Energy Hub Management considering Demand Response, Distributed Generation, and Electric Vehicle Charging Station

Multicarrier energy systems have systematic fexibility for energy and load management. Te integration of diferent types of energy under the title “energy hub” (EH) in multiple energy infrastructures such as electricity, natural gas, and heat in an integrated manner will meet demand at a lower cost. On the other hand, the desire for distributed generation (DG) systems has increased due to their many economic and technical advantages, as well as the technical development of these units and the reduction of environmental pollution. In this paper, a proposed EH system is optimized in terms of cost using linear programming, considering transformer converters, combined heat and power (CHP), heat exchangers, and energy storage, including electric vehicle charging stations (EVCSs) and heat storage. In this regard, four proposed scenarios are defned and evaluated from the aspect of cost in order to mitigate load shedding while considering uncertainty in order to bring the problem closer to reality. Te results show that the total cost in scenarios 1, 2, and 3 (the best case) is reduced by 15.47%, 26.29%, and 27.93% compared to Scenario 0 (the base case), and Scenario 3 is the most economical possible case. Tis method can be a suitable tool for energy hub management, especially for solving possible load shedding.


Motivation.
A recently created idea called an energy hub (EH) is used as an interface between various energy infrastructures and network users, and it is thought to be a viable choice for the best management of multienergy systems (MESs) [1]. Trough modern technological advancements such as energy storage (ES) systems and energy converters, this conceptual idea has been used to describe the interactions between various energy carriers. By arranging converters and storage systems optimally, which increases fexibility in terms of fulflling the various forms of projected demand, it is possible to balance the supply and demand of diferent types of energy [2]. Large building complexes (such as malls, hospitals, and airports), metropolitan regions, tiny confned districts, island power systems (ships, trains), and industrial units like steelworks are some examples of practical EH application areas [3]. Numerous studies have thus far highlighted various facets of MES and EH, including security and dependability [4][5][6], operational optimization [7][8][9], and long-and mid-term planning [10,11]. Demand response (DR) is one part of energy management that has been conceived in recent years when the EH is employed. DR is a very important parameter that can work in concert to improve the efectiveness and functioning of electrical energy systems [12]. A certain amount of fexibility in serving the loads is provided by redundant channels within the hub. When dealing with DR schemes, this supply redundancy can be a useful tool for customers. For instance, the EH can easily stop buying electricity when the incentives for electric load curtailments are high and switch the primary source of electric energy to other energy carriers, such as natural gas, by using CHP to make up for the input electric energy that was not purchased. As a result, the customer buys low-cost gas to cover the necessary electric energy while receiving a larger incentive because the required electric energy was not purchased. In this regard, the aim of this paper is to manage EH (with the presence of energy converters and ES) by considering the electric vehicle charging station (EVCS), the DR program, and distributed generation (DG) simultaneously. Also, the blackout program is included in the objective function of this problem, which is used to solve the blackouts created by the two tools of the DR program and DG.

Literature
Review. Several studies have pointed to diferent aspects of EH use, such as security and reliability [4][5][6], operational optimization [8,9], and long-term and medium-term planning [10,13]. Te optimal performance of energy hubs is modeled as a mixed-integer linear programming optimization problem [14]. Also, a least-cost paradigm is implemented into the model to fulfl the needs of industrial, commercial, and residential energy centers. Multicarrier energy systems are reviewed from diferent perspectives, including mathematical models, integrated components and technologies, uncertainty management, planning objectives, environmental pollution, fexibility, and robustness [15]. Considering the uncertainty in this study, the working point is closer to the optimal point. Te nonlinear problem of energy management in interconnected EHs is solved using goal programming [16]. Te analyst can determine the distributed energy hubs' ideal operating conditions by using this solution technique. Te energy interaction of hubs is formulated using game theory, considering the uncertainty in demands [17]. Researchers may examine the presence and distinctiveness of the Nash equilibrium using the potential game technique as well as create an online distributed algorithm to reach that equilibrium. A generalized heuristic method is proposed to solve the problem of optimal power fow in EHs [18]. Tis method does not need any additional variables, such as dispatch factors or dummy variables, which are necessary for traditional methods. All evolutionary algorithms can be used with the unifed suggested strategy. A mathematical model for residential EHs is presented in which automatic decision-making technology is incorporated into the smart grid [7]. A two-level stochastic model is presented, where the customer is placed at the lower level and the center manager is placed at the upper level [19]. Te performance of existing ES systems and multienergy systems has been investigated and compared, and the results show the superiority of multienergy systems [20]. Te above papers have not considered factors such as DR, DG, and electric vehicle charging station (EVCS) in the problem of hub energy, which is the purpose of our paper to consider these issues.
A new steady-state method is presented for optimal energy management of EHs that helps overcome the limitations of the dominant EH management models [21]. A centralized operation framework is developed with the goal of minimizing system operating costs while addressing EH constraints [22]. Te suggested methodology enables the microgrid operator to reduce microgrid operation costs while ensuring that certain operational restrictions on the EH are met. Te optimal energy planning of multiple hubs with multiple energy infrastructures is studied [11]. Te model also takes emission reduction incentive programs into account when installing distributed generation. Studying the design of future energy networks with connected energy hubs can be aided by this concept. A particle swarm optimization is implemented to solve the energy operation problem of residential hubs [23]. Some studies are presented on the design of a new approach for exploiting the energy of residential hubs in the electricity market [24]. Te mentioned papers have only investigated the DG and DR factors and have not considered the EVCS factor. In addition, the robust planning of the combination of cooling, heat, and power EHs in the presence of demand uncertainty, photovoltaic systems, and wind power generation, as well as the price of electricity, has been investigated [25][26][27]. Reference [28] presents a two-level structure for the best EH operation and planning based on demand unpredictability and renewable energy sources. At the primary level, stochasticprobability models are used to show optimal planning, while at the secondary level, stochastic-probability models are used to present optimal operation. Te proposed technique for problem-solving in continuous and discrete space is the foundation for the method. In order to manage the electricity and natural gas networks in an economical manner, reference [29] suggests a novel paradigm. In this model, an energy hub supplies electric and thermal loads by using water, electricity, and gasoline oil as inputs. In addition, the energy hub produces natural gas using power-to-gas technology and sells it to a gas network to lower costs and reduce congestion in gas pipelines. Tis model also incorporates a demand response program to move loads from peak to ofpeak hours. A strong self-scheduling of a virtual energy hub for participation in the energy and reserve markets is proposed in [30]. It is suggested that the thermal reserve market be used to preserve the thermal power balance in real time and account for the consequences of thermal demand unpredictability. In [31], a brand-new EH operating model is put forth, using which the best responsive load changes may be made in relation to the calculated EH output energy prices as well as the EH schedules. A three-step strategy is suggested to reach this objective. Reference [32] develops a model-free, safe deep reinforcement learning approach for the best control of a renewable-based energy hub that operates on multiple energy carriers while satisfying the physical restrictions in the energy hub operation model. By taking into account numerous energy components, this work's major goal is to reduce the system's energy costs and carbon emissions.
Te EH concept has also been used for other purposes in the power system and energy delivery infrastructure, such as its application in DR. An EH system, including a wind turbine, photovoltaics, and electric vehicles, which are exchanged with the energy, thermal, and gas sectors, the DR program, and the energy storage market, is proposed. Terefore, the uncertainty of wind turbine, photovoltaic, load, and electricity market prices is modeled by the Monte Carlo method [33]. An energy optimization management method for the interaction between the smart EH and users 2 International Transactions on Electrical Energy Systems in the energy system is proposed [34]. Te outcomes demonstrate that the technique is applicable for efcient energy management in an integrated energy system. A new EH operation optimization model is proposed [35], which determines the optimal response of responsive loads to the associated hub energy generation costs to minimize the total EH operation cost as well as customer payments. Various research studies have been conducted to date on the concept of integrated DR. Te changes in energy consumption are examined through smart energy hubs combining electricity and natural gas [36]. In this study, customer payments are minimized while service profts are maximized. By proposing a new perspective on DR programs in the feld of multienergy systems, a systematic review of DR programs is presented in multienergy systems to thoroughly examine the basic concept and value of integrated DR programs [37]. A residential EH is proposed where the customer's beneft is maximized due to optimal load management considering movable and limited loads [38]. An optimization problem is developed to minimize the total energy cost of a residential EH operation while optimally scheduling home appliances in time-based DR programs [9]. Te integrated DR concept is developed to model optimal interactions between smart energy poles and utility companies [39]. An integrated demand-side management-based (IDSM) contemporary energy management approach for electricity and natural gas networks is presented [40]. In traditional research, energy usage is optimized from the viewpoint of each individual user without taking into account how they interact with one another. Te investigation of the energy hub system's emission issue is found [41]. Te cost-environmental functioning of the energy hub system in the presence of the demand response program (DRP) has therefore been presented using a multiobjective optimization model. Additionally, a mixed-integer linear programming method has been employed to analyze the energy hub system's costenvironmental performance issue. A stochastic model for real-time electricity, natural gas, and demand is presented [42]. A smart energy hub works to reduce a weighted sum function made up of the cost of energy and the penalty for emissions. When electricity and natural gas are transformed into electrical, heating, and cooling energy at an intelligent energy hub's output port, researchers apply the conditional value at risk (CVaR) approach to control the operational risk in order to have a more exact model. Tese papers only examined the DR factor and did not analyze the efects of the DG and EVCS factors.

Contribution.
As mentioned, the impact of the factors EVCS, DRP, and DG on the energy hub has been investigated in a few articles. In this regard, the contribution of this paper, as shown in Figure 1, is as follows: (i) Investigate the efect of the DR program and DG on EH costs while considering EVCS simultaneously (ii) Consider energy storage (EVCS and heat storage) (iii) Consider DG in the electric load section and the DR program to solve the blackouts (iv) Consider the uncertainty of DG's capacity to bring the problem closer to reality (v) Consider the entire state space to reach the optimal solution, unlike smart methods Te internal components of the energy hub under study are three converters: a transformer, a combination of heat and power, and a heat exchanger. Also, two electric and thermal storages are used to respond to the load in peak load conditions, which are usually charged in low load conditions and discharged in peak load conditions.

Paper Organization.
Te whole paper is divided into four sections. Te basic and proposed formulas are presented in Section 2. In addition, the results of the proposed formulas are examined in Section 3. Finally, the conclusion of the paper is presented in Section 4. In the following, the formulas related to the problem will be presented.

Problem Formulation
In this section, the basic and proposed formulas related to EH are presented. First, the mathematical formulation of linear optimization is presented to fnd the optimal point by considering equality and inequality constraints. Ten, the objective function of the problem is proposed according to the linear formulation.

Linear Optimization Formulation.
In general, the mathematical formulation for a linear problem considering equality and inequality constraints is as in the following equation [43]: International Transactions on Electrical Energy Systems where symbol ⊕ specifes the type of constraint and is defned as follows: (i) �: equality constraint (ii) ≥ : greater than or equal to constraint (iii) ≤ : less than or equal to constraint 2.2. Objective Function. Te objective function of the optimization problem is the EH cost function. Te objective function of the energy hub, with the presence of energy input carriers (electricity, gas, and heat) and output loads (electricity and heat) for a period of 24 hours a day, is as follows: where λ e (t), λ g (t), and λ h (t) are the prices of electricity, gas, and heat carriers in hour t, respectively. Up e (t) and Up h (t) are equal to unsupplied electric power at hour t and unsupplied thermal power at hour t, respectively. In addition, the generation power of electricity, gas, and heat energy carriers in hour t is introduced by P e (t), P g (t), and P h (t). Also, the amount of penalty for unsupplied power is indicated by Pen. Furthermore, the limitations related to energy carriers are as follows: where P e Min (t) and P e Max (t) are, respectively, the minimum and maximum generation power of the electrical carrier at hour t. Similarly, it is the same for gas and heat carriers. In general, the relationship between the generation power of energy carriers (P (t)) and the output load (L (t)) with the presence of energy storage at the input (M (t)) and output (Q (t))of the hub is defned as follows [3]: 2.3. Formulation of the Basic Case. In this paper, the energy storage devices are placed at the outlet of the hub. Since all the storage devices are located in the output part of the energy hub and no storage is considered in the input part, M (t) is equal to zero. In this regard, (3) is rewritten as follows: where Q (t) is the rate of charging or discharging of the energy storage in hour t, which is defned as follows: where C Q ch (t) and C Q dis (t) are the matrix of charging and discharging coefcients of energy storage devices, respectively. Also, Q ch (t) and Q dis (t) are the charging and discharging rates of energy storage devices in hour t. Terefore, by inserting (6) into (5), we have As a result, the energy hub load matrix form is defned based on the input of energy carriers and energy storage devices as follows: where NL and NP are the number of energy hub loads and the number of energy carriers, respectively. C i,j (t) is the power obtained from the energy carrier j for load i at hour t, which for an energy hub with three input carriers (electricity, gas, and heat) and two output loads (electricity and heat) is as follows: where C ee (t), C eg (t), and C eh (t) are the electric power obtained from the transformer, CHP, and heat at hour t, respectively, and since there is no connection between the thermal carrier and the electric load, C eh (t) is equal to zero. Similarly, the heat obtained from electrical carriers, CHP, and EH is represented by C he (t), C hg (t), and C hh (t), respectively, where C he (t) is equal to zero due to the lack of connection between the electrical carrier and the thermal load.

Formulation of Demand Response.
Demand response means the ability of customers to improve their energy consumption patterns to reach appropriate prices and improve network reliability. In this regard, based on [44,45], load shifting from peak load to low load is formulated as follows, if possible: where P is the prespecifc peak at peak load (Ω) and P(t) is the modifed load model. A is the amount of load added to low load hours (Ψ) according to the incentive program.
where α is the percentage of the energy reduced at peak load that is recovered at low load. In other words, the network operator encourages consumers to reduce consumption during peak periods. Instead, with incentive schemes, it encourages them to increase consumption during low loads to improve the consumption load pattern. At peak load, the network operator encourages users to reduce consumption with incentive plans. Assume the peak load at hour t is equal to P T (t). As a result of using incentive plans, a percentage of peak network load is reduced and added to low load. So, we have where ξ is a coefcient of the total load at peak load. TP (t) is the amount of load that is reduced at hour t and is added to the time of low load.

Formulation of Distributed Generation.
In order to bring the problem closer to reality, the capacity of the DG is randomly considered on the electric load side. Terefore, the random capacities to determine the total capacity are defned as follows: where C s (t) is the matrix of random DG capacities at hour t. ρ is the initial value of the capacity, and μ s (t) is a random value at hour t per sample s. Terefore, the fnal capacity of DG at hour t (FC (t)) is defned as follows: Because the DG on the load side serves as a generator, the capacity of that electric load is reduced.
where L e Total (t) is the matrix of fnal electric load at hour t. In the following, the results of the proposed method in diferent scenarios will be examined.

Case Study
In this section, the results of the proposed method are presented. Te results are analyzed in four scenarios defned in Table 1. In addition, the prices of energy carriers (electricity, gas, and heat) are fxed in all scenarios and shown in Figure 2. Te price of gas and heat is constant at all hours and is equal to 3 and 7 p.u., respectively. Also, the price of electricity varies by hour during the day [46]. In addition, transformers, CHP (gas to electricity and gas to heat), and HE are defned as hub energy converters in this paper, and the conversion factor of these converters is represented by C ee , C eg , C hg , and C hh , and their values are, respectively, equal to 0.985, 0.37, 0.43, and 0.9. Also, the charging and discharging losses of storage devices are omitted.

Energy Hub without DR and DG (Basic Scenario).
In the basic mode, there is no DRP or DG, and the generation of energy carriers and costs (generation cost and blackout, if any) are calculated. Te electrical and thermal loads of EH in the basic mode at diferent hours of the day are shown in Figure 3. It can be seen that in some hours there is a peak load and that in other hours there is a low load. Terefore, it is expected that a part of the load will be shut down during peak hours.
In this regard, the amount of energy carrier generation is obtained based on the load of the EH, as shown in Figure 4. Te gas energy carrier has played a role in supplying loads during peak hours (8 to 16, 18, and 19). In addition, a signifcant part of the electric load power is provided by the electric energy carrier because the transformer in the hub has a high conversion factor while the CHP conversion factor for converting the gas energy carrier into electricity is low and not cost-efective.
Te electric vehicle charging station plays a role in the supply of electric loads as an electric storage hub of energy so that the amount of charging and discharging of the electric vehicle station is shown in Figure 5, which is proportional to the electric load in the basic mode; it is usually charged International Transactions on Electrical Energy Systems (receiving power) during low load hours and discharged (injecting power into the electric load) during peak network hours. Te maximum capacity of this station is 1 p.u., which is specifed in the fgure in the hours when it reaches its maximum and minimum capacity. It is clearly known that during peak hours (15 to 21), the capacity of the EVCS has been reduced to a minimum over time, and in low load conditions, its capacity has reached its maximum. Also, the rate of charging and discharging of the thermal storage to provide part of the thermal load is presented in Figure 6. Te amount of power available in the HS at the beginning of work is equal to 1.5 p.u. It can be clearly seen that usually during the hours when the thermal load is maximum, the storage is discharged, and during the hours when the thermal load is minimum, it is charged. In other words, in hours 1, 10, 15, 16, 18, and 24, when the thermal load is minimal, the thermal storage is at the charging stage with values of 0.8, 0.9, 1.7, 1.9, 1.5, and 0.8 p.u., respectively. In addition, at hours 8, 9, 13, 20, to 22, when the thermal load is at its maximum, the thermal storage is discharging with values of −0.5, −1.2, −1.1, −1.2, −1.3, and −2 p.u., respectively.
Despite the generation of energy carriers and energy storage devices in basic mode, there is a blackout in hours 15, 16, 18, and 19. In this regard, the cost of unsupplied power during these hours is applied to other costs. As shown in     International Transactions on Electrical Energy Systems Figure 7, the fnal cost in hours 15, 16, 18, and 19 is equal to 161.5, 160.7, 86.5, and 86.5 p.u., respectively. In the following, the impact of the DR program and DG on resolving the blackouts will be investigated. It can be clearly seen that the cost caused by the outage of the electrical load has a signifcant impact on the total cost during outage hours.

Energy Hub considering DR Program (Scenario 1).
As mentioned, in the basic mode, there is a blackout at 15, 16, 18, and 19 hours. Terefore, the DR program is used in this scenario to solve the blackout. To fx the blackout, 15% of the peak load (ξ � 0.15) is removed and added to the low load hours during the specifed hours. Terefore, as shown in Figure 8, the fnal electric load has been reduced by applying the DRP during blackout hours, and the removed values have been added to low load hours (1, 3, 4, and 5). As a result, the total load is reduced during peak hours, leading to blackout relief. In addition, the DRP also afects the charge and discharge rates of the HS. So, with the reduction of electric load during peak load hours, the amount of gas energy carrier   International Transactions on Electrical Energy Systems generation changes, and these changes also afect the way thermal load is supplied. Tis phenomenon causes a change in the way the HS is charged and discharged, as illustrated in Figure 9.
Finally, the cost assigned to the energy carriers per generation of each carrier is shown in Figure 10. In this scenario, the outages caused by the DRP are fxed, and the total cost is signifcantly reduced compared to the basic scenario. In the following, the impact of DG on solving the blackout created in the basic scenario will be investigated.

Energy Hub considering DG (Scenario 2).
In this scenario, in order to bring the problem closer to reality, the uncertainty in the capacity of DG is considered. As shown in Figure 11, the average capacity of DG at hour 10 per 1000 samples is equal to 0.598 p.u., which reduces the electric charge in this hour.
Similarly, by averaging the samples obtained in other hours, the capacity of DG is calculated. Finally, according to Figure 12, the fnal electric load is obtained from the difference between the electric charge in the base mode and the capacity of DG.   International Transactions on Electrical Energy Systems Applying DG will eliminate blackouts created in the base mode. Terefore, the total production at diferent hours of the day and night for all factors (electricity, gas, heat, EVCS, and HS) is shown in Figure 13. Te largest share of generation is related to the carrier of electric energy. Also, EVCS and HS devices have played a production role during peak hours.
In addition, the total cost of the production of energy carriers in this scenario is shown in Figure 14, where the electric energy carrier has the largest share due to its high price compared to other energy carriers. Also, the existence of DG has solved the outages in base mode. In this regard, the cost due to load shutdown in this scenario is zero.

Energy Hub considering DR and DG (Scenario 3).
Tis scenario considers both previous scenarios simultaneously. Te cost due to this scenario is shown in Figure 15, where the highest cost is equal to 62 p.u. at hour 16, which is less than other scenarios. In addition, there is no outage cost due to electrical and thermal loads, but it is generally more economical than other scenarios.  International Transactions on Electrical Energy Systems Figure 16, the basic scenario has a higher total cost than the other scenarios due to blackouts in some hours. Te presence of DRP (scenario 1) eliminates the outages caused in the base case and reduces the total cost to 1062.9, but the efect of DG (scenario 2) is better. Te amount of the total cost in scenario 2 is reduced to 926.9 due to the reduction of the fnal electrical load, which is better than other scenarios. Also, the cost assigned to the electric energy carrier is higher than the gas and heat carriers, and the blackout cost in scenarios 1 and 2 is zero. Te power stored in the EVCS can be diferent in various scenarios. Finally, the total cost in scenario 3 (DR and DG) is economically better than the other scenarios, so the cost has reached 906.2 due to the simultaneous use of the DR program and DG. As shown in Figure 17, the stored power in hours 2 to 4, 9, 10, and 17 to 20 in the presence of DG is less than in other scenarios because, in the presence of DG, the fnal electric load is reduced, and since the price of electricity is more expensive than other carriers, in order to optimize the cost, the capacity of the EVCS is used to meet the load. In addition, in some hours, due to the cheaper price of   electricity compared to other hours, the capacity of the EVCS is greater than in other scenarios. In other words, the EVCS is charged.

Comparison of Scenarios. As shown in
Also, the capacity of HS in scenario 1 has decreased compared to the basic scenario in peak hours (16, 17, 18, and 19) in Figure 18. In addition, the power stored in the HS has

Conclusion
Tis paper proposes a method based on linear programming that considers blackouts created by the load to optimize energy costs by considering energy converters and storage devices. In addition, it uses two tools, the DRP and DG, to solve the blackouts, and the capacity of the DG is associated with uncertainty in order to bring the issue closer to reality. Te results are compared in several scenarios, and one of the most important results is the issue of costs. Te cost of Scenario 2 (consideration DG) is more economical than Scenario 1 (consideration DR) and the basic scenario by 136 p.u. and 330.6 p.u., respectively. On the other hand, Scenario 3 is economically more economical than the other scenarios due to the simultaneous use of DRP and DG, so the total cost is equal to 906.2 p.u. In addition, it is possible to point out the correct behavior of ES devices, especially the EVCS, in diferent load conditions, which are charged at low load and discharged at peak load. From the comparison of the results, it can be clearly seen that the behavior of storage devices does not depend only on the decrease or increase of the load but also on the increase or decrease in the price of energy carriers. In addition, in the basic scenario, at hours 15, 16, 18, and 19, the load will be shut down by 0.57, 0.57, 0.07, and 0.07 p.u., respectively, which, with the help of DRP and DG, fxes the blackouts. Te proposed method can be an efective tool for energy hub management, especially when there is a blackout, in order to fx it at the lowest cost.

EH:
Energy hub DR: Demand response DRP: Demand response program DG: Distributed generation UP: Unsupplied power ES: Energy storage NL: Number of energy hub loads NP: Number of energy carriers EVCS: Electric vehicle charging station HS: Heat storage α: Percentage of the energy reduced peak load that is recovered in low load ξ: Coefcient of the total load at peak load ρ: Te initial value of the capacity of DG μ s (t): Random value at hour t per sample s for capacity of DG Ω: Peak load hours Ψ: Low load hours λ e (t): Electricity price at hour t λ g (t): Gas price at hour t λ h (t): Heat price at hour t p (t): Basic load model at hour t P(t): Modifed load model at hour t P: Prespecifc peak P T (t): Peak load at hour t TP (t): Te amount of load that is reduced at hour t and is added to the time of low load P e (t): Generation power of electricity energy carrier at hour t P g (t): Generation power of gas energy carrier at hour t 14 International Transactions on Electrical Energy Systems P h (t): Generation power of heat energy carrier at hour t P e Min (t), P e Max (t): Minimum/maximum generation power of electrical energy carrier at hour t P g Min (t), P g Max (t): Minimum/maximum generation power of gas energy carrier at hour t P h Min (t), P h Max (t): Minimum/maximum generation power of thermal energy carrier at hour t UP e (t): Unsupplied electric power at hour t UP h (t): Unsupplied thermal power at hour t Pen e (t): Penalty for unsupplied electric power at hour t Pen h (t): Penalty for unsupplied thermal power at hour t L (t): Output load matrix at hour t L e Total (t): Te matrix of the fnal electric load at hour t C (t): Coupling matrix at time t C s (t): Te matrix of randomly DG capacities at hour t C ee (t): Electric energy carrier coefcient for electric load at hour t C eg (t): Gas energy carrier coefcient for electric load at hour t C eh (t): Heat energy carrier coefcient for electric load at hour t Q ch (t), Q dis (t): Charge/discharge matrix of energy storage devices at hour t C Q ch (t), C Q dis (t): Matrix of charging/discharging coefcients of energy storage devices at hour t.

Data Availability
Te data used to support the fndings of this study are included within the article.

Conflicts of Interest
Te authors declare that they have no conficts of interest.