Modeling and Optimization of Combined Heating, Power, and Gas Production System Based on Renewable Energies

Abstract

Electrical energy and gas fuel are two types of energy needed that increase environmental pollution by burning fossil fuels in power plants to produce electrical energy and direct combustion of gas fuel. In this research, an attempt has been made to model the electrical energy network in the presence of renewable energy sources and gas production systems. The advantage of this model compared to other models of similar studies can be found in providing a mixed integer linear optimization model of distributed generation sources with gas fuel, energy storage systems, and gas power systems, along with electric vehicles in an integrated electricity and gas system. In addition to the energy consumption of buildings, an electric vehicle is also considered a base load, which is one of the limitations in optimizing the maximum charging of an electric vehicle. Among the important results of this research, it can be mentioned that the investment cost of USD 879,340 in the first scenario, in which 37,374 kW of electric energy was purchased from the network to supply the electric load, and 556,233 m³ was purchased from the gas network to supply the required gas.

1. Introduction

There is a need for an integrated electricity–gas energy system to improve energy efficiency, increase the use of renewable energies, and create a sustainable and low-carbon energy system. It also highlights the risks associated with using distributed natural gas generation units to balance renewable generation systems and energy storage systems [1]. Coordination of gas and electricity networks as an integrated system is desirable to balance renewable energy sources and natural gas generation, improve energy efficiency, and create a sustainable low-carbon energy system [2,3]. Energy analysis of the vacuum tube collector system was also proposed [4]. Optimal distribution of gas with uncertainty in wind power generation was proposed to propose a cost-effective integrated gas/electric system [5]. A study was also proposed on a decentralized optimization framework that considers wind technologies and gas power systems to reduce the operating costs of the entire multi-region electricity and natural gas system [6]. Integrated energy systems were optimized using an integrated stochastic programming-information gap decision theory (IGDT) approach [7]. Exergo-economic analysis of the solar energy-based integrated systems was also proposed [8].

Dynamic simulation for building thermal environments was performed using integrated planning of energy systems [9]. A refined power-to-gas model considering electric vehicles was proposed for DE carbonization scheduling [10], and the electric vehicle future scope is also reviewed by researchers [11]. Flexible resource scheduling of the integrated energy system was also discussed [12]. CVaR-based operation optimization method of community integrated energy system was considered for demand response [13]. A decentralized privacy-preserving operation of natural gas systems with renewable energy resources was studied [14]. There is a problem in assessing the acceptability of wind production in gas systems. Integrated electric-based approximation based on mixed integer linear programming (MILP) to minimize wind generation losses for integrated electric gas systems is reviewed [15,16]. An integrated method for optimizing the smart management of integrated energy systems was defined [17,18]. In addition, a two-stage energy management for a heat-electricity integrated energy system considering dynamic pricing of the Stackelberg game and operation strategy optimization was proposed [19]. A short-term multi-energy load forecasting for integrated energy systems was optimized by an attention mechanism [20]. The coordinated bidding strategy of wind farms and power-to-gas facilities using a cooperative game approach was studied [21]. The optimization method of integrated energy system operation with a multi-subject game is reported in this study [22]. An optimal economic emission planning of multi-energy systems integrated electric vehicles with modified group search optimization was performed [23]. An optimal day-ahead scheduling of power-to-gas energy storage and gas load management in wholesale electricity and gas markets was proposed. Researchers also presented an integrated model and enhanced utilization of power to ammonia for highly renewable penetrated multi-energy systems [24,25]. In reference [26], a steady-state analysis of the integrated natural gas and electric power system with bidirectional energy conversion. A mixed integer linear model is presented to optimize coupled gas power networks by considering a multi-objective function and gas power systems and renewable energy sources [27,28,29]. A possible load distribution framework for integrated electric and gas systems is proposed considering gas generators, electric compressors, and energy hubs integrated with gas power units and wind turbines. Reactive power management in low voltage distribution networks using capability and oversizing of PV smart inverters [30,31]. Distribution systems energy management in the presence of smart homes, renewable energy resources, and demand response programs by considering uncertainties was performed in this study. In reference [32,33], a demand response program modeling in multiple energy and structure management in microgrids equipped by combined heat and power generation.

The proposed model of this study is a mixed integer linear programming model, which guarantees absolute optimal solutions. As seen, the issue of optimization of integrated energy systems has attracted the attention of researchers today. In this paper, a binary linear model is proposed to solve the integrated energy management of electricity/gas networks in the presence of renewable energies, storage, and gas power devices along with electric vehicles. The innovations of the article can be summarized as follows:

• Presenting the effect of modeling the energy storage system, gas power system, scattered gas-burning production sources, and electric vehicles in the electricity and gas distribution network;
• Presenting an integer linear programming model for the integrated operation of electricity and gas distribution networks;
• Presenting the random model of renewable resources, energy cost, and electric cars as a scenario-based method;
• A 24 h dynamic load modeling of IEEE standard 33 bus power distribution network and gas distribution network of 7 nodes.

2. Materials and Methods

Modeling

This section presents an optimization model to reduce operating costs of electricity and gas networks using mixed integer linear programming, with an objective function consisting of penalty factor for unused power, cost of purchasing power from power station, and cost of purchasing gas from gas well.

$$min{{\displaystyle \sum }}_{s\in Ns}{\mathsf{\Omega }}_s\left({{\displaystyle \sum }}_{t=1}^{Nt}{{\displaystyle \sum }}_{i=1}^{Ne}\left(C^r(P_{t,i,s}^r-{\widehat{P}}_{t,i}^r)+C^{ev}(P_{t,i,s}^{ev}-{\widehat{P}}_{t,i}^{ev})\right)+{\lambda }_{t,s}^e{P}_t+{\lambda }_{t,s}^g{g}_t\right)$$

(1)

In this regard, t represents the time index, which is considered as one hour in this study, i, j represents the bus index of the electricity network, and n, m represents the node index of the gas network. The first part of the cost objective function is related to the penalty factor for unused power from renewable sources and electric vehicles.

The following relationships must be established between the power of renewable energy sources and the charging power of electric vehicles:

$$\{\begin{array}{c}{\widehat{P}}_{t,i}^r\ge 0\\ {\widehat{P}}_{t,i}^r\le P_{t,i,s}^r\end{array},\{\begin{array}{c}{\widehat{P}}_{t.i}^{ev}\ge 0\\ {\widehat{P}}_{t,i}^{ev}\le P_{t,i,s}^{ev}\end{array}$$

(2)

The energy storage system designed to increase the flexibility of the network is modeled in this article using Equations (3)–(8). Constraint (3) expresses the logical relationship between charging and discharging states. The battery cannot charge and discharge simultaneously, with limited power and energy, as shown by Equations (4)–(7), and the condition of equality of initial and final states is determined by Equation (8). In Figure 1A, this model is illustrated, and Figure 1B also shows the internal resistance and open-circuit voltage for a sample battery.

$$u_{t,i}^{dis}+u_{t,i}^{ch}\le 1$$

(3)

$$P_i^{dis,min}{u}_{t,i}^{dis}\le P_{t,i}^{dis}\le P_i^{dis,max}{u}_{t,i}^{dis}$$

(4)

$$P_i^{ch,min}{u}_{t,i}^{ch}\le P_{t,i}^{ch}\le P_i^{ch,max}{u}_{t,i}^{ch}$$

(5)

$$e_{t,i}=e_{t-1,i}+{\eta }^{ch}{P}_{t.i}^{ch}-\frac{P_{t,i}^{dis}}{{\eta }^{dis}}$$

(6)

$$e_i^{min}\le e_{t,i}\le e_i^{max}$$

(7)

$$e_{t=0}=e_{t=24}$$

(8)

The power-to-gas system can convert power to natural gas, store and inject it, and provide natural gas consumption, with limitations on the amount of power converted and natural gas stored (Equations (9)–(13)).

$$g_{t,n}^{ch}={\eta }_{P2G}{P}_{t,i}^{P2G}$$

(9)

$$s_{t,n}=s_{t-1,n}+g_{t,n}^{ch}-g_{t,n}^{dis}$$

(10)

$$0\le P_{t,i}^{P2G}\le P_i^{P2G,max}$$

(11)

$$S_n^{min}\le S_{t,n}\le S_n^{max}$$

(12)

$$S_{t=0}=S_{t=24}$$

(13)

The balance of active and reactive power produced in the distribution network must always be equal to the load in each bus, which is modeled in Equations (14) and (15).

$$P_t-P_{t,i}^{ch}-P_{t,i}^{P2G}+P_{t,i}^{dis}-P_{t,i}^P+P_{t,i}^{DG}+{\widehat{P}}_{t,i}^r={\widehat{P}}_{t,i}^{ev}={{\displaystyle \sum }}_{iϵNe\left(j\right)}{P}_{t,ij}-{{\displaystyle \sum }}_{iϵNe\left(j\right)}{P}_{t,ji}$$

(14)

$$q_t-D_{t,i}^q+q_{t,i}^{DG}={{\displaystyle \sum }}_{iϵNe\left(j\right)}{q}_{t,ij}-{{\displaystyle \sum }}_{iϵNe\left(j\right)}{q}_{t,ji}$$

(15)

In Equation (16), the definition of the ith bus voltage of the distribution network at time t is shown. Equation (17) shows the constraint of the ith bus voltage at time t of the distribution network. Equation (18) guarantees the maximum active and reactive power flux passing through line i_j

$$U_{t,i+1}=U_{t,i}-2\left(r_{ij}{p}_{t,ij}+x_{ij}{q}_{t,ij}\right)$$

(16)

$$U_i^{min}\le U_{t,i}\le U_i^{max}$$

(17)

$$p_{t,ij}\le p_{ij}^{max},q_{t,ij}\le q_{ij}^{max}$$

(18)

Equation (19) shows the balance constraint of natural gas in the gas network. Equation (20) shows the minimum and maximum gas injected by P_2G. Equation (21) shows the minimum and maximum flow through the gas pipeline, and Equation (22) shows the minimum and maximum exploitation of the gas well. Equation (23) shows the flow of gas that can be transported through the pipeline depending on the pressure at the beginning and end of each pipeline. Finally, Equation (24) shows the minimum and the maximum pressure of each gas node is showing.

$$g_t+g_{t,n}^{dis}-g_{t,n}^1-{{\displaystyle \sum }}_{nϵNg}\left(p_{t,n}^{DG}{\rho }_n\right)={{\displaystyle \sum }}_{nmϵNg\left(m\right)}{h}_{t,nm}-{{\displaystyle \sum }}_{nmϵNg\left(m\right)}{h}_{t,mn}$$

(19)

$$g_n^{dis,min}\le g_{t,n}^{dis}\le g_n^{dis,max}$$

(20)

$$h_{nm}^{min}\le h_{t,nm}\le h_{nm}^{max}$$

(21)

$$g^{min}\le g_t\le g^{max}$$

(22)

$$h_{t,nm}^2=a_{nm}[{\psi }_{n,t}^2-{\psi }_{m,t}^2]$$

(23)

$${\psi }_n^{min}\le {\psi }_{n,t}\le {\psi }_n^{max}$$

(24)

Equations (23) and (24) are non-linear and need to be converted into a linear relationship by linearizing the quadratic and non-linear variables of gas flow and gas node pressure. A simple variable change can convert the gas node pressure variable into a linear one.

$${\psi }_{n,t}^2=k_{n.t}$$

(25)

$$h_{t,nm}^2={\alpha }_{nm}\left[k_{n,t}-k_{m,t}\right]$$

(26)

$${\left({\mathsf{\Psi }}_n^{min}\right)}^2\le k_{n,t}\le {\left({\psi }_n^{max}\right)}^2$$

(27)

The pressure of the gas node was linearized using a new variable, resulting in Equations (26) and (27), and the nonlinearity of Equation (26) was converted into a set of linear terms using the piecewise linear method. The piecewise method can accurately approximate a parabolic curve using multiple straight lines, as shown in Figure 2. The second power flow variable of a gas line can also be obtained using five sets of linear terms, as per Equation (28) and Figure 3.

$$h_{t,nm}^2={{\displaystyle \sum }}_{z=1}^{10}\left[NF{Z}_{t,nm,z}m{f}_z\right]$$

(28)

In Equation (29), the limit between the line and the binary variable associated with each piece is shown.

$${\gamma }_{t,nm,z+1}\ast df\le NF{Z}_{t,nm,z}\le {\gamma }_{t.nm,z}\ast df$$

(29)

Finally, the limit between pipeline flow and binary variable and large number B will be according to Equation (30).

$$h_{t,nm}-\left[\left(z-1\right)\ast df\right]\le {\gamma }_{t,nm,z}\ast M$$

(30)

Figure 3 shows the connection between the integrated energy system modeling of gas and electricity in this article. As can be seen, the power grid is connected to the gas grid through the gas power system, and the gas grid is also connected to the electricity grid through the distributed generation sources of burning gas. It can be seen that electric loads, renewable energy sources, energy storage systems (batteries), and electric car parking, along with the power-to-gas system that converts electric power into natural gas, are considered in the power grid. In addition, in the natural gas network, in addition to the natural gas load and natural gas sources, there are scattered sources of gas production that burn the gas received from the gas network and convert it into electricity. By introducing the proposed model in this section, which includes the objective function of the problem as well as the constraints related to it, in the form of a standard mathematical problem, it can be solved using powerful commercial solvers such as Groobi and Syplex. In the next section, the simulation results are analyzed.

Table 1. Specifications of the equipment used.

Components	Characteristics	Value
PV array [1, 2]	Size	100 kW
	Initial cost	USD 3882/kW
	Replacement fee	USD 3000/kW
	Maintenance cost	USD 100/year
	Lifetime	25 year
Inverter [1, 2]	Size	100 kW
	Connection mode	Three-phase
	Initial cost	USD 429/kW
	Network voltage and frequency	415 VAC50/60
	Maintenance cost	USD 0/year
	Lifetime	16 year
	Efficiency	95%
Battery [1, 2]	Rated voltage	6 v
	Nominal capacity	1156 Ah
	Initial cost	USD 1200
	Replacement fee	USD 1000
	Maintenance cost	USD 10/year

shows the specifications of the equipment used. Figure 4 shows a block diagram of the methodology.

3. Results

In this section, the simulation results are also presented and analyzed. The implementation of the proposed model has been implemented using the Julia programming language and solved using the powerful business solver Grooby. A proposed integrated system with batteries and gas production units was modeled on a 33-bus IEEE standard distribution network, with batteries having a maximum energy storage capacity of 1000 kW and an efficiency of 75% (Figure 5) [26,27]. The study suggests installing batteries on buses 10, 20, and 30 of the distribution network and distributed production sources on buses 6, 14, and 32 connected to gas nodes 5, 6, and 7, with a maximum capacity of 1.2 MW, while the gas power system in Node 4 has a capacity of 4500 m³/h and a discharge/charging capacity of 1500 m³/h. The gas power system is connected to bus 14, while electric car parking lots are in buses 18 and 25, and a photovoltaic renewable energy source is on bus 12. Different scenarios are analyzed to evaluate the proposed model.

The first scenario—operation without changes in the network. In this scenario, the amount of energy production and the required amount of load are considered constant, the system is without renewable energy sources, and there is no gas production in this state.

The second scenario—without considering the gas power system. In this scenario, there is also a photovoltaic system and distributed generation sources of fuel gas in the network to produce electric power, but it is without gas production equipment, and all the gas needed to produce energy is purchased, and the power produced by the photovoltaic cell is used for the required load, and the excess power is stored in the battery.

The third scenario—without taking into account scattered sources of gas-burning production. In this scenario, there is also a photovoltaic system in the network to produce electric power, but it is without gas production equipment, and all the gas needed to produce energy is purchased, and the power produced by the photovoltaic cell is used for the required load, and the excess power is stored in the battery.

The fourth scenario—is without considering the battery. In this scenario, there is also a photovoltaic system and distributed generation sources of fuel gas in the network to produce electric power, but it is without a battery, and the power produced by the photovoltaic cell is used for the required load, and the excess power produced by the cell is injected into the network.

The cost function is at its lowest level when all available resources, such as batteries, distributed generation, and gas power systems, are considered (

Table 2. The result of comparing different scenarios.

Scenario	Buy Gas (m3)	Power Purchase (kW)	Target Function (USD)	Execution Time (s)
First	580,233	39,052	890,200	57
Second	567,112	38,121	890,112	55
Third	498,525	78,540	899,215	58
Fourth	523,821	36,998	881,425	57

). The first scenario had the lowest cost function of USD 890,200, with 39,052 kW of electricity purchased and 580,233 m³ of gas purchased, while the second scenario saw an increase in the objective function and gas purchased due to the removal of the power-to-gas system (Table 2). Removing scattered gas production sources from the network increases the objective function by 4.7% and reduces the amount of purchased gas, as shown in the third scenario of the proposed model (Table 2). The third scenario had the lowest amount of gas purchased due to the elimination of scattered production sources, while the fourth scenario showed the effect of removing the battery on the objective function (

Table 3. Comparison of different solvents in the first scenario.

Type of Solver	Target Function (USD)	Execution Time (s)
Grooby	879,340	57
Muzak	879,340	63

). Figure 6 shows the amount of electricity purchased in 24 h after optimization in the first scenario. It can be seen that in the early hours of the day, when there is no peak electric load, most electricity is purchased and stored in the gas power system or the battery and injected into the grid during the peak hours.

As can be seen in Table 2, the duration of the problem execution (from the moment the program is executed until the results are obtained) is shown in each scenario. For example, the execution time in the first scenario is more than in the other scenarios, which is the reason for considering all the constraints of the problem in this scenario. The time to solve the problem in the first to the fourth scenario is equal to 57, 55, 58, and 57 s, respectively. The execution time includes the duration of model construction by Julia and the duration of solving by Grooby. Figure 7 shows the purchase of gas in 24 h in the first scenario, and Figure 8 shows the state of energy saving in each battery in the first scenario.

It is well known that the batteries start charging from 12:00 to 6:00 a.m. (non-peak time) and inject power into the grid during peak hours. Similarly, Figure 9 shows the state of energy in each battery. It is known that the energy capacity increased as the battery was charged and decreased as the energy in the battery was discharged.

Figure 10 shows the amount of photovoltaic power used in the first scenario. In this figure, the black line with a circle indicates the used photovoltaic power. The rest of the lines shown in this figure represent the scenarios considered for photovoltaics in the scenario-based model.

Figure 11 shows the actual charging power of the electric cars in the parking lot in each scenario with its operating value. In this figure, the black line with a circle represents the used power of electric vehicles in the optimization problem in the first scenario.

Finally, Figure 12 shows the amount of natural gas storage and injection by the power to the gas system. In this figure, the negative y-axis indicates storage, and the positive axis indicates the injection of natural gas by the power-to-gas system into the gas network in the first scenario. By examining the considered results and scenarios, the impact of the proposed model and method can be seen in the simultaneous optimization of integrated energy systems of electricity and gas.

To check and correct the absolute optimality of the results and also to verify the obtained results, in addition to solving the problem with the latest version of Grooby, it has also been implemented with the latest version of Muzak solver, and the results have been compared. Table 3 also shows a comparison between the results obtained by solving with Grooby and Muzak. As can be seen from this table, the answers obtained in both solvers are equal, which shows the correctness of the proposed model and method. However, as it is shown, Grooby was able to solve the problem and obtain the optimal solutions in about 8 s faster than Muzak.

In the following, to check and analyze the proposed random model, the weight of each scenario is changed, and the results are checked in the first scenario. The results of

Table 4. Comparison of results with different weighting coefficients in scenarios.

Coefficient	Execution Time (s)	Target Function (USD)
Equal weighting factor	57	890,200
Unequal weighting factor	63	922,021

were performed according to the equal weight of each scenario so that considering 10 scenarios, the weight of each scenario is also considered equal to 0.15. In this section, to check the performance of the proposed random model, the weight of each scenario is considered equal to 0.17, 0.26, 0.14, 0.16, 0.11, 0.14, 0.09, 0.11, 0.121, and 0.143, respectively.

According to Table 4, it was observed that by changing the weighting coefficient for each scenario, the value of the objective function also changed, which is a sign of the correct performance of the proposed model.

To show the efficiency and superiority of the proposed method, the results have been compared with other meta-heuristic methods. As can be seen in

Table 5. Compare the proposed method with other methods.

Method	Execution Time (s)	Target Function (USD)
Suggested method	57	890,200
Whale optimization algorithm [34]	150	888,500
Particle swarm optimization algorithm [35]	269	892,530
Refrigeration optimization algorithm [36]	191	889,010

, the whale optimization algorithm [34], the particle swarm optimization algorithm [35], and the refrigeration simulation algorithm [36] have been selected for comparison with the proposed method. The solution is performed by meta-heuristic algorithms in the Julia programming language and the “meta-heuristic” optimization package [37]. The parameters of these algorithms are set based on references [34,35,36].

Table 5 shows the results obtained from the simulation of the first scenario, where the proposed model is solved using other available methods, and its results are shown. It is worth mentioning that due to solving non-linear sentences by meta-heuristic algorithms, linearization was not used for solving these algorithms. As we know, evolutionary algorithms have a random nature, and with each execution in low iterations, they obtain various near-optimal solutions. Therefore, to prevent this, 350 (L) has been considered for evolutionary algorithms to ensure the most optimal solution obtained. It can be seen that the result obtained from the proposed method has the lowest value of the objective function. Here, the objective function of the problem in the first scenario using the whale optimization method is equal to USD 890,200, and in the same way, the objective function in the particle swarm optimization algorithm and the refrigeration optimization algorithm is equal to USD 892,530 and USD 889,010. From this table, it can be concluded that the proposed method achieved the best result and was able to solve the problem in the shortest period.

4. Conclusions

In this study, an optimization model is presented to reduce the purchase cost of electricity and gas in integrated energy systems. The proposed model is a mixed integer linear programming model that can be easily solved by all mathematical solvers, and its absolute optimal solutions are guaranteed. The advantage of this model compared to other models of similar studies can be found in providing a mixed integer linear optimization model of distributed generation sources with gas fuel, energy storage systems, and gas power systems, along with electric vehicles in an integrated electricity and gas system. A proposed model for analyzing gas pipeline flow and electricity distribution network, incorporating uncertainty in renewable resource production and energy cost, was tested on a 33-bus electricity distribution network and a 7-node gas network. The proposed integrated energy management for gas and electricity networks can effectively reduce the costs of purchasing electricity and gas energy.

Also, to show the guarantee of absolute optimal solutions, the proposed model was implemented using several different solvers, and the results were checked, which proved this important. On the other hand, to verify the accuracy of the proposed random model, the weight coefficient of the scenarios was changed and the results were analyzed, and the results showed that the weight coefficient of each scenario can have a significant effect on the objective function.

Recommendations for future work include:

Integration of storage of combined heating, power, and gas production system based on renewable energies;

Modeling and optimization of integration of storage of combined heating, power, and gas production system based on renewable energies for minimum energy cost;

Simulation and optimization of stand-alone combined heating, power, and gas production system based on renewable energies.