Multi-objective capacity optimization of a distributed energy system considering economy, environment and energy

Energy Conversion and Management 200 (2019) 112081

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Multi-objective capacity optimization of a distributed energy system considering economy, environment and energy

T

Zhengyi Luo, Sheng Yang, Nan Xie, Weiwei Xie, Jiaxing Liu, Yawovi Souley Agbodjan, ⁎ Zhiqiang Liu School of Energy Science and Engineering, Central South University, Changsha 410083, People’s Republic of China

A R T I C LE I N FO

A B S T R A C T

Keywords: Distributed energy system Multi-objective optimization Optimal equipment capacity Energy management strategy NSGA-II Decision making

With the climate change and depletion of fossil energy, distributed energy systems (DESs) have attracted widespread attention. In this study, a DES driven by solar, geothermal, aerothermal, natural gas and power grid is constructed with energy conversion devices modeled based on part load performance. A novel operation strategy for the DES is presented considering the complementary characteristics of different energy sources. Besides, a multi-objective nonlinear optimization model for the device capacity is proposed with economic, environmental and energy objectives considered simultaneously. To solve the optimization model, an integrated solution method combining Non-dominated Sorting Genetic Algorithm-II, Technique for Order Preference by Similarity to an Ideal Solution and Shannon entropy approach is developed. A case study of an indoor swimming pool in Changsha city of China is undertaken. Optimal equipment capacity and corresponding energy management strategies of the case are obtained. The final number and capacity of air source heat pump (ASHP) are determined via improving its part load ratio. Additionally, three schemes are set to investigate the effects of constant efficiency/COP of energy conversion devices and operation strategies on the capacity optimization of DESs. Results indicate that constant efficiency/COP of equipment yields an 11.7% drop in annual total cost (ATC), a 10.4% increment in annual total CO2 emission (ATE) and a 12.5% reduction in coefficient of energy performance (CEP). ATC and ATE of the optimal solution acquired under a conventional operation strategy increase by 6.8% and 3.7%, while CEP decreases by 66.9%. This work provides a guidance for the future application of DESs.

1. Introduction Nowadays, energy crisis and global warming are becoming urgent issues around the world [1]. In China, renewable energy such as solar energy and geothermal energy is playing a more and more significant role in energy conservation and emission reduction [2]. As an important way to utilize renewable energy, distributed energy systems (DESs) have attracted attention worldwide recently [3]. A DES can be described as a multi-input and multi-output energy system, including diverse small-scale technologies incorporating traditional techniques, renewable ones and storage units [4]. And it is capable of providing electricity, cooling and heating to end-users at the same time [5]. Though DESs have the strengths of high overall energy efficiency, low emissions and high reliability [6], the systems are difficult to design, because many aspects need to be considered. Especially, it is a major challenge to determine the proper equipment capacity of a DES, which significantly affects the economy, environment and energy performance

⁎

of the entire system [7]. Considerable research efforts have been devoted to the capacity optimization of DESs. Previous studies primarily focused on the four aspects, including energy device model, operation strategy, optimization model and solution method. Energy equipment model is the basis of the capacity optimization of DESs. In many publications [8,9], the constant efficiency/COP of equipment is adopted to characterize the relationship between energy input and output of devices without considering part load performance. Zhou et al. [10] proposed a superstructure of a DES to optimize the equipment capacity. The device models of the DES are established by using constant efficiency/COP. Wei et al. [11] performed a multi-objective optimization of a DES for the device capacity. Energy equipment was modeled with the assumption of constant device efficiency/COP. However, equipment efficiency cannot keep constant in actual operation and would change with operation conditions [12]. Some literature [13,14] clarified how part load performance of devices affect the operation optimization of

Corresponding author. E-mail address: [email protected] (Z. Liu).

https://doi.org/10.1016/j.enconman.2019.112081 Received 20 June 2019; Received in revised form 14 September 2019; Accepted 16 September 2019 Available online 19 September 2019 0196-8904/ © 2019 Elsevier Ltd. All rights reserved.

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Nomenclature Pk,d Pkin, i, t ,t Pkout ,i Ekt , i αkmax ,c αkmax , di βkmin βkmax A L H G T AM F U fαβ N M Z c* n Cn

s g ae dc dh hw e k d c di min max α β

design capacity of device k (kW or kWh) input power of device k at time slot t (kW) output energy of device k at time slot t (kW) energy state of energy storage device at time slot t (kWh) maximal charging ratio of energy storage device k maximal discharging ratio of energy storage device k minimal stored energy ratio of energy storage device k maximum stored energy ratio of energy storage device k area (m2) load (kWh) amount of consumed energy solar irradiation (W/m2) temperature (°C) air mass (kg) heat removal factor heat loss coefficient value of objective function population size evolution generation Euclidian distance relative proximity number of air source heat pump with heating capacity of n th air source heat pump with heating

Abbreviations DES CCHP NSGA-II TOPSIS REDC ATC ATE CEP GB PV SC ASHP APC APH GSHP GPC GPH HE ES TS COP PLF PLR FTL FEL

Greek symbols κ τ η c λ σ θ ϑ νβ+/ νβ−

absorptivity ratio transmissivity ratio efficiency unit capital cost ($/kW or $/kWh) unit price of energy ($/kWh or $/m3) unit maintenance cost ($/kW) CO2 emission factor (kgCO2/kWh) weight positive ideal/negative ideal solutions

Superscript/subscripts in out t 0 pg ng

solar energy geothermal energy aerothermal energy from air district cooling district heating hot water electricity device design condition charge discharge minimum value maximum value the α th design plan the β th objective

input output time reference state power grid natural gas

distributed energy system combined cooling, heating, and power system Non-dominated Sorting Genetic Algorithm-II Technique for Order Preference by Similarity to an Ideal Solution Renewable Energy Demonstration Center annual total costs annual total CO2 emissions coefficient of energy performance gas boiler photovoltaic solar thermal collector air source heat pump air source heat pump with cooling air source heat pump with heating ground source heat pump ground source heat pump with cooling ground source heat pump with heating heat exchanger electricity storage thermal storage coefficient of performance part load factor part load ratio following thermal load following electricity load

different energy sources is necessary for DESs. Additionally, as the most critical part of the optimization model, objective functions are promising to optimize the device capacity of DESs as well. Previous studies focused on two primary types of objectives: economy and environment. The minimization of the annual total costs for capital, energy and maintenance is the top priority in the capacity optimization of DESs [21,22]. Emissions of greenhouse gas, especially CO2, are also the focus of attention in several studies [23,24]. However, the system energy performance that needs to be concerned in the context of energy crisis was overlooked. After optimization model is established, solution method is indispensable to solve the model. Wu et al. [25] formulated the objectives as a weighted sum of the minimum total costs and carbon emissions to obtain the optimal devices capacity of a DES. To optimize equipment capacity of a DES, Eriksson et al. [26] established an overall normalized multi-objective function through constant weights. Generally, in most previous literature, multiple objectives have been transformed into a single objective by adopting a weighted sum of

DESs. Unfortunately, few studies have been conducted to explore the effect of equipment part load performance on the capacity optimization of DESs. Moreover, operation strategy is also of great importance for the capacity optimization of DESs. The operation strategies such as following electricity load (FEL) [15] and following thermal load (FTL) [16], have garnered extensive attention. Li et al. [17] optimized the capacity of the prime mover of a biomass gasification-integrated combined cooling, heating and power system (CCHP) under different operation strategies, including FEL and FTL. Wang et al. [18] optimized the device capacity of a DES integrated with a CCHP under the FTL operation strategy. The optimal design plan and corresponding energy management strategy were obtained. Whereas, the operation strategies mentioned above are not applicable to the DESs that do not comprise power generation units, for instance internal combustion engines. Research has shown that different types of energy can be combined to compensate for the deficiency of each type energy [19,20]. The operation strategy considering the complementary characteristics of

2

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2. Methods

objective functions. Weights are usually assumed to be constants. However, there are generally conflict between different objectives [27]. Transforming multiple objectives into a single objective by using a weighted sum of each objective may not match the actual situation. And the value of weight has a great impact on the optimal equipment capacity [25,28]. Based on the above literature review, it can be seen that the existing research has already involved the capacity optimization of DESs. However, some problems are still not addressed. Firstly, energy devices of DESs are modeled with low accuracy. Secondly, the operation strategies for DESs considering the complementary coordination characteristics between different types of energy are rarely investigated yet. Thirdly, optimization objectives primarily considers the economic and environmental performance of DESs. Energy performance is involved in few publications. Finally, the present solution method to tackle multiobjective optimization problems is mainly to convert multiple objectives into a single objective by employing a weighted sum of objective functions without considering the confliction between different objectives. Therefore, the main purpose of this study is to perform a multiobjective capacity optimization of a DES. The novelties of this work are as follows. The energy conversion devices of the constructed DES are modeled with part load performance considered, which improves the accuracy of the models. A novel operation strategy is put forward based on the complementary coordination characteristics between different types of energy. A multi-objective nonlinear capacity optimization model is proposed considering economic, environmental and energy aspects simultaneously. Besides, a comprehensive solution method combining Non-dominated Sorting Genetic Algorithm-II (NSGA-II), Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS) and Shannon entropy approach is developed, where the conflictions between various objectives are fully considered.

An optimization framework is proposed for the multi-objective capacity optimization of DESs. As shown in Fig. 1, the framework is divided into four parts: (1) pre-process inputs, (2) distributed energy system, (3) optimization model, (4) solution method. Given the required parameters, the final design plan of a DES can be obtained by solving the optimization model with an integrated solution method. 2.1. Distributed energy system Generally, a DES consists of four parts, which are energy input, energy conversion section, energy storage section along with energy output, respectively. As presented in Fig. 2, a DES is modeled in this paper. The DES takes electricity from power grid, aerothermal energy from air, geothermal, solar and natural gas as energy sources to provide electricity, district cooling, district heating and hot water to users. In the DES, photovoltaic (PV) is employed to generate electricity, air source heat pump (ASHP) and ground source heat pump (GSHP) for producing district cooling, gas boiler (GB), solar thermal collector (SC), GSHP and ASHP for providing heat including district heating and hot water, and electricity storage (ES) as well as thermal storage (TS) for storing energy. 2.1.1. Energy device model In this paper, energy conversion equipment models consisting of PV, SC, GB, GSHP and ASHP are established considering the part load performance of devices. Especially, a constant efficiency of 80% is adopted to heat exchanger (HE) [28]. In terms of energy storage devices, constant conversion efficiencies are assumed to simplify the actual models.

Fig. 1. The capacity optimization framework of DESs. 3

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Fig. 2. The structure of the DES. in in Pkout , j = ηk Pk , i , COPk Pk , i

2.1.1.1. Thermodynamic-based model 2.1.1.1.1. PV panels. According to solar irradiation, PV areas and generation efficiency, the output electricity generated by PV panels, out Ppv , e , can be calculated as follows [29]: out Ppv , e = GAPV ηPV

where i and j are the indexes of different energy types. Pkout , j represents the output with type j of device k, Pkin, i denotes the input energy carrier i of device k. ηk and COPk are efficiency and coefficient performance of device k, respectively. Besides, efficiency/COP of a device depends on operation conditions, nominal or part load, which can be determined as a function of part load ratio (PLR) and PLF, defined as follows [22,32]:

(1)

where G is solar irradiation, APV is the area of the installed PV panels, ηPV is PV generation efficiency, which is adopted:

G ηPV = a1 ⎡ ⎛ ⎞ ⎢ ⎝ G0 ⎠ ⎣ ⎜

⎟

a2

G T AM ⎞ ⎤ + a3 ⎛ ⎞ ⎤ ⎡1 + a4 ⎛ ⎞ + a5 ⎛ ⎢ ⎥ ⎥ G T AM 0 0 0 ⎠⎦ ⎝ ⎠ ⎝ ⎝ ⎠⎦⎣ ⎜

⎟

⎜

⎟

⎜

(4)

⎟

PLR =

Pk Pk, d

(5)

PLF =

ηk COPk , ηk, d COPk, d

(6)

(2)

where G0, T0, and AM0 are solar irradiation, cell temperature and air mass at standard test conditions (irradiation 1000 W/m2, cell temperature 25 °C air mass 1.5 kg), T is ambient temperature, AM is air mass, a1, a2, a3, a4, and a5 are constants, which are 0.28204, 0.39668, −0.44730, −0.092864, 0.016010, respectively. 2.1.1.1.2. Solar collector. Similar to PV, the heating generated by SC can be estimated based on solar irradiation, solar panels area and generation efficiency. The generation efficiency of SC can be expressed as [29]:

where Pk, ηk and COPk denote the output energy, efficiency and coefficient performance of device k at partial load operation condition, respectively. Pk, d, ηk, d and COPk, d denote the output energy, efficiency and coefficient performance of device k at nominal operation condition, respectively. ηk, d and COPk, d assumed as constants are known. Additionally, PLF is a function of PLR, defined as Eq. (7) [32].

PLF = f (PLR) ηsc = F (τκ ) −

FU (Tin − T ) G

(7)

(3) 2.1.1.3. Energy storage device model. Energy storage devices in the DES are ES and TS. An energy storage device can be regarded as a load when it charges. On the contrary, it can be considered as an energy source when it discharges. Energy storage models can be expressed with Eq. (8). It indicates that the total amount of energy stored at the beginning of each time equals to the one at the beginning of previous time plus net energy flow during the time interval (the quantity of stored energy minus the one released to satisfy user energy demands).

where F is heat removal factor, (τκ) is the efficient fraction of the incident solar energy ultimately absorbed by absorber plate (τ is transmissivity, and κ is absorptivity), U is the whole heat loss coefficient, Tin and T are the inlet water flow and ambient temperatures, respectively. 2.1.1.2. Correlation-based model. In this paper, the models of GB, ASHP and GSHP are built with the “part load factor” (PLF) method. PLF, defined as the energy efficiency/COP at part load divided by the fullload energy efficiency/COP, is a correction coefficient applied to the steady-state performance of device (0 ≤ PLF ≤ 1) [30]. This method is later adopted in the ASHRAE standard [31], which is also widely employed to evaluate the device performance at part load and has proved valid in previous studies [12,32,33]. The output-input coupling relationship of GB, ASHP and GSHP can be formulated as:

di ,t Ekt +, i 1 = Ekt , i + (ηkc, i Pkin, i, t − Pkout , i / ηk , i )

(8)

ηkc, i

ηkdi, i

and represent the where i is the index of energy type. charging efficiency and discharging efficiency of energy storage ,t equipment k, respectively. Pkin, i, t and Pkout are charging and discharging ,i energy of energy storage device k, respectively. Ekt +, i 1 and Ekt +, i 1 are the amount of stored energy of energy storage equipment k at time slots t and t + 1, respectively. 4

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water load. If hot water load could be satisfied by SC, the unconsumed hot water produced by SC and the heating discharged by TS are used to satisfy district heating demand which would be covered by GPH, APH and GB when the heating provided by SC and TS is not enough to furnish hot water load. Otherwise, the district heating and surplus hot water load would be met by TS, GPH, APH and GB. If surplus hot water load could be covered by TS, TS would continue discharging to provide district heating. District heating load would be firstly supplied by GPH when the heating from TS is not enough, then met by APH when capacity of GPH is not adequate. It is satisfied by GB at the time of capacity of APH still is insufficient. If surplus hot water load could not be covered by TS, district heating would be furnished by GPH, APH and GB, of which the free capacity is utilized to supply surplus hot water load.

2.1.2. Operation strategy In this paper, an operation strategy for the constructed DES is proposed on account of the complementary coordination characteristics among electricity purchased from power grid, natural gas, solar, geothermal and aerothermal energy. The operation strategy is presented in Fig. 3, programmed and implemented in MATLAB. And the operation constraints to ensure that the system operates normally will be introduced in Section 2.2.3. 2.1.2.1. Electricity supply strategy. The electricity generated by PV is primarily transported to users, and the excessive electricity is stored in ES. ES discharges to supplement the shortage at the time of the power produced by PV is not enough to meet electricity demand, and the lack of electricity would be purchased from power grid. 2.1.2.2. District cooling supply strategy. District cooling supply strategy is relatively simple. District cooling load is firstly supplied by GSHP, then the remainder is covered by ASHP when the cooling provided by GSHP is insufficient to satisfy district cooling demand. Besides, the free capacity of GSHP and ASHP is utilized to meet hot water load when the hot water furnished by SC and TS cannot satisfy hot water demand.

2.2. Optimization model A multi-objective nonlinear capacity optimization model is established considering economic, environmental and energy benefits. The formulation of the optimization model composed of decision variables, objective functions as well as constraints has been implemented, which will be described in detail in the following subsections.

2.1.2.3. Hot water and district heating supply strategy. Similar to electricity supply strategy, hot water and district heating loads are firstly satisfied by SC. If the heating provided by SC is sufficient, the excessive hot water would be stored in TS. When SC could not meet hot water and district heating demands, SC is firstly used to supply hot

2.2.1. Decision variables In this paper, decision variables are the design capacity of energy devices, which are continuous variables, presented below:

Fig. 3. The complementary operation strategy for the DES. 5

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follows: T

CE =

∑ λpg Hpgt + λng Hngt t=1

(12)

where λpg and λng are the unit price of electricity purchased from t t power grid and natural gas. Hpg and Hng denote the consumption of electricity purchased from power grid and natural gas at time t. 2.2.2.1.3. The maintenance cost CM. The maintenance cost can be formulated as: T

CM =

,t ∑ ∑ σk Pkout ,j t=1

(13)

where σk and represent the unit maintenance cost and the output energy of device k, respectively. ,t Pkout ,j

2.2.2.2. Environmental objective. Greenhouse gas emissions are a growing concern with global warming. The environmental objective is to minimize the emissions of greenhouse gas, especially CO2. Annual total CO2 emissions, ATE, of the DES is selected as the environmental objective function in this paper, expressed as follows: T

min ATE =

∑ (θpg Hpgt + θng Hngt ) t=1

(14)

where θpg and θng are 0.968 kgCO2/kWh, 0.22 kgCO2/kWh [24], representing the equivalent emission coefficient of electricity purchased from power grid and natural gas, respectively. 2.2.2.3. Energy objective. Energy performance is also a promising index which can’t be ignored when designing a DES. The energy objective is to maximize the overall coefficient of energy performance, CEP, of the DES defined as the ratio of the annual total energy output E out to the annual total non-renewable energy input E in .

max CEP =

E out E in

(15)

At the supply side, annual total non-renewable energy input is the sum of electricity purchased from power grid and natural gas consumed in the whole year, evaluated as:

Fig. 4. The process of the solution method.

T

E in =

X = [APV , d , ASC , d , PGB, d, PAPC, d, PAPH , d, PGPC , d, PGPH , d, PHE , d, PTS, d, PES, d]

t=1

(9)

T

E out =

(17)

2.2.3. Constraints Regarding the capacity optimization problem of a DES, constraints primarily include the energy balance restraints in the net, the limits to the designed capacity of energy devices and equipment operation constraints [24]. The details are described as follows.

(10)

2.2.3.1. Energy balance constraints. Electricity, district cooling, district heating and hot water balance in the DES are described in the following, which consists of both “input balance” and “output balance”, evaluated as Eqs. (18) and (19), respectively. In Eq. (18), the left items represent the sum of input rate of energy carrier i of diverse devices at time slot t, and the right side denotes the input to the DES with the energy carrier i at time slot t. In terms of the Eq. (19), the left side and the right side denote the sum of rate of output type l of all corresponding equipment and the total demand l of user at time slot t, respectively.

2.2.2.1.1. The capital cost CC. The annual capital cost can be defined as [24,32]:

r (r + 1) yk

∑ (r + 1) yk − 1 ck Pk,d

t ) ∑ (Let + Ldct + Ldht + Lhw t=1

2.2.2.1. Economic objective. Economy has always been a concern, significantly related to the interests of users. The economic objective is to minimize the annual total costs, ATC, of the DES, represented as the sum of capital cost CC, fuel cost CE and maintenance cost CM:

CC =

(16)

At the demand side, annual total energy output is the sum of electricity, district cooling, district heating and hot water demands in the whole year, expressed as:

2.2.2. Objective functions In this study, the performance of the DES in economic, environmental and energy aspects is investigated. The aims are to decrease annual total costs, annual total CO2 emissions as well as the usage of non-renewable energy.

min ATC = CC + CE + CM

∑ (Hpgt + Hngt )

(11)

where k denotes energy device; r represents the interest rate which is specified as 6% in this paper [34]; y is the lifetime; ck denotes the specified capital cost; and Pk, d is the design capacity of device k. 2.2.2.1.2. The energy cost CE. The energy cost is computed by multiplying the unit price of energy with energy consumption as 6

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∑ Pkin,i,t = Hit , ,t = Llt , ∑ Pkout ,l

∀ i ∈ {pg , ns, g , s, ae}

(18)

∀ l ∈ {e,dc,dh,hw}

(19)

max in, t ⎧ 0 ≤ Pk, j ≤ αk, c Pk, d/3600 ⎪ ,t ≤ αkmax 0 ≤ Pkout , di Pk , d /3600 ,j ⎨ min ⎪ βk Pk, d ≤ Ekt , i ≤ βkmax Pk, d ⎩

αkmax ,c

(24)

αkmax , di

2.2.3.2. Design constraints. The design capacity of energy device k in the DES, Pk, d, should keep within the minimum and maximum capacity, which can be defined as [24,35]:

and denote the maximum charging and discharwhere ging ratio of energy storage equipment k, respectively. βkmin and βkmax represent the minimal and maximal stored energy ratio of energy storage equipment k, respectively.

max Pkmin , d ≤ Pk , d ≤ Pk , d

2.3. Solution method

Pkmin ,d

(20)

Pkmax ,d

and represent the minimum and maximum capacity of where device k. Particularly, for SC and PV, another constraint that has to be satisfied is that the sum of design installed area of SC and PV is lower than the maximum available installed area, Amax, which is related to the available installed area of design object for SC and PV. Similar to SC and PV, The design capacity of GSHP is also limited by the maximum max available capacity, PGSHP , due to the restriction of available space for the installation of underground heat exchangers. These constraints can be formulated as:

0 ≤ ASC, d + APV , d ≤ Amax

(21)

max 0 ≤ PGSHP, d ≤ PGSHP

(22)

In general, for multi-objective optimization problems, a solution that satisfies all constraints and enables all objectives to achieve global optimality may not exist, due to the conflicting nature of various objectives. Usually, Pareto solution set containing many optimum solutions is obtained by solving multi-objective optimization models, in which each solution denotes a compromise considering different objectives. The final optimal solution can be picked out from Pareto solution set by some specific decision-making approaches. As presented in Fig. 4, an integrated solution method, decomposed into two stages, is developed to solve the optimization model. In the optimization stage, NSGA-II [36] is selected as the optimization tool to acquire Pareto solution set owing to its high running speed and good solving convergence [37]. As one of the most popular intelligence algorithms, NSGA-II was widely applied to multi-objective optimizations [38], which depends on biological evolution and employs three fundamental operators to search for the global optimal solutions. The detailed processes of NSGA-II are introduced as follows [36] and the procedures are programmed and implemented in MATLAB.

2.2.3.3. Operating constraints. The output energy of energy conversion device, Pkout , j , would not exceed their design capacity during operating. ,t 0 ≤ Pkout ≤ Pk, d ,j

(23)

• Initialization: the parameters, including population size N, evolution

In terms of energy storage equipment, the output has to be within the charging and discharging energy capacity. Furthermore, the energy stored at time slot t should stay within permitted range. These constraints can be evaluated as:

generation M, crossover probability and mutual probability, are set up. The initial population, Pm, containing N chromosomes is randomly created under constraints. Then the objective functions of

Fig. 5. Energy demands of REDC during the opening time on four seasonal typical days. 7

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Fig. 6. Solar radiation and ambient temperature on four seasonal typical days.

approach is applied to assign weights for each objective of the optimization model. TOPSIS is a primarily useful decision-making approach with simple, direct and effective features [39]. It finds out the optimal solution according to the relative proximities of each solution in Pareto solution set computed by the distance from positive ideal solution and the distance from the negative ideal solution [40]. The solution with the maximal relative proximity is picked out as the final optimal solution. Shannon entropy is an impersonal means to compute weights based on the objective message without the preferences of the decision makers [29]. An objective with small entropy value signifies the objective is vital and has a large weight [23]. TOPSIS method comprises the following processes [39] and the procedures are programmed and implemented in MATLAB.

Table 1 The economic parameters of equipment. Equipment

Capital cost

Maintenance cost

Life time

GB [24] PV [9] SC [9] GSHP [10] ASHP [10] HE [24] ES [24] TS [24]

123.7 $/kW 364.5 $/m2 243.0 $/m2 436.0 $/kW 174.4 $/kW 880.8 $/m2 281.0 $/kWh 14.0 $/kWh

0.0003 $/kWh 0.0015 $/kWh 0.0015 $/kWh 0.0014 $/kWh 0.0014 $/kWh 0.0003 $/kWh 0.0291 $/kWh 0.0063 $/kWh

20.0 years 15.0 years 30.0 years 20.0 years 20.0 years 20.0 years 13.5 years 20.0 years

each individual are calculated.

• Non-dominated sorting and crowding distance calculation: the fast • • • • • •

• Normalized decision matrix construction

non-dominated sorting is implemented for the population Pm, and the rank of each individual is assigned. Then the crowding distance of each individual is computed. Tournament selection: the approach of tournament selection is employed. Two individuals are randomly selected from population Pm at a time, of which the individual with top ranking and high crowding degree is picked out. Genetic operators: simulated binary crossover and polynomial mutation operators are used for genetic operation to produce the new offspring, Qm. Recombination: population Rm is created via combining the father population Pm with offspring population Qm. Non-dominated sorting and crowding distance calculation is performed for the population Rm. Produce new generation population: select the top N individuals from Rm as the new generation population Pm+1 based on their rank and crowding distance. Judge whether the terminal condition is met. If terminal condition is satisfied, output Pareto solution set. Otherwise, turn to step 3 (Tournament selection) and repeat until terminal condition is met.

fαβ

rαβ =

N 2 ∑α = 1 fαβ

,

α = 1, 2, ⋯, N ;

β = 1, 2, 3 (25)

where fαβ is the member of decision matrix denoting the β th objective value of the α th optimum solution. N is the total number of optimum solutions in Pareto solution set, which equals to population size. rαβ represents the element of normalized decision matrix.

• Weights allocation based on Shannon entropy approach [23] Oβ = −

ϑβ =

1 ln N

N

fαβ N 2 α = 1 ∑α = 1 fαβ ⎝ ⎛

∑⎜

ln

fαβ N 2 ∑α = 1 fαβ

⎞ ⎟ ⎠

(26)

1 − Oβ 3

3 − ∑β = 1 Oβ

(27)

• Weighted normalized decision matrix creation vαβ = rαβ ϑβ

(28)

• Positive ideal (v ) and negative ideal (v ) solutions determination

In the decision-making stage, a decision-making method combining TOPSIS with Shannon entropy is proposed, where Shannon entropy

+ β

− β

Table 2 The technical parameters of correlation-based model [10,12,32,33]. Device

Type

PLF

GB

Stander gas boiler

GSHP

Ground source heat pump with vertical single U tube heat exchanger

PLF = 0.8337 + 0.1967PLR −

ASHP

Variable-speed air-to-water heat pumps

Cooling: PLF = PLR/(0.83PLR + 0.17) Heating: PLF = PLR/(0.73PLR + 0.27)

PLF =

8

Design efficiency or COP −0.0756 + 1.397 × PLR − 7.001 × PLR2 + 21.754 × PLR3 0.0348 + 0.6777 × PLR − 5.342 × PLR2 + 20.667 × PLR3

0.03778PLR2

+

0.0073PLR3

0.90 Cooling:4.50 Heating:5.00 Cooling:3.00 Heating:3.00

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Table 3 The technical parameters of energy storage equipment [24]. Equipment

Maximum charging ratio

Maximum discharging ratio

Minimal stored energy ratio

Maximal stored energy ratio

Charge efficiency

Discharge efficiency

ES TS

0.30 0.40

0.35 0.40

0.20 0.20

0.90 0.90

0.96 0.98

0.96 0.98

Table 4 The prices of natural gas and electricity purchased from power grid [41,42]. Item

Unit

Peak hours 19–21

High hours 8–10,15–18

Flat hours 7,11–14,22

Valley hours 1–6,23–24

Electricity tariff Nature gas price

$/kWh $/m3

0.1279 0.4840

0.1134 0.4840

0.0916 0.4840

0.0625 0.4840

3

Table 5 The NSGA-II parameter settings.

Zα− =

∑ (vαβ − vβ−)2 β=1

Parameters

Value

Remarks

Decision variable Objective function Population size Generation time Crossover probability

10 3 200 2000 0.9

Mutual probability

0.1

Crossover distribution Mutual distribution

20 20

Operating data Optimal objective Number of individuals of each population Iteration time Probability of crossover operation during evolutionary process Probability of mutual operation during evolutionary process Distribution index for crossover operator Distribution index for mutation operator

(32)

• Relative proximities calculation and sorting Cα∗ =

Zα+

Zα− + Zα−

(33)

3. Case study A case study has been conducted in this section. The Renewable Energy Demonstration Center (REDC), located in Changsha, China, is taken as the design object. The REDC project is a public indoor swimming pool built in a one-floor building including a flat roof with a height of 6.61 m and an area of 1143 m2. The indoor swimming pool has a total surface area of 450 m2 and a volume of 540 m3. The opening time is from 9 am to 10 pm every day. The case is calculated in a typical year divided into four seasonal typical days including spring, summer, autumn and winter season. Each seasonal typical day is further subdivided into 24 h. And the durations of spring, summer, autumn and winter typical days are 77 days, 102 days, 61 days and 125 days, respectively. 3.1. Input data As shown in Fig. 1, the required parameters are composed of energy demands, environmental, economic and technical data. The energy demands of the project are electricity, district cooling, district heating and hot water demands. District cooling and district heating loads exist in summer and winter, respectively, while electricity and hot water loads are present throughout the year. The hourly energy demands of REDC during the opening time on four seasonal typical days are simulated by EnergyPlus and are presented in Fig. 5. Loads out of opening time are ignored. Notably, the water in the swimming pool must be regularly replaced absolutely due to the hygiene requirements. The heating to heat the replaced water is provided by GB out of the opening time and is not contained in the hot water load. Electricity load does not include the power consumption of ASHP and GSHP that would be calculated during the implementation of the complementary operation strategy in MATLAB. The outdoor hourly ambient temperature and solar radiation on four seasonal typical days in Changsha, China, is shown in Fig. 6. The economic and technical parameters including the relevant parameters of energy device models, unit investment cost, maintenance cost and life time of each device are listed in Tables 1 and 2. Especially, the technical data of PV and SC has been introduced in Section 2.1.1.1, and the technical parameters of energy storage equipment are displayed in Table 3. Furthermore, the prices of natural gas and electricity purchased from power grid are presented in Table 4. Besides, the maximum available installed area for SC and PV is 372 m2, and the maximum permitted installed capacity of GSHP is 70 kW,

Fig. 7. The obtained Pareto frontier by NSGA-II algorithm.

vβ+ = {(max vαβ |β ∈ J1), (min vαβ |β ∈ J2)}

(29)

vβ− = {(min vαβ |β ∈ J1), (max vαβ |β ∈ J2)}

(30)

α

α

α

α

where J1 and J2 are the benefits and the cost indicators, respectively.

• Euclidian distance computation 3

Zα+ =

∑ (vαβ − vβ+)2 β=1

(31)

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Fig. 8. The connections between the three indicators of objective functions.

presented in Table 6. The relative proximity of each solution in Pareto solution set is computed. Consequently, the optimal solution with the maximum relative proximity of 0.9772 is selected from Pareto solution set, which is presented in Table 7. In the optimal solution shown in Table 7, ATC, ATE and CEP reach 4.12 × 104 $, 1.34 × 105 kg and 5.10, respectively. The design capacity of GB is 370 kW, the minimum value under the capacity limits of GB. The optimal installed areas of PV and SC are 198 m2 and 174 m2, respectively. The way to decline consumption of non-renewable energy is to increase the usage of renewable energy. Hence, the maximal installed areas for SC and PV in REDC, which is 372 m2, are all utilized to make the full use of solar energy. Considering the trade-off relationship of ATC, ATE and CEP, the optimal results of PV and SC are determined. Furthermore, the optimal capacity of ES is 87 kWh and TS for 271 kWh, which depend on the optimal installed areas of PV and SC and the characteristic of energy demands. Additionally, both the design cooling and heating capacity of GSHP reach 70 kW, the maximum permitted installed capacity under the restriction of available space for the installation of underground heat exchangers. It results from that the average annual COP of GSHP in heating and cooling mode, reaching 4.88 and 4.50, are higher than these parameters of ASHP which are 2.75 and 2.35. Devices, such as ASHP and GSHP, with higher COP imply that less electricity would be consumed when meeting the same cooling or heating load. Correspondingly, the design cooling and heating capacity of ASHP are 80 kW and 163 kW, respectively, which are limited by cooling and heating energy balances. And the optimal capacity of HE is 371 kW.

Table 6 The weights, positive ideal solution and negative ideal solutions based on TOPSIS and Shannon entropy method. Item

ATC

ATE

CEP

Weight Positive ideal solution Negative ideal solution

0.051 3.69 × 104 $ 4.25 × 104 $

0.483 1.34 × 105 kg 1.94 × 105 kg

0.466 5.12 3.51

restricted by the available space for the installation of underground heat exchangers. 3.2. Results and discussions The multi-objective nonlinear capacity optimization model is solved by NSGA-II, TOPSIS and Shannon entropy method in MATLAB 2017 software. The parameter settings of NSGA-II algorithm are shown in Table 5. The optimal equipment capacity and energy management strategy of the DES in REDC are obtained simultaneously. 3.2.1. Pareto frontier The Pareto frontier, displayed in Fig. 7, is obtained by solving the optimization model with NSGA-II algorithm. Considering the three objectives (ATC, ATE and CEP) of the optimization model, the Pareto frontier is a three-dimensional space curve, presenting the trade-off relationship of optimal objectives. The connections between the three indicators of objective functions are illustrated in Fig. 8. It can be found that ATE would decrease with the increasing of ATC, which implies that ATC and ATE can’t be minimized simultaneously during the multi-objective optimization progress. In terms of ATC and CEP, expected to be minimized and maximized, respectively, could not be optimized at the same time as well. It can be concluded that ATC, ATE and CEP are the objectives that couldn’t be optimized simultaneously. Hence, a decision-making approach is indispensable to ascertain the optimal solution from Pareto solution set containing 200 alternative design plans in this case.

3.2.3. Energy management strategy The energy management strategies during the swimming pool opening hours are illustrated in Figs. 9–11. In the three figures, the upper part of the abscissa axis denotes the output energy of diverse devices to satisfy corresponding energy demands, while the lower part represents energy demands of users and charging of energy storage devices. Especially, the power consumption of main equipment is also included in the lower part of the abscissa axis in Fig. 10. Fig. 9 for hot water management strategy displays that hot water demand is covered by SC and ASHP, while GSHP does not bear hot water load. In spring, the hot water generated by SC is sufficient to satisfy hot water load, even there is excess during the periods of high solar radiation. The excessive hot water is stored in TS and discharged when SC could not meet hot water demand. Hot water load is satisfied by ASHP when the hot water generated by SC is not enough to cover hot

3.2.2. Optimal capacity of device The optimal solution is picked out from Pareto solution set by TOPSIS method. The weights of ATC, ATE and CEP, calculated by Shannon entropy approach, are displayed in Table 6. The positive ideal solutions and negative ideal solutions of the three objectives are also Table 7 The optimal solution selected via TOPSIS method. GB

PV

SC

GPH

GPC

APH

APC

HE

ES

TS

ATC

ATE

CEP

kW

m2

m2

kW

kW

kW

kW

kW

kWh

kWh

104 $

105 kg

/

370

198

174

70

70

163

80

371

87

271

4.12

1.34

5.10

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Fig. 9. The hot water management strategy on four seasonal typical days.

Fig. 10. The electricity management strategy on four seasonal typical days.

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Fig. 11. District cooling and heating management strategies on summer and winter typical days.

Fig. 14. Distribution of PLR of APH.

Fig. 12. Part load performance of GB, ASHP and GSHP.

Fig. 10 for electricity management strategy presents that electricity demand, composed of electricity load and the power consumption of main equipment including GSHP and ASHP, is provided by PV and power grid. In spring, the power generated by PV is enough to meet electricity demand and the excessive power is stored in ES from 11:00 to 15:00. At other times in spring, the power produced by PV could not cover electricity demand. Hence, it is necessary to purchase electricity from power grid. In summer, electricity load and the power consumption of GSHP and ASHP are jointly supplied by PV and power grid. The electricity management strategy in autumn is similar to that in spring. The difference is that fewer power is purchased from power grid in autumn, because ASHP consumes less power to generate hot water which results from the fact that the hot water produced by SC is sufficient to provide hot water demand due to the stronger solar radiation in autumn. In winter, electricity demand is mainly satisfied by power grid for the weak solar radiation. As for the cooling and heating management strategies presented in Fig. 11, these are relatively simple with respect to hot water and electricity management strategies. The district cooling and heating demands are firstly satisfied by GSHP, then the surplus loads are met by ASHP when the cooling or heating produced by GSHP is insufficient.

Fig. 13. PLR of GSHP and ASHP.

water demand. In summer, the hot water produced by SC is sufficient to provide hot water load due to the strong solar radiation in the daytime. The unconsumed hot water is stored in TS and discharged in the evening. However, only part of excessive hot water is stored in TS due to the limitation of capacity of TS. The hot water management strategy in autumn is similar to that in summer. As for the hot water management strategy in winter, ASHP undertakes most of hot water load owing to the weak solar radiation.

3.2.4. Final design plan Regarding GB, GSHP and ASHP, part load ratio (PLR) has a great impact on their efficiencies/COPs, which is displayed in Fig. 12 obtained based on Eqs. (4)–(7) and the technical parameters in Table.2. It can be observed that efficiency/COP of GB, ASHP and GSHP would increase with the increment of PLR. Hence, increasing PLR would make 12

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be a valid way to improve PLR of equipment in literature [43,44]. Assuming the determined number of APHs is n. The capacity of n APHs is C1, C2 … Cn and satisfies Eq. (34), where 163 is the optimal capacity of APH.

⎧C1 + C2+…+Cn = 163 ⎨ ⎩ C1 ≤ C2 ≤ …≤Cn

(34)

From the distribution of PLR of APH, presented in Fig. 14, it can be seen that the PLR of APH is mainly distributed in the range of 0.35 to 0.7 with a percentage of 62.9%, and 29.7% of PLR of APH is distributed in the range of 0 to 0.25. PLR of APH within the range of 0.75 to 1 only accounts for 7.4%. Therefore, the determined number of APH is 2, which also reduces the complexity of operation strategy presented in Fig. 15 with respect to the situation that more APHs are used. Then, the relationship between the capacity of APHs and their power consumption is investigated. As presented in Fig. 16, the connection between the two is a symmetric relationship. Thus, the relationship when the capacity of APH varies from 0 to 81.5 is examined which can be described with the function in Fig. 16. It can be observed that the power consumption of APHs reaches the minimum value 71,564 kWh, when one of APH capacity is 72 kW. Therefore, the determined capacity of the two APHs is 72 kW and 91 kW with power consumption of APHs reducing by 13.1% compared with the plan that there is only one APH with heating capacity of 163 kW. Besides, as illustrated in Table 8, it can also prevent APHs from starting and stopping frequently when the capacity of the two APHs is 72 kW and 91 kW, the plan that proves valid from the insight of APHs on-off time. Consequently, the final design plan is GB with capacity of 370 kW, PV with area of 198 m2, SC with area of 174 m2, GSHP with both cooling and heating capacity of 70 kW, ASHP with cooling/heating capacity of 80/91 kW, another ASHP with heating capacity of 72 kW, HE with capacity of 371 kW, ES with capacity of 87 kWh, TS with capacity of 271 kWh.

Fig. 15. Operation strategy of APHs.

3.2.5. Effects of constant efficiency/COP of devices and operation strategies on the capacity optimization of DESs In order to illustrate the effects of constant efficiency of energy conversion devices and operation strategies on the capacity optimization of DESs, three schemes are set. Scheme 1 is the base scheme, which is the discussed scheme in Sections 3.2.1–3.2.4. The three schemes are described as follows:

• Scheme 1: In scheme 1, energy device models are built with the • Fig. 16. The relationship between power consumption and capacity of APH.

•

equipment more efficient due to reduction of power consumption. According to the acquired management strategies presented in Figs. 9–11, GB do not work during the opening time. PLR of GSHP and ASHP in cooling or heating mode during the opening time on four seasonal typical days are presented in Fig. 13. Both PLR of GPH and PLR of GPC have relatively high values that the former reaches 1 during most of the operation time and the latter exceeds 0.45, which imply that there is no need to improve PLR of GSHP. Besides, PLR of APC is higher than 0.4 during most of the running time, while PLR of APH is low. Therefore, it is necessary to improve PLR of APH. According to Eq. (5), PLR is the ratio between the actual power provided by device and equipment design power. The share of load from device is determinate, which is presented in Figs. 9–11. Hence, the way to increase PLR of equipment is to determine the proper number and capacity of device based on the distribution of PLR, which proves to

method introduced in Section 2.1.1, and the system operates under the developed complementary operation strategy in Section 2.1.2. Scheme 2: In Scheme 2, the efficiency/COP of energy conversion device is constant, which doesn’t change with operation conditions. The efficiencies of PV, SC and GB are 12%, 40% and 90% [8], respectively, The COPs of GSHP and ASHP are set as Table 2. The operation strategy and other parameters are the same as Scheme 1. Scheme 3: In Scheme 3, the conventional operation strategy is employed, where electricity, district cooling, district heating and hot water networks operate independently. PV, ES and power grid supply electricity demand, SC, GB and TS for hot water demand, ASHP and GSHP for district cooling and heating demands. The method of modeling energy conversion devices and other parameters are the same as Scheme 1.

3.2.5.1. Analysis of the effect of equipment constant efficiency/COP on capacity optimization of DESs. From Eqs. (10)–(17), it can be seen that non-renewable energy consumption would affect the value of objective function. Due to the difference in device efficiency/COP between Scheme 1 and Scheme 2, energy consumption is variable when meeting the same load, accordingly leading to diverse objective function values and resulting in the differences in Pareto frontiers 13

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Table 8 The start and stop status of APHs on four seasonal typical days. Time

9 10 11 12 13 14 15 16 17 18 19 20 21 22

Typical spring day

Typical summer day

Typical autumn day

Typical winter day

APH1

APH2

APH1

APH2

APH1

APH2

APH1

APH2

0 1 1 0 0 0 0 1 1 1 1 1 1 0

1 0 0 0 0 0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 0 0 0 0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 0 0 0 1 1 1 1 1

1 1 0 0 0 0 1 1 1 1 1 1 1 1

APH1 and APH2 denote the ASHP with heating capacity of 72 kW and 91 kW, respectively. 1 and 0 denote the start and stop status of APH.

limits of GB from an economic point of view. Since the COP of GSHP is higher than ASHP, the capacity of GPH and GPC is the maximum permitted installed values from the perspective of energy saving and environmental protection. Correspondingly, the capacity of APH and APC is the same for the restriction of cooling and heating energy balances. As compared with Scheme 1, ATC and CEP of Scheme 2 decrease by 11.7% and 12.5%, respectively. ATE of Scheme 2 increases by 10.4%. However, the efficiency/COP of energy device doesn’t remain constant in actual operation and would change with operation conditions. In order to analyze the performance of the optimal design obtained in Scheme 2 in actual operation, ATC, ATE and CEP of the optimal solution are investigated. However, it is hardly possible to carry out an experiment due to the particularity of the problem. Thus, an operation condition close to reality, which is the same as the operation condition in Scheme 1, is programed and simulated in MATLAB. The calculated ATC is 3.81 × 104 $, ATE for 1.61 × 105 kg and CEP for 4.25. As compared with the results acquired in Scheme 2, ATC and ATE increase by 4.7% and 8.8%, respectively, while CEP reduces by 4.7%. Therefore, it is completely necessary to consider the part load performance of devices in the capacity optimization of DESs.

3.2.5.2. Analysis of the effect of operation strategies on capacity optimization of DESs. Similarly, different operation strategies also lead to variable energy consumption, accordingly resulting in diverse Pareto frontiers, weights of objectives, positive ideal solutions and negative ideal solutions. The differences are presented in Fig. 17 and Table 9. The optimal solutions of Scheme 1 and Scheme 3 shown in Table 10 are also different, which are mainly reflected in the difference of capacity of PV, SC, ES, TS and APH. The energy management strategy of Scheme 3 differs from that of Scheme 1 due to the diverse operation strategies in the two schemes. The obtained energy management strategies of Scheme 3 on four seasonal typical days are presented in Figs. 18–20. As illustrated in Fig. 18, hot water load is supplied by GB when the hot

Fig. 17. The Pareto frontiers of the three schemes.

shown in Fig. 17. Then different Pareto frontiers lead to the various weights of objectives, positive ideal solutions and negative ideal solutions of the two schemes presented in Table 9. Consequently, the optimal solutions of Scheme 1 and Scheme 2, displayed in Table 10, are different. It is also why the capacity of PV, SC, ES and TS is variable. The reason that capacity of GB, GPH, GPC, APC and APH in the two schemes is the same is explained as follows. Due to the same operation strategy of the two schemes that GB doesn’t bear load during the open time, the capacity of GB is the minimum values under the capacity

Table 9 The weights of objectives, positive ideal solutions and negative ideal solutions of the three schemes based on TOPSIS and Shannon entropy method. Scheme

1 2 3

Positive ideal solution

Negative ideal solution

Weight

ATC

ATE

CEP

ATC

ATE

CEP

ATC

ATE

CEP

104 $

105 kg

/

104 $

105 kg

/

/

/

/

3.69 3.38 4.12

1.34 1.27 1.39

5.12 5.33 1.69

4.25 4.38 4.45

1.94 1.93 1.87

3.51 3.53 1.49

0.05 0.29 0.04

0.48 0.41 0.83

0.47 0.30 0.13

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Table 10 The optimal solutions of the three schemes. Scheme

1 2 3

GB

PV

SC

kW

m

2

2

370 370 370

198 118 144

m

174 208 228

GPH

GPC

APH

APC

HE

ES

TS

ATC 4

ATE 5

CEP

kW

kW

kW

kW

kW

kWh

kWh

10 $

10 kg

/

70 70 70

70 70 70

163 163 31

80 80 80

371 370 375

87 12 55

271 216 313

4.12 3.64 4.40

1.34 1.48 1.39

5.10 4.46 1.69

Fig. 18. The hot water management strategy of Scheme 3 on four seasonal typical days.

Fig. 19. District cooling and heating management strategies of Scheme 3.

APC. From the comparison between Figs. 10 and 20, it can be seen that the primarily distinction of electricity management strategy of the two schemes is the difference in spring and winter. The power generated by PV in the two schemes is similar, but more electricity needs to be purchased from power grid in Scheme 1 since ASHP consumes more

water produced by TC is not enough, which is met by ASHP in Scheme 1 because the complementary characteristics among solar, geothermal, aerothermal and natural gas are considered in the operation strategy. District cooling and heating strategy of Scheme 1 and Scheme 3, shown in Fig. 11 and Fig. 19, is similar for the same capacity of GPH, GPC and 15

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Fig. 20. The electricity management strategy of Scheme 3 on four seasonal typical days.

(2) It is necessary to consider the part load performance of devices in the multi-objective capacity optimization of DESs. With respect to the optimal solution obtained by considering the part load performance of equipment, the assumption of constant efficiency/COP would yield an 11.7% drop in ATC, a 10.4% increment in ATE and a 12.5% reduction in CEP. (3) Different operation strategies would result in diverse design plans and energy management strategies. As compared with the optimal solution acquired under the complementary operation strategy, ATC and ATE of the optimal result obtained under the conventional operation strategy improve by 6.8% and 3.7%, while CEP decreases by 66.9%. The comparison also proves the economic, environmental and energy advantages of the complementary operation strategy.

power to supply hot water load in spring and winter. As compared with Scheme 1, ATC and ATE of Scheme 3 increase by 6.8% and 3.7% respectively, while CEP of Scheme 3 decreases by 66.9% owing to more consumption of natural gas. From the above analysis, it can be concluded that the complementary operation strategy is more economic, environmental and efficient than the conventional one, and operation strategy has a great impact on the capacity optimization of DESs. 4. Conclusions A DES driven by solar, geothermal, aerothermal, natural gas and power grid is proposed in this paper, where part load performance is considered in the energy conversion device models. A novel operation strategy is developed for the DES based on the complementary coordination characteristics of different energy sources. To determine the energy devices capacity of the DES, a multi-objective nonlinear capacity optimization model is established considering economic, environmental and energy aspects. The optimization model is solved by NSGA-II and TOPSIS method, where the weights of optimal objectives are assigned by Shannon entropy approach. An indoor swimming pool in Changsha, China is selected as a case study. The final design plan of this case is obtained. Besides, the effects of constant efficiency/COP of energy conversion devices and operation strategies on the capacity optimization of DESs are clarified. The conclusions are summarized as follows:

This study mainly focuses on the capacity of energy devices and corresponding energy management strategy under the complementary operation strategy. The energy resources as well as energy devices considered in this study are limited. Further studies are expected to improve the optimization model for a more comprehensive design of DESs, where other energy resources, such as biomass and wind, and corresponding energy devices are considered. The types and capacity of energy devices as well as the energy management strategy would be optimized simultaneously in our next publication.

Declaration of Competing Interest

(1) The proposed model and method can be applied to the design of DESs in actual engineering. The optimal capacity of energy devices and corresponding energy management strategy under specified input parameters can be obtained by the optimization model. The terminal design plan can be determined through improving PLR of energy devices.

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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