Optimal Design of a Distributed Energy System Using the Functional Interval Model That Allows Reduced Carbon Emissions in Guanzhong, a Rural Area of China

Nowadays, rural power supply in China plays an important role in restricting the economic development and improvement of residential living standards. In this study, an interval full-infinite programming rural energy model (IFIP-REM) was developed for supporting distributed energy system (DES) optimal design under uncertainties in rural areas. By affecting the upper and lower bounds of the interval by complex and variable external conditions, IFIP-REM could simulate the influence of external systems. To validate the model, a real case study of DES optimal design in Guanzhong, a rural area of China, was tested and aimed to minimize system cost and constraints of resources, energy supply reliability, and carbon emission mitigation. The data revealed generation of reasonable optimization schemes to obtain interval solutions of IFIP-REM. Compared to centralized energy system (CES), DES reduced electricity purchasing of the municipal grid by 47.5% and extended carbon emission of both upper and lower bounds to [17.13, 44.51] % and [12.42, 36.02] %, respectively. Overall, the proposed model could help managers make decisions of DES optimal design by coordinating conflicts among economic cost, system efficiency, and carbon emission mitigation.


Introduction
According to the United Nations Environment Program, energy consumption has increased dramatically in the past few decades, where the architecture sector accounts for about 40% [1][2][3].The situation is particularly striking in China since the building energy consumption has doubled in just 20 years with average growth rate of 3.70% [4] and an average energy consumption growth rate of 8.56% in rural areas [5,6].Rural power supply has become an important factor in restricting current rural economic development and improvement of residential living standards.To solve the problem of power supply in rural areas, the government has performed several developments of agricultural networks via upgrading and reconstruction projects.However, many economy and technology-related issues still require solutions, particularly in remote areas and border defense islands.Therefore, it is urgent to seek new solutions while upgrading rural grids [7].
Distributed energy systems (DES) can handle small-scale, short-distance energy supply problems to compensate for the shortcomings of the municipal grid [8].It is increasingly becoming an attractive option worldwide due to its high efficiency, superior reliability, less investment, and reduced transportation distance [9,10].Compared to centralized energy systems (CES), DES uses a wider range of technologies containing prime movers, waste heat recovery, energy storage, heat pumps, solar photovoltaic, small wind turbines, and other equipment with renewable energy resources [11][12][13][14][15][16].DES use renewable energy to reduce dependence on fossil fuel resources and reduce carbon emissions to the atmosphere [17].Therefore, the development of DES according to local conditions would be an effective measure to promote large-scale utilization of rural renewable energy, maintaining rural natural ecological environment, and improving rural power supply capacity.This, in turn, will be of great significance to the rapid development of the rural economy.
Therefore, numerous studies have been devoted for optimal design of DES.For instance, Liu et al. calculated the optimal operation and explored the economic, energy saving, emission reduction, and peak modulating characteristics of DES under different configurations [18].Doagou-Mojarrad et al. proposed an optimized method to reduce power loss, energy cost, and pollutant emissions [19].Falke et al. applied an optimized model on investment planning and operation management of DES to examine the impact of different efficiency measures on total cost and emissions of carbon dioxide equivalent [20].Liu et al. presented the feasibility of different renewable energy penetrations in DES by establishing the Energy PLAN model in Chongming County, China [21].Yeşim and Mehmet established a mixed integer linear programming model to improve efficiency and reduce the cost of DES by decreasing energy distribution losses and the transportation cost of energy distribution networks [22].Wu et al. also considered optimal neighborhoods-scale DES for both supply and demand, as well as optimal combination of building structures within community [23].However, only a few of them focused on uncertainties in the DES optimal design process.Besides, these studies did not consider carbon emission reductions in the same framework, especially in rural areas.In an actual DES optimal design process, uncertainties exist in various aspects, such as energy-supply activities and economic, technical, equipment cost, system stability, fluctuation of energy demand, and political indicators [24][25][26].First, renewable energy represented in the literature is affected by uncertainties linked to climate [27].For example, the output powers of photovoltaic plants suffer from randomness and volatility, which will greatly impact the stable operation of the system.Second, the complexity of DES results from a wide variety of system components, such as equipment associated with power generation, refrigeration, heating, and energy storage.Furthermore, management of energy in DES often requires consideration of economic and reliability constraints in actual situations, and the system must be controlled by constraints of electricity balance to maintain the system's cold and heat balances.Meanwhile, an energy management for DES should not only consider real-time satisfaction of load requirements, but also premeditate the economics of power supply cost.
Several methods have been developed to study such complexities and uncertainties.For example, interval linear programming (ILP) is employed to deal with problems of interval uncertainties in system analysis [28][29][30][31].This defines uncertainties in an interval range, determining the upper and lower bounds, and performing system analysis [32][33][34][35].However, ILP is based on assumptions of the upper or lower determination number (crisp interval) [36].The crisp interval is then input into the model and the relationship between the final values of the upper and lower bounds with external impact factors cannot be intuitively reflected.In practical problems, the upper and lower of intervals are affected by complex and variable external factors, requiring function intervals to characterize the factors.For example, the purchase cost of equipment would change according to discount rate, defining the ratio of future limited-period expected income in present value.The change in discount rate will then be affected by external uncertainty factors.An effective way for describing these uncertainties would be through function intervals, in which the upper and lower bounds are functions related to several influencing factors.When uncertainty factors are expressed as crisp intervals and function intervals, they lead to the interval full-infinite programming method (IFIP) [37].IFIP is based on extension of uncertainties as crisp and function intervals with infinite numbers of objective function and constraints.These reflect decision-making objectives directly in the optimization process and solution results, and prevent shortcomings of various methods running separately [38].
Therefore, in this study, an interval full-infinite programming rural energy model (IFIP-REM) was developed to optimize the design of DES in rural areas.IFIP-REM was then applied to a real case in Guanzhong for optimal design of DES, where both energy load and system equipment matching problems were taken into account to identify the optimal energy-using pattern of local energy systems.The IFIP method for dealing with uncertainties was represented as crisp intervals and functional intervals.The determination of outcomes for desired policy associated with rural environment, economic, and energy supply reliability were discussed.The solutions obtained from the case study provide a foundation for decision-making related to the optimal design of DES in rural areas, such as promoting capacity expansion and/or dynamic changes in development planning.The obtained solutions would also be utilized for coordinating economic cost, system efficiency, and solving conflicts among pollutant mitigation and energy supply security interactions under uncertainty.

Problem Description
To promote China's Western Development Policy and construct the Belt and Road Initiatives, the Ministry of Housing and Urban-Rural Development of the People's Republic of China issued the act of "Development Plan for the Guanzhong Plain Urban Agglomeration" in 2018.The Guanzhong Plain urban agglomeration is located in the center of inland China (104 • 57 E~112 • 56 E, 33 • 09 N~36 • 93 N), centered on Xi'an [39].It is located in a cold temperate zone (National Building Thermal Design Division) with average elevation of about 500 m, and is characterized by four distinct seasons with large temperature differences, long, cold winters, and dry weather [40,41].The Guanzhong area is the second urban agglomeration in western China with administrative space of around 107.10 × 10 3 km 2 , population of 29.80 million, and 1958.42 billion RMB regional GDP as of 2017.It is the most important birthplace of Chinese civilization and the starting point of the ancient Silk Road.Hence, it has a unique strategic position in national modernization and acts as an important fulcrum of the Eurasian Continental Bridge and a gateway to the western region.
Currently, renewable energy usage in Guanzhong is still relatively simple and inefficient.Though solar energy, biomass, and geothermal energy are employed, their efficiency does not exceed 10% [42].Therefore, it is necessary to reform energy use in rural areas and introduce new energy-saving and environmentally-friendly resources, such as DES, to improve energy problems in rural areas.This would promote the development of rural life and economy.Meanwhile, renewable energy storage space in rural areas will enable better use of distributed energy to move away from fossil energy.Thus, it is necessary to optimize the design of DES in this region according to the actual situation.
The region suffers from several problems.First, the introduction of national strategy of Guanzhong Plain urban agglomeration and the Belt and Road Initiatives have both increased energy consumption in Guanzhong, with a gap in energy supply and demand.Second, the large distance separating the Guanzhong rural area from power stations has led to a relatively fragile distribution network that is vulnerable to emergencies and low power supply reliability.Third, the Guanzhong area suffers from adverse natural conditions and a fragile ecological environment and has high environmental protection requirements.Fourth, it is located deeply inland and is sandwiched between the Qinling Mountains and Loess Plateau.The air dispersion is weak, resulting in increased constraints between the atmospheric environment and air pollution [43].
When designing DES in Guanzhong with a hybrid sustainable energy system, including a multisource, multitechnology, and multiuser environment, many uncertainties and interactions must be systematically evaluated.Many uncertainties exist in various factors and processes associated with power production, conversion, transmission and supply, and energy conversion.If local energy supply does not meet energy demand, some degree of energy supply uncertainties may emerge, and addressing these uncertainties and complexities is important.

Modeling
The Guanzhong area is rich in renewable energy resources, such as biomass, geothermal sources, and solar.These features can establish the DES study [44].The target function of the proposed system was to minimize the total system cost.This could be formulated as interval full-infinite programming rural energy model (IFIP-REM) described through Equations ( 1)-( 24): Objective function: (1) Equipment cost: (2) Equipment operating cost: (3) Fuel cost: (4) Purchasing electricity cost: (5) Subsidy: (6) Carbon emissions: a. Carbon emissions of this system: b. Carbon emissions from centralized energy system: Subject to: (1) Maximum power constraints for equipment: (2) Solar photovoltaic panel generation constraints: (3) Energy conversion equipment conversion constraints: (4) Storage equipment constraints: Energy storage equipment at h = 1: Energy storage equipment at h>1: (5) Energy balance constraints: (6) Carbon emission reduction constraint: The meanings of symbols and letters are shown in Appendix A, and the methodology is shown in Appendix B.
The construction area was set to 50.66 × 10 3 m 2 in Guanzhong rural area.The cold and heat loads were simulated by EnergyPlus 8.6 software, and the electrical load was obtained through a questionnaire survey in search region (Tables 1-3).There is little difference in electrical load in spring and autumn, as shown in Table 1.EnergyPlus is a full-featured building energy analysis software that can be used to simulate the building's running energy consumption.In this study, the following data was entered into the software: (1) Building-related data, including geographic location, time zone, and structure; (2) Equipment data within the building; (3) Meteorological data for the study area.Finally, the energy requirements of the research objective can be obtained through software calculation.The solar energy resources data were extracted from the "Standard Meteorological Data Book of Buildings" [45].The biomass boiler and ground source heat pump could only be employed in winter and summer seasons [46].DES equipment is complex, so the equipment is divided into the following categories: (a) energy production equipment: internal combustion generating set (NG), solar photovoltaic panel (PV), biomass boiler (BB); (b) energy conversion equipment: ground source heat pump (GP), absorption chiller (AC), heat exchanger (HE); (c) energy conversion equipment: cold storage equipment (CS), heat storage tank (HS), and battery (BA).This categorization was performed according to government reports, statistical yearbooks, previous research results, equipment product research, and other means to summarize the technical parameters and financial characteristics of various equipment (Table 4).
In this DES, NG and PV generate electricity to supply power to users, and heat generated by NG is recovered by the waste heat storage equipment to directly supply heat load or to supply cold load.When the amount of electricity generated by this system cannot meet the demand of electric load, it is supplemented by the power grid, and when electricity generation is greater than electricity load, it is stored in a battery.In the supply of cold or heat load in this system, when the supply of cold and heat energy of system is greater than the demand of cold or heat load, the excess cold or heat is stored in energy storage equipment; if the cold or heat load requirements are not met, the energy storage equipment is used according to operating criteria, or supplemented by electrical energy conversion.The energy flow chart of the DES designed in this paper is shown in Figure 1.In addition, we compare and analyze the carbon emissions of the CES and DES constructed in this study combined with renewable energy.The compositions of these two systems are shown in Table 5.The optimal design process first determines the availability of local resources and selection of corresponding equipment to establish the optimal design model with total cost of the system as the goal.Then, LINGO13.0 (Linear Interactive and General Optimizer 13.0) is used to program and solve the model.The optimal solution of the model is obtained by inputting relevant parameters.LINGO is a comprehensive tool that makes building and solving linear, nonlinear, and integer-optimized models faster, easier, and more efficient, which provides a fast solver engine to illustrate and solve optimization models [47].In general, using LINGO to solve an operational research problem can be  Total/W 220 220 220 220 220 220 520 3320 120 320 490 4620 320 120 120 320 320 320 6020 1920 2020 1020 1050 620    The optimal design process first determines the availability of local resources and selection of corresponding equipment to establish the optimal design model with total cost of the system as the goal.Then, LINGO13.0 (Linear Interactive and General Optimizer 13.0) is used to program and solve the model.The optimal solution of the model is obtained by inputting relevant parameters.LINGO is a comprehensive tool that makes building and solving linear, nonlinear, and integer-optimized models faster, easier, and more efficient, which provides a fast solver engine to illustrate and solve optimization models [47].In general, using LINGO to solve an operational research problem can be divided into the following two steps: (1) According to the actual problem, establish a mathematical model, use mathematical modeling method to establish an optimization model; (2) LINGO is used to solve the optimization model.Mainly based on LINGO, we translate the mathematical model into a computer language, and solve it with a computer [48].Figure 2 shows the optimal configuration of IFIP-REM flow.The optimization process would first determine the available resources and corresponding equipment to establish the optimal configuration model with minimum total system cost as the objective function.
The model provides the optimal solution and configuration model of the DES.

Results and Discussion
The interval full-infinite programming rural energy model (IFIP-REM) was developed to first optimize design the DES and was then applied to Guanzhong.The relationship between the cost of cold, heat, electricity, and system equipment configuration required by users of IFIP-REM solution was established.The interactions among system equipment configuration, cost, and system energy consumption were considered.In this paper, a typical daily load for each season was selected.Because there are 8760 hours in a year, if the energy supply and demand balance of each hour is used as a constraint in optimal design model, after inputting other nonlinear constraints and objective functions, the model will become a complex nonlinear optimization problem.Solving such a model with a general computer will result in difficulty, i.e., the solution efficiency is low, and the solution quality is unstable.Therefore, in the modeling process, it is necessary to appropriately compress the input quantity of energy supply and demand.The compression time dimension is closely related to the energy load characteristics of the system's terminal.Because the terminal energy load of the study area has a large quarterly difference, the time scale is compressed into four representative seasons, thus reducing the calculation points to 96.

Energy Consumption
The energy supply scheme's lower and upper bounds are presented in Figure 3.The energy resources used in Guanzhong can be divided into five types.Natural gas and solar energy ranked first and second with proportions of 36.98% and 50.34%, respectively.In the lower bound supply, geothermal energy accounts for 5.49%.In contrast, geothermal energy of the upper bound supply only accounts for 0.05% of the total energy.This is due to the optimal design meeting the minimum total cost of the system.In addition, because the DES uses natural gas as an energy source in internal combustion engines, natural gas consumption showed the greatest values.These data suggested that imported renewable energy played a significant role in IFIP-RES, particularly when energy demand reached the lower bound.

Results and Discussion
The interval full-infinite programming rural energy model (IFIP-REM) was developed to first optimize design the DES and was then applied to Guanzhong.The relationship between the cost of cold, heat, electricity, and system equipment configuration required by users of IFIP-REM solution was established.The interactions among system equipment configuration, cost, and system energy consumption were considered.In this paper, a typical daily load for each season was selected.Because there are 8760 hours in a year, if the energy supply and demand balance of each hour is used as a constraint in optimal design model, after inputting other nonlinear constraints and objective functions, the model will become a complex nonlinear optimization problem.Solving such a model with a general computer will result in difficulty, i.e., the solution efficiency is low, and the solution quality is unstable.Therefore, in the modeling process, it is necessary to appropriately compress the input quantity of energy supply and demand.The compression time dimension is closely related to the energy load characteristics of the system's terminal.Because the terminal energy load of the study area has a large quarterly difference, the time scale is compressed into four representative seasons, thus reducing the calculation points to 96.

Energy Consumption
The energy supply scheme's lower and upper bounds are presented in Figure 3.The energy resources used in Guanzhong can be divided into five types.Natural gas and solar energy ranked first and second with proportions of 36.98% and 50.34%, respectively.In the lower bound supply, geothermal energy accounts for 5.49%.In contrast, geothermal energy of the upper bound supply only accounts for 0.05% of the total energy.This is due to the optimal design meeting the minimum total cost of the system.In addition, because the DES uses natural gas as an energy source in internal combustion engines, natural gas consumption showed the greatest values.These data suggested that imported renewable energy played a significant role in IFIP-RES, particularly when energy demand reached the lower bound.

Optimized Model
The optimal configuration pattern of the proposed system in terms of cold, heat, and electricity was obtained by calculating IFIP-REM.With four distinct seasons in the Guanzhong area, no requirement for heat or cold in spring and autumn were required, but it is necessary to provide heat and cold in summer and winter to improve residents' living conditions.The three possible patterns were discussed below.
Figure 4 depicts cold balance patterns for the DES on a typical summer day.During operation of the system, the cold load was mainly met by internal combustion engines, solar photovoltaic panels, and grid power conversion to absorption chillers.User's cold demand peaked between 13:00 and 20:00 with no fluctuations at other times.As user's energy demand decreases, the number of ground source heat pump units configured by the system was small.This occurred due to automatic system optimization.Through ground source heat pump and absorption chiller processes, electricity was converted into cold load and then supplied to users.In some periods, excess cold load was generated and stored by ice-storage devices.Since the energy storage efficiency was 0.65, the cold energy released by ice storage devices was lower than the stored cold energy, and no energy was wasted during the cold conversion process.Figure 5 represents the heat balance pattern for the DES on a typical winter day.The heat generated by biomass boilers and internal combustion engines was supplied to users after conversion by heat exchangers.Since the conversion efficiency was 0.98, no energy was wasted during the heating process.The optimal configuration pattern of the proposed system in terms of cold, heat, and electricity was obtained by calculating IFIP-REM.With four distinct seasons in the Guanzhong area, no requirement for heat or cold in spring and autumn were required, but it is necessary to provide heat and cold in summer and winter to improve residents' living conditions.The three possible patterns were discussed below.
Figure 4 depicts cold balance patterns for the DES on a typical summer day.During operation of the system, the cold load was mainly met by internal combustion engines, solar photovoltaic panels, and grid power conversion to absorption chillers.User's cold demand peaked between 13:00 and 20:00 with no fluctuations at other times.As user's energy demand decreases, the number of ground source heat pump units configured by the system was small.This occurred due to automatic system optimization.Through ground source heat pump and absorption chiller processes, electricity was converted into cold load and then supplied to users.In some periods, excess cold load was generated and stored by ice-storage devices.Since the energy storage efficiency was 0.65, the cold energy released by ice storage devices was lower than the stored cold energy, and no energy was wasted during the cold conversion process.Figure 5 represents the heat balance pattern for the DES on a typical winter day.The heat generated by biomass boilers and internal combustion engines was supplied to users after conversion by heat exchangers.Since the conversion efficiency was 0.98, no energy was wasted during the heating process.The optimal configuration pattern of the proposed system in terms of cold, heat, and electricity was obtained by calculating IFIP-REM.With four distinct seasons in the Guanzhong area, no requirement for heat or cold in spring and autumn were required, but it is necessary to provide heat and cold in summer and winter to improve residents' living conditions.The three possible patterns were discussed below.
Figure 4 depicts cold balance patterns for the DES on a typical summer day.During operation of the system, the cold load was mainly met by internal combustion engines, solar photovoltaic panels, and grid power conversion to absorption chillers.User's cold demand peaked between 13:00 and 20:00 with no fluctuations at other times.As user's energy demand decreases, the number of ground source heat pump units configured by the system was small.This occurred due to automatic system optimization.Through ground source heat pump and absorption chiller processes, electricity was converted into cold load and then supplied to users.In some periods, excess cold load was generated and stored by ice-storage devices.Since the energy storage efficiency was 0.65, the cold energy released by ice storage devices was lower than the stored cold energy, and no energy was wasted during the cold conversion process.Figure 5 represents the heat balance pattern for the DES on a typical winter day.The heat generated by biomass boilers and internal combustion engines was supplied to users after conversion by heat exchangers.Since the conversion efficiency was 0.98, no energy was wasted during the heating process.Figure 6 illustrates the electricity balance for a typical day of DES operation in the four seasons.The electricity consumption increased as residents used more appliances during the morning, day, and night.Before these periods, the system stored electricity in advance to reduce the exertion of power generation equipment.The storage efficiency of the battery was 0.85.Hence, the amount of electricity flowing into the batteries was also higher than that flowing out of batteries.Solar radiation intensity reached photovoltaic panel power generation requirements and could operate at full capacity.The electricity generated by photovoltaic panels was directly regulated and controlled by the controller and sent directly to DC and AC loads.When solar photovoltaic panels were excessively utilized at certain periods, electricity other than load power will temporarily be stored in the battery.When the system is short of electricity, the battery can be invoked as a power source to supply the load.Considering the limited capacity of the battery, the system was connected to the municipal grid to ensure normal load.On the other hand, the system stores electric energy during low-voltage periods and sells electricity to the grid during peak hours while ensuring self-use.This might increase the economics of the system while cutting peaks and filling valleys.As we can see in Figure 6, during spring and autumn, because of the lack of cold and heat load demand, only solar photovoltaic panels, storage batteries, and municipal power grids provide electricity for users in this system.The supply and demand are relatively balanced, and basically no power loss occurs.In summer without heat load, absorption chillers can use internal combustion.The refrigeration efficiency of absorption chillers is high, and the coefficient of performance value can reach 1.30, which ensures high system energy utilization efficiency.In the winter, we can see that there is an excess of electric energy.This is because the system needs to generate heat by using internal combustion generators to meet users' heat demands.Therefore, the power consumption of the system is increased by taking extra power from the grid.electricity flowing into the batteries was also higher than that flowing out of batteries.Solar radiation intensity reached photovoltaic panel power generation requirements and could operate at full capacity.The electricity generated by photovoltaic panels was directly regulated and controlled by the controller and sent directly to DC and AC loads.When solar photovoltaic panels were excessively utilized at certain periods, electricity other than load power will temporarily be stored in the battery.
When the system is short of electricity, the battery can be invoked as a power source to supply the load.Considering the limited capacity of the battery, the system was connected to the municipal grid to ensure normal load.On the other hand, the system stores electric energy during low-voltage periods and sells electricity to the grid during peak hours while ensuring self-use.This might increase the economics of the system while cutting peaks and filling valleys.As we can see in Figure 6, during spring and autumn, because of the lack of cold and heat load demand, only solar photovoltaic panels, storage batteries, and municipal power grids provide electricity for users in this system.The supply and demand are relatively balanced, and basically no power loss occurs.In summer without heat load, absorption chillers can use internal combustion.The refrigeration efficiency of absorption chillers is high, and the coefficient of performance value can reach 1.30, which ensures high system energy utilization efficiency.In the winter, we can see that there is an excess of electric energy.This is because the system needs to generate heat by using internal combustion generators to meet users' heat demands.Therefore, the power consumption of the system is increased by taking extra power from the grid.

Carbon Emission
Figure 7 depicts carbon emissions of the DES and CES during the four seasons.The DES showed better reduction in carbon emission than CES, and the carbon reduction rate is [22.73, 31.05]%.In the CES, carbon emissions in spring and autumn were less (4.90 × 10 3 and 6.83 × 10 3 kg, respectively).The largest values were obtained during winter ([9.45, 13.10] × 10 3 kg).In spring and autumn, users utilized no heating or cooling.Carbon discharge increased due to users' cold and heat loads in summer and winter.However, the carbon emission curve of the DES appeared relatively stable with discharge amounts fluctuating between 3.81× 10 3 kg and 5.98 × 10 3 kg in spring, summer, and autumn.Carbon discharge amounts rose sharply and discharge was recorded as [6.27, 10.06] × 10 3 kg in winter.Because heat generated by biomass boilers and internal combustion engines supplied users after conversion by heat exchangers, carbon discharge rose rapidly.The purchased electricity from the municipal grid by the DES and CES reached [12.49, 17.38] × 10 3 kWh and [23.80, 33.16] × 10 3 kWh, respectively.Hence, the DES purchased less electricity from the municipal grid to ensure the reliability of energy usage.The CES consumed more electricity because it required electricity for providing users with heat and cold loads, and thus consumed more power than the DES.Based on the latest coal consumption rates of 6 × 10 3 kW and above (326 g/kWh) published by the National Energy Administration, the DES reduced the burning of standard coal by

Carbon Emission
Figure 7 depicts carbon emissions of the DES and CES during the four seasons.The DES showed better reduction in carbon emission than CES, and the carbon reduction rate is [22.73, 31.05]%.In the CES, carbon emissions in spring and autumn were less (4.90 × 10 3 and 6.83 × 10 3 kg, respectively).The largest values were obtained during winter ([9.45, 13.10] × 10 3 kg).In spring and autumn, users utilized no heating or cooling.Carbon discharge increased due to users' cold and heat loads in summer and winter.However, the carbon emission curve of the DES appeared relatively stable with discharge amounts fluctuating between 3.81× 10 3 kg and 5.98 × 10 3 kg in spring, summer, and autumn.Carbon discharge amounts rose sharply and discharge was recorded as [6.27, 10.06] × 10 3 kg in winter.Because heat generated by biomass boilers and internal combustion engines supplied users after conversion by heat exchangers, carbon discharge rose rapidly.

Carbon Emission
Figure 7 depicts carbon emissions of the DES and CES during the four seasons.The DES showed better reduction in carbon emission than CES, and the carbon reduction rate is [22.73, 31.05]%.In the CES, carbon emissions in spring and autumn were less (4.90 × 10 3 and 6.83 × 10 3 kg, respectively).The largest values were obtained during winter ([9.45, 13.10] × 10 3 kg).In spring and autumn, users utilized no heating or cooling.Carbon discharge increased due to users' cold and heat loads in summer and winter.However, the carbon emission curve of the DES appeared relatively stable with discharge amounts fluctuating between 3.81× 10 3 kg and 5.98 × 10 3 kg in spring, summer, and autumn.Carbon discharge amounts rose sharply and discharge was recorded as [6.27, 10.06] × 10 3 kg in winter.Because heat generated by biomass boilers and internal combustion engines supplied users after conversion by heat exchangers, carbon discharge rose rapidly.The purchased electricity from the municipal grid by the DES and CES reached [12.49, 17.38] × 10 3 kWh and [23.80, 33.16] × 10 3 kWh, respectively.Hence, the DES purchased less electricity from the municipal grid to ensure the reliability of energy usage.The CES consumed more electricity because it required electricity for providing users with heat and cold loads, and thus consumed more power than the DES.Based on the latest coal consumption rates of 6 × 10 3 kW and above (326 g/kWh) published by the National Energy Administration, the DES reduced the burning of standard coal by The purchased electricity from the municipal grid by the DES and CES reached [12.49, 17.38] × 10 3 kWh and [23.80, 33.16] × 10 3 kWh, respectively.Hence, the DES purchased less electricity from the municipal grid to ensure the reliability of energy usage.The CES consumed more electricity because it required electricity for providing users with heat and cold loads, and thus consumed more power than the DES.Based on the latest coal consumption rates of 6 × 10 3 kW and above (326 g/kWh) published by the National Energy Administration, the DES reduced the burning of standard coal by [3687.06,5144.28]kg per year through clean energy use.Relevant data suggested that natural gas power generation had obvious advantages over coal-fired power in terms of reducing carbon emissions.

System Cost
The purpose of this study is to minimize the system cost based on optimized results.By optimizing the configuration of the proposed system, total system costs in terms of equipment, operation, fuel, procurement, and subsidies should be minimized.Figure 8 presents the composition of the system cost's lower and upper bounds under optimal design circumstances.The total cost of this system hovered between RMB 24.82 × 10 3 and 39.85 × 10 3 .Equipment, fuel, and electricity purchase costs accounted for the majority (RMB [12.92, 16.70] × 10 3 , [6.30, 12.65] × 10 3 and [6.22, 8.66] × 10 3 , respectively).In addition, the cost of renewable energy equipment accounted for [19.82, 63.95] % of total equipment cost.The high price of renewable energy equipment coupled with the lack of subsidies related to DES policies led to the inability of the system to prioritize renewable energy equipment during optimization.The use of renewable energy DES in rural areas would not only solve the daily power needs of thousands of people, but also maximize environmental protection.
[3687.06, 5144.28]kg per year through clean energy use.Relevant data suggested that natural gas power generation had obvious advantages over coal-fired power in terms of reducing carbon emissions.

System Cost
The purpose of this study is to minimize the system cost based on optimized results.By optimizing the configuration of the proposed system, total system costs in terms of equipment, operation, fuel, procurement, and subsidies should be minimized.Figure 8 presents the composition of the system cost's lower and upper bounds under optimal design circumstances.The total cost of this system hovered between RMB 24.82 × 10 3 and 39.85 × 10 3 .Equipment, fuel, and electricity purchase costs accounted for the majority (RMB [12.92, 16.70] × 10 3 , [6.30, 12.65] × 10 3 and [6.22, 8.66] × 10 3 , respectively).In addition, the cost of renewable energy equipment accounted for [19.82, 63.95] % of total equipment cost.The high price of renewable energy equipment coupled with the lack of subsidies related to DES policies led to the inability of the system to prioritize renewable energy equipment during optimization.The use of renewable energy DES in rural areas would not only solve the daily power needs of thousands of people, but also maximize environmental protection.

Comparison of DES Model Results Using Linear Programming (LP) and IFIP Method
Figure 9 shows the composition of system cost under optimal design conditions after optimizing DES in Guanzhong rural area of China using the LP and IFIP methods.As seen from Figure 9, equipment cost and fuel cost fluctuate in the optimization process.Combined with the above resultbased analysis, it can be indirectly concluded that under the constraints of carbon emissions, in order to meet the demand for energy, the system uses renewable energy equipment for energy supply, which leads to a sharp increase in the equipment cost of this system.The carbon emission reduction effect of the distributed energy system is obvious.Compared with CES, the carbon emission reduction rate optimized by the LP method is 28.50%, while that by IFIP method is [22.71, 31.05]%. Results calculated by the LP method are just within the range of the results calculated by the IFIP method, indicating that the IFIP method can reflect the problems in the process of system optimization more comprehensively and completely than the LP method.IFIP addresses multiple uncertainties in costs, impact factors, and system objectives (decisions, crisp interval and function interval).This model inherits the advantages of LP and full-infinite programming and can directly reflect the objective function and uncertainty in the optimization process and solution.

Comparison of DES Model Results Using Linear Programming (LP) and IFIP Method
Figure 9 shows the composition of system cost under optimal design conditions after optimizing DES in Guanzhong rural area of China using the LP and IFIP methods.As seen from Figure 9, equipment cost and fuel cost fluctuate in the optimization process.Combined with the above result-based analysis, it can be indirectly concluded that under the constraints of carbon emissions, in order to meet the demand for energy, the system uses renewable energy equipment for energy supply, which leads to a sharp increase in the equipment cost of this system.The carbon emission reduction effect of the distributed energy system is obvious.Compared with CES, the carbon emission reduction rate optimized by the LP method is 28.50%, while that by IFIP method is [22.71, 31.05]%. Results calculated by the LP method are just within the range of the results calculated by the IFIP method, indicating that the IFIP method can reflect the problems in the process of system optimization more comprehensively and completely than the LP method.IFIP addresses multiple uncertainties in costs, impact factors, and system objectives (decisions, crisp interval and function interval).This model inherits the advantages of LP and full-infinite programming and can directly reflect the objective function and uncertainty in the optimization process and solution.

Conclusions
An interval full-infinite programming rural energy model (IFIP-REM) was developed and applied to the distributed energy system (DES) in a rural area.IFIP-REM was applied to Guanzhong to minimize the system cost, meet the high energy demand, achieve better system equipment capacity, moderate equipment cost, and elevate efficiency.The optimal allocation model of the DES considered the abundant renewable energy in rural areas.
Comparison between the traditional centralized energy system and distributed energy system indicated that the allocation pattern of the DES in Guanzhong could be defined according to IFIP-REM calculations.On the other hand, the DES could reduce electricity purchasing from the municipal grid by 47.5% when compared to the CES.The upper and lower bounds of reduction in carbon emissions of the DES reached [17.13, 44.51] % and [12.42, 36.02]%, respectively.The DES was found to be suitable for rural areas rich in renewable energies.The government should introduce relevant laws and policies to support the construction of new energy conversion sources and invest in new projects.In the long run, this should enhance economic and environmental benefits.These findings would help decision makers in discussing alternative energy supplies, renewable energy matching, system cost, and carbon emission mitigation.
For the first time, we attempted to apply the IFIP method to the optimal design of a rural DES considering the increasingly complex and volatile situation in Guanzhong.However, this method still has space for improvement.A major limitation of this model consisted of only using functional intervals linearly related to its factors.Nonlinearities in the DES optimization process exist, so it is necessary to investigate these systems in more depth.Coping with IFIP problems and multiple independent variables will be the focus of future research.

Figure 2 .Figure 2 .
Figure 2. The framework of the interval full-infinite programming rural energy model (IFIP-REM) for Guanzhong's DES.

Figure 4 .
Figure 4. Cold balance pattern for a typical summer day.

Figure 5 .
Figure 5. Heat balance pattern for a typical winter day.

Figure 6
Figure6illustrates the electricity balance for a typical day of DES operation in the four seasons.The electricity consumption increased as residents used more appliances during the morning, day, and night.Before these periods, the system stored electricity in advance to reduce the exertion of power generation equipment.The storage efficiency of the battery was 0.85.Hence, the amount of

Figure 4 .
Figure 4. Cold balance pattern for a typical summer day.

Figure 4 .
Figure 4. Cold balance pattern for a typical summer day.

Figure 5 .
Figure 5. Heat balance pattern for a typical winter day.

Figure 6 Figure 5 .
Figure6illustrates the electricity balance for a typical day of DES operation in the four seasons.The electricity consumption increased as residents used more appliances during the morning, day, and night.Before these periods, the system stored electricity in advance to reduce the exertion of power generation equipment.The storage efficiency of the battery was 0.85.Hence, the amount of

Figure 6 .
Figure 6.Electricity balance pattern for a typical day.

Figure 7 .
Figure 7.Comparison of systems carbon discharge.

Figure 6 .
Figure 6.Electricity balance pattern for a typical day.

Figure 6 .
Figure 6.Electricity balance pattern for a typical day.

Figure 7 .
Figure 7.Comparison of systems carbon discharge.

Figure 7 .
Figure 7.Comparison of systems carbon discharge.

Figure 9 .
Figure 9. System cost composition obtained by the linear programming (LP) and IFIP methods.

Table 5 .
Description of the energy systems.

Table 1 .
Hourly operation table of household electrical appliances in spring or autumn.

Table 2 .
Hourly operation table of household electrical appliances in summer.

Table 3 .
Hourly operation table of household electrical appliances in winter.
Notes: Numbers in this table are the number of operating pieces equipment at each time.

Table 4 .
Summary of equipment parameters.
Notes: L, R, and D represent cold, heat, and electricity, respectively.α is energy production equipment depreciation rate.β is energy conversion/storage equipment depreciation rate.ϕ is energy production equipment discount rate.τ is energy conversion/storage equipment discount rate.

Table 5 .
Description of the energy systems.