ðtÞ: The electrical energy carrier input to the jth hub in hour t (p.u.) P h,j ðtÞ: The heat energy carrier input to the jth hub in hour t (p.u.) UP j ðtÞ: The unsupplied power of energy hub j at hour t (p.u.) UP j,k ðtÞ: The amount of unsupplied energy in the consumption load of the kth type in the jth hub in hour t (p.u.) E i,j ðtÞ: The amount of energy stored in the ith energy carrier in the jth hub (p.u.) Q i,j ðtÞ: Charge and discharge of type i energy carrier in hub j in hour t (p.u.).
Parameters C j : The coefficients of the coupling matrix of the jth hub L e ðtÞ: Electrical load in hour t (p.u.) L h ðtÞ: Thermal load in hour t (p.u.) VOLL k : The amount of damage specified for unsupplied energy in the kth type load (p.u.) π i : The price of ith energy carrier (p.u.).

Data Availability
The data used to support the findings of this study are included within the article.

Conflicts of Interest
The authors declare that they have no conflicts of interest.